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<h1>信息不完整（CFR）游戏中的遗憾最小化</h1>
<p>论文《<a href="http://martin.zinkevich.org/publications/regretpoker.pdf">信息不完整的游戏中的遗憾最小化</a>》介绍了反事实的遗憾，以及如何利用通过自我游戏最大限度地减少反事实遗憾来达到纳什平衡。该算法称为反事实遗憾最小化（<strong>CFR</strong>）。</p>
<p>论文《<a href="http://mlanctot.info/files/papers/nips09mccfr.pdf">在广泛游戏中尽量减少遗憾的蒙特卡洛抽样</a>》介绍了蒙特卡洛反事实遗憾最小化（<strong>MCCFR</strong>），我们在其中从游戏树中进行采样并估计遗憾。</p>
<p>我们试图让我们的Python实现像教程一样易于理解。我们在<a href="kuhn/index.html">一个名为 Kuhn poker 的非常简单的不完美信息游戏</a>上运行它。</p>
<p><a href="https://colab.research.google.com/github/labmlai/annotated_deep_learning_paper_implementations/blob/master/labml_nn/cfr/kuhn/experiment.ipynb"><img alt="Open In Colab" src="https://colab.research.google.com/assets/colab-badge.svg"></a></p>
<p><a href="https://twitter.com/labmlai/status/1407186002255380484"><img alt="Twitter thread" src="https://img.shields.io/twitter/url?style=social&url=https%3A%2F%2Ftwitter.com%2Flabmlai%2Fstatus%2F1407186002255380484"></a>推特话题</p>
<h2>简介</h2>
<p>我们通过机会抽样（CS）实施了蒙特卡洛反事实遗憾最小化（MCCFR）。它通过尝试所有玩家动作来反复探索游戏树的一部分，但采样机会事件。机会事件就像发牌一样；每次迭代都会对它们进行一次采样。然后，它会计算每个动作遵循当前策略而不是采取该操作的<em>遗憾</em>。然后，它会根据这些遗憾更新策略，使用遗憾匹配进行下一次迭代。最后，它计算整个迭代期间策略的平均值，如果我们进行了足够的迭代，则该平均值与纳什均衡非常接近。</p>
<p>我们将首先介绍数学符号和理论。</p>
<h3>玩家</h3>
<p>玩家用表示<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdn" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqen" style="">i</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeg" style="margin-right:0.10903em">N</span></span></span></span></span></span></span>，其中<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqeg" style=""><span class="mord mathnormal" style="margin-right:0.10903em">N</span></span></span></span></span></span>是玩家集合。</p>
<h3><a href="#History">历史</a></h3>
<p>历史<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span>是包括偶然事件在内的一系列动作，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqee" style=""><span class="mord mathnormal" style="margin-right:0.08125em">H</span></span></span></span></span></span>是所有历史的集合。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8193em;vertical-align:-0.13597em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqee" style=""><span class="mord mathnormal" style="margin-right:0.08125em">H</span></span></span></span></span></span>是一组终端历史记录（游戏结束）。</p>
<h3>行动</h3>
<p>操作<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span>，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqce" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span></span>其中<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span>是非终端<a href="#History">历史记录</a>。</p>
<h3><a href="#InfoSet">信息集<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></a></h3>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqct" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcw" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>为玩家<strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>设置的信息</strong>与历史记录类似，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span>但仅包含玩家可见的动作<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>。也就是说，历史记录<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span>将包含动作/事件，例如发给对方玩家的牌，而<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>不会有。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqct" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcw" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>被称为玩家<strong>的信息分区</strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdl" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span></span></span>是属于给定信息集的所有历史记录的集合；也就是说，所有这些历史在玩家眼中看起来都是一样的。</p>
<p><a id="Strategy"></a></p>
<h3>策略</h3>
<p><strong>玩家的策略</strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.6891em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdg" style=""><span class="mord" style=""><span class="mord" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>是对动作的分布<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdo" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span>，其中<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdg" style=""><span class="mord" style=""><span class="mord" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>是玩家所有策略的集合<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord coloredeq eqep" style=""><span class="mord mathnormal" style="">t</span></span></span></span></span></span>第-th 次迭代的策略用表示<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.943556em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>。</p>
<p>策略被定义为针对给定信息集采取行动的<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span>概率<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>，</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span>是<strong>策略配置文件</strong>，由所有玩家的策略组成<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.638891em;vertical-align:-0.208331em;"></span><span class="mord coloredeq eqcx" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span></span></span></span></span></span>是所有玩家的策略<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p><a id="HistoryProbability"></a></p>
<h3>历史概率</h3>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span>是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span>使用策略配置文件进入历史记录的概率<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span>。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.108031em;vertical-align:-0.358031em;"></span><span class="mord coloredeq eqcm" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.358031em;"><span></span></span></span></span></span></span></span></span></span></span></span>是在没有玩家贡献<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span>的情况下到达<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>的概率；即玩家采取<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>了要遵循的动作<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span>概率为<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span></span>。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0497em;vertical-align:-0.29969999999999997em;"></span><span class="mord coloredeq eqco" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29969999999999997em;"><span></span></span></span></span></span></span></span></span></span></span></span>是仅通过玩家的贡献达到目标<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>的概率。也就是说，<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.108031em;vertical-align:-0.358031em;"></span><span class="mord coloredeq eqco" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29969999999999997em;"><span></span></span></span></span></span></span></span><span class="mord coloredeq eqcm" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen coloredeq eqcu" style="">(</span><span class="mord coloredeq eqcu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqcu" style="">)</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.161964em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.358031em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>达到信息集的概率<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>是<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.3794880000000003em;vertical-align:-1.329483em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdl" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.329483em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<h3>公用事业（还清）</h3>
<p>终<a href="#terminal_utility">端实用程序</a>是玩家<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>对终端历史记录的实用程序（或回报）<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span>。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span>在哪里<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span></p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="mclose" style="">)</span></span></span></span></span></span>是具有策略配置文件的玩家<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>的预期效用（回报）<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span>。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcy" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.3794880000000003em;vertical-align:-1.329483em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdm" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.329483em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mord coloredeq eqcu" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<p><a id="NashEquilibrium"></a></p>
<h3>纳什均衡</h3>
<p>纳什均衡是一种状态，在这种状态下，任何参与者都无法仅通过改变策略来增加预期的效用（或回报）。</p>
<p>对于两个参与者来说，纳什均衡是一种<a href="#Strategy">策略配置文件</a>，其中</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0648000000000004em;vertical-align:-1.2824000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span></span>-纳什均衡是，</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0648000000000004em;vertical-align:-1.2824000000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.7824000000000002em;"><span style="top:-3.9424em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.52566em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7416285714285715em;"><span style="top:-2.188485714285714em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3924000000000001em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8018919999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.2824000000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><h3>遗憾最小化</h3>
<p>遗憾的是玩家因为没有遵循最佳策略或采取最佳行动而没有获得的效用（或回报）。</p>
<p>玩家的平均总体遗憾<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>是在所有<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqei" style=""><span class="mord mathnormal" style="margin-right:0.13889em">T</span></span></span></span></span></span>回合迭代中都没有遵循最佳策略的平均遗憾。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.138331em;vertical-align:-0.247em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqcz" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.177714285714286em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqdg" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3756em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span></span></span></p>
<p>迭代中所有玩家的策略配置文件在哪里<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.7935559999999999em;vertical-align:0em;"></span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.61508em;vertical-align:0em;"></span><span class="mord coloredeq eqep" style=""><span class="mord mathnormal" style="">t</span></span></span></span></span></span>，以及</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<p>是将玩家的策略替换为<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>的策略配置文件<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.7935559999999999em;vertical-align:0em;"></span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.94736em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.688696em;"><span style="top:-2.441336em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>。</p>
<p>平均策略是所有回合中遵循的策略的平均值<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqcw" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span></span></span></span></span></p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eqn" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>
<p>这是没有使用最佳策略的卑鄙遗憾。</p>
<p>如果<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span></span>对所有玩家来说都<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqcf" style=""><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span></span>-Nash 均衡。</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:8.342229em;vertical-align:-3.9211145000000007em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.4211145em;"><span style="top:-7.3581195em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-4.8697835em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span></span></span><span style="top:-1.4743344999999994em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9211145000000007em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.4211145em;"><span style="top:-7.3581195em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span><span style="top:-4.8697835em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqcz" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.177714285714286em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqdg" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3756em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span><span style="top:-1.4743344999999994em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqcz" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.177714285714286em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqdg" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3756em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord coloredeq eqcb" style=""><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqcz" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mbin mtight" style="">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&lt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqdh" style=""><span class="mord mathnormal" style="">ϵ</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.9211145000000007em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>因为这是一场零和博弈，所以我们可以加<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0894389999999998em;vertical-align:-0.24810799999999997em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.4518920000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>和，第二个术语将被取消。</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.395449em;vertical-align:-1.4477245em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9477245em;"><span style="top:-3.9477245em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4477245em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9477245em;"><span style="top:-3.9477245em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4477245em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>一组策略的公用事业平均值等于平均策略的效用。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>因此，</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.5593910000000002em;vertical-align:-0.5296955em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0296955000000003em;"><span style="top:-3.1383645000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5296955em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0296955000000003em;"><span style="top:-3.1383645000000002em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5296955em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>从定义来看<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span></span></span></span></span>，<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.259391em;vertical-align:-0.36806em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">2</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≥</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>那么，</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.1187820000000004em;vertical-align:-1.3093910000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8093910000000002em;"><span style="top:-3.91806em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span><span style="top:-2.358669em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3093910000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8093910000000002em;"><span style="top:-3.91806em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">−</span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span></span></span></span></span><span class="mclose">)</span></span></span><span style="top:-2.358669em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.328331em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdp" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6771428571428572em;"><span style="top:-2.188485714285714em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-2.8448em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.31151428571428574em;"><span></span></span></span></span></span></span><span class="mrel mtight">∈</span><span class="mord mtight"><span class="mord mtight">Σ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31731428571428577em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.36806em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7386959999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">∗</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.3093910000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>这是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqdd" style=""><span class="mord" style="">2</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdh" style="">ϵ</span></span></span></span></span></span></span>纳什均衡。同样，你可以为超过 2 名玩家的游戏进行证明。</p>
<p>因此，我们需要最小化<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>才能接近纳什均衡。</p>
<p><a id="CounterfactualRegret"></a></p>
<h3>反事实的遗憾</h3>
如果@@ <p>玩家<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>尝试到达<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>（采取了行动），<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>则<strong>反事实值</strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.03588em">σ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span></span></span></span>是玩家的预期效用导致<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>的概率为<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span></span>）。</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.03588em">σ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.444646em;vertical-align:-1.394641em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqeq" style=""><span class="mord mathnormal mtight" style="margin-right:0.04398em">z</span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqea" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.394641em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714392em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqcx" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32808571428571426em;"><span style="top:-2.357em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span><span class="mclose">)</span><span class="mord coloredeq eqcj" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen coloredeq eqdx" style="">[</span><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqdx" style="">]</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>wh<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqea" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span> ere 是可从中访问的终端历史记录集<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>，并且<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span></span></span></span></span>是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span></span></span></span></span> up 的前缀<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcj" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen coloredeq eqdx" style="">[</span><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqdx" style="">]</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span>是从到达 z 的概率<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span></span></span></span></span>。</p>
<p><strong>直接的反事实遗憾</strong>是，</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord"><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">A</span><span class="mord mtight"><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.17737em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>在哪里</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mord"><span class="delimsizing size2">)</span></span></span></span></span></span></span></p>
<p>其中<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mrel mtight">→</span><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span>是修改为始终<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span>在信息集中采取行动的策略配置文件<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqdp" style=""><span class="mord mathnormal" style="margin-right:0.03588em">σ</span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>。</p>
<p>本<a href="http://martin.zinkevich.org/publications/regretpoker.pdf">文</a>证明（定理3），</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.138331em;vertical-align:-0.247em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.3717110000000003em;vertical-align:-1.321706em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mrel mtight">∈</span><span class="mord mtight coloredeq eqcw" style=""><span class="mord mtight" style=""><span class="mord mathcal mtight coloredeq eqef" style="margin-right:0.07382em">I</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.321706em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight">,</span><span class="mord mtight">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span></span></span></span></span></span>在哪里<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.372211em;vertical-align:-0.412972em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight">,</span><span class="mord mtight">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.412972em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.274439em;vertical-align:-0.383108em;"></span><span class="mord coloredeq eqdv" style=""><span class="mop" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.00773em;">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mpunct mtight">,</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span><span class="mord mathnormal mtight">mm</span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.383108em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">0</span><span class="mclose">)</span></span></span></span></span></span></p>
<p><a id="RegretMatching"></a></p>
<h3>遗憾的配对</h3>
<p>该策略是使用后悔匹配计算得出的。</p>
<p>对每个信息集和行动对的遗憾<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>仍然存在，</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:4.899005000000001em;vertical-align:-2.1995025000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>策略是用遗憾匹配来计算的，</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.9000399999999997em;vertical-align:-1.70002em;"></span><span class="mord coloredeq eqb" style=""><span class="mord" style=""><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2000199999999994em;"><span style="top:-4.200019999999999em;"><span class="pstrut" style="height:4.05002em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="minner" style=""><span class="mopen" style=""><span class="delimsizing mult" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em;"><span style="top:-2.49999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.49199em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15001em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292009999999999em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.30002em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2158449999999998em;"><span style="top:-2.509755em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mop mtight coloredeq eqbi" style=""><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28539000000000003em;"><span style="top:-2.2853899999999996em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.6068285714285713em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8495600000000001em;"><span style="top:-2.84956em;margin-right:0.1em;"><span class="pstrut" style="height:2.55556em;"></span><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5000100000000001em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.19516666666666668em"></span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.32228571428571423em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5356em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8402519999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mopen mtight coloredeq eqcl" style="">∣</span><span class="mord mathnormal mtight coloredeq eqcl" style="">A</span><span class="mopen mtight coloredeq eqcl" style="">(</span><span class="mord mtight coloredeq eqcl" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight coloredeq eqcl" style="">)∣</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">f</span></span><span class="mord" style=""><span class="mord coloredeq eqbg" style=""><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel coloredeq eqbg" style="">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord coloredeq eqbg" style="">0</span></span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style="">o</span><span class="mord" style=""><span class="mord coloredeq eqep" style="">t</span></span><span class="mord" style=""><span class="mord coloredeq eqem" style="">h</span></span><span class="mord" style="">erw</span><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">se</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.70002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span><p>在哪里<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel coloredeq eqz" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqz" style=""><span class="mop coloredeq eqdv" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">(</span></span></span><span class="mord coloredeq eqz" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span><span class="mpunct coloredeq eqz" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqz" style="">0</span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">)</span></span></span></span></span></span></span></span></p>
<p>这篇论文《<a href="http://martin.zinkevich.org/publications/regretpoker.pdf">信息不完整的游戏中的遗憾最小化</a>》论文证明，如果根据上述方程式<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqdt" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span></span></span></span></span>选择策略，则与<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.383108em;vertical-align:-0.538em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.537397em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord sqrt mtight"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9322900000000001em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mtight coloredeq eqei" style="padding-left:0.833em"><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span><span style="top:-2.89229em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail mtight" style="min-width:0.853em;height:1.08em;"><svg height="1.08em" preserveaspectratio="xMinYMin slice" viewbox="0 0 400000 1080" width="400em" xmlns="http://www.w3.org/2000/svg"><path d="M95,702
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<p><a id="MCCFR"></a></p>
<h3>蒙特卡洛 CFR (MCCFR)</h3>
<p>计算<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.05222em;vertical-align:-0.258664em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.02778em">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-2.441336em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>需要在每次迭代时扩展完整的游戏树。</p>
<p>论文《<a href="http://mlanctot.info/files/papers/nips09mccfr.pdf">在广泛游戏中尽量减少遗憾的蒙特卡洛抽样</a>》表明，我们可以从游戏树中进行采样并估计遗憾。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.78055em;vertical-align:-0.09722em;"></span><span class="mord mathcal">Q</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span> (<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqdz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">⊆</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span>) 的一组子集，我们在迭代<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqdz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span>中只查看单个块。所有子集的联合 spans<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span> (<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.55556em;vertical-align:0em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∩</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.02778em;">r</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqej" style=""><span class="mord mathnormal" style="margin-right:0.07153em">Z</span></span></span></span></span></span>)。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqec" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span>是选取区块的概率<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqdz" style=""><span class="mord" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span>。</p>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span>是当前迭代<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span></span></span></span></span>中选取的概率；即<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.24703em;vertical-align:-0.49703em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999997em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqeo" style=""><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span><span class="mrel mtight">:</span><span class="mord mtight coloredeq eqde" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdz" style=""><span class="mord mathnormal mtight" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.49703em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqec" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span>-where<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.716668em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqec" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span> 的总和<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.969438em;vertical-align:-0.286108em;"></span><span class="mord coloredeq eqde" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord coloredeq eqdz" style=""><span class="mord mathnormal" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span></span>。</p>
<p>然后我们得到区块的<strong>反事实值采样</strong><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.85396em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqeo" style=""><span class="mord mathnormal" style="margin-right:0.05724em">j</span></span></span></span></span></span>，</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord accent" style="color:pink"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord mathnormal" style="margin-right:0.03588em;color:pink">σ</span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mord" style="color:pink">∣</span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqeo" style="margin-right:0.05724em">j</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.813096em;vertical-align:-1.491656em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8556639999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqde" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdz" style=""><span class="mord mathnormal mtight" style="">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.491656em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714392em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqcx" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32808571428571426em;"><span style="top:-2.357em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.20252142857142857em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqdx" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen" style="">[</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">]</span></span><span class="mclose">)</span><span class="mord coloredeq eqcj" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mopen coloredeq eqdx" style="">[</span><span class="mord coloredeq eqdx" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqdx" style="">]</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqeq" style=""><span class="mord mathnormal" style="margin-right:0.04398em">z</span></span><span class="mclose">)</span></span></span></span></span></span></p>
<p>这篇论文表明</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord mathbb">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166400000000005em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqeo" style=""><span class="mord mathnormal mtight" style="margin-right:0.05724em">j</span></span><span class="mrel mtight">∼</span><span class="mord mtight coloredeq eqec" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight" style="margin-right:0.03588em">q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3280857142857143em;"><span style="top:-2.357em;margin-left:-0.03588em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeo" style="margin-right:0.05724em">j</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.2818857142857143em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.34731999999999996em;"><span></span></span></span></span></span></span><span class="mord"><span class="delimsizing size2">[</span></span><span class="mord accent" style="color:pink"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord mathnormal" style="margin-right:0.03588em;color:pink">σ</span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mord" style="color:pink">∣</span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqeo" style="margin-right:0.05724em">j</span></span><span class="mclose" style="color:pink">)</span><span class="mord"><span class="delimsizing size2">]</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">v</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord mathnormal" style="margin-right:0.03588em">σ</span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
<p>用一个简单的证据。</p>
<p>因此，我们可以对游戏树的一部分进行采样并计算遗憾。我们计算遗憾的估计值</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord" style="color:coral"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="color:coral">(</span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:coral">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:coral">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.093556em;vertical-align:-0.25em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span></span></span></span></p>
<p>然后用它来更新<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>和计算每次迭代<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>的策略。最后，我们计算整体平均策略<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:cyan"><span class="mord" style=""><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span></span></span></span></span>。</p>
<p>这里有一个<a href="kuhn/index.html">库恩扑克</a>实现，可以在库恩扑克上试用CFR。</p>
<p><em>让我们深入研究一下代码！</em></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">328</span><span></span><span class="kn">from</span> <span class="nn">typing</span> <span class="kn">import</span> <span class="n">NewType</span><span class="p">,</span> <span class="n">Dict</span><span class="p">,</span> <span class="n">List</span><span class="p">,</span> <span class="n">Callable</span><span class="p">,</span> <span class="n">cast</span>
<span class="lineno">329</span>
<span class="lineno">330</span><span class="kn">from</span> <span class="nn">labml</span> <span class="kn">import</span> <span class="n">monit</span><span class="p">,</span> <span class="n">tracker</span><span class="p">,</span> <span class="n">logger</span><span class="p">,</span> <span class="n">experiment</span>
<span class="lineno">331</span><span class="kn">from</span> <span class="nn">labml.configs</span> <span class="kn">import</span> <span class="n">BaseConfigs</span><span class="p">,</span> <span class="n">option</span></pre></div>
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</div>
<div class='section' id='section-1'>
<div class='docs'>
<div class='section-link'>
<a href='#section-1'>#</a>
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<p>一个玩家<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.72243em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdn" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqen" style="">i</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeg" style="margin-right:0.10903em">N</span></span></span></span></span></span></span><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqeg" style=""><span class="mord mathnormal" style="margin-right:0.10903em">N</span></span></span></span></span></span>，玩家集合在哪里</p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">334</span><span class="n">Player</span> <span class="o">=</span> <span class="n">NewType</span><span class="p">(</span><span class="s1">&#39;Player&#39;</span><span class="p">,</span> <span class="nb">int</span><span class="p">)</span></pre></div>
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</div>
<div class='section' id='section-2'>
<div class='docs'>
<div class='section-link'>
<a href='#section-2'>#</a>
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<p>操作<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span></span></span></span></span>，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqce" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">:</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span></span>其中<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span>是非终端<a href="#History">历史记录</a></p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">336</span><span class="n">Action</span> <span class="o">=</span> <span class="n">NewType</span><span class="p">(</span><span class="s1">&#39;Action&#39;</span><span class="p">,</span> <span class="nb">str</span><span class="p">)</span></pre></div>
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</div>
<div class='section' id='section-3'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-3'>#</a>
</div>
<p><a id="History"></a></p>
<h2>历史</h2>
<p>历史<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdk" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqee" style="margin-right:0.08125em">H</span></span></span></span></span></span></span>是包括偶然事件在内的一系列动作，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqee" style=""><span class="mord mathnormal" style="margin-right:0.08125em">H</span></span></span></span></span></span>是所有历史的集合。</p>
<p>这个类应该使用游戏特定的逻辑进行扩展。</p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">339</span><span class="k">class</span> <span class="nc">History</span><span class="p">:</span></pre></div>
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</div>
<div class='section' id='section-4'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-4'>#</a>
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<p>无论是终端历史；即游戏结束。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span></p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">351</span> <span class="k">def</span> <span class="nf">is_terminal</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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<div class='section' id='section-5'>
<div class='docs'>
<div class='section-link'>
<a href='#section-5'>#</a>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">356</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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</div>
<div class='section' id='section-6'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-6'>#</a>
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<p><a id="terminal_utility"></a>终端历史记录<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>的播放器实用程序。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span>在哪里<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">358</span> <span class="k">def</span> <span class="nf">terminal_utility</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">Player</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">float</span><span class="p">:</span></pre></div>
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<div class="highlight"><pre><span class="lineno">364</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<p>获取当前玩家，用表示<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdu" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span>，其中<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqeh" style=""><span class="mord mathnormal" style="margin-right:0.13889em">P</span></span></span></span></span></span>称为<strong>玩家函数</strong>。</p>
<p>如果<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span>这意味着当前事件是偶然<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord coloredeq eqel" style=""><span class="mord mathnormal" style="">c</span></span></span></span></span></span>事件。比如发牌，或在扑克中打开普通牌。</p>
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<div class="highlight"><pre><span class="lineno">366</span> <span class="k">def</span> <span class="nf">player</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Player</span><span class="p">:</span></pre></div>
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<div class="highlight"><pre><span class="lineno">373</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<p>下一步是否是机会步骤；比如发一张新牌。<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">375</span> <span class="k">def</span> <span class="nf">is_chance</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">bool</span><span class="p">:</span></pre></div>
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<div class="highlight"><pre><span class="lineno">380</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<p>在什么时候抽样机会<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span>.</p>
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<div class="highlight"><pre><span class="lineno">382</span> <span class="k">def</span> <span class="nf">sample_chance</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">Action</span><span class="p">:</span></pre></div>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">386</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<p>在历史记录中添加操作。</p>
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<div class="highlight"><pre><span class="lineno">388</span> <span class="k">def</span> <span class="fm">__add__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">action</span><span class="p">:</span> <span class="n">Action</span><span class="p">):</span></pre></div>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">392</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<p><a href="#InfoSet">获取当前玩家的信息集</a></p>
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<div class="highlight"><pre><span class="lineno">394</span> <span class="k">def</span> <span class="nf">info_set_key</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">str</span><span class="p">:</span></pre></div>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">398</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span></pre></div>
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<p>为当前玩家创建新的<a href="#InfoSet">信息集</a></p>
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<div class="highlight"><pre><span class="lineno">400</span> <span class="k">def</span> <span class="nf">new_info_set</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="s1">&#39;InfoSet&#39;</span><span class="p">:</span></pre></div>
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<a href='#section-19'>#</a>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">404</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<div class='section' id='section-20'>
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<p>人类可读的表示</p>
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<div class="highlight"><pre><span class="lineno">406</span> <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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<div class="highlight"><pre><span class="lineno">410</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<p><a id="InfoSet"></a></p>
<h2>信息集<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></h2>
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<div class="highlight"><pre><span class="lineno">413</span><span class="k">class</span> <span class="nc">InfoSet</span><span class="p">:</span></pre></div>
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<div class='section' id='section-23'>
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<a href='#section-23'>#</a>
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<p>标识信息集的唯一密钥</p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">421</span> <span class="n">key</span><span class="p">:</span> <span class="nb">str</span></pre></div>
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<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>，玩家的<a href="#Strategy">策略</a><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">423</span> <span class="n">strategy</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="n">Action</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span></pre></div>
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<div class='section' id='section-25'>
<div class='docs'>
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<a href='#section-25'>#</a>
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<p>对没有采取每项行动感到非常遗憾<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdo" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span>，</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:4.899005000000001em;vertical-align:-2.1995025000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.6995025000000004em;"><span style="top:-5.6842825em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span><span style="top:-3.1959465em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqed" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="color:coral">(</span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:coral">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:coral">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:2.1995025000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p>我们坚持<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqbt" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mord" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span></span></span></span></span></span>而不是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>因为计算策略时<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.190108em;vertical-align:-0.345em;"></span><span class="mord coloredeq eqcv" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.6550000000000002em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.345em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span>术语无论如何都会被取消<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0913309999999998em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">438</span> <span class="n">regret</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="n">Action</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span></pre></div>
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<div class='section' id='section-26'>
<div class='docs'>
<div class='section-link'>
<a href='#section-26'>#</a>
</div>
<p>我们维持累积策略<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqbd" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span></span></span>来计算整体平均策略</p>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eqn" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">445</span> <span class="n">cumulative_strategy</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="n">Action</span><span class="p">,</span> <span class="nb">float</span><span class="p">]</span></pre></div>
</div>
</div>
<div class='section' id='section-27'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-27'>#</a>
</div>
<p>初始化</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">447</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">key</span><span class="p">:</span> <span class="nb">str</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-28'>
<div class='docs'>
<div class='section-link'>
<a href='#section-28'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">451</span> <span class="bp">self</span><span class="o">.</span><span class="n">key</span> <span class="o">=</span> <span class="n">key</span>
<span class="lineno">452</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">actions</span><span class="p">()}</span>
<span class="lineno">453</span> <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">0</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">actions</span><span class="p">()}</span>
<span class="lineno">454</span> <span class="bp">self</span><span class="o">.</span><span class="n">calculate_strategy</span><span class="p">()</span></pre></div>
</div>
</div>
<div class='section' id='section-29'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-29'>#</a>
</div>
<p>行动<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdo" style=""><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord coloredeq eqdy" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">456</span> <span class="k">def</span> <span class="nf">actions</span><span class="p">(</span><span class="bp">self</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="n">List</span><span class="p">[</span><span class="n">Action</span><span class="p">]:</span></pre></div>
</div>
</div>
<div class='section' id='section-30'>
<div class='docs'>
<div class='section-link'>
<a href='#section-30'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">460</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
</div>
</div>
<div class='section' id='section-31'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-31'>#</a>
</div>
<p>从保存的字典中加载信息集</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">462</span> <span class="nd">@staticmethod</span>
<span class="lineno">463</span> <span class="k">def</span> <span class="nf">from_dict</span><span class="p">(</span><span class="n">data</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">])</span> <span class="o">-&gt;</span> <span class="s1">&#39;InfoSet&#39;</span><span class="p">:</span></pre></div>
</div>
</div>
<div class='section' id='section-32'>
<div class='docs'>
<div class='section-link'>
<a href='#section-32'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">467</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
</div>
</div>
<div class='section' id='section-33'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-33'>#</a>
</div>
<p>将信息集保存到字典中</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">469</span> <span class="k">def</span> <span class="nf">to_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-34'>
<div class='docs'>
<div class='section-link'>
<a href='#section-34'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">473</span> <span class="k">return</span> <span class="p">{</span>
<span class="lineno">474</span> <span class="s1">&#39;key&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">key</span><span class="p">,</span>
<span class="lineno">475</span> <span class="s1">&#39;regret&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span><span class="p">,</span>
<span class="lineno">476</span> <span class="s1">&#39;average_strategy&#39;</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="p">,</span>
<span class="lineno">477</span> <span class="p">}</span></pre></div>
</div>
</div>
<div class='section' id='section-35'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-35'>#</a>
</div>
<p>从保存的字典中加载数据</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">479</span> <span class="k">def</span> <span class="nf">load_dict</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">data</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="nb">any</span><span class="p">]):</span></pre></div>
</div>
</div>
<div class='section' id='section-36'>
<div class='docs'>
<div class='section-link'>
<a href='#section-36'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">483</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="s1">&#39;regret&#39;</span><span class="p">]</span>
<span class="lineno">484</span> <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span> <span class="o">=</span> <span class="n">data</span><span class="p">[</span><span class="s1">&#39;average_strategy&#39;</span><span class="p">]</span>
<span class="lineno">485</span> <span class="bp">self</span><span class="o">.</span><span class="n">calculate_strategy</span><span class="p">()</span></pre></div>
</div>
</div>
<div class='section' id='section-37'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-37'>#</a>
</div>
<h2>计算策略</h2>
<p>使用<a href="#RegretMatching">后悔匹配</a>计算当前策略。</p>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.9000399999999997em;vertical-align:-1.70002em;"></span><span class="mord coloredeq eqb" style=""><span class="mord" style=""><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.2000199999999994em;"><span style="top:-4.200019999999999em;"><span class="pstrut" style="height:4.05002em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="minner" style=""><span class="mopen" style=""><span class="delimsizing mult" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.05002em;"><span style="top:-2.49999em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎩</span></span></span><span style="top:-2.49199em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-3.15001em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎨</span></span></span><span style="top:-4.292009999999999em;"><span class="pstrut" style="height:3.15em;"></span><span style="height:0.016em;width:0.889em"><svg height="0.016em" preserveaspectratio="xMinYMin" style="width:0.889em" viewbox="0 0 889 16" width="0.889em" xmlns="http://www.w3.org/2000/svg"><path d="M384 0 H504 V16 H384z M384 0 H504 V16 H384z"></path></svg></span></span><span style="top:-4.30002em;"><span class="pstrut" style="height:3.15em;"></span><span class="delimsizinginner delim-size4"><span>⎧</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.55002em;"><span></span></span></span></span></span></span><span class="mord" style=""><span class="mtable"><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.2158449999999998em;"><span style="top:-2.509755em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mop mtight coloredeq eqbi" style=""><span class="mop op-symbol small-op mtight" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.28539000000000003em;"><span style="top:-2.2853899999999996em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.6068285714285713em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8495600000000001em;"><span style="top:-2.84956em;margin-right:0.1em;"><span class="pstrut" style="height:2.55556em;"></span><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.5000100000000001em;"><span></span></span></span></span></span></span><span class="mspace mtight" style="margin-right:0.19516666666666668em"></span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.32228571428571423em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.5356em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9717785714285714em;"><span style="top:-2.177714285714286em;margin-left:-0.00773em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-2.9836857142857145em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.3222857142857143em;"><span></span></span></span></span></span></span><span class="mopen mtight" style="color:orange">(</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct mtight" style="color:orange">,</span><span class="mord mtight" style="color:orange"><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="mclose mtight" style="color:orange">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.8402519999999999em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.845108em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mopen mtight coloredeq eqcl" style="">∣</span><span class="mord mathnormal mtight coloredeq eqcl" style="">A</span><span class="mopen mtight coloredeq eqcl" style="">(</span><span class="mord mtight coloredeq eqcl" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight coloredeq eqcl" style="">)∣</span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.394em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct" style="">,</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:1em;"></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.0420485em;"><span style="top:-4.0420485em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">f</span></span><span class="mord" style=""><span class="mord coloredeq eqbg" style=""><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mrel coloredeq eqbg" style="">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em"></span><span class="mord coloredeq eqbg" style="">0</span></span></span></span><span style="top:-2.1937965em;"><span class="pstrut" style="height:3.215845em;"></span><span class="mord" style=""><span class="mord text" style=""><span class="mord" style="">o</span><span class="mord" style=""><span class="mord coloredeq eqep" style="">t</span></span><span class="mord" style=""><span class="mord coloredeq eqem" style="">h</span></span><span class="mord" style="">erw</span><span class="mord" style=""><span class="mord coloredeq eqen" style="">i</span></span><span class="mord" style="">se</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.5420485em;"><span></span></span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.70002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span><p>在哪里<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel coloredeq eqz" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqz" style=""><span class="mop coloredeq eqdv" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">(</span></span></span><span class="mord coloredeq eqz" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span><span class="mpunct coloredeq eqz" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqz" style="">0</span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">)</span></span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">487</span> <span class="k">def</span> <span class="nf">calculate_strategy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-38'>
<div class='docs'>
<div class='section-link'>
<a href='#section-38'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.80002em;vertical-align:-0.65002em;"></span><span class="mord"><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9592389999999998em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.180908em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel coloredeq eqz" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqz" style=""><span class="mop coloredeq eqdv" style=""><span style="">m</span><span style="">a</span><span style="">x</span></span></span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">(</span></span></span><span class="mord coloredeq eqz" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span><span class="mpunct coloredeq eqz" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord coloredeq eqz" style="">0</span><span class="mord coloredeq eqz" style=""><span class="delimsizing size2" style=""><span style="">)</span></span></span></span></span></span></span></span></span></p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">506</span> <span class="n">regret</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="nb">max</span><span class="p">(</span><span class="n">r</span><span class="p">,</span> <span class="mi">0</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
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</div>
<div class='section' id='section-39'>
<div class='docs'>
<div class='section-link'>
<a href='#section-39'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.5660100000000003em;vertical-align:-1.516005em;"></span><span class="mord"><span class="mop op-limits coloredeq eqbi" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9592389999999998em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.180908em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">508</span> <span class="n">regret_sum</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">regret</span><span class="o">.</span><span class="n">values</span><span class="p">())</span></pre></div>
</div>
</div>
<div class='section' id='section-40'>
<div class='docs'>
<div class='section-link'>
<a href='#section-40'>#</a>
</div>
<p>如果<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.433949em;vertical-align:-0.47471em;"></span><span class="mord coloredeq eqbg" style=""><span class="mord" style=""><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="">0</span></span></span></span></span></span>，</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">510</span> <span class="k">if</span> <span class="n">regret_sum</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span></pre></div>
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</div>
<div class='section' id='section-41'>
<div class='docs'>
<div class='section-link'>
<a href='#section-41'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.141331em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.960188em;vertical-align:-1.323949em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.636239em;"><span style="top:-2.150761em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mop coloredeq eqbi" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.22528999999999993em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6828285714285715em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mrel mtight" style="">∈</span><span class="mord mathnormal mtight" style="">A</span><span class="mopen mtight" style="">(</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose mtight" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.47471em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6778919999999999em;"><span style="top:-2.9890000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">′</span></span></span></span></span></span></span></span></span><span class="mclose" style="color:orange">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.6769999999999996em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.959239em;"><span style="top:-2.4231360000000004em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.1809080000000005em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mpunct mtight" style="">,</span><span class="mord mtight" style="">+</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.323949em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">513</span> <span class="bp">self</span><span class="o">.</span><span class="n">strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="n">r</span> <span class="o">/</span> <span class="n">regret_sum</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">regret</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
</div>
</div>
<div class='section' id='section-42'>
<div class='docs'>
<div class='section-link'>
<a href='#section-42'>#</a>
</div>
<p>否则，</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">515</span> <span class="k">else</span><span class="p">:</span></pre></div>
</div>
</div>
<div class='section' id='section-43'>
<div class='docs'>
<div class='section-link'>
<a href='#section-43'>#</a>
</div>
<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span></span></span></p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">517</span> <span class="n">count</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">regret</span><span class="p">))</span></pre></div>
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</div>
<div class='section' id='section-44'>
<div class='docs'>
<div class='section-link'>
<a href='#section-44'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.141331em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdj" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mbin mtight" style="">+</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">520</span> <span class="bp">self</span><span class="o">.</span><span class="n">strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">count</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">regret</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
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<div class='section' id='section-45'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-45'>#</a>
</div>
<h2>获取平均策略</h2>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eql" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">522</span> <span class="k">def</span> <span class="nf">get_average_strategy</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-46'>
<div class='docs'>
<div class='section-link'>
<a href='#section-46'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">531</span> <span class="n">cum_strategy</span> <span class="o">=</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="bp">self</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="o">.</span><span class="n">get</span><span class="p">(</span><span class="n">a</span><span class="p">,</span> <span class="mf">0.</span><span class="p">)</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">actions</span><span class="p">()}</span></pre></div>
</div>
</div>
<div class='section' id='section-47'>
<div class='docs'>
<div class='section-link'>
<a href='#section-47'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.344341em;vertical-align:-1.516005em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">A</span><span class="mopen mtight">(</span><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">535</span> <span class="n">strategy_sum</span> <span class="o">=</span> <span class="nb">sum</span><span class="p">(</span><span class="n">cum_strategy</span><span class="o">.</span><span class="n">values</span><span class="p">())</span></pre></div>
</div>
</div>
<div class='section' id='section-48'>
<div class='docs'>
<div class='section-link'>
<a href='#section-48'>#</a>
</div>
<p>如果<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.2809409999999999em;vertical-align:-0.29971000000000003em;"></span><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">&gt;</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span>，</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">537</span> <span class="k">if</span> <span class="n">strategy_sum</span> <span class="o">&gt;</span> <span class="mi">0</span><span class="p">:</span></pre></div>
</div>
</div>
<div class='section' id='section-49'>
<div class='docs'>
<div class='section-link'>
<a href='#section-49'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.841882em;vertical-align:-1.170941em;"></span><span class="mord coloredeq eql" style=""><span class="mord" style=""><span class="mord" style="color:cyan"><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.670941em;"><span style="top:-2.128769em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8525400000000001em;"><span style="top:-2.4231360000000004em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.0448000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7253428571428571em;"><span style="top:-2.786em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.27686399999999994em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em"></span></span><span style="top:-3.6897100000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord" style=""><span class="mop coloredeq eqbd" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.981231em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.29971000000000003em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqbd" style="">(</span><span class="mord coloredeq eqbd" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqbd" style="">)</span><span class="mord coloredeq eqbd" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.170941em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">541</span> <span class="k">return</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="n">s</span> <span class="o">/</span> <span class="n">strategy_sum</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">s</span> <span class="ow">in</span> <span class="n">cum_strategy</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
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<p>否则，</p>
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<div class="highlight"><pre><span class="lineno">543</span> <span class="k">else</span><span class="p">:</span></pre></div>
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<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">545</span> <span class="n">count</span> <span class="o">=</span> <span class="nb">len</span><span class="p">(</span><span class="nb">list</span><span class="p">(</span><span class="n">a</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">cum_strategy</span><span class="p">))</span></pre></div>
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<a href='#section-52'>#</a>
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<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord" style="color:cyan"><span class="mord" style=""><span class="mord coloredeq eqcf" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.56778em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.25em;"><span class="mord" style="">ˉ</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose coloredeq eqcf" style="">)</span><span class="mopen coloredeq eqcf" style="">(</span><span class="mord coloredeq eqcf" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose coloredeq eqcf" style="">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqcl" style=""><span class="mopen" style="">∣</span><span class="mord mathnormal" style="">A</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)∣</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">548</span> <span class="k">return</span> <span class="p">{</span><span class="n">a</span><span class="p">:</span> <span class="mi">1</span> <span class="o">/</span> <span class="n">count</span> <span class="k">for</span> <span class="n">a</span><span class="p">,</span> <span class="n">r</span> <span class="ow">in</span> <span class="n">cum_strategy</span><span class="o">.</span><span class="n">items</span><span class="p">()}</span></pre></div>
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<p>人类可读的表示</p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">550</span> <span class="k">def</span> <span class="fm">__repr__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">554</span> <span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span></pre></div>
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<h2>反事实遗憾最小化 (CFR) 算法</h2>
<p>我们进行机会采样（<strong>CS</strong>），其中对所有机会事件（节点）进行采样，并探索所有其他事件（节点）。</p>
<p>我们可以忽略该术语，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdw" style=""><span class="mord mathnormal" style="margin-right:0.03588em">q</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span>因为它对所有终端历史都是相同的，因为我们正在进行机会抽样，并且在计算策略时它会被抵消（在分子和分母中很常见）。</p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">557</span><span class="k">class</span> <span class="nc">CFR</span><span class="p">:</span></pre></div>
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<p><span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqcw" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span></span></span></span></span>所有信息集的集合。</p>
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<div class="highlight"><pre><span class="lineno">570</span> <span class="n">info_sets</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">InfoSet</span><span class="p">]</span></pre></div>
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<ul><li><code class="highlight"><span></span><span class="n">create_new_history</span></code>
创建新的空白历史记录</li>
<li><code class="highlight"><span></span><span class="n">epochs</span></code>
是要训练的迭代次数<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqei" style=""><span class="mord mathnormal" style="margin-right:0.13889em">T</span></span></span></span></span></span></li>
<li><code class="highlight"><span></span><span class="n">n_players</span></code>
是玩家的数量</li></ul>
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<div class="highlight"><pre><span class="lineno">572</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="o">*</span><span class="p">,</span>
<span class="lineno">573</span> <span class="n">create_new_history</span><span class="p">:</span> <span class="n">Callable</span><span class="p">[[],</span> <span class="n">History</span><span class="p">],</span>
<span class="lineno">574</span> <span class="n">epochs</span><span class="p">:</span> <span class="nb">int</span><span class="p">,</span>
<span class="lineno">575</span> <span class="n">n_players</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">2</span><span class="p">):</span></pre></div>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">581</span> <span class="bp">self</span><span class="o">.</span><span class="n">n_players</span> <span class="o">=</span> <span class="n">n_players</span>
<span class="lineno">582</span> <span class="bp">self</span><span class="o">.</span><span class="n">epochs</span> <span class="o">=</span> <span class="n">epochs</span>
<span class="lineno">583</span> <span class="bp">self</span><span class="o">.</span><span class="n">create_new_history</span> <span class="o">=</span> <span class="n">create_new_history</span></pre></div>
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<p>包含所有信息<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqcw" style=""><span class="mord" style=""><span class="mord mathcal coloredeq eqef" style="margin-right:0.07382em">I</span></span></span></span></span></span></span>集合的字典</p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">585</span> <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span> <span class="o">=</span> <span class="p">{}</span></pre></div>
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<p>分析追踪器</p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">587</span> <span class="bp">self</span><span class="o">.</span><span class="n">tracker</span> <span class="o">=</span> <span class="n">InfoSetTracker</span><span class="p">()</span></pre></div>
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<p>返回给<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span></span></span></span></span>定历史记录中当前玩家的信息集<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">589</span> <span class="k">def</span> <span class="nf">_get_info_set</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">:</span> <span class="n">History</span><span class="p">):</span></pre></div>
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<div class="highlight"><pre><span class="lineno">593</span> <span class="n">info_set_key</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">info_set_key</span><span class="p">()</span>
<span class="lineno">594</span> <span class="k">if</span> <span class="n">info_set_key</span> <span class="ow">not</span> <span class="ow">in</span> <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">:</span>
<span class="lineno">595</span> <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">[</span><span class="n">info_set_key</span><span class="p">]</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">new_info_set</span><span class="p">()</span>
<span class="lineno">596</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">[</span><span class="n">info_set_key</span><span class="p">]</span></pre></div>
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<h3>走树</h3>
<p>此函数遍历游戏树。</p>
<ul><li><code class="highlight"><span></span><span class="n">h</span></code>
是当前的历史<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></li>
<li><code class="highlight"><span></span><span class="n">i</span></code>
是我们正在计算后悔<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>的那个玩家</li>
<li><a href="#HistoryProbability"><code class="highlight"><span></span><span class="n">pi_i</span></code>
</a>是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.2309039999999998em;vertical-align:-0.258664em;"></span><span class="mord coloredeq eqck" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></li>
</ul><li><a href="#HistoryProbability"><code class="highlight"><span></span><span class="n">pi_neg_i</span></code>
</a>是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.289235em;vertical-align:-0.316995em;"></span><span class="mord coloredeq eqcg" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9722399999999999em;"><span style="top:-2.441336em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.316995em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span></li>
<p>它返回预期的实用程序，对于历史记录，<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span>其中<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.83333em;vertical-align:-0.15em;"></span><span class="mord coloredeq eqeb" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>是一组带有前缀的终端历史记录<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></p>
<p>在行走发球台时，它会更新总的遗憾<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.0999949999999998em;vertical-align:-0.258664em;"></span><span class="mord" style="color:orange"><span class="mord" style=""><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-2.441336em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.258664em;"><span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span>。</p>
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<div class="highlight"><pre><span class="lineno">598</span> <span class="k">def</span> <span class="nf">walk_tree</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">h</span><span class="p">:</span> <span class="n">History</span><span class="p">,</span> <span class="n">i</span><span class="p">:</span> <span class="n">Player</span><span class="p">,</span> <span class="n">pi_i</span><span class="p">:</span> <span class="nb">float</span><span class="p">,</span> <span class="n">pi_neg_i</span><span class="p">:</span> <span class="nb">float</span><span class="p">)</span> <span class="o">-&gt;</span> <span class="nb">float</span><span class="p">:</span></pre></div>
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<p>如果是终端历史记录，则<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqdm" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqej" style="margin-right:0.07153em">Z</span></span></span></span></span></span></span>返回终端实用程序<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdq" style=""><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span></span>。</p>
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<div class="highlight"><pre><span class="lineno">619</span> <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">is_terminal</span><span class="p">():</span>
<span class="lineno">620</span> <span class="k">return</span> <span class="n">h</span><span class="o">.</span><span class="n">terminal_utility</span><span class="p">(</span><span class="n">i</span><span class="p">)</span></pre></div>
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<p>如果是偶然事件<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord coloredeq eqdf" style=""><span class="mord" style=""><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqeh" style="margin-right:0.13889em">P</span></span><span class="mopen coloredeq eqdu" style="">(</span><span class="mord coloredeq eqdu" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose coloredeq eqdu" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel" style="">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqel" style="">c</span></span></span></span></span></span></span>示例 a 然后转到下一步。</p>
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<div class="highlight"><pre><span class="lineno">622</span> <span class="k">elif</span> <span class="n">h</span><span class="o">.</span><span class="n">is_chance</span><span class="p">():</span>
<span class="lineno">623</span> <span class="n">a</span> <span class="o">=</span> <span class="n">h</span><span class="o">.</span><span class="n">sample_chance</span><span class="p">()</span>
<span class="lineno">624</span> <span class="k">return</span> <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="n">h</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">pi_i</span><span class="p">,</span> <span class="n">pi_neg_i</span><span class="p">)</span></pre></div>
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<p>获取当前玩家的信息集<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">627</span> <span class="n">I</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">_get_info_set</span><span class="p">(</span><span class="n">h</span><span class="p">)</span></pre></div>
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<p>要存储<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.15557em;vertical-align:-0.40557em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop" style=""><span class="mop op-symbol small-op" style="position:relative;top:-0.0000050000000000050004em">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.17862099999999992em;"><span style="top:-2.40029em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.40557em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>
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<div class="highlight"><pre><span class="lineno">629</span> <span class="n">v</span> <span class="o">=</span> <span class="mi">0</span></pre></div>
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<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbe" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord mtight" style=""><span class="mord mtight" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span>为每个动作存储<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.5782em;vertical-align:-0.0391em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">A</span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mclose">)</span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">633</span> <span class="n">va</span> <span class="o">=</span> <span class="p">{}</span></pre></div>
</div>
</div>
<div class='section' id='section-69'>
<div class='docs'>
<div class='section-link'>
<a href='#section-69'>#</a>
</div>
<p>遍历所有操作</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">636</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span></pre></div>
</div>
</div>
<div class='section' id='section-70'>
<div class='docs'>
<div class='section-link'>
<a href='#section-70'>#</a>
</div>
<p>如果当前玩家是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>，</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">638</span> <span class="k">if</span> <span class="n">i</span> <span class="o">==</span> <span class="n">h</span><span class="o">.</span><span class="n">player</span><span class="p">():</span></pre></div>
</div>
</div>
<div class='section' id='section-71'>
<div class='docs'>
<div class='section-link'>
<a href='#section-71'>#</a>
</div>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.3644800000000004em;vertical-align:-1.4322400000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqck" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcg" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">643</span> <span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="n">h</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">pi_i</span> <span class="o">*</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">],</span> <span class="n">pi_neg_i</span><span class="p">)</span></pre></div>
</div>
</div>
<div class='section' id='section-72'>
<div class='docs'>
<div class='section-link'>
<a href='#section-72'>#</a>
</div>
<p>否则，</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">645</span> <span class="k">else</span><span class="p">:</span></pre></div>
</div>
</div>
<div class='section' id='section-73'>
<div class='docs'>
<div class='section-link'>
<a href='#section-73'>#</a>
</div>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.3644800000000004em;vertical-align:-1.4322400000000002em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.9322400000000002em;"><span style="top:-3.93224em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqck" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span></span></span><span style="top:-2.25em;"><span class="pstrut" style="height:3.02224em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord coloredeq eqcg" style=""><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style="">−</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.4322400000000002em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p></p>
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<div class='code'>
<div class="highlight"><pre><span class="lineno">650</span> <span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="n">h</span> <span class="o">+</span> <span class="n">a</span><span class="p">,</span> <span class="n">i</span><span class="p">,</span> <span class="n">pi_i</span><span class="p">,</span> <span class="n">pi_neg_i</span> <span class="o">*</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">])</span></pre></div>
</div>
</div>
<div class='section' id='section-74'>
<div class='docs'>
<div class='section-link'>
<a href='#section-74'>#</a>
</div>
<p><span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.266005em;vertical-align:-1.516005em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.808995em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqek" style=""><span class="mord mathnormal mtight" style="">a</span></span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">A</span><span class="mopen mtight">(</span><span class="mord mtight coloredeq eqef" style=""><span class="mord mathnormal mtight" style="margin-right:0.07847em">I</span></span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.516005em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size4">[</span></span><span class="mord coloredeq eqda" style=""><span class="mord" style=""><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqef" style=""><span class="mord mathnormal" style="margin-right:0.07847em">I</span></span><span class="mclose">)</span><span class="mopen">(</span><span class="mord coloredeq eqek" style=""><span class="mord mathnormal" style="">a</span></span><span class="mclose">)</span><span class="mord coloredeq eqbe" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord mtight" style=""><span class="mord mtight" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="delimsizing size4">]</span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">655</span> <span class="n">v</span> <span class="o">=</span> <span class="n">v</span> <span class="o">+</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">*</span> <span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span></pre></div>
</div>
</div>
<div class='section' id='section-75'>
<div class='docs'>
<div class='section-link'>
<a href='#section-75'>#</a>
</div>
<p>如果当前玩家是<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>，更新累积策略和总遗憾</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">659</span> <span class="k">if</span> <span class="n">h</span><span class="o">.</span><span class="n">player</span><span class="p">()</span> <span class="o">==</span> <span class="n">i</span><span class="p">:</span></pre></div>
</div>
</div>
<div class='section' id='section-76'>
<div class='docs'>
<div class='section-link'>
<a href='#section-76'>#</a>
</div>
<p>更新累积策略<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:3.0954490000000003em;vertical-align:-1.267113em;"></span><span class="mord coloredeq eqbd" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span><span class="mrel mtight" style="">=</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqed" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord" style="color:lightgreen"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:lightgreen">)</span><span class="mopen" style="color:lightgreen">(</span><span class="mord" style="color:lightgreen"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:lightgreen">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.157819em;vertical-align:-1.329483em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="delimsizing size2">[</span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.8478869999999998em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdl" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.329483em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mclose">)</span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="delimsizing size2">]</span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">664</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span>
<span class="lineno">665</span> <span class="n">I</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">cumulative_strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">+</span> <span class="n">pi_i</span> <span class="o">*</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">]</span></pre></div>
</div>
</div>
<div class='section' id='section-77'>
<div class='docs'>
<div class='section-link'>
<a href='#section-77'>#</a>
</div>
<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:7.749200999999999em;vertical-align:-3.6246005em;"></span><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1246005em;"><span style="top:-7.1093805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord" style="color:coral"><span class="mord" style=""><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span><span style="top:-5.2993805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"></span></span><span style="top:-1.7708485em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord coloredeq eqbt" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqei" style="margin-right:0.13889em">T</span></span><span class="mord" style=""><span class="mord" style="color:orange"><span class="mord coloredeq eqdt" style=""><span class="mord mathnormal" style="margin-right:0.00773em">R</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.00773em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqei" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span></span><span class="mopen" style="color:orange">(</span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:orange">,</span><span class="mspace" style="margin-right:0.16666666666666666em"></span><span class="mord" style="color:orange"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:orange">)</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6246005em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.1246005em;"><span style="top:-7.1093805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord" style="color:pink"><span class="mord" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord" style="color:pink"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.03588em">v</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.22222em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="color:pink">(</span><span class="mord" style="color:pink"><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.843556em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mpunct" style="color:pink">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:pink"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="color:pink">)</span></span></span><span style="top:-5.2993805em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">π</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.02224em;"><span style="top:-2.4530000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight coloredeq eqen" style=""><span class="mord mathnormal mtight" style="">i</span></span></span></span></span><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.305331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord coloredeq eqem" style=""><span class="mord mathnormal" style="">h</span></span><span class="mclose">)</span><span class="mord"><span class="delimsizing size2">(</span></span><span class="mord coloredeq eqbe" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0222399999999998em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mtight coloredeq eqdi" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8703428571428571em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mord mtight" style=""><span class="mord mtight" style="">∣</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3567071428571427em;margin-left:0em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mrel mtight" style="">→</span><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqek" style="">a</span></span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.14329285714285717em;"><span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span><span class="mord"><span class="delimsizing size2">)</span></span></span></span><span style="top:-1.7708485em;"><span class="pstrut" style="height:3.828336em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.882887em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight coloredeq eqep" style=""><span class="mord mathnormal mtight" style="">t</span></span><span class="mrel mtight">=</span><span class="mord mtight coloredeq eqed" style=""><span class="mord mtight" style="">1</span></span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight coloredeq eqei" style=""><span class="mord mathnormal mtight" style="margin-right:0.13889em">T</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.267113em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord accent" style=""><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6678599999999999em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal" style="margin-right:0.02778em">r</span></span><span style="top:-3.35em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.19444em;"><span class="mord" style="">~</span></span></span></span></span></span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8435559999999999em;"><span style="top:-2.4530000000000003em;margin-left:-0.02778em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mopen" style="color:coral">(</span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mpunct" style="color:coral">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style="color:coral"><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="color:coral">)</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:3.6246005em;"><span></span></span></span></span></span></span></span></span></span></span></span></span><p></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">678</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span>
<span class="lineno">679</span> <span class="n">I</span><span class="o">.</span><span class="n">regret</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">+=</span> <span class="n">pi_neg_i</span> <span class="o">*</span> <span class="p">(</span><span class="n">va</span><span class="p">[</span><span class="n">a</span><span class="p">]</span> <span class="o">-</span> <span class="n">v</span><span class="p">)</span></pre></div>
</div>
</div>
<div class='section' id='section-78'>
<div class='docs'>
<div class='section-link'>
<a href='#section-78'>#</a>
</div>
<p>更新策略<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.043556em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">682</span> <span class="n">I</span><span class="o">.</span><span class="n">calculate_strategy</span><span class="p">()</span></pre></div>
</div>
</div>
<div class='section' id='section-79'>
<div class='docs'>
<div class='section-link'>
<a href='#section-79'>#</a>
</div>
<p>返回玩家的预期效用<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.65952em;vertical-align:0em;"></span><span class="mord coloredeq eqen" style=""><span class="mord mathnormal" style="">i</span></span></span></span></span></span>，<span ><span class="katex-display"><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:2.450201em;vertical-align:-1.400196em;"></span><span class="mord coloredeq eqbp" style=""><span class="mop op-limits" style=""><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.050005em;"><span style="top:-1.855664em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mrel mtight" style="">∈</span><span class="mord mtight" style=""><span class="mord mtight coloredeq eqeb" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqej" style="margin-right:0.07153em">Z</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3448em;"><span style="top:-2.3487714285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqem" style="">h</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15122857142857138em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.0500049999999996em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op" style="">∑</span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:1.400196em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal" style="margin-right:0.03588em">π</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqdp" style="margin-right:0.03588em">σ</span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqem" style="">h</span></span><span class="mpunct" style="">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span><span class="mord" style=""><span class="mord mathnormal" style="">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqen" style="">i</span></span></span></span></span><span class="vlist-s">​</span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqeq" style="margin-right:0.04398em">z</span></span><span class="mclose" style="">)</span></span></span></span></span></span></span></p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">686</span> <span class="k">return</span> <span class="n">v</span></pre></div>
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</div>
<div class='section' id='section-80'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-80'>#</a>
</div>
<h3>迭代更新<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:1.043556em;vertical-align:-0.25em;"></span><span class="mord" style="color:lightgreen"><span class="mord" style=""><span class="mord coloredeq eqdi" style=""><span class="mord" style=""><span class="mord mathnormal coloredeq eqdp" style="margin-right:0.03588em">σ</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7935559999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight" style=""><span class="mord mtight" style=""><span class="mord mathnormal mtight coloredeq eqep" style="">t</span></span></span></span></span></span></span></span></span></span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqef" style="margin-right:0.07847em">I</span></span><span class="mclose" style="">)</span><span class="mopen" style="">(</span><span class="mord" style=""><span class="mord mathnormal coloredeq eqek" style="">a</span></span><span class="mclose" style="">)</span></span></span></span></span></span></h3>
<p>这更新了<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord coloredeq eqei" style=""><span class="mord mathnormal" style="margin-right:0.13889em">T</span></span></span></span></span></span>迭代策略。</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">688</span> <span class="k">def</span> <span class="nf">iterate</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
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</div>
<div class='section' id='section-81'>
<div class='docs'>
<div class='section-link'>
<a href='#section-81'>#</a>
</div>
<p>循环几<code class="highlight"><span></span><span class="n">epochs</span></code>
次</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">696</span> <span class="k">for</span> <span class="n">t</span> <span class="ow">in</span> <span class="n">monit</span><span class="o">.</span><span class="n">iterate</span><span class="p">(</span><span class="s1">&#39;Train&#39;</span><span class="p">,</span> <span class="bp">self</span><span class="o">.</span><span class="n">epochs</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-82'>
<div class='docs'>
<div class='section-link'>
<a href='#section-82'>#</a>
</div>
<p>Walk tree 并更新每位玩家的遗憾</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">698</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">n_players</span><span class="p">):</span>
<span class="lineno">699</span> <span class="bp">self</span><span class="o">.</span><span class="n">walk_tree</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">create_new_history</span><span class="p">(),</span> <span class="n">cast</span><span class="p">(</span><span class="n">Player</span><span class="p">,</span> <span class="n">i</span><span class="p">),</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span></pre></div>
</div>
</div>
<div class='section' id='section-83'>
<div class='docs'>
<div class='section-link'>
<a href='#section-83'>#</a>
</div>
<p>跟踪数据以进行分析</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">702</span> <span class="n">tracker</span><span class="o">.</span><span class="n">add_global_step</span><span class="p">()</span>
<span class="lineno">703</span> <span class="bp">self</span><span class="o">.</span><span class="n">tracker</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">)</span>
<span class="lineno">704</span> <span class="n">tracker</span><span class="o">.</span><span class="n">save</span><span class="p">()</span></pre></div>
</div>
</div>
<div class='section' id='section-84'>
<div class='docs'>
<div class='section-link'>
<a href='#section-84'>#</a>
</div>
<p>每次<span ><span class="katex"><span aria-hidden="true" class="katex-html"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord coloredeq eqed" style=""><span class="mord" style="">1</span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">000</span></span></span></span></span>迭代都保存检查点</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">707</span> <span class="k">if</span> <span class="p">(</span><span class="n">t</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="o">%</span> <span class="mi">1_000</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="lineno">708</span> <span class="n">experiment</span><span class="o">.</span><span class="n">save_checkpoint</span><span class="p">()</span></pre></div>
</div>
</div>
<div class='section' id='section-85'>
<div class='docs'>
<div class='section-link'>
<a href='#section-85'>#</a>
</div>
<p>打印信息集</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">711</span> <span class="n">logger</span><span class="o">.</span><span class="n">inspect</span><span class="p">(</span><span class="bp">self</span><span class="o">.</span><span class="n">info_sets</span><span class="p">)</span></pre></div>
</div>
</div>
<div class='section' id='section-86'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-86'>#</a>
</div>
<h3>信息集追踪器</h3>
<p>这是一个小帮助类，用于跟踪信息集中的数据</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">714</span><span class="k">class</span> <span class="nc">InfoSetTracker</span><span class="p">:</span></pre></div>
</div>
</div>
<div class='section' id='section-87'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-87'>#</a>
</div>
<p>设置追踪指示器</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">720</span> <span class="k">def</span> <span class="fm">__init__</span><span class="p">(</span><span class="bp">self</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-88'>
<div class='docs'>
<div class='section-link'>
<a href='#section-88'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">724</span> <span class="n">tracker</span><span class="o">.</span><span class="n">set_histogram</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;strategy.*&#39;</span><span class="p">)</span>
<span class="lineno">725</span> <span class="n">tracker</span><span class="o">.</span><span class="n">set_histogram</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;average_strategy.*&#39;</span><span class="p">)</span>
<span class="lineno">726</span> <span class="n">tracker</span><span class="o">.</span><span class="n">set_histogram</span><span class="p">(</span><span class="sa">f</span><span class="s1">&#39;regret.*&#39;</span><span class="p">)</span></pre></div>
</div>
</div>
<div class='section' id='section-89'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-89'>#</a>
</div>
<p>跟踪所有信息集中的数据</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">728</span> <span class="k">def</span> <span class="fm">__call__</span><span class="p">(</span><span class="bp">self</span><span class="p">,</span> <span class="n">info_sets</span><span class="p">:</span> <span class="n">Dict</span><span class="p">[</span><span class="nb">str</span><span class="p">,</span> <span class="n">InfoSet</span><span class="p">]):</span></pre></div>
</div>
</div>
<div class='section' id='section-90'>
<div class='docs'>
<div class='section-link'>
<a href='#section-90'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">732</span> <span class="k">for</span> <span class="n">I</span> <span class="ow">in</span> <span class="n">info_sets</span><span class="o">.</span><span class="n">values</span><span class="p">():</span>
<span class="lineno">733</span> <span class="n">avg_strategy</span> <span class="o">=</span> <span class="n">I</span><span class="o">.</span><span class="n">get_average_strategy</span><span class="p">()</span>
<span class="lineno">734</span> <span class="k">for</span> <span class="n">a</span> <span class="ow">in</span> <span class="n">I</span><span class="o">.</span><span class="n">actions</span><span class="p">():</span>
<span class="lineno">735</span> <span class="n">tracker</span><span class="o">.</span><span class="n">add</span><span class="p">({</span>
<span class="lineno">736</span> <span class="sa">f</span><span class="s1">&#39;strategy.</span><span class="si">{</span><span class="n">I</span><span class="o">.</span><span class="n">key</span><span class="si">}</span><span class="s1">.</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">:</span> <span class="n">I</span><span class="o">.</span><span class="n">strategy</span><span class="p">[</span><span class="n">a</span><span class="p">],</span>
<span class="lineno">737</span> <span class="sa">f</span><span class="s1">&#39;average_strategy.</span><span class="si">{</span><span class="n">I</span><span class="o">.</span><span class="n">key</span><span class="si">}</span><span class="s1">.</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">:</span> <span class="n">avg_strategy</span><span class="p">[</span><span class="n">a</span><span class="p">],</span>
<span class="lineno">738</span> <span class="sa">f</span><span class="s1">&#39;regret.</span><span class="si">{</span><span class="n">I</span><span class="o">.</span><span class="n">key</span><span class="si">}</span><span class="s1">.</span><span class="si">{</span><span class="n">a</span><span class="si">}</span><span class="s1">&#39;</span><span class="p">:</span> <span class="n">I</span><span class="o">.</span><span class="n">regret</span><span class="p">[</span><span class="n">a</span><span class="p">],</span>
<span class="lineno">739</span> <span class="p">})</span></pre></div>
</div>
</div>
<div class='section' id='section-91'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-91'>#</a>
</div>
<h3>可配置 CFR 模块</h3>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">742</span><span class="k">class</span> <span class="nc">CFRConfigs</span><span class="p">(</span><span class="n">BaseConfigs</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-92'>
<div class='docs'>
<div class='section-link'>
<a href='#section-92'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">746</span> <span class="n">create_new_history</span><span class="p">:</span> <span class="n">Callable</span><span class="p">[[],</span> <span class="n">History</span><span class="p">]</span>
<span class="lineno">747</span> <span class="n">epochs</span><span class="p">:</span> <span class="nb">int</span> <span class="o">=</span> <span class="mi">1_00_000</span>
<span class="lineno">748</span> <span class="n">cfr</span><span class="p">:</span> <span class="n">CFR</span> <span class="o">=</span> <span class="s1">&#39;simple_cfr&#39;</span></pre></div>
</div>
</div>
<div class='section' id='section-93'>
<div class='docs doc-strings'>
<div class='section-link'>
<a href='#section-93'>#</a>
</div>
<p>初始化 <strong>CFR</strong> 算法</p>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">751</span><span class="nd">@option</span><span class="p">(</span><span class="n">CFRConfigs</span><span class="o">.</span><span class="n">cfr</span><span class="p">)</span>
<span class="lineno">752</span><span class="k">def</span> <span class="nf">simple_cfr</span><span class="p">(</span><span class="n">c</span><span class="p">:</span> <span class="n">CFRConfigs</span><span class="p">):</span></pre></div>
</div>
</div>
<div class='section' id='section-94'>
<div class='docs'>
<div class='section-link'>
<a href='#section-94'>#</a>
</div>
</div>
<div class='code'>
<div class="highlight"><pre><span class="lineno">756</span> <span class="k">return</span> <span class="n">CFR</span><span class="p">(</span><span class="n">create_new_history</span><span class="o">=</span><span class="n">c</span><span class="o">.</span><span class="n">create_new_history</span><span class="p">,</span>
<span class="lineno">757</span> <span class="n">epochs</span><span class="o">=</span><span class="n">c</span><span class="o">.</span><span class="n">epochs</span><span class="p">)</span></pre></div>
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