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<a href="." >9. 混合策略：网球比赛</a>
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<h1 id="lecture-9">Lecture 9</h1>
<p>上一节课我们引入了一个新的概念：mixed strategies。这也是接下来几节课的重点。</p>
<p><script type="math/tex; ">p_i</script>表示使用每一个策略的概率</p>
<ul>
<li><script type="math/tex; ">p_i(s_i)</script>表示对于pure strategy <script type="math/tex; ">s_i</script>的概率是<script type="math/tex; ">p_i</script></li>
<li><script type="math/tex; ">p_i(s_i)</script>可以是0，例如(0.5, 0.5, 0); 也可以是1，这个时候表示pure strategy.也就是说我们之前所有的博弈策略都是它的子集。</li>
</ul>
<p>payoffs from the mixed strategy:</p>
<p>The expected payoff of the mixed strategy 就是期望。</p>
<h2 id="计算混合策略的期望例子">计算混合策略的期望例子</h2>
<table>
<thead>
<tr>
<th></th>
<th>a</th>
<th>b</th>
<th>A, B选择的概率</th>
</tr>
</thead>
<tbody>
<tr>
<td>A</td>
<td>2, 1</td>
<td>0, 0</td>
<td><script type="math/tex; ">\frac{1}{5}</script></td>
</tr>
<tr>
<td>B</td>
<td>0, 0</td>
<td>1, 2</td>
<td><script type="math/tex; ">\frac{4}{5}</script></td>
</tr>
<tr>
<td>a, b选择的概率</td>
<td><script type="math/tex; ">\frac{1}{2}</script></td>
<td><script type="math/tex; ">\frac{1}{2}</script></td>
</tr>
</tbody>
</table>
<p>因此玩家1的mixed strategy p = (1/5, 4/5)</p>
<p>玩家2的mixed strategy q = (1/2, 1/2)。</p>
<p>那么如何计算p的payoffs呢？</p>
<p>分成两部，首先计算pure strategy的期望，然后计算混合期望：</p>
<ol>
<li><script type="math/tex; ">Eu_1(A, q) = 2*0.5+0*0.5 = 1</script></li>
</ol>
<p><script type="math/tex; ">Eu_1(B, q) = 0*0.5+1*0.5 = 0.5</script></p>
<ol>
<li>计算混合期望：</li>
</ol>
<p><script type="math/tex; ">Eu_1(p, q) = 0.2*1+0.8*0.5 = 0.6</script></p>
<p><div class="alert alert-success" style="display: table"><i class="fa fa-check" style="float: left; padding: 5px"></i><div style="margin-left: 32px"><p> 如何混合策略是最佳对策，那么混合策略中的每一个pure strategy 也必须是最佳对策。也就是说，每一个都必须产生相同的payoffs。</p>
</div></div></p>
<p>解释：如果有一个pure strategy不是最佳对策，那么就会拉低其他的平均期望。</p>
<p>定义：A mixed strategy profile <script type="math/tex; ">(p_1^*, p_2^*, \dots, p_n^*)</script>在NE 当且仅当对每一个i，<script type="math/tex; ">p_i^*</script>都是最佳对策。</p>
<h2 id="网球比赛">网球比赛</h2>
<table>
<thead>
<tr>
<th style="text-align:center"></th>
<th style="text-align:center">l</th>
<th style="text-align:center">r</th>
<th style="text-align:center">L, R选择的概率</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">L</td>
<td style="text-align:center">50, 50</td>
<td style="text-align:center">80, 20</td>
<td style="text-align:center">p</td>
</tr>
<tr>
<td style="text-align:center">R</td>
<td style="text-align:center">90, 10</td>
<td style="text-align:center">20, 80</td>
<td style="text-align:center">1-p</td>
</tr>
<tr>
<td style="text-align:center">l, r选择的概率</td>
<td style="text-align:center">q</td>
<td style="text-align:center">1-q</td>
</tr>
</tbody>
</table>
<p>找到纳什均衡点，那么玩家1的两个pure strategy的期望必须相等！</p>
<p><script type="math/tex; ">Eu_1(L, q) = 50q+80(1-q)</script></p>
<p><script type="math/tex; ">Eu_1(R, q) = 90q+20(1-q)</script></p>
<p>令两者相等（也就是对于任意一种pure strategy，它们的payoffs必须相等）：</p>
<p><script type="math/tex; ">50q+80(1-q) = 90q+20(1-q)</script>, 得到<script type="math/tex; ">q = 0.6</script></p>
<p>同理可以求得p = 0.7</p>
<p>因此<script type="math/tex; ">NE = [(0.7, 0.3), (0.6, 0.4)]</script></p>
<p>如果发现玩家2打左边的概率大于0.6，那么玩家1应该往右打，这样可以增加得分的概率。（可以通过计算获得）</p>
<h2 id="证明上面状态是纳什均衡点">证明上面状态是纳什均衡点</h2>
<p>我们将上面的概率带入：</p>
<p>玩家1的payoffs: L -> 50*0.6+80*0.4 = 0.62</p>
<p>R -> 0.62</p>
<p>因此mixed strategy payoffs = 0.62*0.7+0.62*0.3 = 0.62</p>
<p>我们改变(p, 1-p)，发现无论怎么改变，总期望的收益都是不变的，因此没有<strong>严格大于</strong>其期望的策略，使得它能够大于0.62。</p>
<h2 id="微小扰动">微小扰动</h2>
<p>现在玩家2请了教练，增加了自己左手的能力，新的收益矩阵如下：</p>
<table>
<thead>
<tr>
<th style="text-align:center"></th>
<th style="text-align:center">l</th>
<th style="text-align:center">r</th>
<th style="text-align:center">L, R选择的概率</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">L</td>
<td style="text-align:center">30, 70</td>
<td style="text-align:center">80, 20</td>
<td style="text-align:center">p</td>
</tr>
<tr>
<td style="text-align:center">R</td>
<td style="text-align:center">90, 10</td>
<td style="text-align:center">20, 80</td>
<td style="text-align:center">1-p</td>
</tr>
<tr>
<td style="text-align:center">l, r选择的概率</td>
<td style="text-align:center">q</td>
<td style="text-align:center">1-q</td>
</tr>
</tbody>
</table>
<p>通过计算，p,q都会下降，这表明大家都会更多的通过右手进行比赛。</p>
<h2 id="mixed-strategy-examples">mixed strategy examples</h2>
<ul>
<li>足球射门</li>
<li>911后安检仪数量不足，机场放安检仪：他们就是随机进行检查。</li>
</ul>
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