江鸿助手（五十）——多重均衡博弈和混合策略（1）

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分享兴趣，传播快乐，增长见闻，留下美好。亲爱的您，这里是Learningyard新学苑。今天，小编给大家带来：江鸿助手（五十）——多重均衡博弈和混合策略（1）。

Share interest, spread happiness, increase knowledge, leave beautiful. Dear, this is Learningyard New Academy. Today, Xiaobian brings to you: Assistant Jiang Hong (50) -- Multiple Equilibrium Games and Mixed Strategies（1）.‍

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混合策略概念和混合策略纳什均衡分析方法，不仅适用于分析没有纯策略纳什均衡的博弈，也适用于分析有多个纯策略纳什均衡的博弈。本篇文章通过对夫妻之争博弈的分析进行解释。

The concept of mixed strategy and the method of mixed strategy Nash equilibrium analysis are applicable not only to the game with no pure strategy Nash equilibrium, but also to the game with multiple pure strategy Nash equilibrium. This article through the husband and wife dispute game analysis to explain.‍

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夫妻之争：已知，一对夫妻中妻子喜欢看时装秀，而丈夫喜欢足球比赛，则如果两人商议后决定去看时装秀，妻子会非常高兴，丈夫也会因此高兴，而如果决定去看足球比赛，则丈夫会十分高兴，妻子也会因此高兴，但是如果两人并没有达成共识，则两人都不高兴。下图为此博弈得益矩阵：

Marital conflict: It is known that in a couple where the wife likes to watch fashion shows and the husband likes football matches, if they decide to go to a fashion show after discussion, the wife will be very happy, so will the husband; if they decide to go to a football match, the husband will be very happy, so will the wife; but if they do not agree, neither of them will be happy. The following figure shows the benefit matrix of this game:‍

丈夫

时装

足球

妻子

时装

2,1

0,0

足球

0,0

1,3

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而利用划线法和箭头法我们可以得出结论：该博弈有两个纳什均衡，夫妻二人都选择时装秀或者足球赛。在这个博弈中，如果一方知道另一方选择的策略，明智的选择是与对方选择相同的策略。因此，两博弈方的利益有较大的一致性，他们希望对方了解自己的选择。在这个有纳什均衡的博弈汇总博弈方的决策思路和原则与没有纳什均衡的严格竞争博弈显然不同。

Using the line and arrow methods, we can conclude that there are two Nash equilibria in this game, with both spouses choosing the fashion show or the soccer game. In this game, if one side knows the strategy chosen by the other, it is wise to choose the same strategy as the other side. Therefore, the interests of the two players are more consistent, and they want the other side to know their choice. In this game with a Nash equilibrium, the players are going to have different ideas and different principles than in a strictly competitive game without a Nash equilibrium.‍

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但是，在这个博弈中夫妻双方对两个纳什均衡的偏好不同，因此，当两人从自身最大利益出发独立同时决策时，也不能确定他们会选择的策略，进而也无法知道博弈的结果。也就是说，在纯策略的范围内，这个博弈无法对各个博弈方的选择提出明确的建议，需要考虑混合策略的可能性。

However, in this game, couples have different preferences for the two Nash equilibria, so when two couples make independent decisions in their own best interests, there is no way to know which strategy they will choose, and thus the outcome of the game. That is to say, in the range of pure strategy, this game cannot make clear suggestions on the choice of each player, so the possibility of mixed strategy should be considered.‍

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如果其中一方不想让另外一方因为自己的偏好而占上风，那自己随机选择纯策略的概率分布应该让对方选择两种策略时的期望收益相等。而如果双方都这样思考并进行策略选择，那双方最终的期望得益都会比较低，很显然，这样的结果并没有双方能够沟通时的结果更好，因为在双方能够沟通的情况下，任何一方迁就另一方的选择时，前者最低得益就是1。

If one side doesn't want the other side to have the upper hand because of his preferences, then his probability distribution of choosing pure strategies at random should give the other side the same expected payoff of choosing both strategies. If both sides think this way and make strategic choices, then both sides will end up with a lower expected benefit. Obviously, this outcome is not as good as the outcome when both sides can communicate, because when both sides can communicate, the lowest benefit of the former is 1.‍

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Today's share is here, if you have a unique view of today's article, welcome to leave us a message, let us meet tomorrow, I wish you a happy today!‍

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参考文献：《经济博弈论》（第四版）谢识予，复旦大学出版社

图片来源于凡科快图和iThoughts

翻译来源于有道在线翻译

文案--闫庆红

排版--闫庆红

审核--吴金育