江鸿助手（四十七）——混合策略博弈和纳什均衡
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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（四十七）——混合策略博弈和混合策略纳什均衡。
Share interest, spread happiness, increase knowledge, leave beautiful. Dear, this is Learningyard New Academy. Today, Xiaobian brings to you: Jiang Hong assistant - mixed strategy game and mixed strategy Nash equilibrium.
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在上一期的猜硬币博弈中，两个博弈方都以1/2的相同概率随机选择正面或反面时，并且此时双方都无法根据对方的策略，选择或调整自己的策略，从而在双方对可选策略随机选择概率分布上达到了一种稳定或均衡。下面是混合策略的具体定义。
In the coin guessing game of the last period, both players randomly chose heads or tails with the same probability of 1/2, and at this time, both players could not choose or adjust their own strategies according to the strategies of the other side, thus reaching a stability or equilibrium in the probability distribution of random selection of alternative strategies by both sides. The following is a specific definition of the hybrid strategy.
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定义：在一个博弈中，有n个博弈方，每个博弈方有k个可以选择的策略，博弈方以一个特定的概率分布（所有可选策略被选择的概率之和为1）从k个可选策略中随机选择出来的策略就被称为一个“混合策略”。与此相对的，原来意义上的策略则称为“纯策略”。纯策略也可以看作混合策略：选择该纯策略的概率为1，选择其余纯策略的概率为0。
Definition: In a game, there are n players, and each player has k strategies to choose. The strategy randomly selected by the players from k alternative strategies with a specific probability distribution is called a "mixed strategy". In contrast, a policy in its original sense is called a "pure policy". A pure strategy can also be regarded as a mixed strategy: the probability of selecting the pure strategy is 1, and the probability of choosing other pure strategies is 0.
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在一个博弈中，如果每个博弈方的纯策略空间有不同的概率分布，这就形成了不同的混合策略。所有混合策略构成一个混合策略的策略空间。而此时可以把这个混合策略的策略空间中的选择行为看作一个博弈，这个扩展出来的博弈可称为原博弈的“混合策略扩展博弈”。
In a game, if the pure strategy space of each player has a different probability distribution, this results in different mixed strategies. All mixed strategies constitute a mixed policy space. At this time, the selection behavior in the strategy space of the mixed strategy can be regarded as a game, and the expanded game can be called the "mixed strategy expanded game" of the original game.
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把策略扩展到包括混合策略后，纳什均衡仍然是成立的。只不过，此时纳什均衡博弈中的策略既可能是纯策略，也可能是混合策略。这时候的纳什均衡意味着，任何博弈方单独改变各个纯策略的概率分布都不能保证增加自身的得益。
The Nash equilibrium is still valid when the strategy is extended to include mixed strategies. But in this case, the strategy in the Nash equilibrium game could be either pure strategy or mixed strategy. The Nash equilibrium at this point means that there is no guarantee that any player will increase his own profit by changing the probability distribution of each pure strategy alone.
