江鸿助手（五十三）——混合策略和严格下策反复消去法
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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（五十三）——混合策略和严格下策反复消去法。
Share interest, spread happiness, increase knowledge, leave good. Dear you, here is Learningyard New Academy, today, Xiaobian brings you: Jiang Hong assistant (53) – Mixed strategy and strict strategy repeated elimination method.
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前面三期我们一起学习了多重均衡博弈和混合策略，本期我们一起来学习混合策略情形下的严格下策反复消去法。首先，即使是在包括混合策略的情况下，以下关于严格下策反复消去法的三条结论也依然是成立的：
1、任何博弈方都不会采用任何严格下策；
2、严格下策反复消去法不会消去任何纳什均衡；
3、如果经过反复消去后留下的策略组合是唯一的，那其一定是纳什均衡。因此在考虑混合策略的情况时，我们仍然这个方法进行分析。
In the previous three installments, we studied multiple equilibrium games and mixed strategies together. In this installment, we will study the strict iterative elimination method for mixed strategies. First, even when mixed strategies are included, the following three conclusions about strict iterative elimination hold true:
1. No player will adopt any strict strategy;
2, the strict elimination method will not eliminate any Nash equilibrium;
3. If the combination of strategies left after repeated elimination is unique, it must be a Nash equilibrium. Therefore, when considering the case of mixed strategies, we still use this method for analysis.
03
在下图所示博弈中，博弈方1有三种可选策略，博弈方2有两种可选策略。很显然，在只考虑纯策略时，此博弈对任何一个博弈方来说混合策略纳什均衡，都不存在任何严格下策，因为其中一方的选择会让另一方不同策略对应的得益情况发生变化。因此如果只考虑纯策略混合策略纳什均衡，分析这个博弈不能使用严格下策反复消去法。
In the game shown below, player 1 has three alternative strategies and player 2 has two alternative strategies. It is clear that when only pure strategies are considered, the game does not have any strict disadvantages for either player, because the choice of one side will change the benefits of the other side's different strategies. So if we're just looking at pure strategies, we can't analyze this game using strict optimal iterative elimination.
博弈方2
甲
乙
博弈方1
3,1
0,2
0,2
3,3
1,3
1,1
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然而，如果这个博弈中能够采取混合策略，情况就会发生变化。假设博弈方1采取以概率分布（1/2，1/2，0)随机选择A、B、C的混合策略，那么相对于这个混合策略，纯策略D就是博弈方1的严格下策。因为通过计算期望得益我们就可以发现，无论博弈方2采用哪种所有可能的纯策略或混合策略，博弈方1采用上述混合策略的期望得益始终为3/2，大于采用纯策略C的得益1。
However, if a mixed strategy can be adopted in this game, the situation changes. Suppose that player 1 adopts A mixed strategy in which A, B, and C are randomly selected with a probability distribution (1/2, 1/2, 0), then relative to this mixed strategy, pure strategy D is the strict worst strategy of player 1. Because by calculating the expected payoff, we can see that regardless of all possible pure or mixed strategies adopted by player 2, the expected payoff for player 1 of the above mixed strategies is always 3/2, which is greater than the payoff 1 of pure strategy C.
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因此，根据严格下策反复消去法，可以把博弈方1的纯策略C剔除出去。此时我们就发现，剔除策略C之后的博弈就可以轻易地找到纳什均衡（B,乙）。但是，不是考虑混合策略以后，博弈中一定会存在可以剔除出去的的纯策略严格下策。
Therefore, according to the strict optimal repeated elimination method, the pure strategy C of game 1 can be eliminated. Now we see that the game without strategy C can easily find the Nash equilibrium (B, b). But instead of thinking about mixed strategies, there must be pure strategies that can be eliminated from the game.
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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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