In this work we propose a kinetic formulation for evolutionary game theory for zero sum games when the agents use mixed strategies. We start with a simple adaptive rule, where after an encounter each agent increases by a small amount $ h $ the probability of playing the successful pure strategy used in the match. We derive the Boltzmann equation which describes the macroscopic effects of this microscopical rule, and we obtain a first order, nonlocal, partial differential equation as the limit when $ h $ goes to zero.We study the relationship between this equation and the well known replicator equations, showing the equivalence between the concepts of Nash equilibria, stationary solutions of the partial differential equation, and the equilibria of the replicator equations. Finally, we relate the long-time behavior of solutions to the partial differential equation and the stability of the replicator equations.

中文翻译：

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混合策略中的演化博弈论：从微观相互作用到动力学方程

在这项工作中，当代理商使用混合策略时，我们为零和博弈的演化博弈理论提出了一种动力学公式。我们从一个简单的自适应规则开始，在该规则中，每位特工在遭遇比赛后会增加$ h $的小概率，以发挥比赛中使用的成功纯策略的概率。我们推导出描述该微观规则的宏观效应的玻尔兹曼方程，并获得一阶非局部偏微分方程作为$ h $变为零时的极限。我们研究了该方程与众所周知的复制子之间的关系。方程，显示纳什均衡的概念，偏微分方程的固定解和复制器方程的均衡之间的等价关系。最后，