混合策略中的进化对策论：从微观相互作用到动力学方程
Evolutionary game theory in mixed strategies: from microscopic interactions to kinetic equations
摘要：
在这项工作中，我们提出了一个动力学制定的进化对策论的零和对策时，代理人使用混合策略。我们从一个简单的自适应规则开始，在遭遇战之后，每个智能体都增加了在比赛中使用的成功纯策略的概率。我们推导出玻尔兹曼方程，它描述了这种微观规则的宏观效应，我们得到一个一阶，非局部，偏微分方程作为极限时的概率变化为零。我们研究了这个方程和著名的复制者方程之间的关系，证明了纳什均衡、偏微分方程的定态解和复制者方程的均衡之间的等价性.最后，我们将偏微分方程解的长时间行为与复制子方程的稳定性联系起来。
英文摘要：
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 the probability of play 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 the probability change 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.
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