The Letter presents a novel way to connect random walks, stochastic differential equations, and evolutionary game theory. We introduce a new concept of a potential function for discrete-space stochastic systems. It is based on a correspondence between one-dimensional stochastic differential equations and random walks, which may be exact not only in the continuous limit but also in finite-state spaces. Our method is useful for computation of fixation probabilities in discrete stochastic dynamical systems with two absorbing states. We apply it to evolutionary games, formulating two simple and intuitive criteria for evolutionary stability of pure Nash equilibria in finite populations. In particular, we show that the $1/3$ law of evolutionary games, introduced by Nowak et al. [Nature, 2004], follows from a more general mean-potential law.

中文翻译：

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演化博弈中的均势定律

这封信提出了一种新颖的方式来连接随机游走，随机微分方程和进化博弈论。我们介绍了离散空间随机系统潜在功能的新概念。它基于一维随机微分方程和随机游动之间的对应关系，不仅在连续极限中而且在有限状态空间中都可能是精确的。我们的方法对于计算具有两个吸收状态的离散随机动力系统中的注视概率很有用。我们将其应用于演化博弈，为纯Nash均衡在有限种群中的演化稳定性制定了两个简单直观的标准。特别是，我们表明$\mathrm{1个}/3$Nowak等人介绍的进化博弈定律。[ Nature，2004]，来自更一般的均值-电位定律。