  The Letter presents a novel way to connect random walks, stochastic
differential equations, and evolutionary game theory. We introduce a new
concept of 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.
