Mean-potential law in evolutionary games

TL;DR

This paper introduces a new potential function concept for discrete stochastic systems, linking random walks and differential equations, to analyze evolutionary game stability and generalize the 1/3 law.

Contribution

It presents a novel potential function framework for discrete stochastic systems, enabling analysis of fixation probabilities and stability in evolutionary games.

Findings

Derived a mean-potential law generalizing the 1/3 rule

Provided criteria for evolutionary stability in finite populations

Linked stochastic differential equations with random walks in game theory

Abstract

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.