Stochastic Evolutionary Game Dynamics and Their Selection Mechanisms

Abstract

An underlying assumption of deterministic evolutionary game dynamics is that all individuals interact with each other in infinite populations, which seems unrealistic since in reality populations are always finite in size and even disturbed by stochastic effects and random drift. Developed in the context of finite populations and described by finite state Markov processes, stochastic evolutionary game dynamics have received much attention recently. However, the relationship between two types of evolutionary dynamics so far has failed to be thoroughly understood. In this paper, we establish several classes of selection mechanisms in large populations under which corresponding stochastic evolutionary dynamics approach the imitative dynamic (including the replicator dynamic), the impartial pairwise comparison dynamic (including the Smith dynamic) and the separable excess payoff dynamic (including the Brown---von Neumann---Nash dynamic) respectively in adjusted forms. In other words, we present intuitive interpretations from a statistical perspective for these deterministic dynamics by constructing their microscopic foundations in settings with finite but large populations.