Almost sure exponential stability of two-strategy evolutionary games with multiplicative noise

Abstract

In this paper, stochastic two-strategy evolutionary games in which the players are interfered by multiplicative noise are studied. We propose a stochastic replicator dynamic model to investigate the almost sure exponential stability (ASES) of general two-strategy games. Compared with the deterministic games and stochastic games with additive noise, our most interesting results concern the impact of multiplicative noise on the cooperation behavior in large well-mixed populations. We establish sufficient conditions on stochastic noise which can ensure the stable equilibrium points of general two-strategy games are indeed ASES. The stochastic Lyapunov framework is used to prove the stochastic stability, where finding the suitable Lyapunov function is the key and challenging in our case. Three different types of two-strategy game models (snowdrift games, hunt stag games and prisoner’s dilemmas) are simulated to verify our theoretical results.