Two‐strategy evolutionary games with stochastic adaptive control

Summary We investigate the stochastic stability of replicator dynamics for evolutionary games with multiplicative noises and time delay in this article. We study evolutionary games in infinite well‐mixed populations, among which each individual has two strategies, to cooperate or defect, each time when it is matched up with other individuals to play the two‐strategy game together. This article is concerned with the extension of the well‐known two‐strategy evolutionary games to the stochastic case with adaptive control. We analyze the stochastic replicator dynamics with time delays and provide conservative bounds on the strength of noise and the time delay for the almost sure exponential stability of stochastic systems. Finally, numerical simulations based on triple classical game models (snowdrift game, hunt stag game, and prisoner's dilemma game) are given to illustrate the preceding theoretical results.