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.

中文翻译：

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具有随机自适应控制的两策略进化博弈

我们在本文中研究了具有乘性噪声和时间延迟的进化博弈的复制器动力学的随机稳定性。我们研究了无限混合种群中的进化博弈，其中每个个体有两种策略，合作或背叛，每次与其他个体匹配时一起玩两种策略的博弈。本文关注将著名的双策略进化博弈扩展到具有自适应控制的随机情况。我们分析了具有时间延迟的随机复制器动力学，并为几乎确定的随机系统的指数稳定性提供了噪声强度和时间延迟的保守界限。最后，基于三重经典博弈模型（雪堆博弈、猎鹿博弈和囚徒）的数值模拟