Noise-induced stochastic Nash equilibrium in evolutionary game dynamics

In order to better understand the impact of environmental stochastic fluctuations on evolutionary game dynamics, we introduce the concept of a stochastic Nash equilibrium (SNE) that extends the classical concept of a Nash equilibrium (NE). Based on a stochastic stability analysis of a linear evolutionary game with temporally varying payoffs, we address the question of the existence of a SNE, either weak when the geometric mean payoff against it is the same for all other strategies or strong when it is strictly smaller for all other strategies, and its relationship with a stochastically evolutionarily stable (SES) strategy. While a strong SNE is always SES, this is not necessarily the case for a weak SNE. We give conditions for a completely mixed weak SNE not to be SES and to coexist with at least two strong SNEs. More importantly, we show that a pair of two completely mixed strong SNEs can emerge as the noise level increases. This not only indicates that a noise-induced SNE may possess some properties that a NE cannot possess, such as being completely mixed and strong, but also illustrates the complexity of evolutionary game dynamics in a stochastic environment.