It is considered that the deterministic replicator model in the studies of evolutionary dynamics does not help in selecting between strict equilibria. This paper modifies the usual continuous-time replicator model by supposing that the payoff functions are subject to population- level or aggregate shocks that are modeled using Wiener processes. It is considered that it may be possible to discriminate between strict Nash equilibria by considering evolutionary models with stochastic shocks. The authors take into account that stochastic models can have ergodic distributions. They identify a class of games in which the limit distribution is concentrated at equilibrium which is “risks dominant” in the sense of J. C. Harsanyi and R. Selten [A general theory of equilibrium selection in games (1988; Zbl 0693.90098)].
In the absence of “mutation” the system need not have an ergodic distribution. With mutation the system does have an ergodic distribution. In the limit as the mutation rate and the variance of the shocks converge to zero, this distribution concentrates on the risk-dominant equilibrium. However, this result is not robust to changes in the underlying deterministic dynamics.