Stochastic imitation in finite games

In this paper we model an evolutionary process with perpetual random shocks, where individuals sample population-specific strategy-payoff pairs and imitate the most successful behavior. For finite n-player games we prove that in the limit, as the perturbations tend to zero, only strategy-tuples in minimal sets closed under single better replies will be played with positive probability. If the strategy-tuples in one such minimal set have strictly higher payoffs than all outside strategy-tuples, then the strategy-tuples in this set will be played with probability one in the limit, provided the minimal set is a product set and the sample is sufficiently large.