Stochastic Better-Reply Dynamics in Games

In Young (1993, 1998) agents are recurrently matched to play a finite game and almost always play a myopic best reply to a frequency distribution based on a sample from the recent history of play. He proves that in a generic class of finite n-player games, as the mutation rate tends to zero, only strategies in certain minimal sets closed under best replies will be played with positive probability. In this paper we alter Young's behavioral assumption and allow agents to choose not only best replies, but also better replies. The better-reply correspondence maps distributions over the player's own and her opponents' strategies to those pure strategies which gives the player at least the same expected payoff against the distribution of her opponents' strategies. We prove that in finite n-player games, the limiting distribution will put positive probability only on strategies in certain minimal sets closed under better replies. This result is consistent with and extends Ritzberger's and Weibull's (1995) results on the equivalence of asymptotically stable strategy-sets and closed sets under better replies in a deterministic continuous-time model with sign-preserving selection dynamics.