Time Averages , Recurrence and Transience in the Stochastic Replicator Dynamics

We investigate the long-run behavior of a stochastic replicator process, which describes game dynamics for a symmetric two-player game under aggregate shocks. We establish an averaging principle that relates time averages of the process and Nash equilibria of a suitably modified game. Furthermore, a sufficient condition for tran-sience is given in terms of mixed equilibria and definiteness of the payoff matrix. We also present necessary and sufficient conditions for stochastic stability of pure equilibria. 1. Introduction. The present paper deals with a stochastic variant of the continuous-time replicator dynamics. We begin with a brief review of the deterministic model. For a comprehensive discussion, see Hofbauer and payoff matrix A = (a ij). Thus both players have the same set of strategies and for either player, a ij is the payoff from using strategy i if the opponent uses strategy j. There is no symmetry or skew-symmetry assumption on A. The replicator dynamics describes how the proportions of strategies in a population evolve. Consider a large population where every member is programmed to play one pure strategy. Let ζ i (t) denote the size of the subpopulation of i-players at time t, and let ξ i (t) = ζ i (t)/[ζ 1 (t) + · · · + ζ n (t)] denote its proportion. If the population is in state ξ(t) = (ξ 1 (t),. .. , ξ n (t)) T , then, under random matching, {Aξ(t)} i is the expected payoff to individuals playing i. Suppose that this payoff represents the per capita growth rate in the ith subpopulation. Thus ˙