Evolution and equilibrium under inexact information

We study a general model of stochastic evolution in games, assuming that players have inexact information about the game's payoffs or the population state. We show that when the population is large, its behavior over finite time spans follows an almost deterministic trajectory. While this result provides a useful description of disequilibrium behavior adjustment, it tells us little about equilibrium play. We establish that the equilibrium behavior of a large population can be approximated by a diffusion. We then propose a new notion of stability called local probabilistic stability (LPS), which requires that a population which begins play in equilibrium settle into a fixed stochastic pattern around the equilibrium. We use the diffusion approximation to prove a simple characterization of LPS. While LPS accords closely with standard deterministic notions of stability at interior equilibria, it is significantly less demanding at boundary equilibria.