Long-Run Behavior of Stochastic Evolutionary Games with Many Players

We investigate similarities and differencies between systems of many interacting players maximizing their individual payoffs and particles minimizing their interaction energy. To address the problem of equilibrium selection in spatial games with many players we introduce a concept of ensemble stability. The standard stochastic stability describes a longrun behavior of systems with a fixed number of players in the zero-noise limit. On the contrary, the ensemble stability is concerned with a fixed (but nevertheless low) noise level in the limit of the infinite number of players. We present examples of games with globally risk-dominant and Pareto-efficient strategies which are stochastically stable but in the long run are played with arbitrarily small probabilities if the noise level is low and the number of players is big enough. We also discuss effects of the presence of strictly dominated strategies on the equilibrium selection. To show our results we use methods of statistical physics.