From evolutionary stability to stochastic stability

Recently, evolutionary games has analyzed the way evolutionary dynamics is modified when the system is subjected to random perturbations This approach has led to a new concept of stability, the one of stochastic stability proposed by Foster and Young. For 2 x 2 symmetric games with two symmetric strict Nash equilibria, there is only one stochastically stable equilibrium, the one which satisfies Harsanyi and Selten's criterion of risk dominance. This paper reviews the recent literature on this domain and highlights some limits of the stochastic stability. In particular, if the process is non-ergodic, the later criterion cannot be applied. We finish by studying some non-ergodic stochastic evolutionary game dynamics.