Stochastic stability in finite extensive-form games of perfect information

We consider a basic stochastic evolutionary model with rare mutation and a best-reply (or better-reply) selection mechanism. Following Young's papers, we call a state stochastically stable if its long-term relative frequency of occurrence is bounded away from zero as the mutation rate decreases to zero. We prove that, for all finite extensive-form games of perfect information, the best-reply dynamic converges to a Nash equilibrium almost surely. Moreover, only Nash equilibria can be stochastically stable. We present a `centipede-trust game', where we prove that both the backward induction equilibrium component and the Pareto-dominant equilibrium component are stochastically stable, even when the populations increase to infinity. For finite extensive-form games of perfect information, we give a sufficient condition for stochastic stability of the set of non-backward-induction equilibria, and show how much extra payoff is needed to turn an equilibrium stochastically stable.