Stochastic stability in finite extensive-form games of perfect information - Zibo Xu (Stockholm School of Economics, The Economic Research Institute (EFI), 2013)
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.
Keywords: C61; C62; C73; 330; evolutionary game theory; Markovs chains; equilibrium selection; stochastic stability; games in extensive form; games of perfect information; backward induction equilibrium; Nash equilibrium components; best-reply dynamics