Subgame Perfection in Evolutionary Dynamics with Recurrent Perturbations

A fundamental property of evolutionary systems governed by any monotone dynamic such as the replicator equation (Taylor and Jonker 1978) is that every dynamically stable equilibrium is a Nash equilibrium of the underlying game (Nachbar 1990, Samuelson and Zhang 1992). Conversely, when the underlying game in extensive form has perfect information and generic payoffs, Cressman and Schlag (1998) show that every pure strategy Nash equilibrium is Lyapunov stable under the replicator equation. Moreover, they show that any asymptotically stable set must contain the subgame perfect Nash equilibrium, thereby giving a dynamic justification for the classical concept of subgame perfection. The main purpose of this paper is to analyze these fundamental properties when recurrent perturbations are added to the replicator equation. Specifically, we consider perturbations that lead players to shift from one strategy to any other with positive probability and we study what happens in the limit as these perturbation rates go to zero. Related to this question is the article in this journal by Hart (2002), who showed that with a particular non-monotonic dynamic, there is always convergence to the subgame perfect equilibrium as population size increases to infinity and the perturbation rate goes to zero in such a way that the per-period number of perturbations is bounded away from zero. Hart speculated that a similar result ∗Herbert Gintis: Santa Fe Institute and Central European University, [email protected], http://www-unix.oit.umass.edu/g̃intis; Ross Cressman: Department of Mathematics, Wilfrid Laurier University, email: [email protected]; Thijs Ruijgrok: Mathematics Department, Utrecht University, email: [email protected] thank Ken Binmore, Drew Fudenberg, and Sergiu Hart for helpful comments.