Chapter 13: Subgame Perfection in Evolutionary Dynamics with Recurrent Perturbations
Herbert Gintis, Ross Cressman, and Thijs Ruijgrok* 13.1 Introduction A fundamental property of evolutionary systems governed by the replicator dynamic (Taylor and Jonker, 1978) is that every stable equilibrium is a Nash equilibrium of the underlying stage game (Nachbar, 1990; Samuelson and Zhang, 1992). Since some Nash equilibria are dynamically unstable (Samuelson, 1997), stability with respect to the replicator dynamic serves as an equilibrium selection technique. However, this technique is not very effective when the game is given in extensive form. For instance, for every perfect information game (that is, an extensive form where each player decides what action to take with complete knowledge of the previous decisions by all players), Cressman and Schlag (1998) (see also Cressman, 2003) show that every pure Nash equilibrium is Lyapunov stable in the replicator dynamic. They also show that, if this pure Nash equilibrium is not subgame perfect, then its Nash equilibrium component is never stable in the replicator dynamic since the subgame perfect equilibrium is contained in any interior asymptotically stable set. This latter result suggests that subgame perfection can be justified on dynamic grounds. Unfortunately, the subgame perfect Nash equilibrium need not be stable either, as elementary examples exist (Cressman and Schlag, 1998) where the subgame perfect component is not even locally attracting in the replicator dynamic. A heuristic argument why the replicator dynamic fails to select subgame perfection in perfect information games is that this is a monotonic selection dynamic (Samuelson and Zhang, 1992) and so is characterized by negligible...
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