Static Stability in Games

Igal Milchtaich

Abstract

Static stability in games differs from dynamic stability in only considering the players’ incentives to change their strategies. It does not rely on any assumptions about the players’ reactions to these incentives, so it is not necessarily linked with any particular dynamics. This paper introduces a general notion of static stability of strategy profiles, which is applicable to any asymmetric N-player game, and static stability of strategies, which is applicable to any symmetric game and population game. It examines several important, large classes of games, with strategy spaces or payoff functions that have special structures (such as unidimensional strategy spaces or multilinear payoff functions), where this general notion takes a simple, concrete form. In particular, evolutionarily stable strategy (ESS) and continuously stable strategy (CSS) are shown to be essentially special cases of the general static stability concept for symmetric games. As an application, the paper identifies a connection between static stability and comparative statics of altruism. In general, increasing internalization by all players of the aggregate payoff or some other kind of social payoff may paradoxically result in a decrease of that payoff. But this is never so if static stability holds for the equilibria involved.

JEL classification

C72

Keywords

Static stability; Evolutionarily stable strategy; Continuously stable strategy; Risk dominance; Potential games; Comparative statics; Altruism