Essential stability in games with infinitely many pure strategies

Abstract

We study the notion of essential stability of equilibria in games with compact metric spaces of pure strategies and continuous payoff functions. We first prove that the games whose equilibria are all essential form a residual set in the sense of Baire Category. Then we prove that each of the games has one minimal essential set and one essential component of the set of equilibria. Finally, we investigate relations between essential stability and strategic stability (Al-Najjar 1995), and show that every essential equilibrium is perfect and that every minimal essential set contains a stable set. We deduce that each of the games has such an essential component that contains a minimal essential set containing a stable set.