Perfect Conditionalε-Equilibria of Multi-Stage Games With Infinite Sets of Signals and Actions

Econometrica · 2020 · Volume 88 · Issue 2

Abstract

We extend Kreps and Wilson's concept of sequential equilibrium to games with infinite sets of signals and actions. A strategy profile is a conditionalε-equilibrium if, for any of a player's positive probability signal events, his conditional expected utility is withinεof the best that he can achieve by deviating. With topologies on action sets, a conditionalε-equilibrium isfullif strategies give every open set of actions positive probability. Such full conditionalε-equilibria need not be subgame perfect, so we consider a non-topological approach.Perfectconditionalε-equilibria are defined by testing conditionalε-rationality along nets of small perturbations of the players' strategies and of nature's probability function that, for any action and for almost any state, make this action and state eventually (in the net) always have positive probability. Every perfect conditionalε-equilibrium is a subgame perfectε-equilibrium, and, in finite games, limits of perfect conditionalε-equilibria asε→ 0 are sequential equilibrium strategy profiles. But limit strategies need not exist in infinite games so we consider instead the limit distributions over outcomes. We call such outcome distributions perfect conditional equilibrium distributions and establish their existence for a large class of regular projective games. Nature's perturbations can produce equilibria that seem unintuitive and so we augment the game with a net of permissible perturbations.

Keywords

Sequential equilibrium, infinite game, finite consistency, conditional epsilon-equilibrium

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