Perfect Conditional ε‐Equilibria of Multi‐Stage Games With Infinite Sets of Signals and Actions

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 conditional ε‐equilibrium if, for any of a player's positive probability signal events, his conditional expected utility is within ε of the best that he can achieve by deviating. With topologies on action sets, a conditional ε‐equilibrium is full if strategies give every open set of actions positive probability. Such full conditional ε‐equilibria need not be subgame perfect, so we consider a non‐topological approach. Perfect conditional ε‐equilibria are defined by testing conditional ε‐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 conditional ε‐equilibrium is a subgame perfect ε‐equilibrium, and, in finite games, limits of perfect conditional ε‐equilibria as ε → 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.