On the Geometry of Perfect Recall

Abstract

This paper investigates the relationship between the existence of equilibria in behavioral strategies, perfect recall, and notions of convexity in extensive form games. The central result is that players possess perfect recall if and only if their sets of behavioral strategies are behavioral convex, a novel generalized convexity condition. This result implies that perfect recall is the weakest restriction on players’ information partitions under which standard fixed point arguments ensure the existence of equilibria in behavioral strategies in finite games. The equivalence between convexity and perfect recall motivates the adoption of mixed behavioral strategies in games with imperfect recall. By randomizing ex-ante over behavioral strategies, players can generate distributions over outcomes and equilibria that cannot be achieved using mixed or behavioral strategies alone. Since equilibria in mixed behavioral strategies exist in any finite extensive form game, it follows that they are the most natural class of strategies for general extensive form games.