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Toward a Theory of Discounted Repeated Games with Imperfect Monitoring

Dilip Abreu
David Pearce
Ennio Stacchetti

Abstract

This paper investigates pure strategy sequential equilibria of repeated games with imperfect monitoring. The approach emphasizes the equilibrium value set and the static optimization problems embedded in extremal equilibria. A succession of propositions, central among which is "self-generation," allow properties of constrained efficient supergame equilibria to be deduced from the solutions of the static problems. We show that the latter include solutions having a "bang-bang" property; this affords a significant simplification of the equilibria that need be considered. These results apply to a broad class of asymmetric games, thereby generalizing our earlier work on optimal cartel equilibria. The bang-bang theorem is strengthened to a necessity result: under certain conditions, efficient sequential equilibria have the property that after every history, the value to players of the remainder of the equilibrium must be an extreme point of the equilibrium value set. General implications of the self-generation and bang-bang propositions include a proof of the monotonicity of the equilibrium average value set in the discount factor, and an iterative procedure for computing the value set.