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On the Theory of Infinitely Repeated Games with Discounting

Dilip Abreu

Abstract

This paper presents a systematic framework for studying infinitely repeated games with discounting, focussing on pure strategy (subgame) perfect equilibria. It introduces a number of concepts which organize the theory in a natural way. These include the idea of an optimal penal code, and the related notions of simple penal codes and simple strategy profiles. I view a strategy profile as a rule specifying an initial path (i.e., an infinite stream of one-period action profiles), and punishments (also paths, and hence infinite streams) for any deviations from the initial path, or from a previously prescribed punishment. An arbitrary strategy profile may involve an infinity of punishments and complex history-dependent prescriptions. The main result of this paper is that much of this potential strategic complexity is redundant: every perfect equilibrium path is the outcome of some perfect simple strategy profile. A simple strategy profile is independent of history in the following strong sense: it specifies the same player specific punishment after any deviation by a particular player. Thus simple strategy profiles have a parsimonious description in terms of (n+1) paths where n is the number of players. Unlike the undiscounted case there is no need to "make the punishment fit the crime." In particular, a player who has a "myopic" incentive to deviate from his own punishment may be deterred from doing so simply by restarting the punishment already in effect. The key to the above result is that, with discounting, worst perfect equilibria exist for each player. These define an optimal penal code. The notion of a simple penal code yields an elementary proof of the existence of an optimal penal code and leads directly to the theorem on the "sufficiency" of simple strategy profiles.