On the Theory of Infinitely Repeated Games with Discounting

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 stratea 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 historydependent 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 smtegy 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 optimalpenal code. The notion of a simplepenal 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.

Simple coalitional strategy profiles in repeated games

“…Abreu (1988) proves that simple strategy pro…les are simple not only because of their de…n-ition but also because it is easy to check whether they are subgame perfect equilibria. According to his Proposition 1 (p. 391), only one-shot deviations need be checked to ensure subgame perfection, where a one-shot deviation from a strategy consists of a single period deviation followed by sticking to the strategy afterwards.…”

“…Simple strategy pro…les are de…ned to avoid single player deviations (Abreu (1988)). Deviations of two or more players are always ignored.…”

The Arm’s Length Principle and Tacit Collusion

“…In Section 4, we discuss the case where upstream firms cannot exclude any unrelated buyer. 6 Although this punishment strategy is not optimal (Abreu, 1988), we use it for simplicity and tractability. Indeed many previous studies have adopted the grim trigger strategy when analyzing stability of agreements.…”

“…These twin roles of transfer pricing have been the focus of much research. 1 When some divisions of a firm face competition, research has further shown that transfer prices can also serve strategic purposes (Alles and Datar, 1998;Zhao, 2000;Arya and Mittendorf, 2008). Typical reasoning for the strategic use of transfer pricing relies on the firm's ability to exercise price discrimination: by charging transfer price to its own affiliate that is different from what the firm would charge unrelated buyers, the firm is able to improve the affiliate's competitive posture against the unrelated competitors.…”

Putting all One’s Eggs in One Basket: Relational Contracts and the Provision of Local Public Services

“…The comparison of punishment strategies and the determination of the optimal punishment scheme (Abreu [1988]) will be investigated in further works. This is true when p → 0, i.e.…”