Repeated Principal-Agent Games with Discounting

Abstract

In a repeated principal-agent game (supergame) in which each player's criterion is his long-run average expected utility, efficient behavior can be sustained by a Nash equilibrium if it is Pareto-superior to a one-period Nash equilibrium.Furthermore, if the players discount future expected utilities, then for every positive epsilon, and every pair of discount factors sufficiently close to unity (given epsilon), there exists a supergame equilibrium that is within epsilon (in normalized discounted expected utility) of the target efficient behavior.These supergame equilibria are explicitly constructed with simple "review strategies." 1. INTRODUCTION 1.1.Some Background IN A PRINCIPAL-AGENT SITUATION, the agent chooses an action "on behalf of" the principal.The resulting consequence depends on a random state of the environment as well as on the agent's action.After observing the consequence, the principal makes a payment to the agent according to a pre-announced reward function, which depends directly only on the observed consequence.This last restriction expresses the fact that the principal cannot directly observe the agent's action, nor can the principal observe the information on which the agent bases his action.This situation is one of the simplest examples of decentralized decisionmaking in which the interests of the decision-makers do not coincide.2If this action-reward situation occurs only once, I shall call it a short-run principal-agent relationship.The situation can be naturally modeled as a two-move game, in which the principal first announces a reward function to the agent, and then the agent chooses an action (or decision function if he has prior information about the environment).The Nash (or perfect Nash) equilibria of such a game are typically inefficient (unless the agent is neutral towards risk), in the sense that there will typically be another (but nonequilibrium) reward-decision pair that yields higher expected utilities to both players.In order to increase the efficiency of short-run equilibria, the principal could monitor (at least ex post) the information and decision of the agent.However such monitoring would tyically be costly, so that net efficiency need not be increased by monitoring.Another approach to increasing efficiency is suggested by the theory of repeated games.If a game with two or more players is repeated, the resulting situation can be modeled naturally as a game ("supergame") in which the players' actions in any one repetition are allowed to depend on the history of the previous repetitions.In the principal-agent situation, the repetition of the game would l I am grateful to R. A. Aumann, R. W. Rosenthal, and A.