157306When the punishment must fit the crime: Remarks on the failure of simple penal codes in extensive-form games

157306

When the punishment must fit the crime: Remarks on the failure of simple penal codes in extensive-form games

When the punishment must fit the crime: Remarks on the failure of simple penal codes in extensive-form games by George J. Mailath in year 2004.

Abstract:

In repeated normal-form games, simple penal codes (Abreu 1986, 1988) permit an elegant characterization of the set of subgame-perfect outcomes. We show that the logic of simple penal codes fails in repeated extensive-form games. We provide two examples illustrating that a subgame-perfect outcome may be supported only by a profile with the property that the continuation play after a deviation is tailored not only to the identity of the deviator, but also to the nature of the deviation.

Introduction

Game theory is a standard tool in virtually all ﬁelds of economics, including industrial organization, labor, international, macroeconomics, and public and corporate ﬁnance.In a common scenario, there is a socially desirable (or “cooperative”) outcome inconsistent with equilibrium because of opportunistic possibilities. Interest then turns to explore how changes in institutions and/or repetition of the game might allow such desirable outcomes to be supported as equilibria. Some of the most popular applications of game theory involve repeated normal-form games (for example, the repeated Cournot oligopoly model of collusion, ﬁrst studied by Friedman (1971)).

But other applications are more naturally modeled using extensive-form stage games (for example, the interaction between government and the private sector in the time-inconsistency literature (Athey, Atkeson, and Kehoe, 2005), and between principal and agent in the relational contracting literature (Levin, 2003)). It is only when players are suﬃciently patient that the extensive-form structure is irrelevant and standard folk theorem arguments apply (Fudenberg and Maskin (1986), Wen(2002)).

However, the applied theory is often concerned with the impact of a change in institutions or the environment on the set of equilibrium outcomes, and such an analysis is meaningful only for impatient players. Moreover, little attention has been directed towards repeated extensive-form games with impatient players. For applications of repeated normal-form games, the techniques developed by Abreu(1986, 1988) are central.

Abreu (1988) shows that any outcome that can be supported a subgame perfect equilibrium outcome by some set of punishment strategies in a repeated normal-form game can be supported by a set of simple punishment strategies called simple penal codes: If a player, Isay, deviates from the proposed equilibrium playing a given period, in the next period, players switch to the player I’s worst equilibrium (called I’s optimal penal code).

A similar rule applies to any player who deviates from play during an optimal penal code. In other words, the continuation play after a deviation by a player is independent of the nature of the deviation, depending only on the identity of the deviator.

This result vastly simpliﬁes the task of ﬁnding the set of equilibria that can be supported in repeated normal-form games (one need only characterize the worst equilibrium for each player, and from there proceed to ﬁll in all other equilibria that can be supported by using these punishment strategies) and has been correspondingly important for applications. But what about applications that involve extensive-form stage games? In this paper, we ask whether a similar simpliﬁcation is available for characterizing the set of subgame perfect equilibrium outcomes when the stage game has a nontrivial extensive structure: are simple penal codes still suﬃcient to sustain any desired equilibrium outcome in repeated extensive-form games?, …