WHEN AND HOW THE PUNISHMENT MUST FIT THE CRIME

• DOI: http://doi.org/10.1111/iere.12219
• Date: 01 May 2017
• Published date: 01 May 2017
• Author: Lucy White,George J. Mailath,Volker Nocke

INTERNATIONAL

ECONOMIC

REVIEW

May 2017

Vol. 58, No. 2

WHEN AND HOW THE PUNISHMENT MUST FIT THE CRIME∗

BYGEORGE J. MAILATH,VOLKER NOCKE,1AND LUCY WHITE1

University of Pennsylvania, U.S.A., and, Australian National University, Australia; University of

California, Los Angeles, U.S.A., University of Mannheim, Germany, and CEPR, U.K.; Boston

University, U.S.A., and CEPR, U.K.

In repeated normal-form (simultaneous-move) games, simple penal codes (Abreu, Journal of Economic The-

ory 39(1) (1986), 191–225; and Econometrica 56(2) (1988), 383–96) permit an elegant characterization of the set

of subgame-perfect outcomes. We show that in repeated extensive-form games such a characterization no longer

obtains. By means of examples, we identify two types of settings in which 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.

My object all sublime

I shall achieve in time

To let the punishment fit the crime,

The punishment fit the crime;

And make each prisoner repent

Unwillingly represent

A source of innocent merriment,

Of innocent merriment!

W. S. Gilbert (1885), The Mikado

1. INTRODUCTION

Many popular applications of game theory are naturally modeled as simultaneous-move stage

games, such as the Cournot oligopoly model of collusion. But other interesting applications have

a dynamic structure whose stage game interactions are more naturally represented by nontrivial

extensive-form stage games. Examples include the interaction between government and the

private sector in the time-inconsistency literature, between upstream and downstream firms in

the vertical relations literature, between bidders in open-outcry auctions, between firms that first

∗Manuscript received February 2015; revised November 2015.

1Mailath thanks the National Science Foundation for research support (grants SES-0350969). Nocke is grateful for

financial support from the European Research Council (grant no. 313623). This is a major revision of “When the

Punishment Must Fit the Crime: Remarks on the Failure of Simple Penal Codes in Extensive-Form Games,” originally

circulated in 2004. We thank the editor Simon Board and three anonymous referees for helpful comments and Larry

Samuelson for helpful discussions. Please address correspondence to: Volker Nocke, UCLA Department of Economics,

Bunche Hall 8283, 315 Portola Plaza, Los Angeles, CA 90095, U.S.A. Phone: +1 (310) 794-6617. Fax: +1 (310) 825-9528.

E-mail: volker.nocke@gmail.com.

315

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(2017) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

316 MAILATH,NOCKE,AND WHITE

invest or choose standards and then compete, between proposer and responders in bargaining

games, and between principal and agent in contracting games. Such applied models often feature

a socially desirable (or “cooperative”) outcome that is inconsistent with equilibrium in a one-

shot game because of opportunistic actions available to players, but that might be attainable in a

repeated game. Economists are often interested in exploring how changes in institutions and/or

repetition of the game might allow the more desirable outcomes to be supported as equilibria

(see, e.g., the seminal work of Friedman, 1971). But with a few exceptions,2such analysis has

been largely confined to repeated normal-form games, where all players are modeled as moving

simultaneously in the stage game.

When players are sufficiently patient, one may ignore the detailed dynamic structure within

the stage-game interaction and apply standard folk theorem arguments (Wen, 2002). The reason

is that even a small difference in continuation values will dominate any short-term gain for

patient players, so deviations are relatively easy to deter. However, 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. In this article, we show

that for impatient players, the dynamic structure of the stage game can be important and should

not be neglected.

The literature has so far directed surprisingly little attention toward the study of repeated

extensive-form games with impatient players. For applications of repeated simultaneous-move

games, the techniques developed by Abreu (1986, 1988) are central. Abreu (1988) shows that

any pure-strategy subgame-perfect equilibrium outcome can be supported by a set of simple

punishment strategies called simple penal codes: If a player, isay, deviates from the proposed

equilibrium play in a given period, in the next period, players switch to player i’s worst equilib-

rium play (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 devi-

ator. This result vastly simplifies the task of finding the set of equilibria that can be supported

in repeated simultaneous-move games: One needs only to characterize a worst equilibrium for

each player, and from there one can proceed to fill in all other equilibria that can be supported

by using these punishment strategies. Abreu’s result has been correspondingly important for

applications employing normal-form stage games.

But what about applications that involve extensive-form stage games? In this article, we

show that a similar simplification is not available for characterizing the set of subgame-perfect

equilibrium outcomes when the stage game has a nontrivial dynamic structure. We begin by

discussing the appropriate restrictions that simple penal codes should satisfy in this case. We then

present two settings in which simple penal codes can fail to support behavior as an equilibrium

that is supportable with more complicated strategies (for a given discount factor). The forces

driving the equilibrium incentives in these settings are intuitive and natural.

Consider a deviation by some player in a repeated simultaneous-move game. Since the stage

game is a simultaneous-move game, the other players can respond only in the next period.

Abreu’s (1988) results rely on the observation that, in repeated simultaneous-move games,

every subgame is strategically equivalent to the original repeated game.3Hence, the worst

punishment is simply the worst subgame-perfect equilibrium of the original game. Consider

now a deviation in a repeated extensive-form game. In contrast to normal form games, the

other players may be able to respond not only in the next period but also within the same

period. Moreover, the deviation may lead to a subgame that is not strategically equivalent to

the original game. Consequently, the appropriate notion of simple penal code for a repeated

extensive-form game is not obvious. Nonetheless, for any history that ends at the end of a

2Repeated extensive-form games have been analyzed in the relational contracting literature (Levin, 2003), in the

policy games literature (Athey et al., 2005), and in the vertical relations literature (Nocke and White, 2007).

3This observation is a direct implication of the property that every subgame’s initial node is at the beginning of some

period.