IF WE CONFESS OUR SINS

• Date: 01 August 2019
• DOI: http://doi.org/10.1111/iere.12390
• Published date: 01 August 2019
• Author: Francisco Silva

INTERNATIONAL ECONOMIC REVIEW

Vol. 60, No. 3, August 2019 DOI: 10.1111/iere.12390

IF WE CONFESS OUR SINS∗

BYFRANCISCO SILVA1

Ponticia Universidad Catolica de Chile, Chile

I consider a scenario where a social planner suspects that a crime has been committed. There are many suspects

and at most one of them is guilty. I characterize the optimal mechanism for the social planner under different

assumptions with respect to her commitment power. I find that the optimal mechanism is a “confession inducing

mechanism”: Before an investigation, each agent can confess to being guilty in exchange for a reduced pun-

ishment. I find that these mechanisms do better than the traditional trial mechanism because of information

externalities: When an agent credibly confesses his guilt, he reveals everyone else’s innocence.

1. INTRODUCTION

Imagine that there is the suspicion that a crime has been committed. There are Nsuspects

(the agents) and at most one of them is guilty. Each agent knows only whether he is guilty or

innocent. I consider the following question: “What is the best mechanism that a principal can

design if her goal is to punish only the guilty agent?”

The traditional solution is a “trial” mechanism. In a trial mechanism, if the principal suspects

a crime has been committed, she initiates an investigation aimed at obtaining evidence—some

exogenous signal θ, correlated with the agents’ guilt. Based on that evidence θ, the principal

forms beliefs about the guilt of each agent and chooses punishments accordingly: Each agent n

receives a punishment of some xn=dn(θ). However, I argue in this article that these systems

are suboptimal for the principal: She will be able to do better by communicating with the agents

prior to launching an investigation.

The communication takes place as follows: The principal gives each agent n, who is privately

informed of his innocence/guilt, the opportunity to choose some message mnfrom some “mes-

sage set” Mnbefore evidence θhas been collected; then the principal starts an investigation

that produces θ; and, finally, based on message vector m=(m1,...,mN) and on evidence θ,

she chooses a punishment xn=dn(m,θ) for each agent n. Message set M=M1×...×MNand

function d=(d1,...,dN) form a “mechanism” (the trial mechanism is a “special” mechanism

where punishments do not depend on the message mof the agents). The goal of the article is to

study which mechanisms (M,d) are optimal for the principal under two different assumptions

with respect to the principal’s ability to commit to a mechanism: full commitment power and

partial commitment power.

The assumption of full commitment power is the standard one in mechanism design: The

principal can commit to any mechanism (M,d) she chooses. In this setting, I find that the

optimal mechanism is what I call a “confession inducing” mechanism (CIM). A CIM has two

properties. The first property, which is implied by the revelation principle (Myerson, 1979), is

that each agent’s message set is made of only two messages, labeled cand c, that is, Mn={c,c}

∗Manuscript received November 2017; revised November 2018.

1I would like to thank David Abrams, Martin Besfamille, Mustafa Dogan, Selman Erol, Nicolas Figueroa, Nicholas

Janetos, Sangmok Lee, George Mailath, Steven Matthews, Timofiy Mylovanov, Daniel Neuhann, Qiusha Peng, Andrew

Postlewaite, Bernardo S. da Silveira, Rakesh Vohra, and Yanhao Wei as well as the UPenn’s Micro Lunch seminar

participants for their useful comments. Please address correspondence to: Francisco Silva, Department of Economics,

Pontificia Universidad Catolica de Chile, Vicu˜

na Mackenna 4860, Piso 3. Macul, Santiago, Chile (CL). Phone: +56

948544106. E-mail: franciscosilva@uc.cl

1389

C

(2019) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1390 SILVA

for all n. Message cis interpreted as a confession, whereas message cis interpreted as a refusal

to confess: In a CIM, each agent is able to confess to being guilty before an investigation takes

place. The second property is that, whenever an agent confesses, his punishment is independent

of what other agents may have reported and of the evidence, that is, dn(c,m−n,θ) is independent

of m−nand θ(where m−nhas the standard interpretation).

This description is reminiscent of two mechanisms that already exist in American law: “self-

reporting,” which exists in environmental law and “plea bargaining” in criminal law. The idea

behind self-reporting in environmental law is that firms that infringe environmental regulations

can contact the corresponding law enforcement authority and self-report this infringement in

exchange for a smaller punishment than the one they would have received if they were later

found guilty. In plea bargaining, defendants are given the chance to confess to having committed

the crime in exchange for a reduced sentence.

In the literature, there have been papers that have highlighted two advantages of CIMs when

compared to trial systems: Kaplow and Shavell (1994) show that CIMs can do better because

they can reduce costs (in a setting with only a single agent, a credible confession eliminates

the need for a costly investigation), and Grossman and Katz (1983) and Siegel and Strulovici

(2018) show that CIMs can do better because they can reduce risk (for example, the risk of

acquitting guilty agents). In the text, I characterize the optimal CIM under the assumption of

full commitment power and show that it is preferred to the trial mechanism but for a different

reason: information externalities. More precisely, there are information externalities that are

generated by each credible confession: When an agent who is guilty confesses, that confession

informs the principal that all other agents are innocent. Therefore, when the principal decides

the punishments of the agents who did not confess, she will not make the mistake of punishing

them. In other words, the fact that it is commonly known that there is at most one guilty

agent implies that the agents’ guilt is correlated, which is what generates these information

externalities.2

The commitment power of the principal can be used to (i) impose small punishments on

agents who are believed to be guilty, and (ii) to punish agents who are believed to be innocent.

To implement the optimal CIM, the principal uses both: by the revelation principle, following

the optimal CIM, each agent confesses if guilty but refuses if innocent. Therefore, the principal

knows that in equilibrium every agent who confesses is guilty and every agent who refuses is

innocent. So, the principal uses (i) to be able to punish agents who have confessed less than

what she would have preferred and uses (ii) to be able to punish agents who have not confessed

more than what she would have wanted. Assumption (ii) seems particularly problematic: it is

hard to see how a mechanism that punishes agents who are known to be innocent could be

implemented. In light of that, I consider the case of partial commitment power, which is defined

to allow the principal to commit to (i) but not to (ii).3

When the principal has partial commitment power, it is as if she is unable to commit not to

renegotiate: The principal cannot commit to punish those she believes are innocent because

both she and the agent would be better off by renegotiating and selecting a smaller punishment.

In this context, the problem of finding the optimal mechanism is more challenging seeing as,

in general, the revelation principle need not follow (see Bester and Strausz, 2000). The main

technical contribution of this article is to show that, even in this context, CIMs are optimal.

The optimal renegotiation proof CIM is, however, different than the optimal CIM when the

principal has full commitment power in two ways. First, it is such that, in equilibrium, agents do

not completely separate, that is, unlike the case with full commitment power, if the agent is guilty,

he randomizes between confessing and not confessing (while the innocent never confesses). This

means that, upon observing a refusal to confess, the principal is no longer certain that the agent

is innocent, which allows her to impose some punishments on nonconfessing agents, which is

2A similar argument has been made to argue in favor of leniency policies as a way to reduce collusion. I review this

literature on the literature review section.

3In Appendix A.2, I also briefly discuss the case where the principal cannot commit to either (i) nor (ii).