When to Convict Defendants Facing Multiple Accusations?
A Strategic Analysis*
Harry Pei†

Bruno Strulovici‡

August 31, 2024

Abstract: Over the last few decades, prominent legal scholars have proposed to radically alter the rules of adjudication for defendants accused of multiple offenses. In such cases, they have argued that conviction should bebased on the overall probability of guilt of the defendant rather than on the distinct probabilities of guilt for eachpossible offense. This proposal has gained prominence with the rising number of individuals accused of committing serial assaults. We show however that the proposal may undermine the quality of the information available
to adjudicate cases and weaken deterrence once the incentives of potential offenders and witnesses are taken into
account. When conviction entails a large punishment, a potential offender’s decisions to commit distinct offenses
are strategic substitutes, and witnesses’ decisions to report offenses are strategic complements. Taken together,
these strategic considerations tend to weaken the informativeness of witness reports and increase the probabilityof offenses in equilibrium. By contrast, convicting defendants based on the probabilities of distinct offenses leads
to informative testimonies and a low probability of offenses.
Keywords: Strategic communication, coordination, correlated signals, aggregate probabilities principle.
JEL Codes: C72, D82, D83, K42.

* This paper was previously circulated under the title “Crime Aggregation, Deterrence, and Witness Credibility”. We thank S. Nageeb
Ali, Bocar Ba, Arjada Bardhi, Laura Doval, Mehmet Ekmekci, Andreas Kleiner, Anton Kolotinin, Frances Xu Lee, Andrew McClellan,
Aniko Öry, Bobby Pakzad-Hurson, Alessandro Pavan, Daniel Rappoport, Larry Samuelson, Ron Siegel, Teck Yong Tan, and Boli Xu forhelpful comments. We acknowledge financial support from NSF Grants SES-1151410, SES-1947021, and SES-2337566.
†
Department of Economics, Northwestern University. Email: harrydp@northwestern.edu.
‡
Department of Economics, Northwestern University. Email: b-strulovici@northwestern.edu.

11Introduction

When a defendant faces multiple charges, the legal norm is to consider these charges separately and to convict
the defendant for each individual charge for which the evidence of guilt meets the appropriate standard of proof.
However, the desirability of this separation for deterrence and fairness is not obvious. For example, consider
a defendant who may have committed two offenses with probability 0.8 each, independent of each other. If
the conviction threshold for each offense is 0.9, then the defendant is acquitted on both counts, even though
the probability that he is guilty of at least one offense is 1 − 0.2 × 0.2 = 0.96. Under the same threshold, a
defendant accused of a single offense is convicted when his probability of guilt is 0.91 and thus lower than the first
defendant’s.
This issue has received significant attention from prominent legal scholars, starting with Cohen (1977), Bar
Hillel (1984), and Robertson and Vignaux (1993), who explore the possibility of aggregating accusations against
a defendant by considering his overall probability of guilt. Harel and Porat (2009) formalize the Aggregate Probabilities Principle (“APP”), according to which a defendant is convicted if the probability that he has committed
at least one offence exceeds a threshold. Compared to the standard legal norm, which they call the Distinct Probabilities Principle (“DPP”), Harel and Porat advocate the use of APP to varying degrees in both civil and criminalcases, arguing that APP can reduce adjudication errors, improve deterrence, and reduce the cost of enforcement.
This issue has gained even more momentum over the last decade in both the legal and the philosophy literatures(see for example, Pundik (2015), Littlejohn (2017), Smith (2018), Di Bello (2019), Backes (2020), Blome-Tilman(2020), Ross (2020)).1 Schauer (2023) concludes that “....the legal systems ought to be more willing than they now
are to base liability on under-specified wrongs, at least when multiple under-specified acts are relevantly similar
and are alleged to be committed by the same defendant”.
The aforementioned papers rely on two key assumptions. First, the defendant’s guilt for each offense is independently distributed across charges. Second, the informativeness of the evidence available regarding each offense
is exogenously given. In particular, this informativeness is independent of the rule used to adjudicate cases withmultiple accusations. These two assumptions ignore the impact that APP can have on (i) the incentives of potential
offenders to commit offenses and (ii) the incentives of potential witnesses to report offenses. This impact may besignificant when a defendant faces multiple accusations that are hard to prove beyond a reasonable doubt, suchas abuses of power for which the main incriminating evidence consists of witness testimonies. In these cases,
the incentives of witnesses to report their observations truthfully are affected by how likely their report is to be
1

Much of the related work builds on thought experiments such as Cohen’s (1977) Gatecrasher paradox, Pundik’s (2015) serial burglar,and Littlejohn (2017)’s Prison Yard, to argue that adjudication rules based on DDP lead to counter-intuitive and problematic outcomes.
Some of these thought experiments are closer to reality, such as Schauer’s (2023) discussion of repeated assaults and driving infractions.

2corroborated by other witnesses and their fear of retaliation.
This paper investigates the effects of APP and DPP on incentives, informativeness, and deterrence from agame-theoretic perspective.2 Our model features one potential offender and multiple witnesses. The potential
offender’s decisions to commit offenses and the witnesses’ decisions of whether to accuse the potential offender
are both endogenously determined in equilibrium. We show that the strategic implications of APP are such thata potential offender’s decisions to commit distinct offenses are negatively correlated in equilibrium when thepunishment incurred in case of conviction is large relative to the benefits from committing offenses. This negative
correlation, together with witnesses’ coordination motives, lowers the informativeness of testimonies and increases
the expected number of offenses in equilibrium.
In our model, the potential offender—hereafter the defendant—has multiple opportunities to commit offenses,3
each of which is associated with a distinct witness. Each witness observes whether the offense associated with
him takes place. For example, the defendant could be a firm executive with multiple opportunities to commit fraud
or to violate workplace regulations, and the witnesses could be employees who may observe distinct offenses andmay become whistleblowers.4
Once each witness has privately observed whether the offense associated with him has occurred, the witnesses
independently decide whether to accuse the defendant. A witness’s decision is driven by three considerations: (i)a preference for convicting guilty defendants and acquitting innocent ones, (ii) a risk of retaliation if the witness
makes an accusation that fails to result in a conviction (e.g., because no other witness comes forward), and (iii) a
private, idiosyncratic bias for or against getting the defendant convicted, whose distribution is independent fromthe other witness’s bias.5 After observing witnesses’ testimonies, a judge decides whether to convict or acquit thedefendant.6
2

One could perform a more general mechanism design analysis beyond comparing APP and DPP. We undertook this approach in an
earlier version of this paper (Pei and Strulovici 2021) subsumed by the present version. In a somewhat different and more complex modeland using a more involved analysis, we showed that when (i) the defendant may commit at most two offenses and (ii) the punishment incase of conviction is large relative to the benefit from committing an offense, the optimal mechanism is approximated by either APP or
DPP so that focusing on these two rules is, under these conditions, without loss of optimality. This paper focuses on the simpler and morepractical question of comparing the existing rule to the one suggested in the legal and philosophy literature.
3
Formally, the potential offender becomes a defendant only if at least one accusation is made against him. We nonetheless use theshorter terminology for simplicity.
4
In cases that concern abuses of power, police brutality, workplace bullying, and harassment, witnesses may be the victims of theseoffenses. In other cases, they may be mere observers.
5
In our model, some offenses go unreported and, conversely, some false accusations are made. Both features are consistent with theempirical evidence on abuses and reports of abuse. For instance, a 2016 survey conducted by the USMSPB concluded that 21% of womenand 8.7% of men experienced at least one of 12 categorized behaviors of sexual harassment, of which only a small fraction was followedby charges. According to data released by USMSPB, among the harassment charges filed in 2017, only 16% led to merit resolutions. A
similar pattern was found in a study of harassment in the U.S. military by the RAND corporation (2018) and studies of police brutality or
inaction by Ba (2020) and Ba and Rivera (2023) using Chicago data.
6
In an extension, we consider the case in which the defendant’s punishment depends not only on the probability of guilt, but also on theprobability that the defendant has committed multiple offenses. We show that this extension does not affect the results of the model with asimple conviction/acquittal decision.

3

We compare equilibrium outcomes across the adjudication rules APP and DPP. Under APP, a defendant isconvicted if the probability that he has committed at least one offense (i.e., that he is guilty of something) exceeds
some exogenous threshold. Under DPP, the defendant is convicted if there is one specific offense for which theprobability that the defendant committed this offense exceeds some threshold. The APP and DPP criteria are
identical when the defendant may commit at most one potential offense, but may lead to different outcomes when
the defendant may commit multiple offenses.
Our analysis shows that APP and DPP lead to radically different outcomes when the defendant’s loss (e.g.,
disutility) from being convicted is large relative to his benefit from committing an offense. Under this assumption,
we show that (i) under APP, each witness’s accusation is arbitrarily uninformative and the expected number ofoffenses is close to the conviction threshold being used, while (ii) under DPP, a witness’s accusation is highly
informative (in a likelihood-ratio sense) and the expected number of offenses is arbitrarily small.
Turning to judicial errors, we show that APP introduces a tension between deterrence and false convictions:while lowering the guilt-probability threshold used to convict defendants results in a lower expected number ofoffenses, it also increases the conditional probability that a convicted defendant did not commit any offense.
Precisely, we show that when the threshold probability is p ∈ (0, 1), the expected number of offenses under APP
is close to p and the probability that a convicted defendant is innocent is close to 1 − p. This tension, which may
seem intuitive, does not arise under DPP.
We now provide intuition for our results, focusing on settings in which the defendant can commit at most two
possible offenses (the extension to three or more offenses is also analyzed in the paper). We start by explaining
why, under our maintained assumption that the punishment for conviction is large relative to the benefit fromcommitting offenses, the defendant is convicted in equilibrium under APP only if both witnesses accuse him. To
see this, suppose by way of contradiction that only one accusation sufficed to get the defendant convicted withpositive probability, i.e., sufficed to raise the defendant’s probability of guilt to the conviction threshold. Then,
two accusations would raise the posterior probability of guilt even further, strictly above the conviction threshold,and result in the defendant being convicted with probability one. Facing a large punishment in case of conviction,
the defendant would respond by not committing any offense so as to minimize the probability of facing multiple
accusations, contradicting the fact that the defendant is convicted with positive probability in equilibrium.
This observation implies that the defendant’s decisions to commit distinct offenses are strategic substitutes:
the increase in the probability of conviction that results from committing a given offense is higher if the defendantalso commits the other offense than if he does not.
The fact that offenses are strategic substitutes implies, in equilibrium, that witnesses’ private signals are negatively correlated: a witness who observes an offense believes that the other witness is unlikely of also observing4

an offense, and vice versa.
Separately, because each witness faces retaliation if he is the only one to accuse the defendant, witnesses’
decisions to accuse the defendant are strategic complements.
Combining the negative correlation of witnesses’ signals and the complementarity of their accusation decisions
creates a tension between witnesses’ incentives to coordinate on the same reports and the fact that the individual
signals that drive these reports are negatively correlated. This tension has the effect of reducing the dependenceof witnesses’ reports on their private signals, thereby diminishing the informativeness of their testimonies. This
lower informativeness increases the equilibrium probability that the defendant commits offenses and the expected
number of offenses.
DPP is immune to the logic just described. DPP uses each probability that the defendant is guilty of a specificoffense as an input. Crucially, the defendant’s probability of guilt for a specific offence need not increase if he isaccused of another offense. In particular, when the decisions to commit distinct offenses are uncorrelated, a testimony pertaining to a given offense has no bearing on the likelihood of guilt for another offense and thus need notaffect the probability of conviction according to DPP. In fact, we show that this lack of correlation across offenses
must arise in equilibrium when DPP is used. In equilibrium, the judge’s belief about the occurrence of any given
offense reaches the conviction threshold if and only if the witness whose role it is to testify on this specific offense
accuses the defendant, and this belief is independent of other witnesses’ testimonies. In equilibrium, witnesses’
private observations are uncorrelated, which neutralizes the adverse effect that their coordination motive had, under APP, on the credibility of their testimonies. Finally, the defendant’s decisions to commit distinct offenses are
no longer strategic substitutes (and neither are they complements), because the probability of conviction is linear
in the number of accusations made against the defendant.
Our findings are robust with respect to several extensions of our model. First, we consider the case in which thedefendant has a private type that affects his benefit from committing offenses. We consider the case of binary types,
in which the defendant is either a virtuous type who does not benefit from committing offenses or an opportunistic
type who benefits from committing offenses as before. We show that when the probability of the virtuous type isabove some cutoff, the opportunistic type will commit both offenses with positive probability. Nevertheless, ourmain theorems extend after we account for such heterogeneity.
Next, we extend our theorems to the case of three or more offenses maintaining our assumption that each
offense, when it occurs, is observed by a distinct witness. We show that, under APP, the informativeness of
witnesses’ testimonies, pooled together, regarding the defendant’s likelihood of guilt is decreasing in the numberof potential offenses, even though there are more witness reports available to the judge. Moreover, we find thatthe unconditional probability that a given witness accuses the defendant increases in the number of potential
5

offenses. Since each accusation bears a potential cost for the accuser but increases the value of accusations byother witnesses, accusations may be viewed as a form of contribution to a public good between witnesses. Viewed
in this light , the fact that each witness is more likely to accuse the defendant as the number of witnesses increases
distinguishes our analysis from standard theories of public good provision, in which contributions become scarcer
as the number of players increases.
We also consider the following variations of our model: (i) the defendant faces a larger punishment when
the probability that he has committed multiple offenses exceeds some exogenous threshold; (ii) the defendant’s
utility from committing offenses is concave in the number of offenses; (iii) a witness who makes a false accusation
against the defendant may be uncovered and punished ex post with positive probability; (iv) witnesses have an
intrinsic preference for reporting the truth (above and beyond the effect of their testimony on the verdict); (v) each
witness cares not only about the potential offense that he is associated with, but also about all other offenses thatthe defendant may have committed; (vi) the number of opportunities to commit offenses is not publicly known,
and is instead the defendant’s private information; and (vii) the witnesses incur a positive cost when they accuse
the defendant even when the defendant is convicted.
In summary, our comparison between APP and DPP suggests a rationale for using a conviction criterion thattreats each accusation independently of other offenses that the defendant may have committed. Our results echosome critiques of procedures that link accusations across potential witnesses, especially those that are related tothe credibility of witnesses’ reports.7
One caveat for the applicability of our results is that these results are obtained using equilibrium analysis.
While equilibrium analysis is the norm in economics, it rests on the assumption that all players know one another’sequilibrium strategies and are playing mutual best replies to these strategies. This assumption is more plausible
when the environment has remained stable for an extended period of time than after a change in the environment,
which might correspond in our context to a change in the conviction rule (e.g., APP vs. DPP) used for adjudicationor to a discrete jump in the punishment level imposed when convicted of a given offense.
We review the related literature in the remainder of this section. Section 2 sets up our model. Section 3
analyzes a single-offense benchmark in which the defendant cannot commit one of the offenses. Section 4 statesand shows our main results, which compare the equilibrium outcomes under APP and DPP when the defendant cancommit multiple offenses. Section 5 studies several extensions which include settings with three or more agentsand settings in which the defendant has private information about his propensity to commit offenses. Section 6
concludes.
7

Keith Hiatt, the director of the Technology Program at the Human Rights Center at UC Berkeley School of Law notes, concerning themultiple-accusations approach taken by the online platform Callisto that “it may also codify an entrenched attitude that women need tohave corroborating evidence to be believed.” New York Times, “The War on Campus Sexual Assault Goes Digital,” Nov. 13, 2015.

6

Related Literature:

In addition to the debate concerning the potential benefits of aggregating offenses, ourpaper contributes to several literatures that concern strategic communication, voting, and law and economics.
First, our paper is related to the literature on strategic communication with multiple senders (Battaglini (2002,2017), Ambrus and Takahashi (2008), Morgan and Stocken (2008), and Ekmekci and Lauermann (2020, 2022))
and when the sender’s information is endogenous (Pei (2015), Argenziano, Severinov and Squintani (2016), andKreutzkamp (2023)). In all these works, senders communicate information about some exogenous state of theworld. In the present paper, by contrast, witnesses (the senders in our model) communicate information about an
endogenous object, which is the defendant’s action. As a result, the correlation between senders’ private signals
is determined by the defendant’s incentives and is therefore endogenous. Our results shed light on how thisendogenous correlation structure combined with the senders’ coordination motives, affects the informativeness ofcommunication which in turn affects the defendant’s equilibrium behavior.
Second, our paper is related to the literature on strategic voting (Austen-Smith and Banks (1996), Feddersen
and Pesendorfer (1996)). Witnesses’ testimonies under the APP criterion may be viewed as a voting game inwhich witnesses vote on whether the defendant is guilty of having committed at least one offense. In contrast to
the earlier literature, in our model the correlation between voters’ private signals is endogenous and the voting ruleused for conviction is also endogenously determined. These features also distinguish our analysis from Schmitz
and Tröger (2012) and Ali, Mihm, and Siga (2018), who analyze the performances of voting rules when voters’
payoffs are negatively but exogenously correlated, and from Persico (2004), Gershkov and Szentes (2009), andStrulovici (2022), who analyze voters’ incentives to acquire information about some exogenous state of the world.
Third, our paper contributes to the law and economics literature. Our main contribution is to compare theequilibrium outcomes under different conviction rules when potential offender may commit multiple offenses andwhen the offender’s incentives to commit offenses and witnesses’ incentives to testify are both endogenous. In
addition to the works from legal scholarship cited above, our paper is most closely related to the contemporaneousworks of Lee and Suen (2020) and Cheng and Hsiaw (2022). Lee and Suen (2020) study the timing of reports byvictims and libelers in a model in which a criminal commits crimes against each of the two agents with exogenous probability. They provide an explanation for the well-documented fact that victims sometimes delay theiraccusations. Their analysis and ours consider complementary aspects of witnesses’ reporting incentives. Cheng
and Hsiaw (2022) adopt a global game perspective to study the reporting incentives of a continuum of agents who
observe conditionally independent signals of the state. In contrast to those papers, endogenizing the behaviors ofpotential offenders is the main innovation of our model.

7

2

Model

Consider a three-stage game between a defendant, two witnesses (which we will call agents),8 and a judge.
In the first stage, the defendant chooses θ ≡ (θ1 , θ2 ) ∈ {0, 1}2 where θi = 1 stands for the defendantcommitting an offense witnessed by (or against) agent i. In the second stage, each agent i ∈ {1, 2} privately
observes θi and the realization of a preference shock ωi ∈ R. We assume that ω1 and ω2 are independently
distributed, both are drawn from a normal distribution with cdf Φ(·) and pdf ϕ(·).9 Then the agents simultaneously
decide whether to accuse the defendant (ai = 1) or not (ai = 0). In the third stage, the judge observes a ≡(a1 , a2 ) ∈ {0, 1}2 and chooses s ∈ {0, 1}, where s = 1 stands for convicting the defendant and s = 0 stands foracquitting the defendant.

A strategy profile consists of a tuple σo , (σi )2i=1 , q where: σo ∈ ∆ {0, 1}n is the defendant’s strategy andrepresents a probability distribution over vectors θ of offenses; σi : R × {0, 1} → [0, 1] is agent i’s strategy andmaps each realization of ωi and θi to a probability that i accuses the defendant; and q : {0, 1}n → [0, 1] maps
agents’ reports a ∈ {0, 1}2 to the probability of conviction (s = 1).
P
The defendant’s payoff is given by 2i=1 θi − sL, and interpreted as follows: the defendant receives a benefit
normalized to 1 for each committed offense, and loses L > 0 if he is convicted. Agent i ∈ {1, 2}’s payoff is givenbyui (ωi , θi , ai ) ≡ ωi + bθiif s = 1 −ca

if s = 0.

i

(2.1)

This payoff should be understood as follows: if the defendant is convicted, agent i receives an idiosyncratic private
benefit ωi plus an additional benefit b > 0 if i had observed offense θi = 1 committed by the defendant. This
payoff structure implies that, other things being equal, i has a stronger incentive to accuse the defendant when
ωi is higher and when i has observed an offense. In addition, i incurs a cost c if i accuses the defendant but thisaccusation does not lead to a conviction (s = 0). This cost may be interpreted as defendant retaliating against i incase of a failed accusation.
Section 5.3 considers several important extensions, including one in which the defendant has a private type that
determines his benefit from from committing offenses and one in which the defendant faces decreasing marginal
returns from committing offenses. That section also considers the case in which agents have a private benefit
from telling the truth irrespective of the conviction outcome or from convicting defendants who have committed
offenses against other agents, in addition to the one that they get to directly to observe.
8

Section 5.2 studies an extension in which there are three or more witnesses.
Our main result (Theorem 1) and our comparative statics result (Theorem 3) hold regardless of the distribution of agents’ preference shocks. Proposition 1 and Theorem 2 also extend to other distributions as long as their left tails are not too fat in the sense thatlimω→+∞ Φ(ω−b)= 0 for every b > 0. Our condition is satisfied, for example, when the distribution is normal.
Φ(ω)
9

8

As noted in the Introduction, an important consideration to frame our analysis is the debate in legal scholarship
regarding whether one should aggregate accusations faced by a defendant to form an overall probability of the
defendant’s guilt. To engage with this debate, we compare equilibrium outcomes for the two adjudication criteria
used in this literature: the Aggregate Probabilities Principle (or APP) and the Distinct Probabilities Principle(or DPP). Under APP, a defendant is convicted if the probability that he is guilty of committing at least one
offense (i.e., the probability that maxi {θi } = 1) exceeds some threshold, such as 95%, interpreted as the judge’s
standard of proof. Under DPP, by contrast, a defendant is convicted if there is at least one offense for which theprobability that the defendant is guilty of this specific offense exceeds some threshold. Mathematically, DPP leads
to a conviction if maxi Pr(θi = 1) exceeds some threshold.
In practice, DPP is used in many criminal justice systems. For example, when a defendant is charged with anumber of offenses, the court examines each charge individually to decide whether a beyond-a-reasonable-doubt
standard is satisfied. By contrast, APP is more commonly used for punishments meted out outside the criminal
justice system, for example in the case of managers charged with discriminating against minority workers, orwhen supervisors are charged with abusing their subordinates, where the accumulation of charges is sometimes
used as proxy for the probability of guilt of the accused. This perspective is supported by Schauer and Zeckhauser(1996) who write that although sound reasons for the criminal law’s refusal to cumulate multiple low-probability
accusations exist, the reasons for such refusal are often inapt in other settings. Harel and Porat (2009) advocate
for a broader use of APP in judicial settings, arguing that acquitting defendants who are almost surely guilty ofsome unspecified crime is neither just nor efficient (page 263).

We now formally define APP and DPP and their corresponding equilibria. A strategy profile σo , (σi )2i=1 , q isan equilibrium under APP and standard of proof π ∗ ∈ (0, 1) if the following holds: σo maximizes the defendant’s
payoff given (σ1 , σ2 , q); σi maximizes agent i’s payoff given (σo , σ−i , q) for every i ∈ {1, 2}; and q(a) satisfies

q(a) = 1if

Pr(θ = 1|a) > π ∗

 0

if

Pr(θ = 1|a) < π ∗(2.2)

where θ ≡ maxi∈{1,2} θi captures whether the defendant is guilty of at least one offense.

A strategy profile σo , (σi )2i=1 , q is an equilibrium under DPP and standard of proof π ∗ ∈ (0, 1) if thefollowing holds: σo maximizes the defendant’s payoff given (σ1 , σ2 , q); σi maximizes agent i’s payoff given(σo , σ−i , q) for every i ∈ {1, 2}; and q(a) satisfies

q(a) =


 1

if

 0

if


maxi∈{1,2} Pr θi = 1 a > π ∗maxi∈{1,2} Pr θi = 1 a < π ∗ .
9

(2.3)

As is standard in models with discrete action spaces, we allow players to use mixed strategies in order to ensure the existence of equilibrium. In particular, the judge may convict the defendant with any probability when
she is indifferent, which corresponds to the condition Pr(θ = 1|a) = π ∗ under APP and to the condition
maxi∈{1,2} Pr θi = 1 a = π ∗ under DPP.
In our model, each agent i signals his private information about θi and ωi by choosing whether to accuse
the defendant. As in other signaling games, there generally exist numerous equilibria, including some which arearguably unnatural. Fortunately, it is possible to rule out these equilibria using two refinements which are easy tointerpret in our context. First, regardless of the conviction rule and the standard of proof π ∗ ∈ (0, 1), there always
exist equilibria in which the defendant commits all offenses and is convicted even when no agent accuses him.
These equilibria violate the principle–akin to a form of presumption of innocence–that a defendant should not be
convicted based on the sole basis of a judge’s prior belief. This motivates the following refinement:
Refinement 1 (No Conviction Unless Accused). (i) The defendant is acquitted for sure if no agent accuseshim. (ii) The event that no agent accuses the defendant occurs with strictly positive probability.
Refinement 1 implies that the defendant cannot, in equilibrium, be guilty with probability 1. Otherwise, thejudge’s posterior belief would assign probability 1 to the defendant being guilty, regardless of the vector a ofaccusations made against the defendant on the equilibrium path, including when no agent accuses him, which
violates the refinement.
Next, the judge’s conviction decision is a function of her posterior belief about (θ1 , θ2 ), formed based on theset of agents who accuse the defendant. There may exist “contrarian” equilibria in which agents are more likely toaccuse the defendant if they did not observe an offense and making an “accusation” (i.e., ai = 1) runs contrary to
its intended meaning. Refinement 2 rules out this case and ensures that accusations have their intuitive meaningby requiring that each accusation weakly increases the probability of conviction and, hence, that each accusation
is a move against the defendant.
Refinement 2 (Monotonicity). For every i ∈ {1, 2} and a−i ∈ {0, 1}, we have q(1, a−i ) ≥ q(0, a−i ).
In equilibria that satisfy Refinement 2, each accusation is a move against the defendant. This is consistent with
the interpretation that the parameter c appearing in agents’ payoff functions is a retaliation cost borne by an agent
who accused the defendant but failed to secure a conviction. This refinement is consistent with other models of
retaliation. In Chassang and Padró i Miquel (2019), for instance, a defendant commits to retaliate against messages
that increase the defendant’s probability of conviction, but not against messages that do not have this effect.

10

3Single-Offense Benchmark

Before considering the case of multiple potential offenses, we begin the analysis with a single-offense benchmark
in which the defendant can commit an offense only against agent 1, but cannot commit offense against agent 2.
Under this assumption, APP and DPP coincide since Pr(θ = 1) = Pr(θ1 = 1) = maxi∈{1,2} Pr(θi = 1).
We show, in this context, that when the defendant’s loss from conviction L is large relative to his benefit fromcommitting the offense, the probability that the defendant commits the offense (equivalently, since there is only
possible offense, the expected number of offenses) vanishes in equilibrium.
Proposition 1. Suppose that the defendant can only commit an offense against agent 1. For every ε > 0 andπ ∗ ∈ (0, 1), there exists L > 0 such that when L > L, there exists a unique equilibrium that satisfies Refinement
1. In this equilibrium, the defendant commits the offense (i.e., θ1 = 1) with probability less than ε.
Proposition 1 shows that when the punishment in case of a conviction is large relative to the defendant’s
benefit from committing offenses, the probability with which offenses taking place vanishes to zero. Although themechanisms are different, this result is reminiscent of the classic finding in Becker (1968) that larger punishments
are effective at deterring criminal behaviors.10
The proof Proposition 1, provided in Appendix A, has the following intuition. The judge makes her conviction
decision based on her posterior belief about the defendant’s guilt θ after observing agents’ accusations a ≡(a1 , a2 ). According to Bayes rule, the judge’s posterior belief Pr(θ = 1|a) satisfies
Pr(θ = 1|a)Pr(a|θ = 1) Pr(θ = 1)=·.
Pr(θ = 0|a)Pr(a|θ = 0) Pr(θ = 0)

(3.1)

This equation implies that, fixing the defendant’s equilibrium strategy Pr(θ = 1), or equivalently the judge’s priorbelief about θ, the judge’s posterior belief after observing a is pinned down by the following likelihood ratio:
I(a) ≡

Pr(a|θ = 1).
Pr(a|θ = 0)

(3.2)

Equation (3.1) suggests that I(a) measures the informativeness of the agents’ accusations a, in the sense thata larger I(a) implies that the defendant is more likely to be guilty following a. When the defendant does not havethe ability to commit offense against agent 2, I(a) depends only on a1 on the equilibrium path. Since Refinement
1 requires that q(0, 0) = 0, we show that in equilibrium, the defendant is convicted with positive probability only
10
The settings are quite different. For instance, Becker does not consider the possibility of convicting an innocent defendant and treats
the probability of convicting a guilty defendant as an instrument chosen by the regulator, rather than the result of a signal based on witness
with endogenous incentives.

11

if agent 1 accuses him. Let q denote the expected probability that the defendant is convicted after agent 1 accuseshim. From agent 1’s perspective, if he has witnessed an offense, then he prefers to accuse the defendant whenq(b + ω1 ) − (1 − q)c ≥ 0

or equivalently ω1 ≥(1 − q)c
− b.
q

(3.3)

If he has not witnessed any offense, then he prefers to accuse the defendant whenqω1 − (1 − q)c ≥ 0 or equivalently ω1 ≥(1 − q)c
.
q

(3.4)



(1−q)c
)
−
Φ(
−
b)
L. In
Hence, the defendant’s cost of committing an offense against agent 1 equals Φ( (1−q)cqqequilibrium, the defendant’s expected cost from committing an offense must be the same as his benefit 1. Thisimplies that as L becomes large, q must go to 0, and the informativeness of agent 1’s accusation I(a1 = 1, a2 )diverges to +∞.
Since the equilibrium value of q is strictly between 0 and 1, the judge must be indifferent between convicting
and acquitting the defendant after observing a1 = 1, which means that the posterior probability of guilt in thiscase must be equal to the judge’s standard of proof π ∗ . Equation (3.1) then implies that for any fixed π ∗ , the priorprobability of guilt vanishes to 0 as I(1, a2 ) diverges to infinity.

4

Comparisons between APP and DPP

We now turn our focus on the main case of interest, in which the defendant can commit multiple offenses.
As in the case of Proposition 1, we focus on the case in which the punishment meted out to the defendantin case of a conviction is large relative to his benefit from committing offenses and analyze equilibria that satisfy
Refinements 1 and 2. We further focus on equilibria in which the defendant uses symmetric strategies, i.e., chooses(θ1 , θ2 ) = (1, 0) and (θ1 , θ2 ) = (0, 1) with the same probability. As it turns out, this symmetry requirement issatisfied by every proper equilibrium as defined by Myerson (1978). For simplicity and brevity, we directly impose
symmetry rather than define proper equilibrium and prove that it implies symmetry.
From here onwards, we will refer to symmetric equilibria that satisfy Refinements 1 and 2 simply as equilibria.
Section 4.1 establishes preliminary results and properties of equilibria that hold for both APP and DPP. Sections
4.2 and 4.3 present our main results, which show that APP leads to uninformative accusations and a large expected
number of offenses and DPP leads to informative accusations and effective deterrence. Section 4.4 discusses theimplications of our results on the use of APP and DPP, and acknowledges their limitations.

12

4.1

Common Properties of APP and DPP

In this section, we establish properties of equilibria that hold under both APP or DPP. A useful observation isthat the defendant’s incentive to commit offenses and agents’ incentives to accuse the defendant depend on thejudge’s conviction rule (whether it is APP or DPP) only through the mapping from the agents’ accusations to theprobabilities of conviction q : {0, 1}2 → [0, 1]. Hence, changes in the conviction rule affect the defendant’s andthe agents’ incentives only through the mapping q that arises in equilibrium.
We start by showing that regardless of the conviction rule and of the standard of proof π ∗ ∈ (0, 1), the probability that the defendant commits any given offense lies strictly between 0 and 1 in every equilibrium. Furthermore,when L is large enough, each agent’s equilibrium strategy must take the form of two cutoffs–one used when theagent has witnessed an offense and one used when he has not–such that the agent accuses the defendant if and only
if the agent’s realized payoff shock ωi exceeds the cutoff corresponding to agent’s observation.
Lemma 4.1. In every equilibrium, Pr(θi = 1) ∈ (0, 1) for every i ∈ {1, 2}. There exists L > 0 such that inevery equilibrium when L > L, agent i’s strategy is characterized by two cutoffs ωi∗ and ωi∗∗ with −∞ < ωi∗ <ωi∗∗ < +∞ such that
1. When θi = 1, agent i accuses the defendant if and only if his preference shock ωi is greater than ωi∗ .
2. When θi = 0, agent i accuses the defendant if and only if his preference shock ωi is greater than ωi∗∗ .
The proof of this result, provided in Appendix B, has the following intuition. Refinement 1 requires thatq(0, 0) = 0 and that the report vector a = (0, 0) occurs with strictly positive probability. This, together with
Bayes rule, implies that for any equilibrium and under any conviction threshold π ∗ ∈ (0, 1), the defendant must be
innocent of each offense with strictly positive probability, i.e., that Pr(θi = 1) < 1 for every i. The intuition for
the fact that the defendant’s probability of guilt is strictly positive is reminiscent of the equilibrium logic arisingin inspection games: if offenses took place with zero probability, the judge’s posterior belief about the defendant’s
guilt would be equal to zero following every report vector a that arises on the equilibrium path, and the defendant
would not be punished. This, in turn would give the defendant a strict incentive to commit offenses, contradicting
the hypothesis that the defendant commits no crime.
Turning to agents’ behavior, we first observe that agent i chooses ai = 1 with strictly higher probability whenθi = 1. If this were not the case, the defendant would strictly prefer to commit the corresponding offense (θi = 1;
a situation that we have already shown cannot arise in equilibrium) as this would, in addition to providing thebenefit from committing the offense, decrease the probability of being accused by i and lead to a weakly lower
probability of conviction.
13

To understand why agents’ equilibrium strategies take the form of two cutoffs, let us compare their payoffs
when they accuse the defendant to their payoffs when they do not. For j ∈ {1, 2} and k ∈ {0, 1}, let Qk,j denotethe probability that aj = 1 conditional on the event that θi = k, where i ̸= j denotes the agent other than j. If
θ1 = 1, then agent 1 prefers to choose a1 = 1 whennoQ1,2 (q(1, 1) − q(0, 1)) + (1 − Q1,2 )(q(1, 0) − q(0, 0)) (ω1 + b)|{z}

the expected increase in conviction probability when agent 1 accuses the defendant≥cno
1 − Q1,2 q(1, 1) − (1 − Q1,2 )q(1, 0)
{z
}
|

(4.1)

the expected probability that the defendant is acquitted if agent 1 accuses him

where the LHS is his expected payoff when he accuses the defendant and the RHS is his expected payoff when hedoes not accuse the defendant. Similarly, if θ1 = 0, then agent 1 prefers to choose a1 = 1 whennonoQ0,2 (q(1, 1) − q(0, 1)) + (1 − Q0,2 )(q(1, 0) − q(0, 0)) ω1 ≥ c 1 − Q0,2 q(1, 1) − (1 − Q0,2 )q(1, 0) . (4.2)
Refinements 1 and 2 imply that q(1, 1) ≥ q(0, 1) and q(1, 0) ≥ q(0, 0) = 0. Hence, the coefficients in front of ω1
are non-negative for both (4.1) and (4.2). To show that agent 1’s equilibrium strategy takes the form of cutoffs, weonly need to rule out cases where where agent 1 believes that his accusation has no impact on the probability ofconviction, i.e., either Q1,2 (q(1, 1)−q(0, 1))+(1−Q1,2 )q(1, 0) or Q0,2 (q(1, 1)−q(0, 1))+(1−Q0,2 )q(1, 0) is 0.
Intuitively, these cases cannot arise since once agent 1’s accusation has no impact on the probability of conviction,
the defendant will have a strict incentive to commit offenses against against agent 1. This will then contradict ourearlier conclusion that the probability of guilt against each agent must be strictly between 0 and 1.
Next, we show that when the punishment for conviction L is above some threshold, the defendant must be
convicted with probability less than 1 in every equilibrium regardless of the conviction rule and of the standard of
proof used to convict the defendant.
Lemma 4.2. There exists L ∈ R+ such that for every L > L and π ∗ ∈ (0, 1), q(1, 1) ∈ (0, 1) in every
equilibrium under APP and standard of proof π ∗ and in every equilibrium under DPP and standard of proof π ∗ .
The proof of this lemma is provided in Online Appendix A. The key step of the proof is to show that, if
the defendant were convicted for sure when two accusations are made against him, then the increase in conviction
probability when the defendant commits an additional offense would have to be strictly positive and bounded below
away from zero for all punishment levels L large enough. As L increases, the marginal cost from committing
offenses would then have to eventually exceed the marginal benefit from committing each offense (which, we
14

recall, has been normalized to 1), which would give the defendant a strict incentive not to commit any offense andcontradict Lemma 4.1.
Lastly, we provide conditions on q that characterize when the defendant’s choices of θ1 and θ2 are substitutes,
and when they are complements. Conditional on his choice of θj , the defendant’s expected cost of committing an
offense against agent i ̸= j isCi (θj ) ≡ E[q(ai , aj )|θi = 1, θj ] − E[q(ai , aj )|θi = 0, θj ] L.

(4.3)

We say that the defendant’s choices of θ1 and θ2 are strict complements if Ci (1) < Ci (0) for every i ∈ {1, 2},that is, if the defendant’s expected cost of committing an offense against agent i is lower if he commits an offense
against agent j than when he doesn’t. Similarly, say that the defendant’s decision variables θ1 and θ2 are strict
substitutes if Ci (1) > Ci (0) for every i ∈ {1, 2}.
Lemma 4.3. The defendant’s choices of θ1 and θ2 are strict substitutes if and only ifq(1, 1) + q(0, 0) − q(1, 0) − q(0, 1) > 0.

(4.4)

His choices of θ1 and θ2 are strict complements if and only if q(1, 1) + q(0, 0) − q(1, 0) − q(0, 1) < 0.
Proof. Recall from Lemma 4.1 that agent i accuses the defendant with probability Ψ∗i ≡ 1 − Φ(ωi∗ ) when θi = 1,∗∗∗∗∗
and with probability Ψ∗∗i ≡ 1 − Φ(ωi ) when θi = 0, where Ψi > Ψi . If the defendant increases θ1 from 0 to 1

when θ2 = 0, the the probability of conviction increases by



∗∗
∗∗
(Ψ∗1 − Ψ∗∗1 ) (1 − Ψ2 ) q(1, 0) − q(0, 0) + Ψ2 q(1, 1) − q(0, 1) .

(4.5)

If the defendant increases θ1 from 0 to 1 when θ2 = 1, the the probability of conviction increases by



∗
∗
(Ψ∗1 − Ψ∗∗
)
(1
−
Ψ
)
q(1,
0)
−
q(0,
0)
+
Ψ
q(1,
1)
−
q(0,
1)
1
2
2

(4.6)

The defendant’s choices of θ1 and θ2 are strict substitutes if and only if (4.5) is strictly less than (4.6), which is the

∗
∗∗
∗
∗∗
case if and only if (Ψ∗1 − Ψ∗∗
1 )(Ψ2 − Ψ2 ) q(1, 0) + q(0, 1) − q(0, 0) − q(1, 1) < 0. Since Ψi > Ψi for every i,
the above inequality is equivalent to (4.4). A similar argument applies to the statement on strict complements.

15

4.2

Aggregate Probabilities Principle

This section characterizes equilibrium outcomes under APP. Recall the formula for the judge’s posterior belief
about the defendant’s guilt in (3.1) and the measure for the informativeness of agents’ accusations in (3.2).
Theorem 1. For every π ∗ ∈ (0, 1) and ε > 0, there exists L > 0 such that when L > L, in every equilibrium
under APP and standard of proof π ∗ , we have:
1. Negative Correlation Across Offenses: Pr(θi = 1|θj = 1) < Pr(θi = 1|θj = 0) for every i ̸= j.
2. Uninformative Accusations: maxa∈{0,1}2 I(a) < 1 + ε.11
3. Ineffective Deterrence:

P2

i=1 Pr(θi = 1) > π∗ − ε.

Theorem 1 suggests that when the punishment in case of conviction L is large relative to the benefit fromcommitting offenses, APP induces negative correlation between the occurrence of the distinct offenses. This
negative correlation has, in turn, the effect of weakening the credibility of agents’ accusations and results in a highexpected number of offenses. These conclusions stand in contrast to the single-offense benchmark, in which the
expected number of offenses vanishes and the informativeness of the agent’s accusation diverges to infinity as thepunishment for conviction increases.
Two aspects of Theorem 1 are worth emphasizing before presenting its proof. First, the expected number ofoffenses in equilibrium is close to the standard of proof π ∗ . Although a decrease in the standard of proof can lower
the expected number of offenses, such a decrease will also increase the probability that a convicted defendant is
innocent. In fact, we show that when the standard of proof is π ∗ , the probability that a convicted defendant is
innocent equals 1 − π ∗ . For example, if the standard of proof is only 20%, then a convicted defendant is innocent
with probability 80%. From this perspective, Theorem 1 highlights a tension caused by APP between achieving
deterrence and reducing the probability that convicted defendants are innocent. As we will see, this tension does
not exist under DPP, in the sense a high standard of proof π ∗ guarantees that both the expected number of offenses
and the probability that convicted defendants being innocent are low.
Second, since each agent incurs a cost c of accusing the defendant, one might conjecture that ineffective
deterrence is caused by agents’ free-riding motives, as in public good provision games. However, we show in
Theorem 3 of Section 5.2 that this intuition is incorrect in our setting. In fact, each agent accuses the defendantwith strictly higher probability when there are more agents.
Next, we present the proof of Theorem 1, which proceeds in three steps.
11

By definition, maxa∈{0,1}2 I(a) is at least 1. Hence, Theorem 1 implies that I is arbitrarily close to the trivial lower bound.

16

Step 1: The Defendant’s Incentives Since each agent i ∈ {1, 2} is strictly more likely to accuse the defendant
when θi = 1 than when θi = 0, Bayes rule implies that
Pr(θ = 1|ai = 1, aj ) > Pr(θ = 1|ai = 0, aj ) for every aj ∈ {0, 1}.

(4.7)

That is to say, each additional accusation strictly increases the probability that the defendant is guilty of at least
one offense. This observation plays a key role in the analysis of APP, and distinguishes it from the analysis of
DPP, as explained in Section 4.3.
Suppose by way of contradiction that either q(0, 1) > 0 or q(1, 0) > 0. Then, the posterior probability thatθ = 1 is weakly greater than π ∗ either when a = (1, 0) or when a = (0, 1). Inequality (4.7) then implies that theposterior probability that θ = 1 is strictly greater than π ∗ when a = (1, 1). Hence, in equilibrium under APP withstandard of proof π ∗ , we have q(1, 1) = 1. This contradicts Lemma 4.2 that q(1, 1) < 1 when L is large enough.
Therefore, q(1, 0) = q(0, 1) = q(0, 0) = 0 and q(1, 1) ∈ (0, 1) in every equilibrium. Since the equilibrium
conviction probabilities satisfy (4.4), Lemma 4.3 implies that the defendant’s decisions to commit different offenses are strict substitutes. Since Refinement 1 requires that (θ1 , θ2 ) = (0, 0) occurs with positive probability,i.e., committing no offense is optimal for the defendant. This implies that the marginal cost of committing the first
offense is weakly greater than 1. Since the marginal cost of committing a second offense is strictly greater than themarginal cost of committing the first offense, it is strictly suboptimal for the defendant to commit two offenses.
Since the equilibrium probability of guilt is interior, the defendant must be indifferent between committing nooffense and committing one offense. Therefore, his marginal cost of committing the first offense is 1. Since thedefendant’s strategy is required to be symmetric across agents, he will choose (0, 0), (1, 0), and (0, 1) with positive
probability, and choose (1, 0) and (0, 1) with the same probability. This implies that in equilibrium, θ1 and θ2 are
negatively correlated. Let π ≡ Pr(θ = 1) denote the equilibrium probability that the defendant is guilty of atleast one offense. We know that Pr(θi = 1) = π/2 for every i ∈ {1, 2}, and that the conditional on θi = 0, the1−πprobability with which θj = 0 equals β ≡ 1−π/2.

Step 2: Symmetry in Agents’ Equilibrium Strategies Since the defendant chooses both (θ1 , θ2 ) = (1, 0) and(θ1 , θ2 ) = (0, 1) with positive probability in equilibrium, his indifference condition yields:




1
= (1 − Φ(ω1∗∗ )) Φ(ω2∗∗ ) − Φ(ω2∗ ) = (1 − Φ(ω2∗∗ )) Φ(ω1∗∗ ) − Φ(ω1∗ ) .
qL

17

(4.8)

To simplify notation, let q ≡ q(1, 1). From agents’ incentive constraints (4.1) and (4.2), q(0, 0) = q(1, 0) =q(0, 1) = 0, implies that when θi = 1, agent i accuses the defendant if and only if
c
,
qQ1,j

(4.9)

c
.
qQ0,j

(4.10)

ωi ≥ ωi∗ ≡ −b − c +and when θi = 0, agent i accuses the defendant if and only if:
ωi ≥ ωi∗∗ ≡ −c +Since Pr(θi = 1|θj = 1) = 0 and Pr(θi = 1|θj = 0) = 1 − β, we know that Q1,j = 1 − Φ(ωj∗∗ ) andQ0,j = β(1 − Φ(ωj∗∗ )) + (1 − β)(1 − Φ(ωj∗ )). Suppose by way of contradiction that ωj∗∗ > ωi∗∗ . Then (4.8) is true
only when ωj∗ > ωi∗ . However, the expression for ωi∗ is strictly decreasing in Q1,j and Q1,j is strictly decreasingin ωj∗∗ . Therefore, Φ(ωj∗∗ ) > Φ(ωi∗∗ ) together with (4.9) implies that ωi∗ > ωj∗ . This leads to a contradiction
and implies that ω1∗∗ = ω2∗∗ . Equation (4.8) then implies that ω1∗ = ω2∗ . In the remainder of this section, we
can therefore simplify notation by denoting the cutoffs by ω ∗ and ω ∗∗ without subscripts and writing Q1 and Q0
instead of Q1,j and Q0,j .
Step 3: Coordination under Negative Correlation

Since agent i incurs a cost c when he chooses ai = 1 but

fails to convict the defendant and since q(a1 , a2 ) is weakly increasing in both of its arguments, agent i’s incentive
to report ai = 1 is increasing in the probability he assigns to the event aj = 1. That is, agents’ decisions to accuse
the defendant are strategic complements. Since ω ∗ < ω ∗∗ , we know that Q1 < Q0 . The expressions for the cutoffs
then imply that ω ∗∗ − ω ∗ ∈ (0, b). The fact that this difference is strictly less than b stands in contrast to the singleoffense benchmark of Section 3, and it plays an important role in the analysis as it implies that agents’ reports
are less informative relative to the single-offense benchmark. Since the defendant never commits two offenses inequilibrium, we haveI(1, 1) ≡1 − Φ(ω ∗ )Pr(a = (1, 1)|θ = 1)=.
1 − Φ(ω ∗∗ )Pr(a = (1, 1)|θ = 0)

(4.11)∗ππ
Since q(1, 1) ∈ (0, 1), we also have Pr(θ = 1|a = (1, 1)) = π ∗ . From Bayes rule, 1−πI(1, 1) = 1−π∗ . Letting∗πl∗ ≡ 1−π∗ , we have

β=

2I
l∗
and
1
−
β
=
.
l∗ + 2Il∗ + 2I(4.12)

Plugging (4.12) into the expressions for Q1 and Q0 , we obtainβ(1 − Φ(ωj∗∗ )) + (1 − β)(1 − Φ(ωj∗ ))Q0(l∗ + 2)I(1, 1)
=
=
β
+
(1
−
β)I(1,
1)=.
Q1
1 − Φ(ωj∗∗ )l∗ + 2I(1, 1)
18(4.13)

The expressions for ω ∗ and ω ∗∗ imply that ω ∗ + b + c = qQc 1 and ω ∗∗ + c = qQc 0 . Therefore,ω∗ + b + c(l∗ + 2)I(1, 1)
Q0==.
ω ∗∗ + c
Q1l∗ + 2I(1, 1)(4.14)

Since ω ∗∗ − ω ∗ ∈ (0, b), the difference between ω ∗ + b + c and ω ∗∗ + c is at most b. Hence, the LHS of (4.14)
converges to 1 as ω ∗ , ω ∗∗ → +∞. The RHS of (4.14) is strictly increasing in I and equals 1 only if I(1, 1) = 1.
Hence I(1, 1) converges to 1 as ω ∗ , ω ∗∗ → +∞.
The defendant’s indifference condition (4.8) implies that as L → +∞, either q → 0 or at least one of 1 −Φ(ω ∗∗ ) and Φ(ω ∗∗ ) − Φ(ω ∗ ) converges to 0. If q → 0, then the expressions for ω ∗ and ω ∗∗ in (4.9) and (4.10)
imply that ω ∗ , ω ∗∗ → +∞. If 1 − Φ(ω ∗∗ ) converges to 0, then ω ∗∗ → +∞ and ω ∗ → +∞ as well since|ω ∗ − ω ∗∗ | ∈ (0, b). If Φ(ω ∗∗ ) − Φ(ω ∗ ) converges to 0, then either ω ∗∗ → +∞ or −∞ or ω ∗∗ − ω ∗ → 0 or both.
Since (4.10) implies that ω ∗∗ ≥ 0, it cannot be the case that ω ∗∗ → −∞. Our earlier conclusion implies that thevalue of I(1, 1) converges to 1 regardless of whether both ω ∗ and ω ∗∗ diverge to +∞ or ω ∗∗ − ω ∗ → 0 and boththresholds are bounded from above. According to Bayes rule, I(1, 1) → 1 implies that π → π ∗ .

4.3

Distinct Probabilities Principle

Next, we characterize equilibrium outcomes under DPP and compare them to those under APP.
Theorem 2. For every π ∗ ∈ (0, 1) and ε > 0, there exists L ∈ R+ such that when L > L, in every equilibrium
under DPP with standard of proof π ∗ ,1. Uncorrelated Offenses: Pr(θi = 1|θj = 1) = Pr(θi = 1|θj = 0) for every i ̸= j.
2. Linear Conviction Probability: q(1, 1) = 2q(1, 0) = 2q(0, 1) > 0 and q(0, 0) = 0.
Pr(ai =1|θi =1)
3. Effective Deterrence & Informative Accusations: E[θi ] < ε and Pr(a> 1/ε for every i ∈ {1, 2}.
i =1|θi =0)

Theorem 2 implies that, under DPP, agents’ private observations of offenses must be uncorrelated and theconviction probability must be a linear function of the number of accusations. When the conviction probabilities
are linear, the defendant’s incentives to commit different offenses are neither complements nor substitutes, anddeciding to commit each offense independently of one’s decision regarding other offenses is an optimal strategy.
In this case, agents’ private observations of offenses are uncorrelated. Compared to APP, this lack of correlation
restores the credibility of the agents’ accusations, which has the further effect of lowering the expected number ofoffenses in equilibrium for any given standard of proof π ∗ .
The key distinctive feature of DPP is that accusing the defendant of committing an offense under DPP need notincrease the probability that the defendant is guilty of any other offense, and thus need not affect or increase the19maximal probability, over all possible offenses, that the defendant is guilty of a specific offense. If, for instance,
θ1 and θ2 are negatively correlated, then an accusation by agent 2 can lower the posterior probability that thedefendant is guilty of the first offense.
Next, we present the proof of Theorem 2, which leverages the previous observation and sheds light on how
APP and DPP lead to different structures of the conviction probabilities as functions of agents’ reports and on how
these differences affect the defendant’s and the agents’ incentives.
Step 1: Ruling Out Correlations Suppose by way of contradiction that θ1 and θ2 are negatively correlated, i.e.,
Pr(θi = 1|θj = 1) < Pr(θi = 1|θj = 0) for i ̸= j. Then, Pr(θ1 = 1|a = (1, 1)) < Pr(θ1 = 1|a = (1, 0))because an accusation by agent 2 increases the posterior probability that θ2 = 1 and, when θ1 and θ2 are negatively
correlated, his accusation lowers the probability that θ1 = 1. Similarly, one can show that Pr(θ2 = 1|a = (1, 1)) <Pr(θ2 = 1|a = (0, 1)). When conviction is decided according to DPP, the conviction probabilities must satisfyq(1, 0) ≥ q(1, 1) and q(0, 1) ≥ q(1, 1). For these probabilities to satisfy Refinements 1 and 2, it must be the casethatq(1, 1) = q(1, 0) = q(0, 1) = 1 and q(0, 0) = 0.

(4.15)

According to Lemma 4.3, the defendant’s incentives to commit different offenses are strict complements. Since
Refinement 1 requires that the defendant choosing (θ1 , θ2 ) = (0, 0) with positive probability, he has no incentive
to choose (θ1 , θ2 ) ∈ {(1, 0), (0, 1)} since it is dominated by either (θ1 , θ2 ) = (0, 0) or (θ1 , θ2 ) = (1, 1). This
contradicts our earlier hypothesis that θ1 and θ2 are negatively correlated.
Next, suppose by way of contradiction that θ1 and θ2 are positively correlated, i.e., Pr(θi = 1|θj = 1) >Pr(θi = 1|θj = 0) for i ̸= j. In this case, a = (1, 1) is the unique maximizer of both Pr θ1 = 1 a andPr θ2 = 1 a . If the conviction rule is DPP, then it must be the case thatq(1, 1) ≥ max{q(1, 0), q(0, 1)} ≥ q(0, 0) = 0,

(4.16)and q(1, 1) > max{q(1, 0), q(0, 1)} unless q(1, 1) = q(1, 0) = q(0, 1) = 1. If q(1, 1) = 1, this contradictsLemma 4.2 when L is large enough. When q(1, 1) ∈ (0, 1), we know that maxi∈{1,2} {Pr θi = 1 a = (1, 1) } =π ∗ , which implies that maxi∈{1,2},a̸=(1,1) {Pr θi = 1 a } < π ∗ . As a result, q(1, 0) = q(0, 1) = 0. Lemma 4.3
then implies that the defendant’s decisions to commit different offenses are strict substitutes. This contradicts ourearlier hypothesis that θ1 and θ2 are positively correlated.

20

Step 2: Linear Conviction Probabilities Since θ1 and θ2 must be uncorrelated and Refinement 1 implies that(θ1 , θ2 ) = (0, 0) occurs with positive probability, the defendant’s incentives to commit offenses (i) cannot bestrict substitutes since he will have no incentive to choose (θ1 , θ2 ) = (1, 1) and (ii) cannot be strict complements
since he will have no incentive to choose (θ1 , θ2 ) = (1, 0) and (θ1 , θ2 ) = (0, 1). Lemma 4.3 then implies thatq(1, 1) = q(1, 0) + q(0, 1) and q(0, 0) = 0. Lemma 4.1 then implies that the defendant must be playing acompletely mixed strategy. The defendant’s indifference between (θ1 , θ2 ) = (1, 0) and (θ1 , θ2 ) = (0, 1) implies
that q(1, 0) = q(0, 1). Hence, the conviction probability is a linear function of the number of accusations.
Step 3: Informative Accusations & Effective Deterrence:

The fact that θ1 and θ2 are uncorrelated implies thateach agent’s belief about whether the other agent has witnessed an offense is independent of his private observationof offenses. According to the formulas for agents’ reporting cutoffs (4.9) and (4.10), the independence of θ1 andθ2 implies that Q1,j = Q0,j for j ∈ {1, 2}. This together with (4.1) and (4.2) implies that ωj∗∗ − ωj∗ = b for
every j ∈ {1, 2}. Let ∆ ≡ q(1, 1) − q(1, 0) = q(1, 1) − q(0, 1) = q(0, 1) = q(1, 0) denote the increase in
the probability of conviction when there is an additional accusation. Since the defendant is indifferent between
choosing (θ1 , θ2 ) = (1, 0) and (θ1 , θ2 ) = (0, 1), we have:L∆ = Φ(ω1∗ + b) − Φ(ω1∗ ) = Φ(ω2∗ + b) − Φ(ω2∗ ).

(4.17)

Since ω1∗ , ω2∗ are both strictly positive when L exceeds some cutoff, the value of Φ(ω ∗ + b) − Φ(ω ∗ ) is strictly
decreasing in ω ∗ . The defendant’s indifference condition then implies that ω1∗ = ω2∗ and ω1∗∗ = ω2∗∗ . Hence, wewill replace ωi∗ and ωi∗∗ with ω ∗ and ω ∗∗ , respectively. The defendant’s indifference condition also implies thatthe cutoffs ω ∗ and ω ∗∗ diverge to −∞ as L becomes large. Since θ1 and θ2 are uncorrelated, the cutoffs satisfy∗1−Φ(ω )
ω ∗∗ − ω ∗ = b. Hence, the informativeness of each agent’s accusation, measured by 1−Φ(ω∗∗ ) , diverges to +∞ as

L → +∞. Since the judge’s posterior belief about θi = 1 equals π ∗ after observing ai = 1, the informativenessratio converging to +∞ implies that the prior probability with which θi = 1 converges to 0.

4.4

Discussion

The comparison between Theorems 1 and 2 provides a justification for the use of DPP in criminal justice systems.
It also suggests a rationale for several aspects of the law such as the fact that judges are not allowed to aggregate
the probabilities of different criminal behaviors when making conviction decisions and are prohibited from using
character evidence to establish the guilt of defendants.
Relative to earlier writings of legal scholars that discuss the advantages and disadvantages of aggregating prob-

21

abilities across cases (e.g., Schauer and Zeckhauser 1996, Harel and Porat 2009), the contribution of our analysis
is to take potential offenders’ and potential witnesses’ strategic motives into account. Our results suggest that
aggregating probabilities across cases affects the correlations between crimes through the defendant’s incentives,
which can undermine the quality of evidence available to judges when they make conviction decisions.
Nevertheless, implementing DPP in practice requires decision makers (e.g., judges, board of trustees of a firm,etc.) to have commitment power. This is illustrated by the numerical example in the introduction, and is also
clearly explained by the following thought experiment: Consider a setting with two defendants and four potential
victims. The probabilities that each defendant is guilty of committing crime against the potential victims are
Pr(offense against individual 1)

Pr(offense against individual 2)Pr(offense against at least one)49.5 %

49.5 %

99 %Pr(offense against individual 3)Pr(offense against individual 4)Pr(offense against at least one)

50 %

1%

51 %

Defendant 1

Defendant 2

Under DPP, defendant 1 is acquitted whenever defendant 2 is acquitted. This is the case even though defendant 1
is almost surely guilty of at least one crime, and the probability that defendant 2 is guilty is significantly lower.
In practice, firms and organizations may lack the commitment required to implement DPP and yield to socialpressure and aggregate the probabilities across different offenses. For example, firms may face more social pressure to fire a manager whose probabilities of abusing subordinates are given by the first row and political parties
may have a stronger incentive to ostracize party members with bad reputations (e.g., individuals who are believed
to have committed at least some offenses with high probability) even without a thorough investigation of specificevidence related to any given accusation.

5

Extensions

Section 5.1 extends our model in Section 2 to the case in which the defendant has a private type that determines his
benefit from committing offenses. In Section 5.2, we extend the analysis to three or more offenses and agents. We
establish a comparative statics result showing that each agent accuses the defendant with strictly higher probability
as the number of agents increases. Section 5.3 contains several other extensions and variations of our model.

5.1

Uncertain Propensity to Commit Offenses

We have assumed until now that the defendant’s benefit from committing offenses was common knowledge, andfound that when the penalty from conviction is large relative to the benefit from committing offenses, the defendant
never commit both offenses under APP. In practice, individuals have different propensities to commit offenses andanecdotal evidence suggests that some defendants are guilty of multiple offenses.
22Motivated by these observations, we now extend our model in Section 2 to the case in which the defendant hasa private type t ∈ {tv , to }. Specifically, the defendant is either a virtuous type tv whose benefit from committing
offenses is zero (or, more generally, non-positive) or an opportunistic type to whose benefit from committing each
offense is normalized to 1. Let π o denote the prior probability that the defendant has an opportunistic type (themodel in Section 2 corresponds to π o = 1). We show that under APP, a defendant with an opportunistic typecommits multiple offenses with strictly positive probability as long as π o < π ∗ , i.e., as long as the virtuous typeoccurs with high enough probability. This novel feature notwithstanding, we show that the main insight from
Theorem 1 that APP induces negative correlation across offenses still applies.
Proposition 2. For every π ∗ ∈ (0, 1), π o ∈ (0, 1), and ε > 0, there exists L > 0 such that when L > L, inevery equilibrium under APP and standard of proof π ∗ ,1. Endogenous Negative Correlation: Pr(θi = 1|θj = 1) < Pr(θi = 1|θj = 0) for every i ̸= j.
2. Uninformative Accusations & Ineffective Deterrence: Pr(θ = 1) > min{π ∗ , π o } − ε andmax I(a) <a∈{0,1}2n π ∗ . min{π ∗ , π o } o+ ε.
1 − π ∗ 1 − min{π ∗ , π o }(5.1)

3. Multiple Offenses with Positive Probability: Pr(θ1 = θ2 = 1|t = to ) > 0 if and only if π o < π ∗ .
The proof is in Appendix C. Proposition 2 shows that, even when the defendant has private information regarding his benefit from committing offenses, agents’ private observations about offenses are negatively correlated
as in our model of Section 2, the informativeness of agents’ accusations is low as in that model, and the lack of
informativeness still leads to a high probability that offenses take place.
In Proposition 2, the upper bound regarding the informativeness of agents’ accusations and the lower boundregarding the probability of offenses are the same as those in Theorem 1 when π o ≥ π ∗ , i.e., when the probability of the opportunistic type exceeds the conviction threshold. However, these bounds are different when theopportunistic type occurs with probability less than π ∗ . In this case, we show that the defendant commits offenses
with probability π o , which is the highest possible probability of guilt given that the virtuous type never commits
any offense. Given that, in every equilibrium, there exists a ∈ {0, 1}2 such that the defendant is convicted withpositive probability when the judge observes a, the judge’s posterior belief that at least one offense has taken placeis no less than π ∗ after observing a. This suggests a lower bound on the informativeness of agents’ accusations:max I(a) ≥ Imin ≡a∈{0,1}223

π∗ . πo.
1 − π∗ 1 − πo(5.2)

Proposition 2 shows that this lower bound Imin is indeed attained in all equilibria when π o < π ∗ .
The proof of Proposition 2 is similar to that of Theorem 1. We start the proof with the observations that thevirtuous type never commits any offenses and that both Lemma 4.2 and Lemma 4.3 extend to this more general
setting. These observations imply that the opportunistic type’s decisions to commit distinct offenses are strict
substitutes when l is large. As a result, type to cannot be indifferent between committing no offense and committing
two offenses. That is to say, type to is either indifferent between committing no offense and committing only one
offense or indifferent between committing only one offense and committing two offenses. This leads to two
disjoint cases in Proposition 2, whose separation depends on whether the prior probability of the opportunistic
type exceeds π ∗ :1. When π o ≥ π ∗ , the opportunistic type is indifferent between committing no offense and one offense and thequalitative features of the equilibria remain the same as those characterized by Theorem 1.
2. When π o < π ∗ and L is large enough, the opportunistic type commits at least one offense for sure. Otherwise, by the same logic as in Theorem 1, agents’ accusations would be arbitrarily uninformative as L
becomes large. Since the prior probability of guilt is at most π o which is strictly lower than π ∗ , the posterior
probability of guilt is also strictly lower than π ∗ even when both agents accuse the defendant. In this case,the defendant is never convicted under APP, which leads to a contradiction. Hence, the opportunistic typemust be indifferent between committing one offense and committing both offenses.
Nevertheless, we show that θ1 and θ2 remain negatively correlated. To understand why, suppose by way
of contradiction that θ1 and θ2 are independent or positively correlated. In this case, Q0,j ≤ Q1,j , and theexpressions for agents’ reporting cutoffs (4.9) and (4.10) would imply that ωj∗∗ − ωj∗ ≥ b. According toLemma 4.2, q(1, 1) converges to 0 as L → +∞ and agents’ reporting cutoffs both diverge to +∞. As in the
single-offense benchmark, the informativeness of agents’ accusations would then diverge to +∞. Since theopportunistic type commits at least one offense, the prior probability that the defendant commits at least one
offense is π o . Therefore, the posterior probability that the defendant committed an offense when both agentsaccuse the defendant would be strictly greater than π ∗ . This would contradict Lemma 4.2, which requiresthat the judge be indifferent between convicting and acquitting the defendant when both agents accuse him.
Proposition 3 generalizes Theorem 2 to the case in which π o can take any value in (0, 1). It shows that, under
DPP, there is no correlation across different offenses and the conviction probability is a linear function of the number of accusations. These two features result in informative accusations and a low expected number/probability ofoffenses.

24

Proposition 3. For every π o ∈ (0, 1), π ∗ ∈ (0, 1), and ε > 0, there exists L ∈ R+ such that when L > L, inevery equilibrium under DPP with standard of proof π ∗ ,1. Uncorrelated Offenses: Pr(θi = 1|θj = 1) = Pr(θi = 1|θj = 0) for every i ̸= j.
2. Linear Conviction Probability: q(1, 1) = 2q(1, 0) = 2q(0, 1) > 0 and q(0, 0) = 0.
Pr(ai =1|θi =1)> 1/ε for every i ∈ {1, 2}.
3. Effective Deterrence & Informative Accusations: E[θi ] < ε and Pr(ai =1|θi =0)

The proof of Proposition 3 follows from that of Theorem 2 and is omitted in order to avoid repetition.

5.2

Three or More Potential Offenses and Witnesses

This section extends our main results to settings where the defendant can commit n offenses, i.e., choosing(θ1 , .., θn ) ∈ {0, 1}n . For each potential offense, there is a distinct agent who observes whether this offense
has occurred. The defendant’s marginal benefit from committing an offense is still normalized to 1 and the defendant’s loss if convicted is still L > 0. Agent’s payoffs are given by the same formula as in the two-agentcase. Given the standard of proof π ∗ ∈ (0, 1), the defendant is convicted under APP if Pr(θ = 1|a) ≥ π ∗ whereθ ≡ max1≤i≤n θi , and is convicted under DPP if max1≤i≤n Pr(θi = 1|a) ≥ π ∗ .
We show that the takeaways from our main results extend to this multi-agent setting and establish a comparative statics result showing that under APP the informativeness of agents’ accusations about the defendant’s guilt
decreases even when all agents’ reports are aggregated in the number of agents. Moreover, we show that theunconditional probability that each agent accuses the defendant is increasing in the number of agents. We start bygeneralizing the measure of informativeness used in Theorem 1 to three or more agents. According to Bayes Rule,for every a ∈ {0, 1}n ,Pr(a|θ = 1)
Pr(θ = 1)
Pr(θ = 1|a)·=.
Pr(a|θ = 0) 1 − Pr(θ = 1)
1 − Pr(θ = 1|a)(5.3)

Therefore, the ratio
I(a) ≡

Pr(a|θ = 1)
Pr(a|θ = 0)

(5.4)

measures the change in a judge’s posterior belief about θ after observing the agents’ actions a ≡ (a1 , ..., an ). For
tractability, we again focus on symmetric equilibria that satisfy Refinements 1 and 2 (or, equilibrium for short).
Proposition 4 generalizes the insights of Theorem 1 to the case of three or more agents.
Proposition 4. For every n ≥ 2, π ∗ ∈ (0, 1), and ε > 0, there exists Ln,ε > 0 such that for every L > Ln,ε ,in every equilibrium under APP and standard of proof π ∗ ,1. Pr(maxj̸=i θj = 1|θi = 1) < Pr(maxj̸=i θj = 1|θi = 0) for every i ∈ {1, 2, ..., n};25

2. maxa∈{0,1}n I(a) < 1 + ε and Pr(θ = 1) > π ∗ − ε.
The proof is in Online Appendix B.1. According to Proposition 4, an agent who has witnessed an offense
assigns a lower probability to other agents having witnessed offenses. In fact, we show that in every equilibrium,
the defendant commits either no offense or only one offense, which implies that agents’ private observations ofoffenses are negatively correlated. We also show that, as the punishment for conviction becomes large, agents’accusations become arbitrarily uninformative and the expected number of offenses is at least π ∗ .
Next, we establish a comparative statics result with respect to the number of agents, showing that each agent ismore likely to accuse the defendant as the number of agents increases. Recall that in every equilibrium, each agent
i uses a strategy that is characterized by two cutoffs ω ∗ and ω ∗∗ , such that when he has witnessed an offense, heaccuses the defendant if and only if ωi > ω ∗ , and when has not witnessed an offense, he accuses the defendant if
and only if ωi > ω ∗∗ , i.e., lower cutoffs imply higher probabilities of filing accusations.
Theorem 3. For every π ∗ ∈ (0, 1) and k, n ∈ N with k > n, there exists L > 0 such that for every L > L,and compare any equilibrium under APP and standard of proof π ∗ with k agents to that with n agents:
1. Lower Informativeness: maxa∈{0,1}k I(a) < maxa∈{0,1}n I(a).
2. Higher Number/Probability of Offenses: The equilibrium probability of offense Pr(θ = 1) as well as theP
expected number of offenses i Pr(θi = 1) are both strictly higher with k agents than with n agents.
3. Higher Probability of Filing Accusations: Each agent’s reporting cutoffs (ω ∗ , ω ∗∗ ) are both strictly lower
with k agents than with n agents.
The proof is in Appendix D. Theorem 3 shows that as the number of potential offenses (or, equivalently, thenumber of agents) increases, the informativeness of accusations decreases, the probability of offense increases,
and each agent is more likely to accuse the defendant. This last feature distinguishes our result from those on
public good provision, in which inefficiencies arise because agents free ride on other agents’ contributions.
Next, we study the game’s equilibrium outcomes when the judge uses DPP. Proposition 5 establishes theexistence of equilibria in which agents’ private information is uncorrelated and the conviction probability is a linear
function of the number of accusations against the defendant.12 These equilibria feature arbitrarily informative
reports and a vanishing probability of offenses as the punishment L becomes arbitrarily large.
Proposition 5. For every n ≥ 3, π ∗ ∈ (0, 1), and ε > 0, there exists L ∈ R+ such that when L > L, there
exists an equilibrium under DPP with standard of proof π ∗ such that θi is independent of θ−i for every i, theprobability of conviction is a linear function in the number of accusations and
12

Whether this equilibrium is the unique equilibrium remains an open question. This is because when there are three or more θi , one
cannot use the argument in the proof of Theorem 2 that rules out positive and negative correlation between θ1 and θ2 .

26

• Effective Deterrence: E[

Pn

i=1 θi ] < ε.

Pr(ai =1|θi =1)• Informative Accusations: Pr(a> 1/ε for every i ∈ {1, 2, ..., n}.
i =1|θi =0)

The proof is in Online Appendix B.2.

5.3

Other Extensions

Decreasing Marginal Benefits from Committing Offenses:

Our main result, Theorem 1, continues to holdwhen the defendant faces decreasing marginal returns from committing multiple offenses or receives a punishment
larger than L when the probability that he has committed multiple offenses exceeds some threshold. Indeed, thesechanges motivate the defendant to commit fewer offenses and induce negative correlation in the agents’ private
observations. As in the model of Section 2, the agents’ coordination motives undermine the informativeness of
their accusations and lead to a high probability of offenses taking place. For example, our results extend when the
defendant is convicted of a minor crime and receives punishment L if the probability with which he is guilty of atleast one offense exceeds π ∗ , and is convicted of a felony and receives punishment L′ (> L) if the probability withwhich he is guilty of two offenses exceeds some other cutoff π ∗∗ . When L is large, in every equilibrium under
APP and standard of proof π ∗ , the defendant commits at most one offense and the probability with which he does
so approaches π ∗ .
Agents’ Accusation Costs: Our results continue to hold when agents’ costs of accusing the defendant are strictly
positive even when the defendant is convicted, as long as accusation costs are strictly higher when the defendantis acquitted than when he is convicted. Our results also continue to hold if each agent suffers a lower retaliation
cost when there are more accusations filed against the defendant as long as the retaliation cost is strictly positive
whenever the defendant is acquitted. Under these variations of our model, the coordination motives among agents
become stronger and their private observations remain negatively correlated.
Interdependent Preferences:

Agents may have a preference for convicting defendants who have committed

offenses against or witnessed by other agents, even if they have not witnesses or experienced these offenses themselves. Our results extend to the general case where agent i’s payoff is:ui (ωi , θi , ai ) ≡ ωi + bf (θi , θ−i )if s = 1 −ca

if s = 0,

i

27

(5.5)

where f (θi , θ−i ) is strictly increasing in θi and is weakly increasing in θ−i . The model of Section 2 considers the
special case where f (θi , θ−i ) = θi . Equation (5.5) stipulates that agent i has stronger preference for convicting
defendants who have committed offenses, no matter whether the offense is witnessed by himself or by other agents.
In fact, directly caring about offenses committed against or witnessed by other agents, may lower even further the
informativeness of agents’ accusations, by prompting agent i’s action ai to become more responsive to his belief
about the other agent’s private information θj , in effect magnifying agents’ coordination motive. When θi and θjare negatively correlated, ai will become less responsive to his own private information θi .
Agents’ Preferences for Truth-telling: Suppose that each agent receives a direct benefit d (> 0) from filing an
accusation when he has witnessed a crime regardless of the conviction decision. This can arise, for example, when∗

agents have intrinsic preferences for telling the truth. One can show that with two agents and when d < l∗l+2 c, theinformativeness of each agent’s accusation is at mostcl∗,cl∗ − (l∗ + 2)d(5.6)

which converges to 1 when d → 0, i.e., our results are robust when agents receive small benefits from truth-telling.
Ex Post Evidence & Punishing False Accusations: We consider the possibility that evidence may arrive ex postthat exposes false accusations. For example, suppose that when an innocent defendant is convicted, hard evidence
arrives with probability p∗ that reveals his innocence, causing every false accuser to be penalized by some constantℓ ≥ 0. Our analysis is essentially unchanged, because such punishments are equivalent to an increase in the added
benefit b from reporting after witnessing an offense.
Uncertainty about the Number of Potential Victims:

In some applications, the number of potential vic-

tims/witnesses is usually not observed by the judge and the victims. We consider an extension of our model,
e of {1, 2, ..., n}, interpreted as the set of agents against whom the dein which nature randomly selects a subset N
e can accuse the defendant:
fendant has opportunities to commit offenses against. We assume that only agents in Ne filed an accusation, this accusation would be easily refuted by the defendant (e.g., by
if an agent outside of N
e . Agent i privately observes whether i ∈ N
e as well as (θi , ωi ).
using an alibi). Only the defendant observes N
We informally argue that the logic behind our results is preserved, and even stronger, when the judge and thee , whether the defendant isagents face this extra layer of uncertainty. Since the judge does not observe the size of N
convicted or not depends only on the number of accusations, but not on the number of potential witnesses. Since
the defendant is convicted with weakly higher probability when there are more accusations, he has a stronger28

e | is smaller, i.e., fewer agents can accuse him. If an agent witnesses an
incentive to commit offenses when |N
e | is more likely to be small and, hence, the expected number of accusations filed
offense, this agent infers that |Nby other agents is also likely to be small. This effect dampens an agent’s incentive to accuse the defendant when
he has witnessed an offense which lowers the informativeness of accusations in equilibrium.

6

Concluding Remarks

Motivated by the debate among legal scholars regarding whether judges should aggregate the probabilities of distinct offenses when making conviction decisions, we present a game-theoretic model in which potential offenders’
incentives to commit offenses and witnesses’ incentives to report offenses, or the absence thereof, are both endogenous. We show that aggregating the probabilities of distinct offenses into a single overall probability of guilt,
on which the conviction is based, can induce negative correlation in the private observations of distinct witnesses.
This negative correlation, together with witnesses’ coordination motive, lowers the informativeness of accusations
and leads to a high number of offenses taking place. By contrast, the distinct probabilities principle commonlyused in criminal justice systems across the world can restore the credibility of witness testimonies and lower the
probability of offenses.
Our conclusions are obtained under the standard assumption that players have correct expectations about otherplayers’ strategies, and under the further assumption that the punishment received by the defendant in case of aconviction is large relative to defendants’ benefit from committing an offense. Hence, our results are best suited
for settings in which (i) a conviction rule has been in place for a long time and players in the game (defendants,
witnesses, and judges) know each other’s equilibrium strategies and (ii) conviction is effective for deterrence.
Our results are less suitable to analyze behavior following a sudden change in the conviction rule or in the model’s
parameters, after which the assumption that players’ behaviors form an equilibrium is less plausible. The results arealso less suitable when the benefit from committing an offense is significant relative to the punishment associatedwith a conviction. In this case, conviction plays a less central role than in our arguments and the comparison
between different conviction rules will like require a different analysis.

A

Proof of Proposition 1

First, we show that under Refinement 1, Pr(θ1 = 1) ∈ (0, 1) and Pr(a1 = 1) ∈ (0, 1). Refinement 1 requiresthat q(0, 0) = 0 and Pr((a1 , a2 ) = (0, 0)) > 0, which implies that Pr(θ1 = 1), Pr(a1 = 1) < 1. Suppose by
way of contradiction that Pr(θ1 = 1) = 0. Then, the defendant is never convicted on the equilibrium path. If29

a1 = 1 occurs with positive probability, then q(1, 0) = q(1, 1) = 0 and the defendant has a strict incentive tochoose a1 = 1, which leads to a contradiction. If a1 = 1 occurs with zero probability, then agent 1 always chooses
a1 = 0 when θ1 = 0. Hence, the defendant has an incentive to choose θ1 = 0 only if there exists a2 ∈ {0, 1} that
occurs with positive probability such that q(1, a2 ) > 0. However, this implies that when θ1 = 0, a type of agent
1 with a sufficiently high ω1 has a strict incentive to choose a1 = 1, which leads to a contradiction. Given thatPr(θ1 = 1) ∈ (0, 1), suppose by way of contradiction that Pr(a1 = 1) = 0. Then, agent 1 never chooses a1 = 1
regardless of the realization of θ1 . This provides the defendant a strict incentive to choose θ1 = 1, which leads toa contradiction.
Next, we show that when L > (Φ(0))−1 , either q(0, 1) = 0 or agent 2 never chooses a2 = 1. Supposeby way of contradiction that q(0, 1) > 0 and a2 = 1 occurs with positive probability. Let π(a1 , a2 ) denote theposterior probability that θ = 1 after observing (a1 , a2 ). We have π(0, 0) = π(0, 1) and π(1, 0) = π(1, 1). Sinceq(0, 0) = 0 and q(0, 1) > 0, we know that π(0, 0) = π ∗ . If π(1, 0) ≤ π ∗ , then agent 1 accuses the defendant withweakly lower probability when θ1 = 1, which gives the defendant a strict incentive to choose θ1 = 1 and leads toa contradiction. Hence, π(1, 0) = π(1, 1) > π ∗ and, therefore, q(1, 0) = q(1, 1) = 1. Agent 2 has an incentive tochoose a2 = 1 if
ω2 ≥c(1 − q(0, 1))≥0q(0, 1)

which implies that agent 2 chooses a2 = 0 with probability at least Φ(0). Since q(0, 0) = 0 and q(1, 0) = 1,
the defendant’s expected cost of choosing a1 = 1 is at least LΦ(0), which exceeds his benefit from committing
offenses as long as L > (Φ(0))−1 . This leads to a contradiction for L above this threshold.
Let q denote the expected probability of conviction when a1 = 1. We show that q ∈ (0, 1). Suppose by
way of contradiction that q = 0. Then, the defendant has a strict incentive to commit offenses, which leads to acontradiction. Suppose now by way of contradiction that q = 1. Then, the defendant is accused with probability
1 − Φ(−b) when he commits an offense and with probability 1 − Φ(0) when he does not commit an offense.
Hence, he has an incentive to commit an offense only if L(Φ(0) − Φ(−b)) ≤ 1, which cannot be true when1.
L > Φ(0)−Φ(−b)

Hence, when L is large enough, it must be the case that q ∈ (0, 1) in every equilibrium. This implies that thedefendant’s posterior probability of guilt equals π ∗ conditional on a = 1. Since the probability that he commits anoffense is strictly between 0 and 1, we have the following indifference condition: (1 − q)c  (1 − q)c

1
=Φ
−Φ
−b .
L
q
q

30

When L is large enough, the conviction probability q becomes arbitrarily small, which implies that
1−Φ
I=





(1−q)c
−b
q

1−Φ



(1−q)cqdiverges to infinity. Bayes rule then implies that the probability that the defendant chooses θ1 = 1 converges to 0.

B

Proof of Lemma 4.1

Step 1: We show that for every i ∈ {1, 2}, Pr(θi = 1) ∈ (0, 1) in every equilibrium. First, suppose by way ofcontradiction that Pr(θi = 1) = 1 for some i ∈ {1, 2}. Then, regardless of the conviction rule used, the defendantis convicted with probability 1 under every a that occurs with positive probability. This cannot happen in any
equilibrium that satisfies Refinement 1 since q(0, 0) = 0 and a = (0, 0) with positive probability.
Next, suppose by way of contradiction that Pr(θi = 1) = 1. By symmetry, it must be the case that (θ1 , θ2 ) =(0, 0) with probability 1. In this case, the defendant is acquitted with probability 1 under every a that occurs withpositive probability. In order for the defendant to have an incentive to choose θ = (0, 0), there exists a ∈ {0, 1}2such that q(a) > 0. Consider two cases. First, suppose q(1, 0) > 0. By Refinement 1, a2 = 0 with positive
probability, which implies that agent 1 has a strict incentive to choose a1 = 1 when ω1 is sufficiently high. Hence(a1 , a2 ) = (1, 0) occurs with positive probability yet q(1, 0) > 0, which leads to a contradiction. A symmetricargument rules out q(0, 1) > 0. Next, suppose q(1, 1) > 0 and q(1, 0) = q(0, 1) = 0. If a2 = 1 occurs with zeroprobability, then the defendant has a strict incentive to choose θ1 = 1 given that q(1, 0) = q(0, 0) = 0, whichleads to a contradiction. If a2 = 1 occurs with positive probability, then agent 1 has a strict incentive to choose
a1 = 1 when ω1 is large enough, under which (a1 , a2 ) = (1, 1) occurs with positive probability. This contradicts
the requirement that q(a) = 0 for every a that occurs with positive probability.
Step 2: We show that for every i ∈ {1, 2}, agent i accuses the defendant with strictly higher probability whenθi = 1 compared to when θi = 0. To see this, note that if agent i accuses the defendant with weakly higher
probability when θi = 0, the defendant has a strict incentive to choose θi = 1 since the benefit from doing so
is 1 and, according to Refinement 2, reducing the probability that ai = 1 weakly decreases the probability ofconviction. This contradicts the result obtained in the first step that Pr(θi = 1) ∈ (0, 1) for every i ∈ {1, 2}.
Step 3: We rule out cases in which either Q1,2 (q(1, 1) − q(0, 1)) + (1 − Q1,2 )q(1, 0) or Q0,2 (q(1, 1) − q(0, 1)) +(1 − Q0,2 )q(1, 0) is 0, which implies that agent 1’s equilibrium strategy takes the form of two cutoffs.

31Suppose by way of contradiction that Q1,2 (q(1, 1) − q(0, 1)) + (1 − Q1,2 )q(1, 0) = 0, in which case the LHSof (4.1) is 0. Refinement 2 implies that the RHS of (4.1) is weakly positive. If it is strictly positive, then agent 1
never chooses a1 = 1 when θ1 = 1. As a result, the defendant has a strict incentive to choose θ1 = 1, which leads
to a contradiction. If the RHS of (4.1) is 0, then Q1,2 = 1 and q(0, 1) = 1, i.e., conditional on θ1 = 1, agent 2
chooses a2 = 1 with probability 1. We consider two cases separately:
1. Conditional on θ1 = 1, θ2 = 0 with positive probability, and agent 2 plays a2 = 1 when θ2 = 0.
2. Conditional on θ1 = 1, θ2 = 0 with zero probability, and agent 2 plays a2 = 1 when θ2 = 1.
The first case is ruled out since it would give the defendant a strict incentive to choose θ2 = 1, given that doing
so weakly decreases the probability of conviction by decreasing the probability that a2 = 1. In the second case,the symmetry requirement implies that only (θ1 , θ2 ) = (0, 0) and (θ1 , θ2 ) = (1, 1) occur with strictly positive
probability. This implies that Q1,1 = 1 and q(1, 0) = 1. Plugging q(1, 1) = q(1, 0) = q(0, 1) = 1 into agent 1’s
incentive constraint when θ1 = 0, we know that he will accuse the defendant if (1 − Q0,2 )ω1 ≥ 0, and similarly,
we know that when θ2 = 0, he will accuse the defendant if (1 − Q0,1 )ω2 ≥ 0. If Q0,i = 1, the defendant has astrict incentive to choose θi = 1, which leads to a contradiction. If Q0,1 , Q0,2 < 1, then when (θ1 , θ2 ) = (0, 0),agent i accuses the defendant if ωi ≥ 0, which occurs with probability 1 − Φ(0). When L is large enough so thatLΦ(0)2 > 2, the defendant strictly prefers (θ1 , θ2 ) = (0, 0) to (θ1 , θ2 ) = (1, 1), which leads to a contradiction.
Suppose by way of contradiction that Q0,2 (q(1, 1) − q(0, 1)) + (1 − Q0,2 )q(1, 0) = 0. Refinement 2 implies
that Q0,2 (q(1, 1) − q(0, 1)) = 0 and (1 − Q0,2 )q(1, 0) = 0. That is, either q(1, 1) = q(0, 1) or Q0,2 = 0, andeither q(1, 0) = 0 or Q0,2 = 1. We consider three cases separately. First, if q(1, 1) = q(0, 1) and q(1, 0) = 0, thenthe defendant has a strict incentive to choose θ1 = 1, which leads to a contradiction. Second, if q(1, 1) = q(0, 1)
and Q0,2 = 1, i.e., θ1 = 0 implies that a2 = 1. Refinement 1 implies that θ2 = 0 occurs with positive probability
conditional on θ1 = 0. However, in this case agent 2 chooses a2 = 1 with probability 1 when θ2 = 0, which
implies that the defendant has a strict incentive to choose θ2 = 1, leading to a contradiction. Third, if q(1, 0) = 0
and Q0,2 = 0, agent 1 believes that agent 2 chooses a2 = 0 for sure after observing θ1 = 0. If θ2 = 1 occurs withpositive probability when θ1 = 0, then agent 2 chooses a2 = 0 for sure when θ2 = 1. In this case, the defendant
has a strict incentive to choose θ2 = 1, which leads to a contradiction. If θ2 = 0 occurs with probability 1 when
θ1 = 0, then agent 2 chooses a2 = 0 for sure when θ2 = 0. This together with q(1, 0) = 0 implies that thedefendant strictly prefers (θ1 , θ2 ) = (1, 0) to (θ1 , θ2 ) = (0, 0), which leads to a contradiction.

32

C

Proof of Proposition 3

Lemma 4.1, 4.2, and 4.3 apply to this more general setting in which the virtual type occurs with strictly positive
probability. We consider two cases separately depending on which of π o and π ∗ has the higher value. The case inwhich π o ≥ π ∗ is similar to the model in Section 2, while the case in which π o < π ∗ requires a different argument.

C.1

Case 1: π o ≥ π ∗Lemma 4.2 implies that q(0, 0) = q(1, 0) = q(0, 1) = 0 and q(1, 1) > 0. Therefore, q(1, 1) + q(0, 0) −q(1, 0) − q(0, 1) > 0 and the definition of APP implies that Pr(θ = 1|a) ≤ π ∗ for every a ∈ {0, 1}2 . Therefore,Pr(θ = 1) < π ∗ ≤ π o , which implies that the opportunistic type of the defendant chooses θ = (0, 0) with positive
probability. From Lemma 4.3, the defendant’s choices of committing distinct offenses are strategic substitutes, andas argued in the proof of Theorem 1, the opportunistic type cannot be indifferent between committing no offense
and committing two offenses. This implies that the defendant chooses θ = (1, 1) with zero probability.
Let π denote the ex ante probability that the defendant commits at least one offense. Our symmetry requirement
implies that (θ1 , θ2 ) equals (1, 0) and (0, 1) with the same probability. Conditional on θi = 0, the probability that
θ = (0, 0) is
β≡

1−π
.
1 − π/2(C.1)

Let Q1 ≡ Q1,1 = Q1,2 and Q0 ≡ Q0,1 = Q0,2 . Since θ = (1, 1) occurs with zero probability,
Q1 = 1 − Φ(ω ∗∗ )(C.2)

Q0 = 1 − βΦ(ω ∗∗ ) − (1 − β)Φ(ω ∗ ) = βQ1 + (1 − β)(1 − Φ(ω ∗ )).

(C.3)

and

Subtracting (4.10) from (4.9) yields
ω ∗∗ − ω ∗ = b −c−1 + Q0 /Q1·.
q(1, 1)
Q0(C.4)

Lemma C.1. ω ∗∗ − ω ∗ ∈ (0, b).
Proof of Lemma C.1: According to (C.2) and (C.3), ω ∗∗ − ω ∗ > 0 is equivalent to Q0 > Q1 . To see this, supposeby way of contradiction that Q0 ≤ Q1 . Equation (C.4) implies that ω ∗ ≤ ω ∗∗ − b < ω ∗∗ . The comparison between
0 /Q1(4.9) and (4.10) then yields Q0 > Q1 , which leads to a contradiction. Since Q0 > Q1 , the term −1+Q
is
Q0

strictly positive, which shows that ω ∗∗ − ω ∗ < b.
33

Let
I≡

Pr(a1 = a2 = 1| θ = 1).
Pr(a1 = a2 = 1| θ = 0)

Since the opportunistic type of the defendant mixes between (0, 1), (1, 0), and (0, 0), we haveI≡

(1 − Φ(ω ∗ ))(1 − Φ(ω ∗∗ ))1 − Φ(ω ∗ )=.
(1 − Φ(ω ∗∗ ))2
1 − Φ(ω ∗∗ )(C.5)

Since q(1, 1) ∈ (0, 1), the judge assigns probability π ∗ to θ = 1 after observing (a1 , a2 ) = (1, 1). This implies
that
π
l∗
=
1−π
I

where l∗ ≡π∗.
1 − π∗(C.6)

Plugging (C.6) into (C.1), we obtain the following expressions for β and 1 − β:
β=

l∗
2I
and
1
−
β
=
.
l∗ + 2Il∗ + 2I(C.7)

Plugging (C.7) into (C.2) and (C.3) then yields(l∗ + 2)IQ0
= β + (1 − β)I = ∗.
Q1
l + 2I(C.8)

Plugging (4.9) and (4.10) into (C.8), we obtain(l∗ + 2)I|ω ∗ − c − b|=.
|ω ∗∗ − c|l∗ + 2I(C.9)

This leads to the following lemma.
Lemma C.2. If ω ∗ → −∞, then I → 1 and π → π ∗ .
Proof of Lemma C.2: Since ω ∗ − ω ∗∗ ∈ (0, b), the difference between |ω ∗ − c − b| and |ω ∗∗ − c| is at most b. The
LHS of (C.9) converges to 1 as ω ∗ → −∞. Since the RHS of (C.9) is strictly increasing in I and is equal to 1
when I = 1, I must converge to 1 as ω ∗ → −∞. Equation (C.6) then shows that π converges to π ∗ .
The opportunistic type’s indifference condition (4.8) implies that as L → +∞, either q → 0 or at least oneof 1 − Φ(ω ∗∗ ) and Φ(ω ∗∗ ) − Φ(ω ∗ ) converges to 0. If q → 0, then the expressions for ω ∗ and ω ∗∗ in (4.9) and(4.10) imply that ω ∗ , ω ∗∗ → +∞. If 1 − Φ(ω ∗∗ ) converges to 0, then ω ∗∗ → +∞ and ω ∗ → +∞ as well since|ω ∗ − ω ∗∗ | ∈ (0, b). If Φ(ω ∗∗ ) − Φ(ω ∗ ) converges to 0, then either ω ∗∗ → +∞ or −∞ or ω ∗∗ − ω ∗ → 0 or both.
Since (4.10) implies that ω ∗∗ ≥ 0, it cannot be the case that ω ∗∗ → −∞. Our earlier conclusion implies that thevalue of I(1, 1) converges to 1 regardless of whether both ω ∗ and ω ∗∗ diverge to +∞ or ω ∗∗ − ω ∗ → 0 and both34

thresholds are bounded from above by some finite number. According to Bayes rule, I(1, 1) → 1 implies thatπ → π∗.

C.2

Case 2: π o < π ∗First, we show that the opportunistic type commits two offenses with positive probability in every equilibrium.
Suppose by way of contradiction that he never commits two offenses. Since type tv defendant never commits any
offense, the equilibrium probability of that an offense occurs cannot exceed π o , which is strictly less than π ∗ . As
a result,I≡

Pr(a1 = a2 = 1|θ = 1)
πo . π∗≥> 1.
1 − πo 1 − π∗Pr(a1 = a2 = 1|θ = 0)

(C.10)The expressions for Q0 and Q1 in (C.2) and (C.3) also apply to this setting. The derivations contained in Appendix
C.1 imply that for every ε > 0, there exists Lε > 0 such that when L ≥ Lε , the informativeness ratio is less than1 + ε. This contradicts (C.10), which requires that the informativeness ratio be strictly bounded below, away from1.
Since q(1, 1) ∈ (0, 1) and q(1, 1) + q(0, 0) − q(1, 0) − q(0, 1) > 0, Lemma 4.3 implies that the opportunistic
type cannot be indifferent between committing no offense and committing two offenses. Hence, it can only be thecase that the opportunistic type chooses θ = (0, 1), (1, 0) and (1, 1) with strictly positive probability, and choosesθ = (0, 0) with zero probability. This is the case when the expected marginal cost of committing a second offense
equals 1, which implies that the expected marginal cost of committing the first offense is strictly less than 1. Thisimplies that Pr(θ = 1) = π o . The informativeness ratio is pinned down by Bayes Rule:I

Pr(θ = 1)
Pr(θ = 1|a1 = a2 = 1)=.
1 − Pr(θ = 1)
1 − Pr(θ = 1|a1 = a2 = 1)

(C.11)

We now show that θ1 and θ2 are negative correlated, i.e., that Pr(θj = 1|θi = 1) < Pr(θj = 1|θi = 0) for
every i ̸= j. Suppose by way of contradiction that there exist i and j with i ̸= j such that Pr(θj = 1|θi = 1) ≥Pr(θj = 1|θi = 0) for every i ̸= j The definitions of Q1,j and Q0,j imply that Q1,j ≥ Q0,j . The equations (4.9)
and (4.10) for the cutoffs then imply that ωi∗∗ − ωi∗ ≥ b. When L is large enough, we have q(1, 1) → 0 andωi∗ → +∞, and the analysis of the single-agent case shows that I → ∞. This contradicts (C.11) and theconclusion that Pr(θ = 1|a1 = a2 = 1) = π ∗ .

D

Proof of Theorem 3

Our proofs in this section use the following lemma, which is proved in Online Appendix B.
35

Lemma D.1. Fix any n ≥ 2 and suppose that the judge uses APP. There exists L > 0 such that for every
L > L and every symmetric equilibrium that satisfies Refinements 1 and 2,
1. the defendant is convicted with positive probability only if a = (1, 1, ..., 1);
2. the defendant commits at most one offense.
With n ≥ 3 agents, we derive formulas for agents’ reporting cutoffs (ωn∗ , ωn∗∗ ), the informativeness of reports
In , and the equilibrium probability that θ = 1, i.e., that at least one offense occurs, which we denote by πn . Agenti’s reporting cutoffs arecwhen θi = 1
qn Q1,n(D.1)

cwhen θi = 0
qn Q0,n(D.2)

ωn∗ = −b − c +andωn∗∗ = −c +where

Q0,n ≡

n−1
Q1,n ≡ 1 − Φ(ωn∗∗ )
,

(D.3)


n−1

n−2 
(n − 1)l∗
nIn
∗∗
∗∗
∗
1
−
Φ(ω
)
1
−
Φ(ω
)
1
−
Φ(ω
)
+
n
n
n .
(n − 1)l∗ + nIn(n − 1)l∗ + nIn(D.4)

In any symmetric equilibrium, the aggregate informativeness of reports, defined in (5.5), can be written asIn =1 − Φ(ωn∗ ).
1 − Φ(ωn∗∗ )Since the judge is indifferent between convicting and acquitting the defendant when there are n accusations, we
have
In =

π ∗ . πn.
1 − π ∗ 1 − πn(D.5)

When L is large enough, the defendant is indifferent between committing only one offense and committing nooffense, which leads to the following indifference condition


n−1
1
.
= qn Φ(ωn∗∗ ) − Φ(ωn∗ ) 1 − Φ(ωn∗∗ )L
Reporting Cutoffs & Distance Between Cutoffs:(D.6)

We show that ωk∗ < ωn∗ . Suppose by way of contradictionthat ωk∗ ≥ ωn∗ . From (D.1), we know that

k−1

n−1
qk 1 − Φ(ωk∗∗ )≤ qn 1 − Φ(ωn∗∗ )
.
36

(D.7)

Therefore, qk Q1,k ≤ qn Q1,n , which is equivalent tok−1 
n−1 

qk 1 − Φ(ωk∗∗ )Φ(ωn∗∗ ) − Φ(ωn∗ ) ≤ qn 1 − Φ(ωn∗∗ )Φ(ωn∗∗ ) − Φ(ωn∗ )k−1 
= qk 1 − Φ(ωk∗∗ )Φ(ωk∗∗ ) − Φ(ωk∗ ) .
This implies that
Φ(ωn∗∗ ) − Φ(ωn∗ ) ≤ Φ(ωk∗∗ ) − Φ(ωk∗ ).

(D.8)

Under our hypothesis that ωk∗ ≥ ωn∗ , inequality (D.8) is true only ifωn∗∗ − ωn∗ ≤ ωk∗∗ − ωk∗ ,(D.9)

which in turn implies that ωk∗∗ ≥ ωn∗∗ and that qk Q0,k ≤ qn Q0,n . Computing the two sides of (D.9) by subtracting(D.2) from (D.1) for n and k, we obtainωn∗∗ − ωn∗ = b −c Q0,k − Q1,kc Q0,n − Q1,nand ωk∗∗ − ωk∗ = b −.
qn Q1,n Q0,nqk Q1,k Q0,k

Since we have shown that qk Q1,k ≤ qn Q1,n and qk Q0,k ≤ qn Q0,n , (D.9) is true only if
qn (Q0,n − Q1,n ) ≥ qk (Q0,k − Q1,k ).

(D.10)We have
Q0,n − Q1,n =



n−2
(n − 1)l∗
∗∗
∗
∗∗
δ
Φ(ω
)
−
Φ(ω
)
1
−
Φ(ω
)
n
n
n
(n − 1)l∗ + nIn

and



n−2
Φ(ωn∗∗ ) − Φ(ωn∗ ) 1 − Φ(ωn∗∗ )= L−1 qn−1

1.
1 − Φ(ωn∗∗ )Combining this with (D.6) and (D.10) yields(n − 1)l∗(k − 1)l∗


≥



,
(n − 1)l∗ 1 − Φ(ωn∗∗ ) + n 1 − Φ(ωn∗ )(k − 1)l∗ 1 − Φ(ωk∗∗ ) + k 1 − Φ(ωk∗ )which may be rewritten as








(n − 1)(k − 1)l∗ 1 − Φ(ωk∗∗ ) + (n − 1)k 1 − Φ(ωk∗ ) ≥ (n − 1)(k − 1)l∗ 1 − Φ(ωn∗∗ ) + (k − 1)n 1 − Φ(ωn∗ ) .

37

This inequality cannot hold because 1 − Φ(ωk∗∗ ) < 1 − Φ(ωn∗∗ ), 1 − Φ(ωk∗ ) < 1 − Φ(ωn∗ ) and (n − 1)k < (k − 1)n.
This leads to a contradiction and shows that ωk∗ < ωn∗ whenever k > n. The relation that k > n was only used in
the last step. Using the fact that ωk∗ < ωn∗ and repeating our earlier argument up until (D.9), we obtain thatωn∗∗ − ωn∗ > ωk∗∗ − ωk∗ .

(D.11)

This, together with ωk∗ < ωn∗ , implies that ωk∗∗ < ωn∗∗ .
The Informativeness of Accusations and Expected Number of Offense: We show that In > Ik . The posterior
probability that an offense was committed reaches π ∗ after observing a = (1, 1, ..., 1). This, together with (5.3),
implies that the probability of offenses and the expected number of offenses must be ranked as claimed by Theorem
3.
Applying (D.1) and (D.2) to both n and k, we obtainqk Q1,k−ωn∗ + b + c=∗−ωk + b + c
qn Q1,nand

qk Q1,k (βk + (1 − βk )Ik )−ωn∗∗ + c=.
∗∗
−ωk + cqn Q1,n (βn + (1 − βn )In )(D.12)First, we show that−ωn∗ + b + c−ωn∗∗ + c>.
−ωk∗ + b + c−ωk∗∗ + c

(D.13)

Suppose toward a contradiction that the RHS of (D.13) is at least as large as the LHS of (D.13). Then,−ωn∗∗ + c − (−ωn∗ + b + c)−ωn∗ + b + c≥.
∗∗∗−ωk + c − (−ωk + b + c)−ωk∗ + b + c(D.14)

When L is large enough, the RHS of (D.14) is strictly greater than 1 since 0 < ωk∗ < ωn∗ . This implies that the
LHS of (D.14) is greater than 1, which is equivalent tob − (ωn∗∗ − ωn∗ ) > b − (ωk∗∗ − ωk∗ ).
This contradicts (D.11), which was established earlier, and proves (D.13). This, together with (D.12), implies thatβk + (1 − βk )Ik < βn + (1 − βn )In .
Plugging in the expressions of In and Ik obtained in (D.5), we get
Ik k + (k − 1)l∗ nIn + (n − 1)l∗ < In n + (n − 1)l∗ kIk + (k − 1)l∗ .
38

Letting ∆ ≡ Ik − In , the previous inequality reduces to(k − n)In (Ik − 1) = (k − n)In (In + ∆ − 1) < k∆ − l∗ (k − 1)(n − 1) + nk ∆.
Suppose toward a contradiction that ∆ ≥ 0. Then, the LHS is strictly positive since Ik > 1 and k > n. The
RHS is negative since l∗ (k − 1)(n − 1) + nk > k. This leads to the desired contradiction and implies that ∆ < 0or, equivalently, that In > Ik . Equation (5.3) then implies that the probability with which the defendant chooses
θ = 1 increases when the number of agents increases from n to k. It also implies that the expected number ofoffenses increases since the defendant commits at most one offense on the equilibrium path.

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41