Can Society Function Without Ethical Agents?
An Informational Perspective*
Bruno Strulovici
Northwestern University
June 14, 2022
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Abstract
In order to function, society relies on many facts that must be learned throughintermediaries with special expertise or access to information. This paper considers
whether society can learn about such facts when intermediaries are devoid of ethical motives and act sequentially. The answer depends on the severity of information
attrition affecting the amount of discoverable evidence about each fact. Information
attrition is nonexistent in fields based on reproducible scientific evidence but can affectthe evidence in criminal and corruption investigations. Applications to institution enforcement, social cohesion, scientific progress, and historical revisionism are discussed.

Keywords: Disinformation, Ethical Necessity, Fake News, Information Attrition, Institution Design, Mediated Learning, Social Learning.

* This project has benefited from numerous conversations and comments, particularly from Alessandro

Lizzeri, Alex Frankel, Ben Golub, Boli Xu, Doron Ravid, Francisco Poggi, Larry Samuelson, Laura Doval,
Ludvig Sinander, Meg Meyer, Piotr Dworczak, Richard Zeckhauser, Ron Siegel, Simone Galperti, and Xavier
Duran and audiences at various seminars. Early versions of this project were developed while I was visiting
Microsoft Research New England and Harvard whose hospitalities are gratefully acknowledged.

1

1

Introduction

Journalism, science, branches of government, the military, and the corporate sector offer
numerous examples of individuals who risk their life or their career to expose important facts.
Journalists have been silenced, scientists have been condemned and ostracized, public officials
and members of the military have seen their careers derailed, and corporate whistleblowers
have been retaliated against for trying to learn and truthfully reveal the truth about factsof public interest.
To what extent do societies and institutions rely on such ethical agents to acquire information? Can institutions be designed with appropriately calibrated incentives to elicit
the truth when agents lack intrinsic motives to seek and reveal the truth? Or are intrinsic
motives necessary to achieve this objective? This paper introduces a framework to analyze
these questions and provides results suggesting that the answer depends on the nature of
the facts of interest and of the evidence available about these facts. The analysis rests on
two concepts: mediated learning and information attrition, which are introduced next.

1.1

Mediated Learning

Many facts must be learned through agents with specific expertise or access to information.
For example, the net benefits of a vaccine, the results of an election, the historicity of anevent, and the manmade nature of climate change cannot be directly verified by the average
citizen, and there is no public “epiphany” at which the truth is exogenously revealed to all.
In these and many others instances, citizens must rely on intermediaries to learn anything
about the fact of interest, a situation that we will call mediated learning.
To succeed, mediated learning relies on the investigative efforts and truthfulness of agents
who can make false, misleading, or uninformed statements and are subject to biases, pressures, ambitions, cognitive and time costs and other considerations that may distort their
behavior. In philosophy and various social sciences, it is common to consider agents whoserules of behavior prescribe, at least to some extent, to act truthfully.1 By contrast, economists
1

One branch of epistemic philosophy concerns the vulnerability of testimony, i.e., the fact that a speaker

can lie, and resolves this vulnerability through behavioral assumptions driven by norms of truthfulness andprinciples of ethical behavior, such as Grice’s “cooperative principle” (Grice (1975)). Going further, Williams(2002) argues that agents must value truthfulness intrinsically in order to exchange information successfully2usually conceptualize truthful behavior as the result of properly calibrated incentives given
to agents who are devoid of intrinsic motives for such behavior.
This paper examines the feasibility of mediated learning through agents who lack intrinsic
motives to seek and reveal the truth. These agents will be called non-ethical to indicatethat they are devoid of ethical motives to be truthful.2 According to the definition usedin this paper, non-ethical agents may care about material rewards and punishments andinformational costs and biases, but not directly about the facts that they are supposed toinvestigate. For example, a prosecutor who gains from securing the conviction of a defendantand desires this outcome regardless of the defendant’s actual guilt is non-ethical.3

1.2

Information Attrition

The cornerstone of the analysis is the concept of information attrition, which captures theidea that information about a fact may be in limited, fragile, or diminishing supply. Asa first illustration, consider the problem of determining whether a given athlete has useda banned substance during a competition. Drug tests typically rely on two blood or urine
samples, “A” and “B”. If the first sample, “A”, is positive (suggestive of doping), the agency
storing the second sample, “B”, is then asked to test this sample. At this point, the second
sample is the only direct source of evidence left about whether the athlete benefited from abanned substance during the event: the amount of evidence concerning this fact has shrunk,and each test reduces the amount of information available in the case.
(see Section 3). Computer scientists consider the related problem of communication through potentiallyadversarial intermediaries, known in the literature as the Byzantine generals problem (Lamport, Shostak,and Pease (1982)). This literature assumes the existence of “loyal” generals who obediently follow thecommunication protocol set out by the planner. Finally, it is common in sociology to assume the existenceof pro-social norms that facilitate truthful behavior. See Granovetter (2017) for a recent comparison of theparadigms in economics and sociology.
2
The term “non-ethical” emphasizes the lack of an intrinsic disposition to seek and reveal the truth, as
opposed to extrinsic motives to do so. In general, ethical agents may fail to be truthful and vice versa. For
example, a host lying to a murderer about the location of a potential victim may be ethical. Conversely, anindividual who always tells the truth regardless of consequences need not be ethical. This paper abstracts
from this distinction and specifically defines non-ethical agents as having truth-independent preferences.
3
Truth-dependent preferences do not imply ethical behavior. For instance, a prosecutor who wishes toconvict innocent defendants and acquit guilty ones is unethical. This observation has no bearing on themain analysis of this paper, which focuses on truth-independent preferences. Mediated learning with ethicalagents is discussed in Sections 3.1, 3.2, and 5.

3

Information attrition may be caused by various factors. The first one, outlined above, isthat the process of learning about a fact sometimes requires transforming the evidence in a
way that prevents its use by subsequent investigators: the signals are disposable.4 Second,
information attrition may be due to the exogenous degradation of evidence: as time goesby, evidence may deteriorate and become uninformative, for example due to poor storage
conditions. Third, information attrition may be caused purposefully by individuals whotamper with or destroy some evidence, e.g., to dissimulate fraud.
Information attrition may also stem from social considerations. Experiments that requirea large amount of resources (financial or human, as in the case of large-scale experiments)
may become increasingly difficult to organize and unlikely to occur. In some cases, such as
the determination of election results, the fact of interest is time sensitive and the number ofexperiments (e.g., vote recounts) is constrained by a deadline.

1.3

Incentive Unraveling with Information Attrition

To appreciate how information attrition affects mediated learning, let us first consider asituation in which the numbers of agents and of pieces of information are fixed, finite, andcommonly known. Specifically, let us reconsider the sample-testing example, proceeding
backward from the agency in charge of testing the second and last sample.5 Since this agency
possesses the only sample left, it can lie at no risk of being contradicted. If, for instance, theagency stands to gain publicity from incriminating the athlete, it can do so with impunity.
If instead the agency benefits from exculpating the athlete (perhaps due to pressure from a
sport governing organization), it can also do this without risking contradiction. And if theagency is indifferent between incriminating the athlete and absolving him, it has no incentive
to incur the cost of running the test to begin with. In all these cases, the second agency’s
report is untethered to the truth.
4

Other examples of disposable signals include human subjects in experimental psychology: once a person

has been exposed to a particular experiment, this person is irremediably affected and no longer a good subject
for the same experiment. More generally, it is a fundamental principle in physics that measurements are
subject to the observer effect, whereby the observation of a physical system affects the state of the system.
This principle is of particular importance for applications in quantum physics.
5
While sample testing in sports is primarily used here for expositional purposes, unreliable handling ofsamples and the limited amount of samples can have severe consequences for athletes. For a recent example
in boxing, see, e.g., https://www.asahi.com/ajw/articles/14338804 and https://www.japantimes.co.
jp/sports/2021/06/15/more-sports/boxing-2/kazuto-ioka-false-positive-jbc/.

4

Consider now the agency in charge of the first blood sample. Whatever report this agency
produces, the second agency’s report will not be based on the truth, as shown in the previous
paragraph. The first agency’s expected payoff is therefore independent of the content of its
sample. Like the second agency, it has no incentive to report the true content of its sample,
or even to learn this content, and incentives unravel. In this example, mediated learning is
infeasible if agencies are non-ethical. When agencies are non-ethical, mediated learning is
infeasible regardless of agencies’ material incentives.6

1.4

Reproducible Evidence and Incentive Design

Some investigations are not subject to information attrition. Consider the question of determining whether a mathematical proof is correct. The proof remains available for anyone toread no matter how many times it has been checked in the past. The unraveling argument
no longer applies and this paper shows (Theorem 1) that mediated learning by non-ethicalagents is feasible provided that agents’ material incentives are designed appropriately. The
incentives that deliver successful mediated learning have an intuitive structure: an agent isrewarded if his report is vindicated by subsequent findings and punished otherwise, and themagnitude of the reward or punishment depends on how surprising the agent’s report was
relative to the public belief about the state of the world before the agent’s report.7
Information attrition does not arise, either, in scientific inquiries that rely on reproducible
experiments. For instance, no matter how many times physicists measure the weight of anelectron, the experiment can be replicated more times. Facts based on reproducible evidencecan, under the appropriate incentive structure, be learned through non-ethical agents.
As a result, the theory can explain why oaths of truthfulness are used in some contexts, inwhich information attrition is a concern, and not others. For instance, it may explain why
6

In practice, the same laboratory is often in charge of storing and testing both samples, which may leadto even more severe incentive problems, as in the case of nationally organized doping schemes. If there arethree or more agencies, the results are the same when the agencies proceed sequentially, regardless of therule (e.g., majority rule) used to convict the athlete or incentives given to the agencies. Section 5.2 discusses
the case in which agencies move simultaneously as well as other learning structures.
7
Some of these features are similar to the socially optimal policy analyzed by Smith, Sørensen, and Tian
(2021) in the context of herding models. In contrast to some results in the present paper (Theorems 2, 3, and4), in herding models all equilibria are at least somewhat informative: uninformative cascades may occur,
but only after—and, precisely, because—some information has been publicly revealed.

5oaths are used in courts of law but not in “hard” sciences (see Section 5.3).

1.5

General Framework: Uncertain and Endogenous Information
Attrition

To explore the effects of information attrition on mediated learning, this paper considers aframework in which investigators act sequentially over an infinite horizon and communicate
through public reports, and information attrition may be uncertain and endogenous.
Sequentiality. The sequential nature of investigations is a key feature of the analysis.8 This feature may be motivated by various activities, such as journalism and academic
research, in which investigation outcomes often take the form of sequential publications.
Likewise, police investigations typically proceed sequentially, with only one individual in
charge of the investigation at any given time.
Another motivation for modeling the investigatory process as sequential and potentiallyunbounded is to avoid the existence of a terminal investigatory stage whose findings aredefinitive and can never be checked for distortions. For example, an enforcement or militaryagency sending investigators to learn about an event may be subject to biases and try tocover up or misrepresent the findings of its agents, unless the agency may with positive
probability be itself subject to a higher level of monitoring. Sequentiality has the uniquepotential to hold every investigator accountable.9
General Framework. The general model expands the scope of the theory in three directions: (i) exogenous uncertainty about the amount of discoverable evidence, (ii) sequential
accountability, and (iii) endogenous uncertainty concerning how evidence was handled bypast investigators. These features are modeled as follows: (i) The supply of evidence available in a case may be unknown a priori, even to investigators; (ii) The number of potential
investigators may be large and a priori unknown, even to current investigators. Unlike thesecond testing agency in the earlier example, all investigators can a priori be held accountable for their actions and declarations thanks to the existence of subsequent investigators;
8
9

Other mediated learning structures, including non-sequential ones, are discussed in Section 5.2.
Some institutions feature an investigatory level of last resort. Judicial systems in which a highest court

makes unappealable decisions are an example. One may then be concerned that such institutions strongly
rely for their success on the ethical standards of the individuals placed at the terminal investigatory level.
Such a concern is consistent with the main results of this paper.

6

(iii) investigators may not know how past investigators have affected the evidence available.
For example, consider the evidence left on a crime scene. Investigators typically do not knowa priori the amount and nature of evidence left on the scene. Moreover, if an investigatorinherits the case from a previous investigator, he may not know how diligent the previous
investigator has been and, hence, what fraction of the evidence has already been discovered,lost, or destroyed. An investigator thus faces exogenous and endogenous uncertainty about
the amount of discoverable evidence.
This paper provides conditions on the probability distribution of the supply of evidence
under which mediated learning is feasible (Theorem 1), and under which it is not (Theorems 2, 3, and 4). When the necessary conditions are violated, mediated learning by nonethical agents is impossible in a strong sense: even when (i) there is an unbounded sequenceof potential investigators and (ii) investigators’ incentives may be arbitrarily designed withfull commitment of the designer, there does not exist any equilibrium in which at least one
investigator provides an informative report with positive probability.
Analytical Challenges. It is easy to design incentives for which mediated learningalways fails or, given some incentive structure, to construct an equilibrium in which mediatedlearning fails. The main challenge is to show that, in the presence of information attrition,
mediated learning fails for all incentive structures and all equilibria, in the strong sense withprobability one, agents reveal no information at all about the fact.
This result is established in a dynamic game in which the state variables in any given
period include past reports and the public belief about the set of evidence that remains to
be discovered. This set is not a priori bounded and thus lies in an infinite-dimensional space.
Another difficulty is that the public belief about the number of pieces of evidence that remain
to be discovered need not be monotonic: While discovering of a piece of evidence may suggestthat the remaining supply of evidence is now smaller by one piece, this discovery could also
reveal the tip of an iceberg of evidence, pointing to many more pieces to discover. Moreover,a discovery that contradicts past reports can indicate that previous investigators have lied
and that whatever evidence they purported to have discovered (and thus “removed” fromthe supply of evidence) was in fact fabricated, resulting in a more optimistic belief about
the amount of evidence that actually remains to discover.
An agent’s belief about the remaining supply of evidence affects his incentives directly,
through the probability that he will discover new evidence if he looks for it, and indirectly,
through the probability that subsequent agents discover evidence that could be compared to7the first agent’s report. Unpacking this indirect channel, an agent cares about subsequent
agents’ beliefs about the supply of evidence, their beliefs about subsequent agents’ beliefs,
and so on. Moreover, an agent can manipulate subsequent agents’ beliefs about the supplyof evidence by lying in his report.
Information attrition does not preclude the possibility of a long sequence of agents whowork, or the possibility that many pieces evidence remain to be discovered, but it imposes anegative correlation between these two events. To exploit this negative correlation and showthat an informative equilibrium cannot exist, one must compute bounds on agents’ beliefs
along any given sequence. This is achieved by showing that the probability that an agent
discovers evidence is probabilistically linked to the expected impact that this agent’s reporthas on subsequent agents’ relevant beliefs. This link is formalized by Proposition 2.

1.6

OutlineSection 2 introduces the formal model and the main results of the paper. In the baselinemodel, all agents must incur a cost to acquire information. This cost may be arbitrarily small,but strictly positive. A high-level presentation of the proof and its challenges is provided in
Section 2.4. The model is then extended to allow for the existence of witnesses, who receive
information for free and are subject to idiosyncratic, private biases, and analysts, who receive
evidence for free but incur a cost from processing it.
Sections 3 and 4 discusses consequences and illustrations of information attrition. Section 5.1 describes the relation between the concepts of hard evidence, information attrition,and mediated learning. Section 5.2 considers alternative investigation structures. Section 5.3
discusses how to foster truth-dependent preferences among investigators. All results areproved in the Appendix.

82

Formal Analysis

This section contains the formal model and the main results of the paper. In the baselinemodel, agents must incur a cost to look for signals about the fact of interest. Theorem 1
shows that if (i) the supply of signals is unrestricted in a sense formalized by the theorem, and(ii) agents’ rewards and punishments can be designed and taken to be sufficiently large, thenthere are instances in which agents can be incentivized to look for signals and truthfully
report their findings. By contrast, Theorem 2 shows that mediated learning fails if the
supply of signals is limited in a specific probabilistic sense, even when the cost of informationacquisition is arbitrarily small.
The impossibility result is then extended to two other agency problems, in which signal
acquisition is sometimes costless, but those agents who receive costless signals are subject toeither private reporting biases (e.g., a witness holds a private grudge against a defendant) or
to information processing costs (e.g., a laboratory receives a sample that it costly to analyze).
In both cases, mediated learning fails even if private biases and information process costs
are arbitrarily small (Theorems 3 and 4).
In reality, the same individual may be subject to both biases and informational costs,
and the informational costs may include both information acquisition and an information
processing costs.10 For simplicity, the analysis assumes that each agent is subject to only
one agency problem.

2.1

Baseline Model

A fact of interest, ω ∈ Ω, must be inferred from a sequence S = (s1 , . . . , sK̃ ) of signals, eachof which takes values in some finite signal space Σ. The sequence S and its length K̃ ≤ ∞are stochastic. The joint distribution of (ω, S) is arbitrary.11
10

For example, an individual investigating an industrial accident may have to look for forensic evidence

and then process this evidence to extract its meaning.
11
One could impose some restrictions on this joint distribution. For example, one may impose that S
reveals ω perfectly or with some minimum level of precision. Such additional structure is not required
for the general analysis of this paper. Alternatively, one could consider an information structure in which
S always contains infinitely many signals, but the number of signals whose correlation with ω and with
one another exceeds any given positive threshold is finite and decreases with the threshold. The model inthis paper adopts—for tractability—a different structure of dependence, in which discoverable signals are9In each round i ≥ 1, a new agent arrives and makes two decisions: First, the agent
privately chooses between seeking a signal (“working”) at cost c > 0 and doing nothing(“shirking”). Second, the agent publicly sends a message mi from some finite message space
M . The agent can use mixed strategies.
Let Si denote the sequence of signals that remain to discover at the beginning of round i(in particular, S1 = S). If the agent in round i (hereafter, “agent i” or simply “i”) works
and Si 6= ∅, then i discovers some element of Si with probability λ ∈ (0, 1].12 The discovered
signal is denoted si .13 No constraint is imposed on how likely each element of Si is of being
discovered by agent i. For instance, i could always be discovering the first element of Si .
Alternatively, this likelihood of each signal could depend arbitrarily on i’s identity, on Si ,and on past messages.14 With probability 1 − λ, i discovers no signal. If Si is empty, i surely
discovers no signal.
For simplicity, we do not model the possibility that agents destroy signals without seeing
them, or that signals disintegrate exogenously. These additional forms of information attrition would only strengthen the paper’s impossibility results (Theorems 2, 3, and 4).15 The
extension is more complex when some agents, such as witnesses, can discover signals for free.
This extension is analyzed explicitly in Sections 2.5 and 2.6.
After the information-seeking stage, i sends a report mi whose distribution in ∆(M ) can
depend arbitrarily on what (if anything) i has observed during the information-seeking stage
and on the reports mi−1
= (m1 , . . . , mi−1 ) made by previous agents.
1
Entering round i + 1, we have Si+1 = Si if i did not discover any signal. If i discovered asignal, then Si+1 is a subsequence of Si with length |Si+1 | = |Si | − 1.
arbitrarily correlated with ω (and modeled as signals belonging to S) but may be in finite supply.
12
To simplify exposition, we assume that an agent discovers at most one signal, but what is called a“signal” in the paper could stand for multiple pieces of evidence. An earlier version of the paper allowed
the possibility that each agent discovers multiple signals. It yielded the same results at the cost of morenotation.
13
The index notation distinguishes si , which is the signal discovered by agent i (when applicable) and sj
is the j th signal in the sequence S.
14
Moreover, Theorem 2 can be generalized to the case in which the probability λ of discovering a signal isnondecreasing in the length of Si .
15
The possibility result (Theorem 1), which allows a geometrically distributed number of signals, may beinterpreted as allowing an exogenous decaying rate of evidence. Likewise, we do not model the possibility
that signals be generated over time. If new signals can arrive and other can disappear, the feasibility of
mediated learning will likely depend on the relative rates and dynamics of growth and disintegration.

10

Let m = (m1 , m2 , . . .) denote the sequence of reports made by all agents. The realized
utility of agent i is given by
Ui = Vi (m, ω) − c1i works(1)

where Vi takes values in some compact interval [−R, R].
Agent i is non-ethical if the function Vi is independent of ω. This definition captures theidea that i does not care directly about the truth.16 If i is non-ethical Vi may be defined on the
restricted domain M N . Vi can depend arbitrarily on the entire sequence m = (m1 , m2 , . . .).
In particular, the impossibility results presented in this paper hold regardless of whether
Vi is an exogenously given utility function or one that is specifically designed (or, at least,
influenced) by a regulator or social planner.
To illustrate the various forms that Vi may take, note that i could be punished if his
report is contradicted by subsequent investigators or rewarded if his report differs frompast investigators’ (as in a journalistic scoop). Vi may aggregate a discounted stream ofrewards and punishments.17 The general formulation captures situations in which i’s utility
is affected by reports indirectly through the actions that these reports trigger. For example,
i could be a prosecutor in a trial, whose outcome a(m) ∈ {‘guilty’, ‘not guilty’} depends on
the statements made by all agents involved in the case. If i is non-ethical, his utility may
be modeled by Vi (m) = R × 1{a(m) = ‘guilty’}, as in Landes’ (1971) model of prosecutorial
behavior. The model also encompassed the possibility that agent i’s realized utility could
depend stochastically on other agents’ reports. For example, suppose that the number ofinvestigators is stochastic. We can model this by interrupting mediated learning at some
stopping time τ , in which case i’s realized utility depends only on (m1 , . . . , mτ ) and i cares
about his expected utility conditional on the information at the beginning of round i. Other
variations, such as including a private type that affects i’s utility, are also easily encompassedby the model.18
Agents have a common prior about the distribution of S. The equilibrium concept is
16This concept is somewhat similar to the notion of being purely “extrinsically motivated” in Bénabou

and Tirole (2003), and one could say that i is intrinsically motivated by the truth if Vi depends on ω.
17
For example, if i receives utility vi,j (m1 , . . . , mj ) in round j ≥ i and discounts future utility with somefactor δ < 1, thenVi (m) =Xδ j−i vi,j (m1 , . . . mj ).

j≥i18Such a private type is explicitly considered in Section 2.5.

11

(weak) Perfect Bayesian Equilibrium.19

2.2

Positive Result with Reproducible Evidence or Limited Attrition

For each k ≥ 1, let F k = Pr(|S| ≥ k) denote the prior probability that there are at least ksignals to discover at the beginning of the investigation process.
Definition 1 An equilibrium is informative if at least one agent works with positive probability.
When the survival function F k decreases at most at a geometric rate, there are instances ofthe model and utility functions {Vi }i∈N for which mediated learning is feasible, as indicatedby the following theorem, whose proof is in Appendix B.2.
Theorem 1 For any ρ ∈ (0, 1] and λ ∈ (0, 1], there exist a joint distribution of (ω, S) andutility functions {Vi }i≥1 such that the distribution of S satisfies F k = ρk−1 F 1 for all k ≥ 1
and an informative equilibrium exists.
Theorem 1 shows that even if agents are non-ethical and reports are public, mediate
learning may be feasible provided that the supply of evidence is sufficiently large. Even if
agents are non-ethical and can see past reports, it is possible to incentivize them to work aslong as one can use sufficiently high rewards and/or punishments and these incentives areappropriately designed.
To illustrate Theorem 1, let us consider for now the case of in which ρ = 1 and λ = 1. The
assumption that ρ = 1 captures reproducible evidence: it means that the supply of signals is
unlimited.20 The assumption that λ = 1 means that every agent who works surely discovers
a signal.
We construct a joint distribution over state and signals for which mediated learning canbe made arbitrarily precise as long as the rewards and punishments are high enough.
19

Impossibility results clearly hold for stronger equilibrium concepts, such as sequential equilibrium, andhold even for Bayes-Nash equilibria since off-path beliefs play no role in the analysis. Reciprocally, theequilibrium constructed to prove the positive result, Theorem 1, is sequential.
20
The argument is almost identical if ρ < 1, as explained in the proof.

12Thus suppose that S consists of infinitely many signals taking binary value, “H” or “L”.
The signals are conditionally i.i.d.: there is an unknown state of the world ω ∈ {H, L} suchthat each signal s̃ satisfies Pr(s̃ = “H”|ω = H) = Pr(s̃ = “L”|ω = L) = π ∈ (1/2, 1), andthe signals are independently distributed conditional on ω. Agents’ message space is chosen
to be binary: mi ∈ {‘‘H”, “L”} for all i.
In the equilibrium that we consider, it is always in an agent’s interest to follow his signal
if he acquired one. The relevant pure strategies are therefore: (i) to work at cost c > 0 andreport one’s signal (mi = si ), (ii) to shirk and send message “H” at no cost, and (iii) to
shirk and send message “L” at no cost.
Let p1 = P (ω = H) denote the prior about ω. For any given equilibrium, let pi denote theprobability at the beginning of round i that ω = H conditional on past reports m1i−1 and γidenote the probability that i works conditional on past reports m1i−1 .
Proposition 1 For any thresholds p− and p+ such that 0 < p− < p1 < p+ < 1, there
exist R > 0, utility functions {Vi }i≥1 taking values in [−R, R], and thresholds p, p̄ such that¯
0 < p < p− and 1 > p̄ > p+ for which the following strategy profile constitutes an equilibrium:¯
γi = 1 if pi ∈ (p, p̄) and γi = 0 otherwise.
¯
The state ω can thus be learned with arbitrary precision as long as the rewards andpunishments used to incentivize agents can be taken to be high enough. In equilibrium,
agents work with probability 1 until the posterior belief becomes extreme enough, at which
point learning stops.
Since pi is a martingale and each signal has the same level of informativeness, pi must
exit [p, p̄] with probability 1 in the candidate equilibrium. Incentives are provided as follows:¯
if i reported “H”, he gets a reward if p̄ is reached and a punishment if p is reached, and¯
vice versa. These rewards and punishment depends on the belief pi before i’s report. If
pi was very close to one of the boundaries and i’s report takes the posterior away fromthis boundary, i gets a high reward if the belief process ends up exiting through the otherboundary (a low probability event) and a very mild punishment if the belief process ends up
crossing the nearby boundary.
It is beyond the scope of the present paper to describe the optimal way to incentivize
learning. One would have to define an objective function for the designer and the optimalmechanism would depend on details of the distributions. Instead Theorem 1 shows that13

mediated learning is compatible with a large class of instances of the model. The next
sections show that what destroys mediated learning is the presence of information attrition.

2.3

First Impossibility Result

Say that an equilibrium is uninformative if it is not informative in the sense of Definition 1.
According to that definition, uncovering any modicum of information about S (and, hence,ω) with positive probability, no matter how small, suffices to qualify an equilibrium as“informative.” The following theorem shows, however, that informative equilibria fail toexist when information is subject to attrition in a specific sense.
Theorem 2 For any parameters (R, c, λ), there exist strictly positive thresholds {F k }k≥1¯
with the following property:
All equilibria are uninformative unless F k ≥ F k for all k ≥ 1.
¯
Corollary 1 (Bounded Support) For an informative equilibrium to exist, the support of|S| must therefore be unbounded.
This corollary is an immediate consequence of Theorem 2: if the support of |S| is boundedby some constant K, then F K+1 = 0, which violates the threshold condition of Theorem 2
for all k ≥ K + 1 regardless of the parameters (R, c, λ) and utility functions {Vi }i≥1 .
Even if |S| has unbounded support, Theorem 2 implies that mediated learning is feasible
only if the survival function F k = Pr(|S| ≥ k) does not decrease too fast in k, i.e., so fastthat F k drops below the corresponding threshold F k for some k. When this violation occurs,¯even if a social planner could design agents’ payoff functions {Vi }i≥1 arbitrarily subject tothe reward-punishment bound R, learning is guaranteed to fail.21
Intuition: To understand mediated learning failures, suppose for simplicity that there
is a fixed number of discoverable signals. Successful learning requires that at least one
agent discovers one of the signals, which can be incentivized only if a subsequent agent
discovers (with positive probability) a second signal, which can be incentivized only if a
It would be interesting to characterize the rate at which the cutoffs {F k }k≥1 go to zero. This question¯
does not seem to have a simple answer. In particular, while Theorem 1 shows that there are settings in21

which geometric distribution of |S| are compatible with mediated learning, there is no indication that thiscondition is necessary.

14

third agent discovers a third signal with positive probability, and so on. Therefore, agents
must reach with positive probability a round in which (i) the probability that a signal
remains is arbitrarily low and (ii) the probability that some agent makes an informative
report is positive. These conditions are incompatible: no agent wants to work when the
expected benefit is arbitrarily close to zero. This shows that mediated learning must faildown this path, which causes incentives to unravel all the way back to the first agent and leads
to a complete and global failure of mediated learning.22 This intuition ignores important
challenges, which are presented next together with a gist the rigorous proof. The formal
proof is in Appendix A

2.4

Overview of the Proof and Challenges

For any i, k ≥ 1, let Fik = Pr(|Si | ≥ k | mi−1
1 ) denote the probability that there remain atleast k signals to discover at the beginning of round i given past reports mi−1
1 . The priorprobability F k that the initial sequence S contains at least k signals satisfies F k = F1k .
To understand how the proof works, suppose first that S contains at most one signal, i.e.,that Fi2 = 0 for all i. We will show by contradiction that no agent ever works in equilibrium.
When Fi2 ≡ 0, the probability Fi1 that there is a signal to discover in round i is decreasing
in i path by path.23
Agent i works only if two conditions hold: (i) the probability Fi1 that a signal remains
is high enough—above a cutoff F 1 provided by Lemma 1, below—(ii) the probability that¯some agent j > i works after i has worked is high enough.
This creates a tension: On the one hand, the more likely agent i is to work, the larger the1
expected drop from Fi1 to Fi+1
. On the other hand, an agent works only if subsequent agentsalso work with sufficient probability, which requires that Fj1 stay above F 1 . This dynamic¯
22This intuition differs from herding models (Bikhchandani, Hirshleifer, and Welch (1992) and Banerjee(1992)), in which learning failures (“cascades”) occur only after so much public information has been revealed
that agents prefer to forgo their own signals. In the present model, the learning failure occurs from the firstvery agent. One way to visualize this feature is that cascades occurring in the future “ripple back” to thevery beginning of the investigation sequence because agents’ payoffs in early rounds depend on the messages
produced during the cascades.
23
Intuitively, either i shirked, in which case his message is uninformative, or he worked, in which casehe either found the only signal, and there is nothing left, or he found nothing, which makes agents more
pessimistic that there is a signal left to be found.

15

is impossible as Fj1 must go down by a non trivial amount but becomes squeezed above F 1 .
¯
To formalize this tension, consider any history up to round i and let Mi+ denote the setof messages mi such that some j > i works with positive probability. We derive a formulathat combines the following observations: (i) The probability that Mi+ occurs conditionalon i working cannot be too small, because Mi+ contains all continuations for which i’s realfindings matter and, hence, drives i’s incentive to work. (ii) Conditionally on Mi+ , the average
drop in Fi1 is proportional to the probability that i works, because i affects beliefs only if
he works: the likely i is to work ex ante, the larger the expected impact of his message
on subsequent beliefs. These observations are captured by the following Discovery-Belief(DB) formula, derived in Proposition 2. Let βi = Pri (i discovers a signal) and suppose thatFi2 = 0. Proposition 2 shows that there exists Q > 0 s.t.
hi
1
1
Ei (Fi − Fi+1 (mi ))1mi ∈Mi+βi ≤ Q.
Fi1
The numerator captures i’s expected impact on beliefs conditional on sending a message thatyields an informative continuation equilibrium (mi ∈ Mi+ ). The denominator is bounded
below by F 1 .
¯
Now suppose by way of contradiction that there exists an informative equilibrium, and
let F = inf{Fj1 : j works with positive proba} denote the infimum belief over all informative¯
continuation equilibria. To build a contradiction, we choose an informative continuation
equilibrium starting from some round i such that i works and Fi1 ≤ F +ε. Such a continuation¯
equilibrium must exist by definition of F . Summing the DB formula over j > i yields
¯

Q  1
Pr i ( ∃j > i who discovers a signal) ≤ 1 Ei (Fi+1− FJ1 )
(2)F¯
where J (random, possibly infinite) indexes the last person who works.
We have FJ1 ≥ F and, by monotonicity,¯
1
Fi+1≤ Fi1 ≤ F + ε.
¯(3)From (2), the probability that some j > i discovers a signal must thus be of order ε, whichis too small to incentivize i and yields the desired contradiction.
Challenges

16The previous argument relies on two assumptions: Fi2 = 0 and Fi1 is decreasing. In
general, beliefs Fik can be positive for all k and nonmonotonic in i. For example, agent i
could send a message suggesting that previous agents have shirked and, hence, there are
more signals to discover. Or i could send a message that is positively correlated with theexistence of many other signal to discover, i.e., the equivalent of uncovering a “gold mine”
of signals. The proof must address several challenges:
1. Suppose we wish to show that i never works if Fi2 is arbitrarily small (rather thanexactly zero, as in the previous argument). Proposition 2 shows that the DB formula holds
as long as Fi2 is sufficiently small relative to Fi1 . To apply Proposition 2 to all j > i, however,
we need to show that Fj2 remains small for all j > i. Doob’s martingale inequality (Lemma4) allows us to show that this event, denoted A in the proof, is highly likely.
2. However, Doob’s martingale inequality holds with respect to a specific filtration, whichis in the present case the filtration generated by past public reports at the beginning of eachround. By contrast, agent i’s incentives are driven by the probability of event A conditionalon i working. This probability can be very different from the unconditional probability atthe beginning of round i, especially if the probability that i works is very small. This issue
is addressed by Lemma 5.
3. The argument above selected i such that Fi1 was close to the infimum F . When there¯
are more than two signals, we would ideally like to choose i such that (i) Fi2 is arbitrarily
small and (ii) Fi1 is close to F , but we cannot impose both conditions. To address this, the¯
proof uses a boundary function mapping f 7→ F 1 (f ), which defines the infimum of beliefs Fi1
over informative continuation equilibria such that Fi2 ≤ f , for f ∈ [0, 1].
1
4. To exploit Equation (2), the argument above used that Fi+1
was close to F (see¯Equation (3)). This is generally false if Fi1 is nonmonotonic. This issue is addressed by

Lemma 6.

2.5

Witnesses

We now introduce witnesses, who differ from the previous agents along two dimensions: They discover a signal for free.
 They are subject to (possibly, arbitrarily small) private preference shocks that affectwhich report they prefer to send.
17In each round i ≥ 1, agent i can be an investigator, identical to the agents of the baselinemodel, or a witness. Whether i is an investigator or a witness is public information.
If i is an investigator, the structure of i’s round, information, and utility is as in thebaseline model. If i is a witness, he receives at no cost a signal si ∈ Si at the beginning of
the round, where Si is the set of signals remaining at the end of round i − 1. In particular,i can be a witness only if Si is nonempty. If i is a witness, the sequence Si+1 of available
signals at the end of round i satisfies |Si+1 | = |Si | − 1.
At the beginning of round i, the probability ϕi that i drawn to be a witness is equal tozero if Si = ∅, and can take any value in [0, 1] otherwise. ϕ can depend on calendar time24
and on past reports mi−1
1 .

In principle, the content of a witness’ signal could be informative about the number ofsignals that remain to discover and, in particular, about whether information attrition is an
issue for subsequent agents.25 This would make it more difficult to state clear results linking
the feasibility of mediated learning to the prior distribution of S.
We rule out this possibility and focus on the case in which a witness’ signal never increases
expectations about the total number of signals. This is achieved as follows: the sequence
S of signals is obtained by, first, generating an infinite sequence S ∞ of signals, which may
exhibit any arbitrary correlation between one another and, second, by truncating this sequence at some integer-valued random variable K̃ that is independently distributed from S ∞ .
Agents observe signals in the order of the sequence. Thus, writing S = (s1 , . . . , sK̃ ) (using
superscripts to avoid confusion with the signals discovered in round i, which are denotedwith subscripts), suppose that qi signals have been uncovered by the beginning of round i,so that Si = (sqi +1 , sqi +2 , . . . , sK̃ ). If i discovers a signal si , then necessarily si = sqi +1 andSi+1 = (sqi +2 , . . . sK̃ ).
Moreover, we assume that K̃ has an increasing hazard rate, i.e., Pr(K̃ = k)/ Pr(K̃ ≥ k)is increasing in k.
Assumption 1 (i) The total number of signals K̃ has an increasing hazard rate. (ii) For
24

For example, there could be a fixed (finite or infinite) subset N ⊂ N of rounds such that ϕi =ϕ 1i∈N 1Si 6=∅ for some parameter ϕ∗ ∈ (0, 1).
25
For example, the prior probability that |S| = 1 could be high enough that mediated learning is infeasible∗

according to Theorem 2, and infinite with very small probability. However, the first witness’ signal could
reveal that |S| is infinite, in which case mediated learning could become feasible, as described by Theorem 1.

18

any round i, the probability that i is a witness is given by some function ϕ(i, 1Si 6=∅ , mi−1
1 ).
Part (i) guarantees that the more signals have been discovered, the more likely it is that there
are no signals left to discover. Part (ii) implies that the identity (witness or investigator)of i does not reveal information about the amount of evidence left to discover, other than
the fact that Si was non empty. It rules out situations, for instance, in which the mere factthat i is a witness implies that there are many other signals left to discover, or the opposite
inference in which the fact that i is not a witness implies that there are many more signals
to discover.
After observing his signal si , witness i sends a report mi ∈ M . His realized utility has twoparts:Ui (m) = Vi (m) + i (mi )(4)where Vi (m) plays the same role as investigators’ utility function, and i (mi ) is a private
shock affecting i’s preferences.
Assumption 2 The random variables {i (mi )}mi ∈Mi are privately observed by i. Conditional on mi−1
1 , they are independently distributed from one another and from all other exogenous variables in the model. Their density functions {fi,mi }mi ∈Mi are uniformly bounded
above by some arbitrary constant f¯.
Assumption 2 allows the possibility that i have strong public biases, such as i’s ideology,
financial interest or, in a criminal case, i’s relationship with the defendant. All such publicbiases are captured by Vi . Assumption 2 requires that, in addition to any possible publicbias, i is also subject to some (possibly, small) private bias, which may for instance qualify
the extent of i’s public bias.
The bound f¯ appearing in Assumption 2 plays for witnesses the same role as the costinverse 1/c does for investigators. Intuitively, i’s disutility of sending a less preferred message
mi instead of a more preferred message m0 is of order 1/f¯ in expectation, as explained iniLemma 8.
If i is a witness, we will say that i’s message is uninformative if it is statistically independent of i’s signal conditional on mi−1
1 . Otherwise, i’s message is informative. An informativeequilibrium is defined as before: there is at least one agent (investigator or witness) who produces an informative message with positive probability. Continuation informative equilibria
are defined analogously. The following result is proved in Appendix D.
19

Theorem 3 For any parameters (R, c, λ), there exist strictly positive thresholds {F k }k≥1¯
with the following property:
All equilibria are uninformative unless F k ≥ F k for all k ≥ 1.
¯

2.6

Analysts

Finally, we consider analysts, who differ from investigators along two dimensions: Analysts receive a signal for free.
 They incur a (possibly, arbitrarily small) processing cost to learn the content of theirsignal.
Compared to witnesses, analysts have no biases but they need to make some effort to understand (e.g., analyze) their signal. Compared to investigators, analysts know that they havea signal for sure at hand, so they are not concerned about not finding a signal. However,they still need to incur a cost to learn the content of the signal, like the laboratory testinga blood sample mentioned in the Introduction, which gets the sample at no cost but must
process it at some (possibly, small) cost. The model is identical to Section 2.5, except thatwitnesses are replaced by analysts, as follows: If i is an analyst, he receives at no cost asignal si ∈ Si at the beginning of round i. To observe the content of si , i must incur apositive cost ci that is independently distributed from other random variables in the model.
The cost ci may be arbitrarily small, but we assume that its distribution H has a bounded
density. After learning his cost ci , i decides whether to observe si (“work”) or to shirk. ithen chooses some message mi to send. i’s realized utility is given by:
Ui (m) = Vi (m) − ci 1i works
where Vi (m) plays the same role as investigators’ utility function. We maintain Assumption 1.
An equilibrium is informative if there exists at least one agent (investigator or analyst) who
works with positive probability. The following result is proved in Appendix E.
Theorem 4 For any parameters (R, c, λ), there exist strictly positive thresholds {F k }k≥1¯
with the following property:
All equilibria are uninformative unless F k ≥ F k for all k ≥ 1.
¯
20

3

Ethical Necessity

3.1

Ethical Necessity

The arguments presented so far bring us to one of the paper’s key motivating questions: is
ethical behavior necessary for society to function? Ethical necessity is not a salient concern ineconomic analysis, which typically focuses on “selfish” agents evolving within the boundaryof well-defined institutions, such as market or democratic institutions whose enforcement
is taken for granted.26 In reality, institutions cannot be taken for granted: if agents are
unethical, they may attempt to violate these institutions in various ways, and it is unclear
why agents should be presumed to act selfishly within the behavioral boundaries imposed bythese institutions but ethically with respect to these boundaries. This separation amounts
to a “heroic” dichotomy that, at the very least, deserves closer inspection.
By focusing on mediated learning, this paper provides a tractable framework to study
ethical necessity.27 Since mediated learning is required to investigate criminal cases, politicalcorruption, and other possible violations of institutions, successful mediated learning is anecessary condition for society to function: whenever mediated learning requires ethicalbehavior, so does society.28
26

Economists have studied various forms of non-selfish behaviors, such as pro-social behavior arising in

dictator and ultimatum games and various forms of altruism. Unlike earlier works, this paper does not study
ethical behavior per se, but whether ethical behavior, in the form of an intrinsic motive to seek and reveal
the truth, is necessary for society to function. Self-interest remains a central assumption in economic models
and is deeply rooted in the discipline. For example, Edgeworth (1881) observed that “self-interest is the firstprinciple of pure economics.”
27
The present framework relies on a sequential learning structure. One could consider alternative structures. For example, a central agency may ask several intermediaries to seek and report the truth simultaneously and independently of one another. However, the incentives of such a centralized agency must also
be scrutinized by a third party, which reintroduces a sequential element. One example concerns potential
abuses by various law enforcement agencies, which require the oversight of higher authorities. This and otherdesigns are discussed in Section 5.
28
In this sense, the paper is related to the work of the philosopher Bernard Williams, who argues thattruthfulness must emerge as an intrinsic value in order for the transmission of information to be successful(Williams (2002)). Successful transmission requires that the sender be truthful in the sense of being accurate(i.e., actually possess information) and sincere (i.e., intend to fully reveal this information) and that thereceiver believe that the sender is truthful. When agents have material incentives to lie, Williams expresses
the view that agents must value truthfulness intrinsically in order to overcome these material incentives to
lie.

21

While ethical necessity has not played a central role in economic theory, related questions
have been considered in previous work.29 Hurwicz (2007) discusses the existence of “intervenors,” defined as ethical monitors in a monitoring-the-monitor problem, and expresses
his personal belief in the existence of intervenors. Unlike the present paper, however, Hurwicz argues that intervenors are not needed for successful monitoring hierarchies.30 Hurwicz
describes an environment with three agents A, B, C, in which B monitors A’s actions, C
monitors B’s monitoring of A, A monitors C’s monitoring of B’s monitoring A, and so on.
This “Hurwicz triangle,” in which the monitoring hierarchy is folded in a loop going throughthe agents, omits corruption across monitors, a possibility studied by Strulovici (2021).31
Although ethical necessity is not discussed explicitly in economic models of law enforcement, the existence of a reliable monitor is often implicit. In Becker and Stigler’s (1974)
study of wrongdoing and malfeasance by enforcement officers, for instance, the authors assume that wrongdoing may be exogenously detected. When ethical monitors are unavailable,
however, the question of how this detection is generated remains to be answered.32

3.2

Existence and Observability of Ethical AgentsThe arguments developed so far assume that agents are known to be non-ethical. Without
this assumption, mediated learning can be achieved if agents put sufficiently high weight29

Myerson (2006) shows that in a federalist regime, the existence of virtuous politicians can guarantee thesuccess of democracy either at a national or a provincial level, whereas democracy can fail at both levels whenvirtuous politicians are surely absent. Glazer and Rubinstein (1998) describe an information aggregation
environment in which, if all agents are purely concerned with achieving the social optimum, there always
exist equilibria in which the optimum fails to be achieved, whereas if all agents are also concerned with theirindividual recommendation being followed, the social optimum is uniquely selected. Matsushima (2008)
studies the possibility of full implementation when agents prefer truthtelling whenever their material payoffs
are unaffected by their message.
30
Rahman (2012) proposes a different approach to monitoring, well suited for repeated monitoring tasks
such as controls for airport security: a principal asks agents to sometimes violate the rules on purpose tocheck whether these violations are caught by the monitor. Such violations are detected at no cost for the
principal, since he instigates them. The approach is well suited when violations can be faked at little social
cost, the principal has commitment power, and collusion between principal and agents is impossible.
31
Levine and Modica (2016) consider a similar structure, in which agents in a group take some initialaction, then verify with some probability the action previously taken by their neighbor, then verify theearlier verification task of their neighbors, and so forth.
32
Milgrom, North, and Weingast (1990), who study the enforcement of trade institutions by law merchants
in medieval Europe, consider some of the law merchants’ incentives to lie and take bribes.

22

on the probability that other agents are ethical. To see this, let us consider once more theblood-testing example. If everyone believes that the second agency will behave ethically (i.e.,
will correctly test the second sample and truthfully reveal the result of its test), this can beused to induce the first agency to truthfully test and report the content of the first blood
sample. By comparing the two agencies findings, and punishing the first agency for any
discrepancy, the first agency can be held accountable thanks to a second agency’s reliable
source of information.
More generally, any agent can be incentivized to behave truthfully as long as the agent
believes that subsequent agents are likely to provide informative reports.
Conversely, if some agent is ethical but other agents believe that he is not, mediatedlearning can fail just as when all agents are non-ethical, for two reasons: First, the findings ofthe ethical agent are (wrongly) believed to be uninformative. Second, precisely because thisagent’s findings are believed to be uninformative, the findings cannot be used to incentivize
other agents to behave truthfully. Therefore, mediated learning requires that agents believethat other agents behave ethically with high enough probability.33
The theory thus provides a specific mechanism for why eroding trust in institutions isdamaging: society may need everyone to believe in the existence of ethical agents in orderto sustain ethical behavior. Events that erode the strength of this belief can have severe
consequences for the feasibility of mediated learning and the functioning of society.34
Furthermore, it may be empirically difficult to distinguish between agents who have ethicalpreferences and agents who merely behave ethically because they believe in the existence of
ethical agents. Put differently, the belief in ethical behavior can be self-fulfilling.35

33

It would be interesting to study, following the results of the present paper, how likely and concentrated

ethical agents have to be in order to sustain mediated learning.
34
Belief in ethical behavior gets rid of virtual attrition, but not of real attrition. For example, if thesecond blood-testing agency is, in fact, non-ethical, it cannot be incentivized to tell the truth: real attritioninterrupts mediated learning. But if everyone erroneously believes that the second agency is ethical, thenthe first agency may be incentivized to behave truthfully.
35
Bénabou, Falk, and Tirole (2020) study theoretically and empirically the extent to which moral preferences can be elicited.

23

4Social Consequences

4.1

Political Consequences

Consider the perspective of a citizen who is cynical in the sense that he does not trust
information intermediaries to behave ethically. In this citizen’s mind, agents involved in thelearning process do not care about the truth per se and are collectively aware of this.
To see how information attrition affects the views of such a citizen, let us first revisit the
blood-testing example. Given that the blood samples are subject to information attrition, it
is reasonable—indeed, rational—for a cynical individual to treat as uninformative any finding
reported by the blood-testing agencies, for the reasons explained above.36 As a result, twocitizens with cynical beliefs about agencies’ behavior and otherwise different views of theworld may rationally entertain very different beliefs about whether a particular athlete useda banned substance, even after agencies have made their reports public: mediated learning
fails to convince citizens and to bring their views closer.37
Mediated-learning failures can have severe consequences for social cohesion and politicalstability. When, for instance, a politician is accused of corruption, the average citizen must
rely on declarations made by intermediaries, such as officials and journalists, to learn whether
the corruption charge is true or, on the contrary, an attempt to smear and neutralize thispolitician. There is no deus ex machina to lift all confusion and reveal the truth to the
public. Information attrition is a concern in these environments: incriminating documents
can be destroyed, witnesses intimidated or eliminated, and so on.38
36

Blood tests could in principle be certified by a third party, which would check wether the agency did its

job properly and thus increase the trust in its report. One would then have to understand the third party’s
incentives. This “monitoring the monitor” structure is discussed in Section 5.2.
37
Although the mechanism is unrelated, this result echoes failures of belief convergence in social learning
environments. See, e.g., Acemoglu, Chernozhukov, and Yildiz (2016).
38
These observations also apply to government agencies suspected of abusing their power or violatingsome rules. Examples of evidence destruction by governmental agencies abound, even in prominent democracies. In the Unites States, for instance, CIA director Richard Helms ordered in 1973 that all documents pertaining to the CIA’s infamous MK Ultra program on mind-control experiments be destroyed (“Aninterview with Richard Helms”, https://www.cia.gov/static/9845318ed4b2db36bc185604a2c3bc40/
interview-with-richard-helms.pdf). In 2005, the CIA’s Director of Operations ordered the destruction of all interrogation tapes of Abu Zudaydah and Abd al-Rahim al-Nashiri that featured “enhanced”
interrogations (“Tapes by C.I.A. Lived and Died to Save Image,” https://www.nytimes.com/2007/12/30/
washington/30intel.html.)

24

Citizens holding different priors about such a corruption case, perhaps due to differing
ideologies or political perspectives, but holding a similarly cynical view about the lack of
ethical motives underlying public declarations, may therefore both dismiss official statements
and journalistic reports about the case and maintain strong disagreements.
The theory thus explains why rational citizens may remain divided on questions subject
to information attrition, and why the lack of trust in investigative institutions can rationally perpetuate polarization. It also provides a specific meaning and mechanism for theobservation that eroding citizens’ “trust” in institutions harms social cohesion.

4.2

Historical Revisionism and Information Attrition

Historical revisionism is another instance of mediated learning in which information attrition
plays an important role. Understanding historical events is of obvious importance, not only tolearn lessons from the past but also to assess claims based on such events, such as territoriality
and reparation claims. Mediated learning is necessary because citizens cannot directly verify
historical events.39 They must rely on experts and officials to access and correctly interpretarchives, artifacts, and other sources of information. Information attrition is both exogenous(e.g., witnesses die) and endogenous (e.g., documents may be destroyed on purpose).
To give a concrete example,40 consider the fire of the German parliamentary building(Reichstag) on February 27, 1933. The importance of this event can hardly be overstated.
The Nazis, who had lost seats in the previous parliamentary election, claimed that thefire had been caused by communists and used this claim to pressure president Hindenburg
into imposing martial law on Germany (the “Reichstag fire decree”) and arrest and weakencommunists. This allowed the Nazis to form a majority coalition following the March 5,1933 parliamentary elections, consolidated Hitler’s power, and led to the Enabling Act.
While the claim of communist involvement in the fire has long been rejected, there wasuntil recently a consensus among mainstream historians that the Reichstag fire had not been
39

This point is obvious with regard to events that took place before citizens’ lifetime. Even with regard tocontemporaneous events, aggregating information and forming a global picture of an event is a highly complex
task that requires expertise, time, and a special access to information. The pitfalls of such information
aggregation have been famously illustrated by Stendhal’s La Chartreuse de Parme whose protagonist, Fabricedel Dongo, takes part to the Battle of Waterloo with the Napoleonic army and construes a completely
erroneous version of the battle.
40
Huq (2018) discusses similar, very recent examples, in which the possibly false threats of terrorism orcoups were used to weaken democratic institutions and shift power to more authoritarian regimes.

25caused by the Nazis. Fritz Tobias, one of the most respected historians on this subject in
the postwar period, published a series of articles purporting to show that van der Lubbe,
the person who was convicted for the arson, had acted alone.41 Historians accepted Tobias’s
version of the event until 2001, when two historians studying Gestapo archives raised thepossibility that it was a group of SA officers who had set the fire (Bahar and Kugel (2001)).
Hett (2014) used recent scientific advances to convincingly argue that it would have been
impossible for a single individual to set the fire. In 2019, an affidavit written in 1955 byformer SA Hans-Martin Lennings was discovered in Tobias’s personal files and published by
RedaktionsNetzwerk Deutschland, which stated that Lennings and other SAs had driven vander Lubbe from an infirmary to the Reichstag when the fire had already started, effectively
setting up van der Lubbe.
Information attrition took several forms: all but one of the SA officers who were allegedly
involved in the Reichstag fire were killed (and, hence, silenced) during the Night of the Long
Knives; van der Lubbe was beheaded in 1934 for his alleged role in the arson; and any
forensic evidence about the Reichstag fire has been long gone. Furthermore, Fritz Tobias hid
Lennings’ affidavit, which contradicted Tobias’ single perpetrator theory, until his death.42
To this day, the strongest case for Nazi involvement thus seems to come from Lennings’
affidavit, and hence hinges on one man’s statement. It is unclear what Lennings’ motivation
for incriminating the SA may have been, except perhaps for setting the record straight. For
observers who doubt Lennings’ statement, the question of who set the Reichstag on fire
may reasonably represent a failure of mediated learning. Lutjens (2016) believes that “thecontinuous reshaping of the Reichstag fire by those with a stake in the matter has fragmented
the truth beyond recovery.”5

Discussion

Mediated learning fails if the following conditions hold jointly: (i) intermediaries do not careabout the truth, (ii) information is subject to attrition, (iii) there is no exogenous, public
revelation of the truth at any future time, and (iv) intermediaries proceed sequentially.
This result holds for arbitrary utility functions as long as intermediaries have arbitrarily
41
42

The articles were published by Der Spiegel under the title ,,Stehen Sie auf, van der Lubbe!” in 1959-60.
Tobias is now suspected of protecting former Nazi officers after the war and having a private interest in

dissimulating the Nazis’ role in the Reichstag fire.

26

small biases or informational costs. It holds for all equilibria, in the strong sense that nothingat all is learned about the state of the world, and despite the fact that agent incentives canbe administered without any further agency problem: whatever rewards and punishments
are promised to the agents as a function of reporting histories can be perfectly enforced.
This impossibility result may be viewed as a reference point: to succeed, mediated learning
must break at least one of the four conditions above. In particular, mediated learning cansucceed if some intermediaries are motivated by the desire to seek the truth, or by a belief
that other intermediaries have such a motivation. Several ways out are discussed below.

5.1

Escaping Attrition with Hard evidence

In some cases, signals are hard to fabricate and non-disposable. Consider video footage ofa crime, in which the criminal is clearly identifiable. Such footage can be viewed numerous
times with little degradation and is conceptually similar to reproducible evidence.
Even this kind of evidence is need not be perfectly reliable. For example, video footage canfabricated, as exemplified by the emergence of baffling deepfakes, which have raised severe
concerns (see, e.g., the journalistic work of Schick (2020)).43
DNA testing also illustrates this issue. The amount of usable DNA samples on a crime
scene is finite. The procedure of DNA testing involves a replication phase, such as PCRamplification or DNA cloning, which creates more “evidence” that can be stored and verified
by subsequent investigators. Crucially, however, these replications are only as reliable as theoriginal DNA sample. They do not constitute new, independent evidence.
This is all the more important as DNA samples can be synthesized, i.e., literally “fabricated,” to match any desired DNA profile.44 Moreover, DNA samples can be erroneously
or malevolently taken outside of the crime scene and presented as coming from the scene.45
43

Deepfakes can be detected by computer scientists who specialize in the field. However, this kind of

detection increases the reliance on mediated learning. With deepfakes, video evidence no longer constitutes
public, hard evidence. The disappearance of hard video evidence due to deepfakes is emphasized by Schick(2020).
44
Frumkin, Wasserstrom, Davidson, and Grafit (2010) show the possibility of creating saliva or blood
samples with the desired DNA. The authors, as well as subsequent work by other researchers, show that
identifying methylation patterns in DNA samples can help distinguish synthetic and natural DNA, althoughsuch identification is challenging.
45
A famous example is the “Phantom of Heilbronn,” a presumed serial killer whose DNA was found on 40

27

Ultimately, agents joining an investigation can either test the DNA material that previous
laboratories have left them, which have been manipulated or fabricated, or look for genuinely
new DNA samples, which brings us back to the problem of information attrition.
The role of technology on mediated learning is complex and deserves a separate exploration. While, DNA testing, video footage, and other technological advances have increased
the set of reliable evidence, technology can be used to manipulate evidence and do so moreanonymously than before, increasing the reliance on experts and, hence, on mediated learning.

5.2

Alternative Learning Structures

Several remedies may be considered to address the unraveling results of Theorems 2, 3,
and 4. First, agents could be asked to investigate and report their findings simultaneously, a
structure that we will call parallel monitoring. Second, agents could investigate the actionsof past investigators, rather than the initial fact. Third, while after-the-fact investigationsof the type studied in the baseline model are crucial and widely used to maintain ex ante
incentives of would-be offenders, in some applications investigators could be incentivized
by the perspective that their findings will have an influence on subsequent actions andevents, which brings the framework closer to a repeated game setting. Finally, each agent
could intervene multiple times in the investigation process, rather than just once. Thesepossibilities are examined in turn.
1. Parallel Monitoring with a Centralized Authority
In some applications such as journalistic investigations and academic research, it is realistic to assume that agents report their findings sequentially. It is nonetheless natural toconsider, from a mechanism-design perspective, the case in which several agents simultaneously and independently investigate the fact of interest and report their findings to some
central authority. A conceptually similar design is to hide past reports from investigators
until they have made their own report.46
crime scenes over a fifteen-year span in Germany, Austria, and France. The DNA turned out to belong to a
woman working in the factory that made the cotton swabs used to collect DNA samples.
46
Gershkov and Szentes (2009) show that such a design is optimal in a voting model with costly informationacquisition. In their model, voters’ preferences depend on the state of the world and informative equilibria
exist even when all reports are public, but optimal learning requires that past reports be hidden.

28For example, blood-testing laboratories could be asked to test their respective samples
independently from each other and report their findings simultaneously to some overseeing
agency. The laboratories would be rewarded if their findings match and punished otherwise.47
An important first concern with this solution stems from the incentives of the central
authority coordinating the agents. If this authority has a material interest in a specificoutcome (which cannot be ruled out, especially in politically charged investigations), it cansecretly help agents coordinate their reports or influence agents’ reports in various ways.
In order to avoid this, the central agency must itself be monitored, which brings us backto a sequential monitoring problem. In the blood-testing example, if laboratories must
report their findings simultaneously, there is no recourse for an athlete accused of doping if
laboratories conspire to accuse the athlete.
A second issue concerns the robustness of equilibria in which agents are induced to learn.
Agents subject to strategic uncertainty may be harder to incentivize. This question is studied
by Pei and Strulovici (2022), who find that the answer depends on the concept of robustness
used: When one focuses on partial implementation and uses a local, ex ante notion ofrobustness building on Kajii and Morris (1997), it is possible to design robust mechanismsthat prompt agents to acquire costly information and report their findings truthfully. For
the case of full implementation, or for stronger notions of robustness, however, impossibility
results are derived when agents are non-ethical in the sense of the present paper.
In some cases, parallel monitoring may be challenging to implement. It is difficult, forinstance, to send multiple investigators on a crime scene to independently interrogate witness
and collect evidence, without the investigators being able to communicate, either directlyor through witnesses and possibly coordinate. Moreover, when the evidence is in limited
supply (such as the weapon of a crime), such a limitation creates negative correlation in
investigators’ reports, since at most one of them can discover the evidence (weapon).48
2. Monitoring the Monitor
Instead of all agents investigating the same initial fact, agents could investigate one another. For example, agent 1 would investigate the initial fact, agent 2 would investigate
47

The academic refereeing process has reporting features that resemble the parallel-monitoring design,

although the incentives for referees are different and arguably more complex than a mere coordination
motive.
48
The adverse effect of negative correlation on the informativeness of agents with a coordination motive
is a central finding in Pei and Strulovici’s (2020) analysis of strategic crime.

29

agent 1’s investigation of the initial fact, agent 3 wouold investigates agent 2’s investigationof 1, and so on. Such a “monitoring hierarchy” is described by Hurwicz (2007). In his model,
the number of agents is finite and the monitoring chain cycles repeatedly across a fixed setof agents. A first conceptual difficulty with this approach concerns the simultaneity andcomplexity of these monitoring tasks: agents are supposed to conduct an infinite amount ofmonitoring tasks and are indirectly the subject of the tasks that they are investigating.
Even if we considered an unbounded sequence of distinct agents, each of which is tasked
with investigating the previous agent in the sequence, another issue would emerge, which
concerns agents’ ability to collude. For example, if an agent discovers incriminating evidence
about the agent he was monitoring could hide or destroy the evidence in exchange for apayment from the guilty agent. Such a transfer amounts to a local form of corruption among
agents in direct contact. In a separate paper, I show that even such a local form of corruption
may suffice to destroy the possibility of mediated learning (Strulovici (2021)).
3. Repeated Setting
When mediated learning concerns the identification of a criminal and opportunities tocommit crime are repeated over time, it is a priori possible that investigators care about thetruth indirectly, through the impact that their findings have on citizens’ future behavior.
Suppose that a citizen’s decision to commit crime depends on whether his past actions
were accurately called by past investigators: for example, a citizen who was wrongfully
accused in the past or mistakenly acquitted of crimes that he did commit may be more
likely to commit crime in the future. In this setting, investigators could in principle havean endogenous incentive to report accurate findings. Provided that players are sufficiently
patient, this kind of strategy profile could a priori be used to incentivize accurate reporting.
However, for this argument to work, a citizen’s strategy must depend on his private history,
where the public history consists of official findings about citizens’ past actions and a citizen’s
private history records his actual past actions. In order for a citizen’s private history to affecthis decision of whether to commit crime, the citizen must indifferent between committing
crime and abstaining from it, a knife-edge condition that is violated if, for instance, citizens
are subject to small private shocks affecting their benefits from committing crime.49

49

The argument is somewhat similar to the section on witnesses in this paper.

30

4. Alternating Statements
Finally, one could ask a fixed set of agents to take turns investigating and reporting on
the question of interest. The key difference with the baseline model is that agents now haveprivate information about what they did in the past, which affects how they interpret thedeclarations of other agents. If an agent has discovered disposable signals in the past, heknows that other agents have fewer signals to discover. The analysis becomes more complex
because agents’ decisions now depend on their beliefs about the amount of evidence left,
about other agents’ beliefs about the amount evidence, their belief about agents’ beliefs
about their beliefs about the amount of evidence left and so on. While information attritionis likely to have a similar effect as in this paper’s model, confirming this intuition andexploring this question is left for future research.

5.3

Designing Truth-Dependent Preferences: Oaths, Capitalism,
and Popular Juries

A more direct approach to improving mediated learning is to increase the salience of truthdependent preferences.
This may be achieved by fostering agents’ ethical sense, from inculcating an ethical education and culture to strengthening trust in institutions and developing effective vetting andselection processes for key learning responsibilities.
Professional oaths, from the Hippocratic oath in medicine to journalistic oaths such as
Walter Williams’ Journalist’s Creed (Farrar (1998)) aim at eliciting ethical behavior. This
paper suggests a positive correlation between the need for oaths in various professions andthe severity of information attrition in these professions.
Even if a small fraction of agents is swayed by such oaths, this may in principle suffice forincentivizing truthful behavior by other agents. Studying the mechanisms and behavioralfeatures through which ethical agents can incentivize mediated learning is beyond the scopeof this paper, but it is easy to conceive of simple examples:50 suppose that an agent, who is
known to truthfully seek and report the truth is commonly know to appear in round N > 1.
50

There are various approaches to modeling ethical behavior. For instance, Ellingsen and Mohlin (2020)

distinguish three dimensions: decency, integrity, and punitivity. Harsanyi (1980) and Feddersen and Sandroni(2006) study rule-utilitarian agents and Roemer (2019) consider Kantian agents.

31This agent’s report provides reliable information, akin to an exogenous public signal about
the state of the world, which can be used to incentivize all agents coming in rounds i < N .
Even if agent N has only a small probability p < 1 of behaving ethically, his report may
still be used to incentivize agents in earlier around as long as these agents’ rewards andpunishments are of order 1/p.
Another approach is to organize society in a way that increases information mediators’material dependence on the truth, i.e., gives them “skin in the game.” Eliciting informationfrom agents about a scientific fact or the social value of a new product or process is easier
when the agents stand to gain financially from this information, which may broadly interpreted as capitalistic incentives. Thus interpreted, the theory offers a new perspective on
the “virtue” of capitalism relative to systems in which agents have low-powered incentives.51
The theory also emphasizes that violations of the rules of capitalism (or any other system,
for that matter) may be difficult to detect and reveal truthfully, and thus suggests a potential
tradeoff between the incentives provided within a given economic or political system and the
incentives required to guarantee that the system is respected by its participants.
A final angle to attack mediated learning failures is to democratize the learning processby enlarging the pool of potential information intermediaries. Large pools can increase
the alignment—real or perceived—between the intermediaries and society as a whole, incontrast to the baseline model of the paper, in which society’s objective is dissociated fromthe intermediaries’. Large pools of intermediaries are conceivable when the expertise required
to learn the fact of interest is limited. The institution of popular juries may be viewed as
one such application, which trades off intermediaries’ expertise with their representativenessof a more global and diffuse body of stakeholders.

51

A large literature emphasizes capitalism’s ability to reduce moral hazard problems relative to socialistic

systems. See Myerson (2007) and Tirole (2006) for a review of relevant papers and corporate-finance models
capturing this idea. The question of incentive compatibility and its relation to various economic systems is
at the heart of Leonid Hurwicz’s development of mechanism design (Hurwicz (1973)).

32

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35

A

Proof of Theorem 2

A.1

Discovery-Belief Formula

The section presents three lemmas leading to Proposition 2, which contains the discoverybelief formula. Lemma 1 states that an agent works only if the probability Fi1 that there
remains at least one signal to discover is high enough. Lemma 2 states that when an agent
works and finds nothing, he cannot be much better off, ex post, than if he had shirked
instead, especially if Fi1 is close to 1, which means that the working agent’s lack of a discovery is likely attributable to bad luck. Lemma 3 computes an upper bound on an agent’s
expected benefit from working relative to shirking. This upper bound is expressed in termsof the probability that a working agent produces a message that triggers an informative
continuation equilibrium. This set, Mi+ , plays a key role in the analysis. All three lemmas
and Proposition 2 are proved in Appendix C.
We make two simplifications throughout the proof, which are without loss of generality.
First, agents’ decisions are invariant with respect to a uniform translation in their grossutility functions. We can therefore assume that these functions all take values in someinterval [0, R]. Moreover, since R is an upper bound on payoffs, it can be increased toguarantee that R > c, which is assumed from now on.
Let γi denote the probability that i works given m1i−1 .
Lemma 1 γi > 0 only if Fi1 ≥ Rc > 0.
Given any round i and report history m1i−1 such that γi > 0, let:
 Vi∗ denote i’s maximal expected gross utility if he shirks, where the maximum is takenover all possible messages mi ∈ M that i can send after shirking; fi0 = 1 − Fi1 denote the probability that Si = ∅ at the beginning of round i.

Lemma 2 Suppose that λ < 1. If i works, finds nothing, and sends message mi , his expected
gross utility Viw (∅, mi ) satisfies
Viw (∅, mi ) ≤ Vi∗ +

fi0 R
.
1−λ

For any round i and M̃i ⊂ M , we consider the following probabilities conditional on history
mi−1
1 :
36

 gi (M̃i ): probability that i finds a signal and sends a message in M̃i conditional on

working; di (M̃i ): probability that i finds no signal and sends a message in M̃i conditional on

working;
Also let Mi+ denote the set of messages mi followed by an informative continuation equilibrium in round i + 1 given the report history mi−1
until round i.
1
Lemma 3 Agent i’s expected gross utility conditional on working has the following upperbound. If λ < 1, thenViw ≤ Vi∗ + di (Mi+ )fi0 R+ gi (Mi+ )R.
1−λ

If λ = 1, thenViw ≤ Vi∗ + di (Mi+ )R + gi (Mi+ )R.
For each round i, we introduce the following variables.
 βi : probability that i discovers a signal conditional on report history m1i−1 (before

observing whether i works, i.e., viewed from the beginning of round i);k Fi+1(mi ): probability that there remain at least k signals at the beginning of roundi + 1 given reports mi1 = (mi−1
1 , mi ).
Proposition 2 (DB Formula) Let C(λ) = 2R/(cλ(1 − λ)) if λ < 1 and C(1) = 2R/c.
The following inequality holds for all constants C ≥ C(λ), round i, and integer k ≥ 1 suchthat Fik > CFik+1 :βi ≤ C

A.2

h
i
k
k
Ei (Fi − Fi+1 (mi ))1mi ∈Mi+
Fik − CFik+1.

(5)

Proof of Theorem 2

The proof proceeds by induction on k. Lemma 1 already proves the claim for k = 1 with
F 1 = c/R. Now suppose that the claim holds for some k ≥ 1: there exists a threshold¯
F k > 0 such that any informative continuation equilibrium in round i satisfies Fik ≥ F k .52¯¯
52

Theorem 2, which was stated for round 0, also applies to all continuation equilibria: the thresholds{F k }k≥1 depend only on the parameters (R, c, λ), which are constant throughout the game.
¯

37

We will show that there is a constant F k+1 > 0 such that a continuation equilibrium can be¯
informative only if Fik+1 ≥ F k+1 .
¯For any f ∈ [0, 1], let F k (f ) = inf{Fik ∈ [0, 1] : γi > 0, Fik+1 ≤ f }, where the infimum istaken over all on-path histories and all equilibria of the game and is by convention equal
to 1 if no informative equilibrium exists for which Fik+1 ≤ f . By construction, F k (f ) is
nonincreasing in f .
Let F = inf{f : F k (f ) < 1}. F is the smallest value of Fik+1 for which an informative¯¯
continuation equilibrium exists (or the infimum of such values, if the infimum is not achieved).
Our objective is to show that F > 0. Let F̄ k = limω↓F F k (ω), i.e., the right limit of F k (·) at¯¯F . This limit is guaranteed to exist because F k (·) is nonincreasing. Intuitively, F̄ k is defined¯
by looking at all informative equilibria that have the smallest possible probability that there
remain at least k + 1 signals, and taking the smallest probability that there remain at least
k signals among all such equilibria.
kF̄, where G > 0 is
Let ε > 0 denote any small constant such that F̂ k = F k (F + Gε) ≥ 1+η
¯
a large constant and η > 0 is a small constant, both determined at the end of the proof

independently of k.53 Since F̄ k is the right limit of F k (·) at F , such an ε exists. Moreover,¯
since all ε0 ∈ (0, ε) also satisfy the condition, we can choose ε so that
!2
1 F̂ kε≤.
(6)
2 G
By definition of F , there exists at least one informative continuation equilibrium for which¯
Fik+1 ∈ [F , F + ε].
¯ ¯

(7)

Moreover, by definition and monotonicity of F k (·), there exists an informative continuation
equilibrium among those for which Fik ≤ F̄ k + η F̂ k .
Consider such a continuation equilibrium. Since our objective is to prove that F > 0, suppose¯
by way of contradiction thatF = 0.
(8)¯
From (7), this implies that Fik+1 ≤ ε. Combining this with Doob’s martingale inequality
guarantees that Fjk+1 remains small for all j > i with high probability. Formally, let A
denote the event that Fjk+1 ≤ Gε for all j ≥ i. We have the following result.
53

For example, if λ < 1 one can choose G and η such that

explained at the end of this proof.

38

√

√
G = 128R3 /(λ2 (1 − λ)c3 ) and η = 1/ G, asLemma 4 i assigns probability at least 1 − 1/G to A.
Proof.

For j ≥ i, let F̄jk+1 denote the probability assigned at the beginning of round j(i.e., conditional on m1j−1 ) that there remain at least k + 1 signals at the beginning of roundi (fixed). The process {F̄jk+1 }j≥i is nonnegative and bounded above by 1. Moreover, it is amartingale by the law of iterated expectations and the fact that j’s filtration grows finer as
j increases. Moreover, F̄ik+1 = Fik+1 ≤ ε.
We can therefore apply Doob’s martingale inequality, which implies that for any J ≥ i,
Pr(maxi≤j≤J F̄jk+1 ≥ Gε) ≤Ei [F̄ik+1 ]Gε≤ 1/G. The event Ā∞ defined by {maxj≥i F̄jk+1 ≤Gε} is the intersection of the events ĀJ = {maxi≤j≤J F̄jk+1 ≤ Gε}. Therefore, Pr(Ā∞ ) =limJ→∞ Pr(ĀJ ) ≥ 1 − 1/G.
Finally, we note that F̄jk+1 ≥ Fjk+1 for all j ≥ i, because the true number of remaining signals
only decreases over time and thus whatever signals remained at the beginning of round i musthave contained the signals that remain at the beginning of round j, so Pr(A) ≥ Pr(Ā∞ ) ≥1 − 1/G.



Conditional on A, Fjk+1 ≤ Gε for all j ≥ i. Moreover, if round j belongs to an informative
continuation equilibrium, we have Fjk ≥ F k (Fjk+1 ) ≥ F k (Gε) = F̂ k . Given any positive
constant C, this implies that for informative continuation equilibrium starting in round j,
Fjk − CFjk+1 ≥ F̂ k − CGε ≥ F̂ k /2 > 0(9)provided that G ≥ C, where the last weak inequality comes from (6). The condition G ≥ Cwill be satisfied by setting C = C(λ), where C(λ) is defined in Proposition 2, and thenchoosing G large enough. (A specific value of G is given at the end of the proof.)
Let Aj denote the event that Flk+1 ≤ Gε for all integers l such that i ≤ l ≤ j. The
events {Aj }j≥i form a decreasing sequence that converges to A as j → ∞. Moreover, Aj is
measurable with respect to the information available at the beginning of round j.
Proposition 2 implies, conditional on event Aj , that:"#k
C(Fjk − Fj+1
(mj ))1mj ∈Mj+
βj ≤ Ej.
Fjk − CFjk+1(10)

Let Bj denote the event that mj ∈ Mj+ . Bj is adapted to mj1 , i.e., known at the beginning ofround j + 1. Notice that if Bj does not occur, it means by definition that no l > j ever works(i.e., mj1 is followed by an uninformative continuation equilibrium). This implies that the39sequence of events {Bj }j≥i is decreasing path by path and, hence, that the sequence {1Bj }j≥i
is nonincreasing.
Since Aj is measurable with respect to mj−1
1 , Equations (9) and (10) imply that for j ≥ i + 1" "##k2C(Fjk − Fj+1
)1Bj
Ei+1 [1Aj βj ] ≤ Ei+1 Ej 1Aj.
F̂ kThe law of iterated expectations then implies that"#k
2C(Fjk − Fj+1
)1Bj
2C
k
Ei+1 [1Aj βj ] ≤ Ei+1 1Aj=
Ei+1 [1Aj (Fjk − Fj+1
)1Bj ].
kk
F̂
F̂
Summing (11) over all j ≥ i + 1, we obtain
"
"
#
#
X
X
2C
k
Ei+1
1Aj βj ≤ k Ei+1
1Aj 1Bj (Fjk − Fj+1
(mj )) .
F̂
j≥i+1
j≥i+1

(11)

(12)

Let J denote the first (possibly infinite) round for which the product of the indicator functionsin the right-hand side of (12) is zero:J = inf{j ≥ i + 1 : 1Aj 1Bj = 0}
with the convention that J = +∞ if the set is empty. Since both of these indicator functions
are nonincreasing in j for j ≥ i + 1, path by path, their product must be equal to zero for allj ≥ J. In words, J is either the first round in which Fjk+1 exceeds Gε, or the last round atwhich the continuation equilibrium is informative (which implies that γj = 0 for all j > J),
whichever occurs first.54
Consider first the paths for which J is finite. The argument of Ei+1 [·] on the right-hand sideof (12) then reduces to
J−1
X

k
k
(Fjk − Fj+1
) = Fi+1− FJk .

j=i+1

Consider now paths for which J is infinite. In this case, the argument of Ei+1 [·] on theright-hand side of (12) is equal to
J˜
X
k
k
k
k
k
k
k
(Fj − Fj+1 ) = lim(Fjk − Fj+1
) = lim {Fi+1− FJ+1˜ } = Fi+1 − lim FJ˜ .
˜
˜
˜
J→∞
J→∞
J→∞
j=i+1
j=i+1
∞
X

54

J is not a stopping time with respect to the filtration {Fj }j≥i+1 generated by public histories{mj−11 }j≥i+1 , because at the beginning of any round j at which j works with positive probability, it isunknown whether j will be the last round in which the agent works. For this reason, the proofs below do
not rely on any theorems pertaining to stopping times, such as the optional sampling theorem or the strong
Markov property.

40Notice that the limit is well defined because Fjk is a nonnegative supermartingale with respect
to j.55 We will call this limit FJk to be consistent with the case in which J is finite.
Combining these observations with (12), we conclude that
"
#
X

2C
k
Ei+1
1Aj βj ≤ k Fi+1
− Ei+1 [FJk ] .
F̂
j>i
Since βj is the probability that j discovers a signal conditionally on mi−1
1 , it may be decomposed as βj = γj Fj1 λ, because j discovers a signal only if three independent (conditionallyon m1i−1 ) events occur: (i) j works, (ii) Si is non empty, and (iii) j is “lucky” to discoversuch an element from Si . Moreover, Lemma 1 shows that γj > 0 only if Fj1 ≥ c/R. ThisR
implies56 that γj ≤ gβj where g = λcand hence that
"
#
X

2Cg
k
Ei+1
1Aj γj ≤ k Fi+1
− Ei+1 [FJk ] .
F̂
j≥i+1

(13)

Let Z denote the event that at least one agent j > i works and πi+1 (mi ) = Pri+1 (A ∩ Z),
55

The argument is similar to the one used to prove Lemma 4. For any fixed j, let F̄lk+1 denote theprobability assigned by l ≥ j to there remaining at least k + 1 signals at the beginning of round j. The
process {F̄lk+1 }l≥j is a martingale in j, by the law of iterated expectations and the fact that that l’s
filtration grows finer as l increases. Moreover, F̄lk+1 ≥ Flk+1 for all l ≥ j, because the actual number of
k+1
k+1
remaining signals only decreases over time. Therefore, we have Fjk+1 = F̄jk+1 = Ej [F̄j+1] ≥ Ej [Fj+1]. This,together with the fact that Flk+1 is uniformly bounded and measurable with respect to the information at
the beginning round l, shows that it is a supermartingale.
56
The inequality clearly holds if γj = 0.

41

i.e., the probability that A and Z both occur conditional on mi1 . We have!
X
πi+1 (mi ) = Pri+1 1A
1j works ≥ 1
j>i
"
#
X
≤ Ei+1 1A
1j works
j>i
i
h
X
=
Ei+1 1A 1j works
j>i
i
h
X
≤
Ei+1 1Aj 1j works
j>i
h h
ii
X
=
Ei+1 Ej 1Aj 1j works
j>i
h
h
ii
X
=
Ei+1 1Aj Ej 1j works
j>i

=

X



Ei+1 1Aj γj .

j>i

The first equality comes from the fact that Z is identical to the event {P1}. The first inequality comes from the fact that the random variable 1AP

1j works ≥j>i 1j works isj>inonnegative and integer valued, which implies that its expectation exceeds the probability
that it is strictly positive. The second equality is an application of Tonelli’s theorem. The
second inequality comes from the fact that A ⊂ Aj and, hence, 1A ≤ 1Aj . The next equality
comes from the law of iterated expectations and the next one comes from the fact that Aj
is measurable with respect to the information at the beginning of round j. The last equality
holds by definition of γj .
From (13) and Tonelli’s theorem, this implies that
πi+1 (mi ) ≤

2Cg
F̂ k


k
Fi+1
− Ei+1 [FJk ] .

(14)

Let Fik,r (mi ) denote the probability that there are at least k signals left conditional on i
working and reporting mi . For any signal si , let Fik,w (si ) denote the probability that there
are at least k signals left conditional on i working and discovering si . Fik,w (si ) represents i’sbelief after discovering si , while Fik,r (mi ) represents what i + 1 would believe after observing
report mi if he knew for sure that i has worked.
The following lemmas are proved in Appendices C.5 and C.6. Let Ni = {mi : Fik,r (mi ) >(1 + η)Fik }.
42

1
Lemma 5 (i) γi (Ni ) ≤ 2ηG/ Ni , πi+1 (mi ) ≤ 2Cg(1 + η)Fik − Ei+1 [FJk ] .
2 , (ii) for mi ∈F̂ kk+1
Let Ti = {mi : Fi+1
(mi ) ≥

√

Gε}.

√
√
/ Ti , (ii) γi (Ti ) ≤ 1/ G.
Lemma 6 (i) Pri+1 (A) ≥ 1 − 1/ G for all mi ∈Let V w denote i’s expected gross utility if he works and V w (mi ) denote his expected grossutility conditional on working and reporting mi . We haveVw =X

γi (mi )V w (mi )

i∈Mi

X

≤ (γi (Ni ) + γi (Ti ))R +γi (mi )V w (mi )mi ∈N/ i ∪Ti


≤

1
1
+√
2
2ηG
G



X

R+

γi (mi )V w (mi )

(15)

mi ∈N
/ i ∪Ti

Moreover,V w (mi ) = Pr(Z|mi1 )V w (mi |Z) + Pr(Z c |mi1 )V w (mi |Z c ).

(16)

Conditional on mi1 , the event Z is independent of how mi was produced (i.e., whether mi
was obtained by work or fabrication). Indeed, as long as no one works, the distribution ofreports mj made by agents following i depends only on mi1 , not on the signals that remain to
be discovered in the case. And as soon as someone works, then by definition Z has occurred.
Thus, what triggers the event Z (whenever it occurs) is a sequence of uninformative (until Z
occurs) reports mj for agents following i, whose probability distribution is completely pinned
down by mi1 .
kk
From the previous lemmas, we have Pr(A ∩ Z|mi1 ) ≤ 2Cg
(1
+
η)F
−
E
[F
]
for all
i+1
i
k
J
F̂
√
mi ∈
/ Ni and Pr(Ac |mi1 ) ≤ 1/ G for all mi ∈/ Ti . Letting M̂i = Mi \ (Ni ∪ Ti ), this implies
that
Pr(Z|mi1 ) = Pr(Z ∩ A|mi1 ) + Pr(Z ∩ Ac |mi1 ) ≤

2Cg
F̂ k

√

Fik (1 + η) − Ei+1 [FJk ] + 1/ G (17)for all mi ∈ M̂i .
Conditional on A, Fjk+1 ≤ Gε for all j ≥ i. By definition of J, all continuation equilibria
until round J included are informative, which implies that Fjk ≥ F k (Fjk+1 ) for all j ≤ J.
Since F k (·) is nonincreasing, this implies that Fjk ≥ F k (Gε) = F̂ k for all j ≤ J.

43

We thus have for mi ∈ M̂i
Ei+1 FJk = Pri+1 (A) Ei+1 [FJk |A] + Pri+1 (Ac ) Ei+1 [FJk |Ac ]≥ Pri+1 (A)Ei+1 [FJk |A]√≥ (1 − 1/ G)F̂ k .
By construction, Fik ≤ F̄ k + F̂ k η and F̂ k ≥ F̄ k − F̂ k η. Therefore, Fik − F̂ k ≤ (F̄ k + F̂ k η) −(F̄ k − F̂ k η) = 2η F̂ k . Letting B = 8Cg, (17) then implies (for η ≤ 1, which we assume) thatfor all mi in M̂i√
B
Pr(Z|mi1 ) ≤ Bη + √ + 1/ G.
2 G

(18)

For each mi , let Vif (mi |Z c ) denote i’s expected gross utility if he sends message mi conditional
on no j > i working and V f,∗ denote the maximizer of Vif (mi |Z c ) over all messages mi ∈ M̂i .
Notice that i’s expected gross utility conditional on mi and no j > i working does not depend
on whether i worked or shirked: either way, the subsequent reports {mj }j>i are independentof the signals that remain to be discovered. Therefore, i’s conditional expected gross utilities
satisfy V w (mi |Z c ) = Vif (mi |Z c ).
Combining these observations with (16) and (18), we obtain


B
1
w
V (mi ) ≤ Bη + √ + √R + V f,∗ ,2 G
Gfor mi ∈ M̂i . Combining this with (15) yields




1
B
1
1
w
V ≤
+√
R + Bη + √ + √R + V f,∗ .
2ηG2
G
2 G
G
i’s utility from working thus satisfies


1
1
B
1
w
U ≤
+ √ + Bη + √ + √R + V f,∗ − c.
2ηG2
G
2 G
G
If i sends a message m∗i ∈ M̂i that achieves V f,∗ , his utility U f satisfies
U f ≥ Pr(Z|m1i−1 , m∗i ) × 0 + Pr(Z c |m1i−1 , m∗i )V f,∗B1≥ 1 − Bη − √ − √
V f,∗
2 G
G


B
1
f,∗
≥ V − Bη − √ − √
R
2 G
G
44

(19)

where 0 is used as a lower bound on i’s realized gross utility in the first inequality.
Therefore, working is strictly suboptimal if




1
B
1
B
1
1
f,∗
f∗
R + V − c < V − Bη + √ + √R+ √ + Bη + √ + √
2ηG2
G
2 G
G
2 G
G
or
1
c
3
B
√
√
>
+
+
2Bη
+
.
R
2ηG2
G
G

(20)

This inequality is always satisfied as long as η is small enough and ηG is large enough.
In particular, since B = 8Cg, g = R/λc, and C can be taken to equal 2R/c if λ = 1√
and 2R/(cλ(1 − λ)) as noted in Proposition 2, the inequality is satisfied if η = 1/ G and√√
G = 128R3 /c3 for λ = 1 and G = 128R3 /(c3 λ2 (1 − λ)) if λ < 1 (recalling that R > c),in which case each term on the right-hand side of (20) is less than c/4R with some strict
inequalities.

45

ONLINE APPENDIX
B

Proof of Theorem 1

To prove Theorem 1, we need to find, for any given ρ ∈ (0, 1] and λ ∈ (0, 1], some joint
distribution of (ω, S) that satisfies the condition of the theorem and some payoff functions
for which an informative equilibrium exist.
We first prove the theorem when λ = 1 and ρ = 1, which means that any agent who works
finds a signal with probability 1. For this case, it suffices to prove Proposition 1.

B.1

Proof of Proposition 1

We construct payoff functions for which the strategy profile described in the proposition constitutes an equilibrium. Under this strategy profile, as long as pi lies in (p, p̄) all agents work¯
and truthfully report their signal about ω. Moreover, given the symmetric signal structure,pi depends only on the number of “H” and “L” signals as long as all agents j < i work withprobability 1. Therefore, the set of equilibrium posteriors forms a grid {q k } containing p̂ andcontaining a single point on each side of (p, p̄). Let q 0 ≤ p < q 1 , . . . , p̂, . . . , q N < p̄ < q N +1¯¯
denote this grid. Along the candidate equilibrium, the belief pi evolves on this grid until ithits either q 0 or q N +1 , after which the investigation stops.
Let J denote the last investigator who works: we have pJ ∈ {q 1 , q N } and pJ+1 ∈ {q 0 , q N +1 }.
Also let p̃ = pJ+1 denote the value of the belief when learning stops under the candidate
equilibrium.
We construct utility functions in which an investigator’s compensation depends only on his
report and on the posterior p̃.
k
For any i such that pi = q k ∈ (p, p̄), if i reports “H”, he receives a reward RH≥ 0 if p̃ = q N +1¯
and a punishment PLk ≤ 0 if p̃ = q 0 . If i reports “L”, he gets RLk ≥ 0 if p̃ = q 0 and PLk ≤ 0 ifp̃ = q N +1 .
For any p, q on the grid, let π(p, q) denote the probability that the belief sequence ends withp̃ = q N +1 , i.e., exits (p, p̄) through p̄, from the perspective of an agent who assigns probability¯
p to ω but when the prior used by investigators is p0 = q. That is, π(p, q) is the probability
46

that an individual with prior p assigns to the sequence pi converging to q N +1 in equilibrium
given that the public belief, which serves as the state variable for the equilibrium, starts
with prior belief q.
If i sends report “H” starting from prior pi = q k , he assigns a probability π(q k , q k+1 ) to the
public belief converging to q N +1 . If i works and receives report “H”, his belief about thecontinuation equilibrium is π(q k+1 , q k+1 ). Similarly, if i sends “L”, his belief is π(q k , q k−1 )whereas if he works and reports “L” his belief is π(q k−1 , q k−1 ). It is straightforward to verify
the inequalities
π(q k+1 , q k+1 ) > π(q k , q k+1 )(21)

π(q k−1 , q k−1 ) < π(q k , q k−1 ),

(22)

and

for all k ∈ [2, N − 1]. The strictness of the inequalities comes from the fact that conditionalon the true state ω, the path of {pj }j≥i+1 starting any given value of pi+1 is strictly increasingin ω in FOSD, as is easily checked.57 Therefore, the probability of hitting q N +1 before q 0 isstrictly increasing in the belief pi that the state is high.
For k = 1, the investigation stops if i reports “L” so (22) holds as an equality, but (21) is still
strict, because this report triggers further investigation. The reverse is true for k = N : (21)only holds as an equality while (22) is strict.
If i shirks, his maximal utility iskmax{π(q k , q k+1 )RH+ (1 − π(q k , q k+1 ))PHk ; π(q k , q k−1 )PLk + (1 − π(q k , q k−1 ))RLk }.

(23)

The left argument is i’s expected payoff if he sends “H”, and the right one is his payoff if hesends “L”. Since i can send either message at no cost, his highest payoff from fabrication is
the maximum of these two terms. If i works, he getskz k [π(q k+1 , q k+1 )RH+ (1 − π(q k , q k+1 ))PHk ] + (1 − z k )[π(q k−1 , q k−1 )PLk + (1 − π(q k−1 ,k−1 ))RLk ]

(24)
k

k

where z is the probability of receiving signal “H” given belief q , and is equal to z k =Pr(“H 00 |q k ) = q k π + (1 − q k ) × (1 − π).
Working is optimal for i if (24) exceeds (23) by at least c.
57

There is a probability space in which the following property holds: if a working agent gets a high signal

when ω = L, then he must also get a high signal when ω = H. In this probability space, the sequence{pj }j≥i+1 is pointwise higher for ω = H than for ω = L.

47

This condition is obtained as follows: set PHk = PLk = −Q where Q is a strictly positive
k

k

k+1

k−1

π(q ,q),q)k
and RLk = Q 1−π(qThis guarantees that i’sconstant, and let RH= Q 1−π(qk ,q k−1 ) .
π(q k ,q k+1 )expected payoff from fabrication is zero, regardless of the outcome. From (21) and (22), his
payoff from working is of order Q and thus exceeds c, for Q high enough.58
If k = 1 or N , there is one signal that i can send after working which yields a payoff of orderQ, while the other signal yields 0. The signal associated with a positive payoff arises with aprobability that is bounded away from 0, since pi lies in (p, p̄).
¯Moreover this scheme is feasible as long as the maximal reward R and punishment −R
respectively exceed sup{Rθk : θ ∈ {L, H}, k ∈ {1, . . . , N }} and Q.

B.2

Proof of Theorem 1: General case

The argument of Section B extends easily when ρ and/or λ are less than 1.
With ρ < 1 and λ = 1, the informative equilibrium is identical to the one described in
Proposition 1 except that learning stops as soon as an agent fails to report evidence, inwhich case he gets a zero compensation. By construction of the equilibrium in Section B.1,
shirking and reporting that no evidence was found has the same value as fabricating anyother message and can thus be deterred. Since a working agent may find nothing, or thelearning process may be interrupted before the belief process exits (p, p̄), in which case the¯working agent receives 0, the rewards and punishments must be scaled up by 1/πρ (q k ), whereπρ (q k ) is the probability that the belief process exits (p, p̄) in equilibrium, given the current¯
belief q k , so that the expected compensation of a working agent still exceeds the cost c ofworking.
If λ < 1 and ρ = 1, a working agent may fail to find evidence even when there surely exists
some. In this case, we assume once more that the compensation is zero, which deters shirking
and reporting the empty message, and we scale up all rewards and punishments by 1/λ toincentive the agent to work, as in the previous paragraph. The belief process will surely exit(p, p̄) since the amount of evidence is unlimited (only individual agents may be unlucky and¯find nothing with probability 1 − λ).
The case in which both λ and ρ are less than 1 is a convex combination of the previous cases
58

To see this, let π̄ denote a strictly positive lower bound on all inequalities (21) and (22) over all k’s

whenever they hold strictly. Then, the gain from working is of order Qπ̄.

48

and addressed accordingly.

C

Remaining Proofs for Theorem 2

C.1

Proof of Lemma 1

Let Viw denote i’s expected gross utility if he works. Viw may be decomposed in terms of i’s
expected gross utility V̄i if he works and there exists some signal left to be found, and his
expected gross utility Viw,∅ if he works but there is no signal left to be found (Si = ∅):Viw = Fi1 V̄i + (1 − Fi1 )Viw,∅ .
Conditional on Si = ∅, i’s expected gross utility if he works is the same as his expected
gross utility Vis,∅ if he shirks and uses the same reporting strategy as he does after working
and finding nothing: conditional on i’s report (whatever it is), the distribution of reports by
subsequent agents is identical since there is no signal left to be found. Therefore, Viw,∅ = Vis,∅ .
Furthermore, we also have V̄i ≤ R since R is the maximum possible gross utility.
Therefore, i’s net utility Uiw from working, including the cost of working, satisfies
Uiw ≤ Fi1 R + (1 − Fi1 )Vis,∅ − c.
Similarly, i’s utility Uis from shirking satisfies
Uis ≥ Fi1 × 0 + (1 − Fi1 )Vis,∅ = (1 − Fi1 )Vis,∅where the inequality comes from the fact that 0 is a lower bound on i’s realized gross utility.
Comparing the previous two equations shows that shirking strictly dominates working if
Fi1 R − c < 0.

C.2

Proof of Lemma 2

For any sequence S 00 of signals, let ∆i (S 00 ) denote the probability that Si = S 00 conditional∅00
00
on report history mi−1
1 , and ∆i (S ) denote the probability that Si = S conditional on i

working and finding nothing. Bayesian updating implies that for any S 00 6= ∅:∆∅i (S 00 ) = ∆i (S 00 )(1 − λ)(1 − fi0 )(1 − λ) + fi0
49,and for S 00 = ∅:∆∅i (∅) = ∆i (∅)

1(1 − fi0 )(1 − λ) + fi0.

This implies that
λfi0 ∆i (S 00 )(1 − fi0 )(1 − λ) + fi0(25)

λ(1 − fi0 )∆i (∅).
(1 − fi0 )(1 − λ) + fi0(26)∆∅i (S 00 ) − ∆i (S 00 ) = −for any S 00 6= ∅, and∆∅i (∅) − ∆i (∅) =Let Vi (mi , S 00 ) denote i’s expected gross utility conditional on i producing evidence mi andon Si+1 = S 00 . Notice that mi1 = (m1i−1 , mi ) and Si+1 completely determine the distributionof reports {mj }j>i . Therefore, Vi (mi , S 00 ) is the same regardless of whether i has worked or
shirked. Agent i’s expected gross utility conditional on (i) working, (ii) finding no signal,and (iii) producing message mi , isViw (∅, mi ) =X

Vi (mi , S 00 )∆∅i (S 00 ),S 00 ∈S

whereas his expected gross utility if i shirks and sends message mi is
Vis (mi ) =X

Vi (mi , S 00 )∆i (S 00 )S 00 ∈S

because i has learned nothing from shirking and thus holds the same belief as his prior belief
at the beginning of round i. Combining these expressions, we get
Viw (∅, mi ) − Vis (mi ) =X

Vi (mi , S 00 )(∆∅i (S 00 ) − ∆i (S 00 )).

(27)

S 00 ∈S

Since Vi (mi , S 00 ) ∈ [0, R] for all mi and S 00 , combining (27) with (25) and (26) yields
Viw (∅, mi ) − Vis (mi ) ≤R∆i (∅)λ(1 − fi0 ).
(1 − fi0 )(1 − λ) + fi0

Since λ < 1, the denominator is bounded below by 1 − λ. Since ∆i (∅) = fi0 , the numerator
is bounded above by Rfi0 . This yields
Viw (∅, mi ) ≤ Vis (mi ) + fi0

RR≤ Vi∗ + fi0
,
1−λ
1−λ

which proves the lemma. Intuitively, this results means that if fi0 is negligible relative to(1 − λ), then i’s expected gross utility after working and finding nothing cannot be muchhigher than if i had shirked, because finding nothing in this case merely reveals that i was
unlucky and otherwise conveys little else information.
50

C.3

Proof of Lemma 3

For each mi ∈ M , let Viw (mi ) denote i’s expected gross utility conditional on working andsending message mi and Mi− denote the set of messages mi after which no j > i ever works,
so that M = Mi+ ∪ Mi− and Mi+ ∩ Mi− = ∅. Letting γi (M̃i ) denote the probability that isends a message in M̃i conditional on working and on mi−1
1 , we have:
Viw =

X

γi (mi )Viw (mi ) +mi ∈Mi−Xγi (mi )Viw (mi ).

(28)

mi ∈Mi+

For the first term, note that i’s expected utility conditional on reporting mi and on no j > iever producing real evidence does not depend on whether i worked or shirked: either way,
the distribution of the reports {mj }j>i is independent of the set of signals that remain in thecase. Letting, as in the previous lemma, Vis (mi ) denote i’s expected gross utility conditionalon shirking and sending message mi , we thus have Viw (mi ) = Vis (mi ) for all mi ∈ Mi− . SinceVi∗ = maxmi ∈M Vis (mi ), the first term in (28) is bounded above by γi (Mi− )Vi∗ .
For the second term, we have γi (mi ) = di (mi ) + gi (mi ) andγi (mi )Viw (mi ) ≤ di (mi )Viw (∅, mi ) + gi (mi )R,
where we used the fact that i’s expected gross utility conditional on working, finding a signal,
and reporting mi is bounded by R.
Combining these observations yields
Viw ≤ γi (Mi− )Vi∗ + gi (Mi+ )R +X

di (mi )Viw (∅, mi ).

(29)

mi ∈Mi+f 0R

i. Summing over all mi ∈ Mi+ , we get
If λ < 1, Lemma 2 implies that Viw (∅, mi ) ≤ Vi∗ + 1−λXdi (mi )Viw (∅, mi ) ≤ di (Mi+ )Vi∗ + di (Mi+ )mi ∈Mi+

fi0 R
.
1−λ

(30)

Since γi (Mi− ) + di (Mi+ ) ≤ γi (Mi− ) + γi (Mi+ ) = 1, combining (29) and (30) proves the lemma
when λ < 1.
If λ = 1, using in (29) the fact that Viw (∅, mi ) is bounded above by R directly proves thelemma.

51

C.4

Proof of Proposition 2

The right-hand side of (5) is increasing in C over the range of C that satisfy the condition
Fik − CFik+1 > 0. Therefore, if (5) is satisfied for any C such that Fik − CFik+1 > 0, it isalso satisfied for any C 0 ≥ C such that Fik − C 0 Fik+1 > 0. The proposition thus follows if weshow the inequality for C(λ).
First, we show that the claim holds if βi = 0. In this case, i must shirk with probability 1: if
not, Lemma 1 implies that Si is nonempty with positive probability and, hence, that βi > 0.
Since i shirks with probability 1, there is no belief update between rounds i and i + 1.
k
Therefore, Fik = Fi+1
(mi ) for any message mi that i sends in equilibrium. The right-handside of (5) is thus equal to zero and (5) is satisfied.
Now suppose that βi > 0 or, equivalently, that γi > 0: i works with positive probability. We
consider two cases, distinguished by the magnitude of the probability gi (Mi+ ) that i finds asignal and sends a message in Mi+ conditional on working and on history m1i−1 .
cCase 1: gi (Mi+ ) ≥ 2R
. By definition, we have

βi = γi λ(1 − fi0 ),which implies that βi ≤ γi , andβi (Mi+ ) = γi gi (Mi+ )where βi (M̃i ) is the probability that i discovers a signal and sends a message in M̃i . Since
gi (Mi+ ) ≥ c/2R, this implies thatβi ≤ γi ≤2R
βi (Mi+ ).
c(31)

Therefore, the desired
inequality (5) will
follow for C = 2R/c if we prove that βi (Mi+ ) is
bounded above byk (m ))1
Ei (Fik −Fi+1im ∈M +i

Fik −CFik+1i.

For each mi that i may send in equilibrium and k ≥ 1, Bayesian updating implies thatkFi+1(mi ) =Fik (αi (mi ) + γi (1 − λ)δi (mi )) + Φ(mi )αi (mi ) + γi ((1 − Fi1 ) + Fi1 (1 − λ))δi (mi ) + βi (mi )where the following probabilities are defined conditional on mi−11 : αi (M̃i ): probability that i shirks and sends a message in M̃i ;

52

(32)

 δi (M̃i ): probability that i sends a message in M̃i conditional on working and finding

no signal;59 Φ(mi ) is the probability that (i) i works, (ii) i discovers a signal, (iii) i sends reportmi , and (iv) there remain at least k signals at the beginning of round i + 1.
Let pi (mi ) denote the probability that i produces report mi conditional on mi−1
1 : pi (mi ) is
the denominator of (32). Rearranging (32) and simplifying, we have
Fik (βi (mi ) + γi (1 − Fi1 )λδi (mi )) = (Fik − Fik+1 (mi ))pi (mi ) + Φ(mi )Since γi (1 − Fi1 )λδi (mi ) ≥ 0, summing the previous equation over mi ∈ Mi+ yields
h
i
X

k
Φ(mi ).
Fik βi (Mi+ ) ≤ Ei Fik − Fi+1
(mi ) 1mi ∈Mi+ +

(33)

(34)

mi ∈Mi+hiSince Φ(mi ) = Ei 1|Si |≥k+1 1i works, discovers a signal, and reports mi , we haveXmi ∈Mi+hi
Ei 1|Si |≥k+1 1i works and reports mX

Φ(mi ) ≤imi ∈Mi+h= Ei 1|Si |≥k+1 1i works and reports m ∈ M +ii≤ Ei 1|Si |≥k+1 1i works= Fik+1 γi ,

i

(35)

noting, for the last equality, that the event that i works, which has probability γi , depends
only on mi−1
and is thus independent of the event {|Si | ≥ k + 1} conditional on m1i−1 .
1Combining this with (34) yields
Fik β(Mi+ ) ≤ Ei

hk
(Fik − Fi+1
(mi ))1mi ∈Mi+i

+ Fik+1 γi .

(36)

Since gi (Mi+ ) ≥ c/2R, we have β(Mi+ ) = γi gi (Mi+ ) ≥ γi c/2R. Inequality (36) then yields
β(Mi+ ) ≤1k
E [(Fik − Fi+1
(mi ))1mi ∈Mi+ ].
k+1 ikFi − 2R/cFi(37)

Combining this with (31) yields (5) for C(1) = 2R/c.60 Since C(λ) ≥ C(1) for all λ ∈ (0, 1],
the monotonicity noted at the beginning of the proof yields the desired conclusion for C(λ).
5960Note that δi (M̃i ) ≥ di (M̃i ), where di (M̃i ) was defined before Lemma 3.
k+1
Note that the proposition’s assumption that Fik − C(λ)Fik+1 > 0 implies that Fik − 2R
> 0 sincec Fi

C(λ) ≥ C(1) regardless of λ.

53

cCase 2: gi (Mi+ ) < 2R
. We prove that γi is bounded above by the right-hand side of (5) for

C = C(λ). Since γi ≥ βi , this will yield the desired conclusion.
Intuitively, in Case 2 the probability of discovering a signal that, together with i’s equilibrium
message strategy, triggers subsequent work is too low to incentivize i to work. The only way
of incentivizing i to work is therefore for him to signal by his message that he found nothing
through work. For this to happen, the probability fi0 that there remains no evidence mustbe high enough. We will use this fact to obtain a bound on γi .
From Lemma 3, if gi (Mi+ ) < c/2R, i’s utility from working is bounded above by
Uiw = Viw − c ≤ Vi∗ + di (Mi+ )

fi0 R
c
−
1−λ 2

if λ < 1, and by
Uiw ≤ Vi∗ + di (Mi+ )R −c
2

if λ = 1. Therefore, working is optimal only if di (Mi+ )fi0 ≥ c(1 − λ)/2R when λ < 1 andonly if di (Mi+ ) ≥ c/2R when λ = 1.
Summing (33) over Mi+ and using (35) and fi0 = 1 − Fi1 yields
i
h
+
+
k
k
0
k
k
+
Fi βi (Mi ) + γi Fi δi (Mi )fi λ ≤ Ei (Fi − Fi+1 (mi ))1mi ∈Mi + Fik+1 γi .

(38)

For λ < 1, we have di (Mi+ )fi0 ≥ c(1 − λ)/2R. Since δi (mi ) ≥ di (mi ) for all mi (by definitionof these variables) and Fik βi (Mi+ ) ≥ 0, (38) implies thatihkEi (Fik − Fi+1
(mi ))1mi ∈Mi+γi ≤.
Fik cλ(1 − λ)/2R − Fik+1
Multiplying the numerator and denominator by C(λ) yields the result.
For λ = 1, we have di (mi ) = δi (mi )fi0 for all mi and, hence, δi (Mi+ )fi0 λ = di (Mi+ ), which isgreater than c/2R as noted earlier. Therefore, (38) implies thathikEi (Fik − Fi+1
(mi ))1mi ∈Mi+.
γi ≤Fik (c/2R) − Fik+1
Multiplying the numerator and the denominator by C(1) yields the result.

54



C.5

Proof of Lemma 5(i) Let Si0 = {si : Fik,w (si ) > Fik }, and, for each si , let γi0 (si ) (resp. βi0 (si )) denote theprobability that i discovers si given that he works (resp., the probability that i works anddiscovers si ). Also let γi0 (si |k+1) (resp. βi0 (si |k+1)) denote the same probabilities conditional
on |Si | ≥ k + 1.
We have for all si
Fik,w (si ) =Fik+1 γi0 (si |k + 1)
Fik+1 βi0 (si |k + 1)=
γi0 (si )βi0 (si )where the second equality comes from βi0 (si ) = γi γi0 (si ) and βi0 (si |k + 1) = γi γi0 (si |k + 1).
Therefore, Fik,w (si ) > Fik only if βi0 (si ) < βi0 (si |k + 1)Fik+1 /Fik .
k+1 0
βi (si |k + 1) ≤ γi Fik+1 .
si ∈Si0 Fi
and finds a signal in Si0 satisfies

We haveP

βi0 (Si0 ) =X

Therefore, the probability βi0 (Si0 ) that i works

βi0 (si ) ≤ γi

si ∈Si0

Fik+1
.
Fik

Since βi0 (Si0 ) = γi γi0 (Si0 ), we have
γi0 (Si0 ) ≤

ε
Fik+1
≤
,
k
Fi
F̂ k
k

F̂
since Fik+1 ≤ ε and Fik ≥ F̂ k . From (6), the right-hand side is bounded above by 2G2.

For any mi , let q(mi ) denote the probability, conditional on i working and sending reportmi , that i has discovered a signal si ∈ Si0 , and let σ(si |mi ) denote the probability that i
discovered si given that he worked and reported mi . We also let si = ∅ denote the eventthat i did not find anything, σ(∅|mi ) denote the probability that i found nothing given that
he worked and reported mi , Fik,w (∅) denote the probability that there at least k signals
conditional on i working and finding nothing. We haveX
Fik,r (mi ) =σ(si |mi )Fik,w (si )si ∈Si=XX

σ(si |mi )Fik,w (si ) + σ(∅|mi )Fik,w (∅) +si ∈Si0σ(si |mi )Fik,w (si )si 6=∅,si ∈Si \Si0

By construction, Fik,w (si ) ≤ Fik for all si in the last term. Moreover, we have Fik,w (∅) ≤ Fik ,as is easily checked:61 intuitively, finding nothing always increases the probability that there61

F k (1−λ)1−λkFormally, for k ≥ 1, we have Fik,w (∅) = (1−F 1i)+F 1 (1−λ) = Fik (1−F 1 )+F
1 (1−λ) ≤ Fi .
i

i

55

i

i

are no signals remaining to be found. Finally, the first term is bounded above by σ(Si0 |mi ) =q(mi ). Therefore,
Fik,r (mi ) ≥ (1 + η)Fik ⇒ q(mi ) + (1 − q(mi ))Fik ≥ (1 + η)Fik⇒ q(mi ) ≥ ηFik .
To conclude, note thatXγi (mi )q(mi ) = Pr(si ∈ Si0 |m1i−1 , i works) ≤

mi

F̂ k
2G2

The left-hand side is bounded below by γ(Ni )ηFik . Since Fik ≥ F̂ k , this implies thatγ(Ni ) ≤

1
.
2ηG2

k
k
(ii) Fi+1(mi ) is a convex combination62 of Fik and Fik,r (mi ). This implies that Fi+1
(mi ) ≤/ Ni . From (14), this further implies that
(1 + η)Fik for all mi ∈πi+1 (mi ) ≤

2Cg
F̂ k

(1 + η)Fik − Ei+1 [FJk ]

for all mi ∈/ Ni .

C.6

Proof of Lemma 6

k+1(i) If mi ∈/ Ti , we have Fi+1≤√

Gε. Using this inequality in Lemma 4 instead of Fik+1 ≤ εand repeating the argument of that lemma applied to round i + 1, we conclude that i + 1√
assigns probability at least 1 − 1/ G to A whenever mi ∈/ Ti .
(ii) Let Fik+1,r (mi ) denote the probability that there are at least k + 1 signals left at the
k+1
beginning of round i + 1 conditional on i working and reporting mi . Fi+1
is a convex

combination of Fik+1 and Fik+1,r (mi ). This, together with the fact that Fik+1 ≤ ε and the
62

k
Fi+1
(mi ) is the probability that i + 1 assigns to there being at least k signals left upon observing mi .

If i + 1 knew that i didn’t work and simply sent message mi , this belief should be Fik since mi conveys no
additional information. And if i + 1 knew that i produced mi through working and then reporting mi , hiskupdated belief should be Fik,r (mi ). Since i + 1 doesn’t observe i’s action, in general Fi+1
(mi ) is a convex

combination of these two posteriors, where the weights corresponds to the probability assigned by i + 1 toi fabricating or working conditional on observing mi . This fact is straightforward to check using Bayesian
updating.

56

definition of Ti , shows that mi ∈ Ti only if Fik+1,r (mi ) ≥√Gε. Let Ti0 denote the set of

messages mi for which the last inequality holds. As noted, Ti ⊂ Ti0 .
Since i’s prior probability that there are at least k + 1 signals left is Fik+1 ≤ ε, the law of
iterated expectations implies that the probability F̄ik+1,r (mi ) that there were at least k + 1
signals left at the beginning of round i conditional on i working and finding mi must satisfy
X
Ei [F̄ k+1,r |working] =γi (mi )F̄ik+1,r (mi ) = Fik+1 ≤ ε.
i∈Mi√
Using Markov’s inequality, this implies that Pr(mi : F̄ik+1,r (mi ) ≥ Gε| i works) ≤ √εGε =√√1/ G. Since also F̄ik+1,r (mi ) ≥ Fik+1,r (mi ), we get γi (Ti0 ) ≤ 1/ G. Since Ti ⊂ Ti0 , this shows√that γi (Ti ) ≤ 1/ G.

D

Proof of Theorem 3

Without loss of generality, we assume once more that agents’ gross utility functions {Vi }i∈N
all take values in [0, R].
Conditional on the history up to round i , the number of signals discovered until round i
is a random variable whose support depends on mi−1
and the set of past witnesses. Let qi
1denote the lower bound of this support. Put differently, qi is the smallest number of signalsthat have been surely discovered by round i, and it is equal to the number of past witnesses
plus the number of past investigators whose equilibrium strategies and messages imply thatthey have surely discovered signals given the history up to round i. Let Qi denote the eventthat the number of signals discovered up to round i is exactly equal to qi , and F̂ik denotethe probability that |Si | ≥ k conditional on Qi and qi .
We observe that (i) qi is nondecreasing along any equilibrium path and is strictly increasing
whenever a witness arrives, (ii) given the construction of S, the distribution of Si conditionalon mi−1
and Qi is only a function of qi , and (iii) F̂1k = F1k for all k.
1
We will prove that there exist strictly positive thresholds {F k }k≥1 such that an informative¯
continuation equilibrium exists in round i only if F̂ik ≥ F k for all k ≥ 1. Applied to i = 1,¯
this result implies Theorem 3. The proof uses the following lemma.
Lemma 7 Under Assumption 1, the following inequalities hold: (i) Fik ≤ F̂ik for all i, k ≥ 1and (ii) Path by path, F̂jk ≤ F̂ik for all j ≥ i and k ≥ 1.
57

Proof.

Part (i): Let ri denote the number of signals discovered before round i. We haveri ≥ qi and
Fik =

X

Pr(ri = r | mi−1
1 ) Pr(K̃ ≥ r + k | K̃ ≥ r)r∈{qi ,...,i−1}≤

X

Pr(ri = r | mi−1
1 ) Pr(K̃ ≥ qi + k | K̃ ≥ qi )r∈{qi ,...,i−1}=XkPr(ri = r | mi−1
1 )F̂ir∈{qi ,...,i−1}= F̂ik .
The first equality comes from the independence of K̃ from S ∞ : ri is a sufficient statistic forK̃ given all the information produced before round i, and only to the extent that it reveals
that K̃ ≥ ri . The inequality comes from the increasing hazard rate condition, which implies
that for any k ≥ 0, Pr(K̃ ≥ k + q | K̃ ≥ q) is non-increasing in q.63 The second equality is
due to the equality F̂ik = Pr(K̃ ≥ k + qi | K̃ ≥ qi ), which again comes from the independenceof K̃ and S ∞ : the only relevant information about K̃ conditional on Qi is the number ofsignals qi discovered by round i.
Part (ii) For any j ≥ i, we have qj ≥ qi . As explained in the proof of Part (i), we have forall k ≥ 1
F̂jk = Pr(K̃ ≥ k + qj | K̃ ≥ qj )≤ Pr(K̃ ≥ k + qi | K̃ ≥ qi )= F̂ik ,where the inequality comes from the increasing hazard rate property (see Footnote 63). The existence of thresholds {F k }k≥1 in Theorem 3 is proved by induction on k. We start¯with the base case k = 1 and then prove the induction step.
63

See, e.g., Barlow et al. (1963, p. 379). In brief, a random variable X with distribution F has theincreasing hazard rate property if and only if the survival distribution F̄ = 1 − F is log-concave over its
domain (which could be discrete or continuous). This property implies, as is easily checked, that for any
p, q ≥ 0, Pr(X ≥ p + q | X ≥ p) = F̄ (p + q)/F̄ (p) is nonincreasing in p.

58

D.1

Proof for k = 1

Suppose that F̂i1 < c/2R. Combining the two parts of Lemma 7, this implies that Fj1 < c/2R
for all j ≥ i. Lemma 1 still applies: no investigator j ≥ i works because the probability that
he finds something is too small to justify the cost of effort, given the maximal reward R.
We now show that if F̂i1 lies below another threshold, smaller than c/2R, witnesses provide
no informative message, either.
We will use the following lemma:
Lemma 8 Consider L ≥ 2 pairwise independent random variables {Y` } with non-atomic
distributions over R and densities f` that are bounded above by f¯. For any ε ≥ 0, let E L
ε0denote the event that there exist `, ` ≤ L such that |Y` − Y`0 | ≤ ε. Then
Pr(EεL ) ≤ L(L − 1)f¯ε.
Proof. The result is proved by induction on L ≥ 2. For L = 2, we haveZZ
0¯Pr(|Y − Y | ≤ ε) =fY (x)FY 0 [x − ε, x + ε]dx ≤ 2εf
fY (x)dx = 2εf¯.
RR

Now suppose that the claim holds for L−1. Notice that the event EεL is the union of L events:
the event EεL−1 involvingh the first L − 1 random variables, and, for each ` ≤ L − 1, theevent E `,L that the Lth random variable lies within ε of the `th random variable. Therefore,Pr(EεL ) ≤ Pr(EεL−1 ) +X

Pr(|Y` − YL | ≤ ε)`≤L−1

≤ (L − 1)(L − 2)f¯ε + (L − 1) × 2εf¯
= L(L − 1)f¯ε,where the second inequality comes from the induction hypothesis and the fact that Pr(|Y` −YL | ≤ ε) ≤ 2f¯ε, as shown in the first step of the induction for L = 2.

If i is a witness, we will use the following notation:
 βi : probability that i produces an informative message given mi−1
1 ;
 Mi+ : set of messages mi that are followed by an informative continuation equilibrium; Pri (Mi+ ): probability that i produces a message in Mi+ given m1i−1 ;59 γi (mi ): probability that i sends mi given mi−1
1 ; γi (mi |si ): probability that i sends mi after observing si .

Lemma 9 There is a threshold F 1 ∈ (0, c/2R) such if i a witness, then βi > 0 only if¯
F̂i1 ≥ F 1 .
¯
Proof.

Recall that if i is a witness, his message mi is informative (in equilibrium) if itis statistically dependent of S conditional on m1i−1 . Given a vector  = ((mi ) : mi ∈ M )of shocks, say that i’s message is -informative if whenever i has the preference shock  as
defined by (4), there exist two signals si 6= s0i such that the equilibrium distributions of miconditional on i getting signals si versus s0i are different across these two signals.
The following observation is straightforward to prove.
Observation 1 i’s message is informative if and only if the set of preference shocks  forwhich i’s message is -informative has positive probability.
For any equilibrium and history up to some round i in which i is a witness, let νi denote theprobability that i’s preference shock i is such that i’s message is i -informative.
For any F < c/2R, let ν(F ) denote the supremum of νi over all witness rounds i of all
equilibria such that F̂i1 ≤ F . We will show that ν(F ) = 0 for all F below some strictly
positive threshold.
Consider such an equilibrium. For any witness round i and message mi , let z(mi ) denote theprobability that at least some j > i produces an informative message following message mi .
i’s expected utility if he receives signal si and sends message mi is given by
Ui (mi ; si ) = z(mi )Ei [Vi (m) | si , mi1 ] + (1 − z(mi ))V i (mi ) + i (mi )
¯

(39)

where
V i (mi ) = Ei [Vi (m) | mi1 , no j > i produces an informative message].
¯
Notice that V i (mi ) does not depend on the content of signal si since this signal is payoff¯
irrelevant whenever no j > i produces an informative message.
Given i , i sends an informative signal only if there exist mi 6= m0i and signals si 6= s0i suchthat
Ui (mi ; si ) ≥ Ui (m0i ; si )60

andUi (m0i ; s0i ) ≥ Ui (mi ; s0i )From (39) and the fact that Vi (m) ∈ [0, R], this is possible only if:|i (mi ) + V i (mi ) − i (m0i ) − V i (m0i )| ≤ R(z(mi ) + z(m0i )).
¯¯
The random variables Y` = i (m` ) + V i (m` ) satisfy the assumptions of Lemma 8. Letting¯|M | denote the cardinality of the message space M , we thus havePr(i sends an informative message) ≤ |M |2 f¯R

sup (z(mi ) + z(m0i ))(40)

mi ,m0i ∈M

Since no investigator j ≥ i works when F̂i1 < c/2R, we have for any message mi :Xz(mi ) ≤Pr(there are j witnesses in the sequence after round i)jν(F ).

(41)

j≥1

Indeed, by definition of ν(F ) and the fact that F̂j1 ≤ F for all j ≥ i, a witness provides
an informative signal with probability at most ν(F ). Thus jν(F ) is an upper bound on theprobability that at least one witness provides an informative signal given there are j such
witnesses.
The probability that at least j witnesses come after round i is bounded above by F j , where
j is an exponent (not a superscript): To show this for j = 1, notice that by Lemma 7,
1≤ F , and the probability that there is at least one witness after round i is bounded
F̂i+1
1
1
above by the probability Fi+1
that there is at least one more signal, which is less than F̂i+1

again by Lemma 7. For j = 2, note that conditional on the first witness arriving, Lemma 7
implies that the probability that a second witness arrives is again by bounded by F since
the probability that there remains another signal is bounded by F , and a witness can arise
only if such a signal exists. By induction, this shows that the probability of having at least
j witnesses and, hence, the probability of having exactly j witnesses, are bounded above byP
F j . Combining this with (41) and using the standard formula j≥1 jxj = x/(1 − x)2 for allx ∈ (0, 1), we get for all mi :z(mi ) ≤X

F j jν(F ) =j≥1

F ν(F ).
(1 − F )2

Combining this with (40), we obtain
Pr(i sends an informative message | F̂i1 ≤ F ) ≤ 2|M |2 f¯Rν(F )61F.
(1 − F )2(42)

Taking the supremum of the left-hand side over all witness rounds i and equilibria such thatF̂i1 ≤ F , we obtain
ν(F ) ≤ |M |2 f¯R

2F
ν(F ).
(1 − F )2

For 2F/(1 − F )2 < 1/|M |2 f¯R, this relation is possible only if ν(F ) = 0. Since the function
F 7→ F/(1 − F )2 is increasing on [0, 1) and starts at zero, we conclude that there exists a
threshold F 1 > 0 such that ν(F ) = 0 for all F ≤ F 1 .
¯
¯

D.2



Induction Step0

Suppose that there exist strictly positive thresholds {F k }k0 ∈{1,...,k} such that a continuation¯ 00equilibrium starting in round i is informative only if F̂ik ≥ F k for all k 0 ≤ k. We will¯
show that a similar condition holds for k + 1. The proof works by contradiction: we will
suppose that for all ε ∈ (0, 1), there exists an informative continuation equilibrium such thatF̂ik+1 ≤ ε and obtain an impossibility for ε small enough.
Consider any ε < F k × F 1 and any informative continuation equilibrium starting in round¯¯
i such that F̂ik+1 ≤ ε. Then, all continuation equilibria become uninformative as soon assome witness j ≥ i arrives. To see this, suppose that some witness arrives in round j ≥ i.
We have for any message mj sent by this witness:kF̂j+1(mj ) = Pr(K̃ ≥ k + qj + 1 | K̃ ≥ qj + 1)=

Pr(K̃ ≥ k + qj + 1)
Pr(K̃ ≥ qj + 1)

Pr(K̃ ≥ k + qi + 1)
Pr(K̃ ≥ qi + 1)
Pr(K̃ ≥ k + qi + 1)
Pr(K̃ ≥ qi )=×Pr(K̃ ≥ qi )Pr(K̃ ≥ qi + 1)≤=

F̂ik+1F̂i1where the inequality comes for the monotone hazard rate assumption (see Footnote 63) andthe fact that qj ≥ qi . Since the continuation equilibrium from round i is informative, wemust have F̂i1 ≥ F 1 . Therefore, F̂ik+1 ≤ ε < F k F 1 , which implies that
¯
¯ ¯
k
F̂j+1
< Fk
¯

62

and shows by induction hypothesis that all continuation equilibria are uninformative fromround j + 1 onwards.
Moreover, the first witness, j, knowing that continuation equilibria are uninformative regardless of his message, has no incentive to send an informative message.64
Therefore, if F̂ik+1 ≤ ε, the only agents who may send informative messages are the investigators arriving between round i and the arrival of the first witness. The situation is therefore
almost identical to the setting of Theorem 2, in the absence of witnesses, except that any
sequential learning activity is interrupted at the apparition of the first witness. We know
from Theorem 2 that such equilibria can be informative only if Fik+1 exceeds the (k + 1)th threshold given by Theorem 2, which we denote here F̃ k+1 . Since Fik+1 ≤ F̂ik+1 by Part (i) of
Lemma 7, we conclude, letting F k+1 = min{F̃ik+1 , F k F 1 }, that no informative continuation¯¯ ¯equilibrium exists in round i if F̂ik+1 ≤ F k+1 . This concludes the induction step.
¯EProof of Theorem 4

The proof is similar to the proof of Theorem 3. In particular, the preliminaries of Section D
and Lemma 7 still apply. Lemma 9 starts identically. Instead of z(mi ), we use ziw , which
denotes the probability that there exits some agent j > i who works conditional on thehistory up to round i and on the fact that analyst i works. We redefine ν(F ) to denote thesupremum, over all analyst rounds such that F̂i1 ≤ F , of the probability that i works. As in
Lemma 9, we haveziw ≤ F ν(F )/(1 − F )2 .

(43)

Consider i’s incentive to work: if i works, his expected utility equals
Uiw = ziw Viw + (1 − ziw )V i − ci¯(44)where Viw is i’s expected gross utility conditional on i working and some j > i also working,and V i is i’s expected gross utility conditional on i being the last person to work. If i shirks¯and chooses the message mi optimally, his expected utility is
Uis =

X

Ei [Vi (m)|mi1 ].

(45)

mi ∈M
64

Formally, the argument is similar to the proof of Lemma 9 except that here z(mi ) = 0 regardless of themessage.

63

Since Viw ≤ R, Uis ≥ V i and (recalling the normalization, used in previous proofs, that¯Vi (m) ≥ 0 for all m) Uis ≥ 0, we deduce from (44) and (45) thatUiw ≤ Uis + ziw R − ci .
Therefore, i works only if ci ≤ ziw R and the probability that i works at the beginning ofround i is bounded above by H(ziw R). Letting h̄ denote an upper bound on the density of
H near zero and using (43), we conclude that
Pr(i works) ≤ h̄RF ν(F )/(1 − F )2 .
Taking the supremum over all analysts i such that F̂i1 ≤ F , we obtain
ν(F ) ≤ h̄RF ν(F )/(1 − F )2 .
The rest of the proof is identical to Sections D.1 and D.2.

64