ISSN (Print) 0473-453X

Discussion Paper No. 1247

ISSN (Online) 2435-0982

OPTIMAL FEEDBACK DYNAMICSAGAINST FREE-RIDING IN
COLLECTIVE EXPERIMENTATION

Chia-Hui Chen
Hülya Eraslan
Junichiro Ishida
Takuro Yamashita

July 2024

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Optimal Feedback Dynamics Against Free-Riding in Collective
Experimentation
Chia-Hui Chen∗

Hülya Eraslan †Junichiro Ishida‡

Takuro Yamashita§
July 1, 2024

Abstract
We consider a dynamic model in which a principal decides what information to releaseabout a product of unknown quality (e.g., a vaccine) to incentivize agents to experimentwith the product. Assuming that the agents are long-lived and forward-looking, their
incentive to wait and see other agents’ experiences poses a significant obstacle to social
learning. We show that the optimal feedback mechanism to mitigate information freeriding takes a strikingly simple form: the principal recommends adoption as long as she
observes no bad news, but only with some probability; once she does not recommend atsome point, she stays silent forever after that. Our analysis suggests the optimality ofpremature termination, which in turn implies that: (i) false positives (termination in thegood state) are more acceptable than false negatives (continuation in the bad state); (ii)overly cautious mechanisms that are biased toward termination can be welfare-enhancing.

JEL: D82, D83
Keywords: strategic experimentation, information design, free-rider problem, social learning, COVID-19 pandemic.

∗

Institute of Economic Research, Kyoto University; chchen@kier.kyoto-u.ac.jp.
Department of Economics, Rice University, NBER and Institute of Social and Economics Research, Osaka
University; eraslan@rice.edu.
‡
Institute of Social and Economic Research, Osaka University; jishida@iser.osaka-u.ac.jp.
§
Osaka School of International Public Policy, Osaka University; yamashita.takuro.osipp@osaka-u.ac.jp.
†

1

Introduction

The COVID-19 pandemic presented an unprecedented challenge, compelling governments aroundthe world to navigate through uncertainties and introduce many new, often untested, measures
to cope with it. Consider the uptake of mRNA vaccines as an example. While this radicallynew technology had been known for years and considered safe among medical scientists, it
had never been widely used before the pandemic, and politicians and citizens alike were notentirely sure about its possible adverse consequences. Amid this uncertainty, a potential impediment to widespread adoption is citizens’ unwillingness to be among the first to try out
the new technology: Hamel et al. (2021) document that “31% of the public say that when an
FDA-approved vaccine for COVID-19 is available to them for free, they will ‘wait until it has
been available for a while to see how it is working for other people’ before getting vaccinatedthemselves.” Those who stand to gain less from the vaccine may choose to wait and see others’
experiences, but this kind of wait-and-see attitude slows down the adoption process, resulting
in a loss of surplus that can be substantial.
The presence of information free-riding is in fact ubiquitous in many instances of policy
experimentation that require the voluntary participation of citizens; examples include technology adoption in a variety of sectors such as healthcare, education, and manufacturing. In close
inspection, two aspects of this experimentation process are particularly noteworthy. First,
experimentation of this kind is often “feedback-based” in that uncertainty regarding the net
value of a new policy is revealed gradually through the feedback of the citizens who have embraced the experimentation. Second, it is also often “large-scale,” involving a large number ofcitizens who interact anonymously, such that they do not directly observe what others haveexperienced on the whole, e.g., the frequencies of severe side effects, and must instead rely on
the information released by central authorities. The first aspect makes incentivizing the citizens to experiment a central problem for the government, whereas the second aspect provides
a potentially viable tool to achieve this goal. Along with the fact that contingent monetarytransfers are often not feasible in this type of environment, these two aspects give rise to a newclass of information-design problems in which the government determines what information todisseminate to the public, with the aim of overcoming information free-riding and facilitatingsocial learning.
To address this issue, we consider a collective experimentation environment that consistsof a principal (e.g., a government) and a continuum of long-lived and forward-looking agents(e.g., citizens). The principal has a product to be consumed (or a technology to be adopted) by
the agents who are heterogeneous with respect to their valuations of the product. The quality1

of the product is initially unknown to anyone and gradually revealed via the arrival of news
whose arrival rate is proportional to the measure of agents who consume the product. However,reflecting the large-scale nature of the experimentation, news is observable only to the principal,
who therefore faces a problem of what information to disseminate to the agents. Formally, theprincipal designs and commits to a feedback mechanism that specifies the distribution of thesequence of beliefs subject to the Bayes plausibility. Given some feedback mechanism, eachagent independently decides when, if ever, to adopt the product based on his expectation of
how the belief evolves as dictated by the mechanism in place.
We start by analyzing a bad-news case with a benevolent principal, i.e., the case where
bad news (a “breakdown”) arrives only when the state is bad, and the principal maximizes thediscounted sum of the payoffs of all agents. We first observe that in the environment describedabove, the amount of adoption under full disclosure is generally insufficient compared to its firstbest level. This observation stems from the free-rider problem that has been well documented
in the strategic experimentation literature (Bolton and Harris, 1999; Keller, Rady and Cripps,2005): because information is a public good whose benefits are not fully internalized by theagents, each agent has an excessive incentive to wait and see the experimentation outcomes ofother agents. However, in large-scale experimentation where the agents have no direct accessto the outcomes, the extent of the free-rider problem depends crucially on what information
the principal disseminates to the agents. This point can be seen most clearly if we consider an
extreme measure of no disclosure: if the principal commits to revealing no information, there
is no point in waiting for news, and the free-rider problem dissipates completely. Of course,
revealing no information is never optimal because that would imply entirely giving up the gains
from additional information gleaned from the experimentation. The principal thus needs tostrike the right balance between exploiting the gains from the experimentation on one handand managing the free-rider incentive on the other.
The principal’s problem is potentially complicated because the set of mechanisms is large
in our dynamic context, with a myriad of choices available for the principal. Specifically,
the principal can choose not only what information to disclose in a given period but also
when to do so: for instance, she may withhold favorable information by garbling signals insome period and deliver it later with a delay. Despite this potential complication, we showthat the optimal feedback mechanism to mitigate information free-riding takes a strikingly
simple form: the principal recommends adoption as long as she observes no news, but onlywith some probability; once she stops recommending, she will never recommend again even
if she has observed no news. Owing to this result, the principal’s problem effectively reduces
to an optimal-stopping problem, and the resulting optimal mechanism can be characterized2

thoroughly by a sequence of the continuation probabilities conditional on having observed nonews.
This characterization result is a consequence of two observations, which we label as termination and caution for clarity. The termination property suggests that if the principal suspends
the adoption process for a period, she will never resume the process again, even if she hasreceived no bad news. This property alternatively implies that there is no strategic gain fromwithholding information to slow down adoption, and the optimal mechanism entails no delayed
information release, which enables us to substantially reduce the set of mechanisms we need toconsider. The caution property suggests that the optimal mechanism should exercise caution
by being fully revealing when the principal is relatively pessimistic while randomizing when
she is relatively optimistic. This latter result pertains to the fact that the value of information
varies across agents with different valuations: information that distinguishes small risk andno risk is crucial for agents with lower valuations but irrelevant for those with higher valuations. The optimal mechanism can mitigate the free-rider problem and achieve the optimal
outcome because it provides no useful information for agents with higher valuations, similar to
the no-disclosure policy, but still provides enough information to improve the welfare of thosewith lower valuations. Both of these properties are largely detail-free and robust to various
alterations to the underlying setup, as a consequence of which our main characterization result
holds under a more general objective function that accommodates a wider range of social goals(Section 6.1) as well as in an alternative good-news scenario where news (a “breakthrough”)
arrives only when the state is good (Section 6.2).
An important economic insight of our analysis is the optimality of premature termination,i.e., it is welfare-enhancing for the principal to occasionally terminate the adoption process evenwhen she has observed no bad news and is still relatively optimistic. This finding provides some
practical implications for how the government should structure its decision-making process—the process in which it collects and analyzes data to produce evidence and transforms it into a
recommendation. First, the optimality of premature termination suggests that false positives(termination in the good state) are more acceptable than false negatives (continuation in thebad state). This finding in turn implies that if the government can design its experiments todetermine the precision of the information-generating process, it should direct more resources
to minimizing false negatives while accepting false positives to some extent. Second, theoptimality of premature termination also points to a virtue of overly cautious mechanisms thatare excessively biased toward termination. It is worth noting that we obtain this conclusion
even though all the concerned parties are assumed to be risk-neutral.

3

Literature. Our model lies at the intersection of strategic experimentation (Bolton and Harris,
1999) and information design (Kamenica and Gentzkow, 2011). Our learning environmentbuilds on the exponential-bandits model of Keller et al. (2005).1 A crucial departure from thisstrand of literature is that we consider large-scale collective experimentation that involves alarge number of agents, where each individual agent cannot directly observe the experimentation outcomes of other agents and must instead rely on the information released by theprincipal. This aspect of our model draws a clear contrast to canonical models of strategic
experimentation that typically focus on a relatively small group of agents and assume that eachagent can directly observe the actions and/or outcomes of other agents (Keller et al., 2005;
Bonatti and Hörner, 2011). The fact that only the principal can observe the experimentation
outcomes gives her some leeway to control the agents’ belief formation process, amounting to
a new class of information-design problems as noted above.
Our analysis is in spirit most closely related to Kremer et al. (2014) and Che and Hörner(2018) in that they both explore ways to facilitate social learning, with emphasis on agents
as both consumers and generators of information. Kremer et al. (2014) consider a sociallearning environment à la Bikhchandani et al. (1992) and characterize the optimal disclosure
policy to mitigate information herding. Che and Hörner (2018) consider a similar feedbackbased information structure to ours in which information arrives at a rate proportional to theamount of adoption and study how a recommender system can improve the incentives for early
exploration. In both of these models, however, agents are assumed to be myopic and makeonce-and-for-all adoption decisions. By contrast, in our setting, agents are forward-looking andstrategically choose their timing of adoption. Within this framework, we focus on a different
incentive issue—free-riding stemming from the option to wait and see others’ experiences—and
explore the optimal feedback mechanism to alleviate this problem.
Several works explore how to incentivize experimentation via monetary transfers. To namea few, Manso (2011) analyzes a moral-hazard problem with unknown success probability andshows that the optimal incentive contract exhibits substantial tolerance for early failure. Halacet al. (2016) consider a model of long-term contracting with both moral hazard and adverse
selection and obtain a characterization of optimal menu contracts. Halac et al. (2017) consider both monetary rewards and information disclosure (about whether success had occurredor not) in a contest environment where multiple agents compete for a prize. We view ouranalysis as complementary to this strand of literature, where we focus exclusively on the useof information to motivate agents to take a risky alternative without monetary rewards. Our
1More precisely, our baseline model follows the bad-news (“breakdown”) specification of Keller and Rady(2015), although we later extend our analysis to the good-news (“breakthough”) case.

4

analysis is more applicable to situations where contingent monetary transfers are either unconventional or infeasible due to institutional factors, as is often the case in the context of policy
experimentation.
Finally, we also aim to contribute more broadly to the literature on dynamic information
design. There are several recent works that analyze information disclosure in dynamic environments where an agent makes stopping decisions (Au, 2015; Ely, 2017; Nikandrova and Pancs,
2018; Che and Mierendorff, 2019; Ely and Szydlowski, 2020; Orlov et al., 2020).2 There are two
notable differences from this strand of literature. First, while those previous works generally
focus on single-agent cases, we consider an environment where the principal attempts to persuade a continuum of agents who are heterogeneous with respect to their valuations. Second,and more importantly, the principal’s information structure in our setting is endogenous in thatthe precision of the information she acquires depends on the agents’ adoption decisions. This
aspect of our model stands in sharp contrast to the standard setting where the information
structure is exogenously given and fixed over time.

2

Illustrative example

We first provide a simple example with two periods and two agent types to highlight the mainideas. Suppose a government attempts to introduce a new vaccine to a continuum of citizens
with unit measure. Each citizen can take the vaccine in either period or can choose to abstain
from it altogether. The quality of the vaccine, which is initially unknown to anyone, can beeither good (ω = 0) or bad (ω = 1), where the common prior probability that the vaccine isbad is m ∈ (0, 1). The citizens are divided into two preference types, where a half of them
have low valuation (v = vℓ ) and the other half have high valuation (v = vh > vℓ ). If a citizen
with valuation v chooses to take the vaccine, his payoff is v − ω; if not, his payoff is normalized
to 0. We assume vh < 1 so that even the high-valuation citizens would not take the vaccine ifthey knew that the vaccine is bad. For simplicity, we assume that there is no time discounting
between the two periods.
The quality of the vaccine is partially revealed via the arrival of news at the end of period 1.
Here, we consider a bad-news scenario where news (e.g., the occurrence of severe side effects)
arrives only when the state is bad. We make two crucial assumptions regarding the information
structure. First, as news reflects the aggregate outcome, it is observable only to the government
but not to each citizen. Second, conditional on the state being bad, the likelihood of news
2
Ball (2023) and Correia da Silva and Yamashita (2024) consider information disclosure in repeated interactions.

5arriving depends on and is increasing in the measure of citizens who choose to take the vaccine.
Specifically, after a measure n of citizens take the vaccine in period 1, the government observes
bad news with probability p(n)ω, where p : [0, 1] → [0, 1] is some increasing function. Let µp2
denote the government’s belief (that the state is bad) in period 2. If bad news is observed,
the government learns that the vaccine is bad for sure, and hence µp2 = 1. If not, taking n as
given, the government’s belief is updated toµp2 = µ(n) :=m(1 − p(n)),
m(1 − p(n)) + 1 − mwhich is lower than the initial prior for any n > 0, i.e., “no news is good news.”
The government’s problem is to determine the distribution of period-2 beliefs of the citizens
subject to the Bayes plausibility, so as to maximize the sum of the payoffs of all citizens. Letµ2 denote the citizens’ period-2 belief that the vaccine quality is bad, i.e. ω = 1, based on theinformation released by the government. Since period 2 is the last period, the problem facedby the citizens in that period is straightforward: given some belief µ2 , a citizen with valuationv takes the vaccine if v ≥ µ2 . Taking this as given, in period 1, a citizen with valuation v takes
the vaccine ifv − m ≥ Eµ [max{v − µ, 0}].

(1)

Each citizen’s decision in period 1 depends on the continuation payoff, which in turn depends
on the distribution of period-2 beliefs. For illustration, assume 1 > vh > m > vℓ > µ(0.5), sothat the low-valuation citizens would never take the vaccine in period 1 but could be persuaded
to do so in period 2 if enough information were generated in period 1.
To illustrate the value of information design in this context, we begin with two extreme cases
as benchmarks. We first consider the full-disclosure policy where the government mechanicallyreveals everything it observes (so that µ2 = µp2 with probability 1). In this case, if a highvaluation citizen waits until period 2, he takes the vaccine in period 2 if and only if no news isobserved, which occurs with probability m(1 − p(n∗ )) + 1 − m. The expected payoff of taking
the vaccine in this contingency is vh − µ(n∗ ), where n∗ is the equilibrium measure of agents
who take the vaccine in period 1. Since the expected payoff of taking the vaccine in period 1
is vh − m, the IC constraint for the high-valuation citizens to take the vaccine in period 1 can

6be written as
vh − m ≥ [m(1 − p(n∗ )) + 1 − m](vh − µ(n∗ ))= [m(1 − p(n∗ )) + 1 − m]vh − m(1 − p(n∗ )),taking n∗ as given.3 It is easy to verify that the IC constraint does not hold for any n∗ > 0
for any high-valuation citizen, and therefore, it does not hold for any low-valuation citizen
either. As such, no one chooses to take the vaccine in period 1 under full disclosure. Thisis a manifestation of the free-rider problem, and as a consequence, the opportunity for social
learning is entirely lost, leading to a suboptimal outcome.4
In a sense, the full-disclosure policy fails because it is too informative, giving the highvaluation citizens an excessive incentive to wait for news. This implies that the optimal feedback mechanism must be somewhat more obscure than the full-disclosure policy. To illustrate
this point, we now turn to the opposite case of no disclosure where the government commits torevealing no additional information (so that µ2 = m with probability 1). In this case, there isclearly no point in waiting for news, and it is (weakly) optimal for the high-valuation citizens
to take the vaccine in period 1,5 resulting in the maximum amount of information that can begenerated in this environment. However, this no-disclosure policy is also suboptimal because
none of the low-valuation citizens could be induced to take the vaccine in period 2, givenµ2 = m. This argument suggests that the government must find a middle ground between thetwo extreme policies.
So, what is the optimal way to disclose information in this environment? To achieve theoptimal outcome, the government must induce (i) the high-valuation citizens to take the vaccine
in period 1 and (ii) the low-valuation citizens to do so in period 2 occasionally (in case no news
is observed). We have already observed that the no-disclosure policy achieves (i) but not (ii).
Note that the no-disclosure policy is effective for (i) because it provides no valuable information
to the high-valuation citizens in that their behavior does not depend on the information releasedin period 2. This reasoning suggests, however, that any mechanism that has this feature canachieve (i) as well. Specifically, we modify the no-disclosure mechanism by applying a meanpreserving spread to m and splitting it into µ(0.5) and vh . This modified mechanism gives
3

Throughout the analysis, we assume that an agent chooses to adopt if he is indifferent.
Although we consider a simple binary-type distribution and no time discounting, the observation that theamount of adoption is insufficient can be extended to any (continuous) type distribution and discount rate. See
Appendix B for a more formal argument.
5
The high-valuation citizens are indifferent between taking the vaccine and waiting for news because of notime discounting. The incentive can be made strict if they discount future payoffs even slightly.
47

period-2 payoff

period-2 belief

Figure 1: The dashed line at the top represents the value function of the high-valuation citizens,and the dotted line at the bottom represents the value function of the low-valuation citizens. A
mean-preserving spread of m into µ(0.5) and vh raises the expected payoff of the low-valuation
citizens while keeping the expected payoff of the high-valuation citizens constant.
the high-valuation citizens the same continuation payoff and hence continues to achieve (i)because the support of the belief distribution lies entirely to the left of their indifference point
vh ; in other words, the extra information provided by the mean-preserving spread is irrelevant
for the high-valuation citizens. It is, however, beneficial for the low-valuation citizens because
the support spans over their indifference point µ2 = vℓ , allowing them to make more informed
decisions in period 2. Figure 1 graphically illustrates this situation.
This argument suggests that the government must provide the most accurate information,i.e., µ2 = µ(0.5), whenever it chooses to continue (recommending the low-valuation citizens to
adopt) in period 2. This in turn implies that the government must surely terminate if it has
observed bad news. If it has observed no news, it may continue with some probability less than
1 (given that full disclosure is not optimal in this example). To sum up, the government canalleviate information free-riding and achieve the optimal outcome by the following mechanism:• If the government observes bad news, it terminates with probability 1;• If the government observes no news, it continues with some probability β ∈ [0, 1] andterminates with the remaining probability.
This proposed mechanism is characterized by the continuation probability β, which measures8the degree of transparency with β = 1 (β = 0) corresponding to full (no) disclosure. It is fully
revealing when the government is pessimistic (has observed bad news) but randomizes when
it is optimistic (no news). The reason why such a mechanism works pertains to the fact thatthe value of information differs across different preference types: more precise information at
the low end of the belief distribution is valuable for the low-valuation citizens but is irrelevant
for the high-valuation agents. Since the government must persuade the high-valuation citizens
to adopt early to generate the information necessary to persuade the low-valuation citizens,a mechanism that is precise at the low end but obscure at the high end works to achieve theoptimal outcome. Remarkably, this observation can be extended to any (continuous) type
distribution and any number of periods as we will detail below, even though the problem
becomes substantially more complicated.
This illustrative example is simple enough to explicitly obtain the optimal mechanism. We
now consider an arbitrary discount factor δ ∈ (0, 1] and let p(n) = 1 − e−λn for concreteness.
Since the low-valuation citizens always benefit from more information, the optimal mechanism
is such that it provides the most accurate information to the extent that it does not affectthe incentive of the high-valuation citizens. This implies that the high-valuation citizens mustbe held indifferent between adopting and waiting. If we let β ∗ be the optimal continuation
probability, it must solvevh − m = δ[β ∗ (me−0.5λ + 1 − m)vh − me−0.5λ ].
Thus, the optimal continuation probability is given by∗

β = min
vh − m(1 − δe−0.5λ ),1 .
δ(me−0.5λ + 1 − m)vh

Some useful observations can be drawn from this.
1. A higher δ (more patience) leads to less transparency.
2. A lower vh leads to less transparency.
3. An increase in λ (more efficient feedback) leads to less transparency.
4. A higher m (more pessimistic outlook) leads to less transparency.
As is intuitively clear, anything that aggravates the free-rider problem—more patience, lowvaluations, more efficient feedback, and a more pessimistic outlook—makes the optimal mechanism less transparent (a lower β ∗ ). Whenever we have β ∗ < 1, although lowering the con9

tinuation probability in the optimistic state necessarily entails a loss of surplus, the gain fromalleviating information free-riding is more than enough to compensate for the loss.

3

Model

Environment. Consider the same experimentation environment in which a principal (e.g., agovernment) has a product to be consumed by a continuum of agents (e.g., citizens). Time isdiscrete and denoted by t = 1, . . . , T where T is the terminal period. In each period t, eachagent decides whether to “adopt” the product or “wait” for news; if an agent chooses to adopt,
he immediately leaves the game. The cost of adopting the product is determined by the stateof nature ω ∈ {0, 1}, which is initially unknown to anyone, where ω = 0 denotes the good(low-cost) state while ω = 1 denotes the bad (high-cost) state. The common prior of the state
being bad is m ∈ (0, 1).
Information. The true state is gradually revealed via the arrival of a signal (news). We
primarily consider a bad-news (“breakdown”) case where a signal arrives only when the stateis bad; in Section 6.2, we extend the analysis and consider the opposite case where a signal (ora “breakthrough”) arrives only when the state is good.6 Let st ∈ {b, ∅} be the signal realized
at the end of period t, where st = b indicates bad news while st = ∅ indicates no news. The
probability of receiving signal b at the end of period when the state is ω and the measure ofagents who adopt in period t is nt is given by
P[st = b | ω, nt ] = 1 − e−λωnt .
This suggests “no news is good news.” We assume that only the principal can observe signals
while the agents do not. At the beginning of each period t, the principal updates her belief(that the state is bad) upon observing st−1 which we denote by µpt with µp1 = m.
Mechanism. The principal designs and commits to a feedback mechanism P at the outset of
the game. The mechanism specifies the distribution of the sequence of beliefs µt subject to theBayes plausibility and her own information structure, where µt denotes the agents’ period-t
belief. The public history of the game at the beginning of period t can be summarized by thesequence of realized beliefs up to that point, which we denote by µt−1 := (µ1 , . . . , µt−1 ). The
6

We focus on the bad-news case because it is more relevant in many instances of policy experimentation.
For instance, in the context of vaccine uptake, it is more likely that the government would receive feedback
when the vaccine is of bad quality.

10

mechanism can then be written as P = (Pt (· | µt−1 ))Tt=1 , where Pt (· | µt−1 ) is the probability
mass function of period-t beliefs conditional on the history of the game. Let Mt (µt−1 ) denote thePsupport of Pt (· | µt−1 ), i.e., Mt (µt−1 ) is the smallest set M such that µt ∈M P (µt | µt−1 ) = 1.
Payoffs. Each agent is characterized by his valuation v ∈ [0, 1], which is distributed according
to some distribution F with full support over [0, 1]. Let f denote the corresponding density. If
an agent with valuation v (hereafter, simply agent v) adopts in period t, he earns v − ωc andleaves the game; if not, his payoff for the period is normalized to 0. Throughout the analysis,
we maintain the following assumption to focus our attention on relevant cases.
Assumption 1 c ≥ 1 > mc.
The first inequality ensures that learning is essential for all agents,7 while the second inequality
rules out the trivial case where no agents adopt. The agents maximize the discounted sum of
expected payoffs with a discount factor δ ∈ (0, 1). For now, we assume that the principal isbenevolent and maximizes the discounted sum of the payoffs of all agents, which we call the
total surplus for clarity, with the same discount factor δ; in Section 6.1, we extend the analysis
to incorporate a more general objective function that can capture other factors such as payoff
externalities and the welfare of third parties (e.g., firm profit).

4

Analysis

In the two-period example, there is only one possible history, which is µ1 = m, and any
disclosure mechanism is entirely summarized by a single distribution of period-2 beliefs P2 (·; m).
By contrast, when there are more than two periods, the problem becomes substantially more
complicated because the set of possible histories could expand exponentially over time. In thegeneral setting, we need to consider how the current distribution of beliefs potentially affects
their future evolution in the continuation game, thereby giving rise to a much wider array of
feedback mechanisms.

4.1

Preliminaries

Before we proceed to the general analysis, we first provide a more detailed account of three key
constraints in this general setup: IC constraint, Bayes plausibility, and consistency. Let vt∗ be
7

If c < 1, it is optimal for agents v ∈ [c, 1] to adopt regardless of the state. The optimal decision for thoseagents is hence trivial in that they always adopt immediately in period 1 in any mechanism.

11the threshold type who is indifferent between adopting and waiting in period t and let v0∗ = 1.
Once a mechanism P is given, we can pin down a non-increasing sequence of thresholds up toperiod t for any given history µt . More precisely, fix a mechanism P and let Hts be the set of∗∗under P . Then, for a< vt∗ = vt+s−1
all subsequences µt,t+s := (µt+1 , . . . , µt+s ) such that vt+s
given history µt and the corresponding sequence of thresholds up to period t − 1, the thresholdtype in period t is determined byvt∗ − µt c =

T −t
X

X

δ s Pt+1 (µt+1 |µt ) · · · Pt+s (µt+s |µt+s−1 )(vt∗ − µt+s c),(2)

s=1 µt,t+s ∈Hts∗∗
if a solution vt∗ ∈ [0, vt−1
) exists. If no solution exists in [0, vt−1), no agents are induced to∗∗adopt in period t, and we let vt = vt−1 . In what follows, we call (2) the IC constraint forclarity.
The principal’s problem is restricted by two conditions in our setting. First, as usual,
the Bayes plausibility requires that the expectation of posteriors must coincide with the priorconditional on µt−1 , i.e.,Xµt Pt (µt | µt−1 ) = µt−1 .

(3)µt ∈Mt (µt−1 )Second, any feasible mechanism must also be consistent with the principal’s private information. Specifically, at any point in time, the principal is in either one of the two information
states: pessimistic (has observed bad news) or optimistic (has observed no news). In thebad-news case, the principal’s belief is always µpt = 1 in the pessimistic state, and∗∗µpt = µ(vt−1) :=me−λ(1−F (vt−1 ))∗me−λ(1−F (vt−1 )) + 1 − min the optimistic state. Since the principal mixes these two information states to generate a∗belief, any feasible belief must be bounded between µ(vt−1) and 1. For later use, we write thisconsistency restriction more generally as∗∗), µ(vt−1)],
Mt (µt−1 ) ⊂ [µ(vt−1

(4)

∗
∗
where µ(vt−1) = 1 for all vt−1
∈ [0, 1) in the bad-news case.88

We use this slightly more general notation to make it applicable to the good-news case we consider later.

12

4.2

Main result

The principal’s problem is to determine the distribution of the sequence of beliefs to maximize
the expected total surplus subject to the three constraints noted above. Define Wt (µt , x, y) as
the principal’s continuation payoff (or simply the continuation surplus) in period t when the
belief is µt and agents v ∈ [x, y) adopt in period t. Given some mechanism P , the continuation
surplus is given by∗
Wt (µt , vt∗ , vt−1)=

Z vt−1∗(v − µt c)dF (v)
vt∗

+

T −t
X

X

∗
δ s Pt+1 (µt+1 |µt ) · · · Pt+s (µt+s |µt+s−1 )Wt+s (µt+s , vt+s
, vt∗ ).

s=1 µt,t+s ∈Hts

The principal’s problem is then defined as
max W1 (m, v1∗ , 1),P

subject to the Bayes plausibility (3) and consistency (4), where (vt∗ )Tt=1 is a non-increasing∗
sequence determined by the IC constraint (2) whenever vt∗ < vt−1
. Recall that since the principal has no additional information over and above the prior in period 1, the Bayes plausibility
requires P1 (m) = 1. Our focus is therefore on period 2 and beyond.
A few more notations would be helpful in formally stating the main result. First, for t ≥ 2,
we divide the support Mt (µt−1 ) into two disjoint sets MtA (µt−1 ) and MtN (µt−1 ), where∗
MtA (µt−1 ) := {µt ∈ Mt (µt−1 ) : vt∗ < vt−1},∗
MtN (µt−1 ) := {µt ∈ Mt (µt−1 ) : vt∗ = vt−1},

with M1 (µ0 ) = M1A (µ0 ) = {m}. We say that the game is in the adoption phase in period t
if µt ∈ MtA (µt−1 ) and in the no-adoption phase in period t if µt ∈ MtN (µt−1 ). Also, for thesubsequent analysis, let
∗

∗

∗
∗
∗
)e−λ(F (vt−1 )−F (vt )) + 1 − µ(vt−1)
q(vt∗ , vt−1
) := µ(vt−1denote the probability of receiving no news in period t conditional on being optimistic. Given∗∗
this, for any β ∈ [0, 1], let µ̃(vt−1, vt−2; β) denote the belief that satisfies the Bayes plausibility∗∗∗
in period t when the mechanism assigns µt = µ(vt−1) with probability βq(vt−1, vt−2
) and

13

∗
∗
∗
µt = µ̃(vt−1, vt−2; β) with the remaining probability conditional on µt−1 = µ(vt−2). Formally,
it is given by the solution to the following equation:
∗
∗
∗
∗
∗
∗
∗
∗
; β) = µ(vt−2, vt−2)µ(vt−1))µ̃(vt−1, vt−2, vt−2βq(vt−1) + (1 − βq(vt−1).

Definition 1 A mechanism P is a binary-message termination mechanism (or, simply, abinary-message mechanism) if for each t = 2, . . . , T ,A(a) for any µt−1 ∈ Mt−1(µt−2 ), there is βt ∈ [0, 1] such thatβ q(v ∗ , v ∗ ) t t−1 t−2∗∗Pt (µt | µt−1 ) = 1 − βt q(vt−1), vt−20∗
if µt = µ(vt−1),∗∗; βt ),, vt−2
if µt = µ̃(vt−1otherwise,N(b) for any µt−1 ∈ Mt−1(µt−2 ), the mechanism assigns all the probability on µt−1 , i.e.,1 if µ = µ ,
t
t−1
t−1
Pt (µt | µ ) =0 otherwise,
where (vt∗ )Tt=1 is the induced sequence of thresholds when the game stays in the adoption phase
for all t.9
The following facts are useful for interpreting binary-message mechanisms. Throughout the
paper, all the proofs are relegated to Appendix A.
Lemma 1 In a binary-message termination mechanism,∗∗∗∗(i) µ(vt−1) ∈ MtA (µt−1 ) and µ̃(vt−1, vt−2; βt ) ∈ MtN (µt−1 ) if µt−1 = µ(vt−2) ∈ MtA (µt−2 ),N(ii) µt ∈ MtN (µt−1 ) if µt = µt−1 ∈ Mt−1(µt−2 ),

where (vt∗ )Tt=1 is the induced sequence of thresholds when the game stays in the adoption phase
for all t.
Along with Lemma 1, Definition 1 provides two essential properties, which we call termination and caution for clarity. First, part (b) of the definition states the termination property
9Under Definition 1, there is a unique sequence of thresholds as long as the game stays in the adoption
phase.

14

where if the game is in the no-adoption phase for a period, it will never revert back to the adoption phase; as such, µt ∈ MtN (µt−1 ) indicates permanent termination of the adoption process.
∗Second, part (a) requires the principal to provide the most accurate information µt = µ(vt−1)
with some probability in period t if the game is in the adoption phase in period t − 1. This alternatively implies that the principal must be optimistic whenever the game is in the adoption
phase.10 Given this, when the game is in the adoption phase in period t − 1 (implying thatthe principal is optimistic in period t − 1), she remains optimistic (by observing no news) with∗∗), in which case she randomizes between continuation (with probability, vt−2
probability q(vt−1βt ) and termination (with probability 1 − βt ). With the remaining probability, the principal
turns pessimistic (by observing bad news), in which case she surely terminates the adoption
process. We refer to this as the caution property, because such a mechanism is biased towards
termination in that it terminates surely in the pessimistic state but may also terminate withsome probability in the optimistic state (where the principal has received no bad news).
Any binary-message mechanism is characterized solely by a sequence (βt )Tt=1 , where each βt
admits an economically meaningful interpretation: βt = 1 corresponds to full disclosure while
βt = 0 corresponds to no disclosure. The following statement is the main characterization
result of this paper.
Theorem 1 There exists an optimal mechanism that is a binary-message termination mechanism.

4.3

Sketch of the proof

The proof of Theorem 1 is lengthy, and we relegate its technical details to Appendix A. Below, we provide a sketch of the proof to summarize the key steps towards establishing ourcharacterization result and to build intuition for it.
The proof proceeds in two steps. The first step is to show that there is no gain fromadopting a more complicated “stop-and-go” mechanism that switches back and forth between
the adoption phase and the no-adoption phase. This can be seen most clearly by inspecting athree-period example. Consider a mechanism P that induces the lowest possible belief, which
we denote by µ, with probability β2 q(v1∗ , 1) in period 2 and the belief µ̃(v1∗ , 1; β2 ) leading to no
adoption with the remaining probability. Suppose further that in the history with v1∗ = v2∗ (noadoption in period 2), P induces the lowest possible belief µ3 = µ with probability γ3 q(v1∗ , 1)
10

To realize the lowest possible belief, the principal must send a fully separating message in the optimistic∗
state. The conclusion then follows since µt ∈ MtA (µt−1 ) if and only if µt = µ(vt−1) by Lemma 1 .

15

in period 3. The total surplus attained by this mechanism is then given by
W1 (m, v1∗ , 1) =

Z 1
v1∗

(v − mc)dF (v) + β2 q(v1∗ , 1)δW2 (µ, v2∗ , v1∗ )+ (1 − β2 q(v1∗ , 1))γ3 q(v1∗ , 1)δ 2 W3 (µ, v3∗ , v1∗ ),where the IC constraint in period 1 isv1∗ − mc = β2 q(v1∗ , 1)δ(v1∗ − µc) + (1 − β2 q(v1∗ , 1))γ3 q(v1∗ , 1)δ 2 (v1∗ − µc).
On one hand, the IC constraint is unaffected as long as β2 q(v1∗ , 1) + (1 − β2 q(v1∗ , 1))γ3 q(v1∗ , 1)δ
is fixed. On the other hand, we have W2 (µ, v2∗ , v1∗ ) ≥ W3 (µ, v3∗ , v1∗ ) because in period 2, theprincipal could always replicate what she would do in period 3 (by providing no informationin period 3). This means that we can improve the original mechanism by reducing γ3 andincreasing β2 in a way to keep β2 q(v1∗ , 1) + (1 − β2 q(v1∗ , 1))γ3 q(v1∗ , 1)δ constant. In the end,
this implies γ3 = 0, i.e., it is optimal to front-load all the information. This argument directly
implies the termination property that substantially reduces the class of mechanisms we needto consider.
Given this, the next step is to show that we can always improve the total surplus by applying
a mean-preserving spread to any interior belief. Intuitively, such an operation is beneficial forthe agents because their value functions are convex; it is in fact relatively straightforward
to establish this when there are only two periods. However, when there are more than two
periods, the situation becomes more complicated because the game continues after the period
in which we apply a mean-preserving spread. To illustrate this point, suppose we start fromv∗∗
some interior belief µt ∈ (µ(vt−1), t−1) at which a positive measure of agents adopt in periodct < T . To improve the payoffs of those agents by providing more information, their behavior
must be contingent on the new information. This alternatively implies that we must split
the belief µt into two outer beliefs such that they choose to adopt at the lower belief but
not to adopt at the higher belief. While this operation clearly benefits those agents who are
induced to adopt in period t, it necessarily lowers the continuation probability (more frequenttermination), which could adversely affect the remaining agents who may adopt in some future
periods.
Despite this complication, we can still show that the benefit of more precise information
always dominates the cost of more frequent termination. Specifically, we establish this resultv∗∗
by considering a mean-preserving spread that splits an interior belief µt ∈ (µ(vt−1), t−1
) for
c

16

∗
t ≥ 2 into a slightly lower belief µt − ε and the highest possible one µ(vt−1) (which equals 1 for∗11
any vt−1 < 1) and showing that it is Pareto-improving. This observation directly implies thatthe principal must provide the most accurate information in the adoption phase by terminating
surely in the pessimistic state while randomizing in the optimistic state. The reason why thiscaution property holds is the same as in the two-period example: more precise information at
the low end of the belief distribution helps agents with lower valuations make more informed
decisions but is irrelevant for those with higher valuations. The principal can exploit this fact
to alleviate the free-rider problem and achieve the optimal outcome.
In any period t, the principal is either pessimistic with µpt = 1 or optimistic with µpt =∗). The caution property suggests that whenever the game is in the adoption phase,µ(vt−1∗
it must induce the lowest possible belief, i.e., MtA (µt−1 ) = {µ(vt−1)}, which implies that the
principal surely terminates in the pessimistic state. When the game switches to the no-adoption
phase, the mechanism can specify any beliefs resulting in no adoption, but the termination
property suggests that we can merge all these beliefs into a single one without loss of generality,∗∗; βt )}. The resulting mechanism uses only two messages, one, vt−2
i.e., MtN (µt−1 ) = {µ̃(vt−1
indicating the continuation of the adoption process and the other indicating its permanenttermination, which amounts to a simple binary-message termination mechanism.

4.4

When information design matters

Given Theorem 1, the principal’s problem can now be substantially simplified. Note that the
principal’s objective function can be written as
W1 (m, v1∗ , 1) =

T
X
t=1

δ t−1 Bt me∗ ))−λ(1−F (vt−1

Z vt−1∗

Z vt−1∗(v − c)dF (v) + (1 − m)vt∗!
vdF (v) ,vt∗∗
where Bt := Πts=1 βs . Let r(vt∗ , vt−1
) denote the joint probability of state being bad and receiving
no news in period t conditional on being optimistic. Formally,
∗

∗

∗
∗
r(vt∗ , vt−1
) := µ(vt−1)e−λ(F (vt−1 )−F (vt )) .

In the proof, we exploit the fact that the highest possible belief is 1 (for any vt∗ < 1) in the bad-news case,at which point there is no potential loss of surplus (because the state is bad for sure). As it turns out, though,this property also holds in the good-news case where the highest possible belief is strictly less than 1. We
expand on this point in Section 6.2.
11

17

Then, the IC constraint in each period t is simplified to∗∗∗vt∗ − µ(vt−1)c = δβt+1 (q(vt∗ , vt−1
)vt∗ − r(vt∗ , vt−1
)c).

(5)

From these, the principal’s problem is redefined as
max W1 (m, v1∗ , 1),(βt )T
t=1

subject to (5).
Observe that the principal has no private information and has no choice but to set P1 (m) =∗and also vt∗ , . . . , vT∗ through its
1, which implies β1 = 1. For t ≥ 2, a change in βt affects vt−1∗
. Taking derivative with respect to βt yields
effect on vt−1δ

1−t ∂W1

∂βt

=

T
X

δ

s−t

s=t

+

T
X
s=t

δ

Z vs−1
∗
∗
∂vs−1
∗
∗
(v − c)dF (v)Bs λr(vs−1 , 1)f (vs−1 )∂βt vs∗s−t Bs+1

βt∗q(vs−1, 1)

Z vs−1∗vs∗∗(v − µ(vs−1)c)dF (v).

(6)

Notice that the second term in this expression is always positive. Thus, if there in an interior
solution for the optimal βt , then the first term must be negative. This illustrates the key
trade-off faced by the principal: lower transparency, i.e. a lower βt , makes the agents more
willing to experiment now (as captured by the first term) but leads to more distortion andmore premature termination (as captured by the second term). It is easy to see that the firstterm of (6) disappears as βt → 0 while the second term is unaffected. Therefore, it is never
optimal to set βt = 0 (no disclosure) for any t.
By contrast, there may be cases where it is optimal to fully disclose all the information
by setting βt = 1, despite the fact that the amount of adoption is generally insufficient due
to the free-rider problem. This is because the principal must respect the IC constraint and,in some cases, cannot do any better than fully disclosing all the information; formally, thisoccurs when we have a corner solution for some period (i.e., βt = 1 for some t ≥ 2), in whichcase information design is of no use for that period. To illustrate this possibility, consider
the problem of choosing βT . Given that vT∗ = µ(vT∗ −1 )c, the derivative with respect to βT is

18

obtained asδZ ∗∂vT∗ −1 vT −1∗∗= βT λr(vT −1 , 1)f (vT −1 )(v − c)dF (v)∂βT∂βT µ(vT∗ −1 )c

1−T ∂W1+ q(vT∗ −1 , 1)

Z vT∗ −1
∗
µ(vT
−1 )c

(v − µ(vT∗ −1 )c)dF (v)(7)

where, by (5),δ(q(vT∗ −1 , vT∗ −2 )vT∗ −1 − r(vT∗ −1 , vT∗ −2 )c)∂vT∗ −1=.
∂βT
1 − δβT [λr(vT∗ −1 , vT∗ −2 )(vT∗ −1 − c) + q(vT∗ −1 , vT∗ −2 )]∂v ∗Note that ∂βT T−1 → 0 as δ → 0, and hence the first term of the left-hand side of (7) disappears.
Since the second term is always strictly positive, we end up with a corner solution when δis sufficiently small. Intuitively, when the agents are almost myopic, there is no information
free-riding problem, and therefore, there is no gain from lowering transparency.
As we have shown, no disclosure is never optimal but full disclosure can sometimes be
optimal. Against this backdrop, an economic question of practical importance is when full
disclosure is not optimal. As one might expect from our discussions so far, full disclosure is not
optimal when the agents are more forward-looking because they wait more patiently for news,
thereby aggravating the free-rider problem. To see this, suppose the principal sets βt+1 = 1 forsome t and let δ → 1. The period-t IC constraint then converges to∗∗∗)c = q(vt∗ , vt−1
)vt∗ − r(vt∗ , vt−1)c.
vt∗ − µ(vt−1(8)∗,12 i.e., no agents
Provided that c > vt∗ , the only solution that can satisfy this is vt∗ = vt−1
adopt in period t, resulting in an extreme form of procrastination that produces no useful
information along the way. We can then show that setting βt+1 = 1 is not optimal for thesame reason employed in establishing the termination property. This observation suggests that
there is always room for information design to improve welfare when the agents are sufficiently
forward-looking.

Theorem 2 In the optimal mechanism, for t = 2, . . . , T ,(i) no disclosure is never optimal, i.e, βt > 0,∗∗∗∗
To see, note that r(vt∗ , vt−1
) = q(vt∗ , vt−1
) + µ(vt−1) − 1. Substituting for r(vt∗ , vt−1
) and rearranging, the∗∗ ∗only solution that can satisfy this equation when c > vt must have q(vt , vt−1 ) = 1. This is only possible when∗vt∗ = vt−1
.
12

19

(ii) partial disclosure is optimal when agents are sufficiently patient, i.e., βt < 1 if δ issufficiently close to 1.

4.5

Asymptotic outcome and the limit of social learning

We have thus far observed that the optimal disclosure mechanism exercises caution and occasionally terminates even when the principal has observed no news and is relatively optimistic
as a result. One way to illustrate the trade-off generated by the caution property is to look at
the asymptotic outcome as T tends to infinity. To this end, we start with the full-disclosure
policy. It is easy to observe that social learning never ceases under full disclosure as long as
no bad news is observed.13 When the state is bad, the game continues indefinitely until bad
news arrives, and the true state is eventually identified. When the state is good, the principal’s
belief when she is optimistic converges to 0, albeit very slowly, and so does the threshold type.
This means that the true state can be identified almost surely under full disclosure, suggestingthat there is no inherent upper limit of social learning in the bad-news case.
The fact that the true state can be identified asymptotically under full disclosure clarifies
a distortion introduced by the optimal mechanism. For the sake of argument, suppose δ isclose to 1, so that βt < 1 for some t. By the same argument as in the previous paragraphand also the termination property, social learning never ceases in the optimal mechanism aslong as the adoption phase continues. Therefore, when the state is bad, the adoption process
terminates almost surely as under full disclosure, though at a faster rate. However, when thestate is good, the adoption process may terminate prematurely, so that the true state may
not be fully identified even asymptotically; this is an obvious distortion associated with thecaution property, giving rise to the essential trade-off between the cost of premature terminationand the gain of faster social learning. As we can see from Theorem 2, the cost arising fromthis distortion is always outweighed by faster social learning when the agents are sufficiently
forward-looking.

5

Policy implications

Our analysis suggests that it is ex ante optimal to terminate the adoption process with someprobability, even when the principal has observed no news. There are, however, some concep∗∗∗∗∗
A strictly positive measure of agents adopt in period t if vt−1
− µ(vt−1)c > δ(q(vt∗ , vt−1
)vt−1
− r(vt∗ , vt−1
)c).
∗
∗
∗
∗
∗
As vt−1 → vt , this reduces to vt−1 > µ(vt−1 )c if δ ∈ [0, 1), which holds for any vt−1 that satisfies the IC
constraint.
13

20

tual issues in implementing this outcome in practice. First, it may be deemed inappropriate,
or even unethical, for the government to knowingly manipulate information; this is especially
so in health-related issues such as vaccine uptake. Second, the implementation of this outcome
requires credible randomization, but it is certainly inappropriate for the government to flip acoin to determine when to terminate.
In the context of collective experimentation, it is more reasonable to interpret our model
environment as the one in which the principal designs and commits to the way she collects
and analyzes data to produce evidence and transforms it into a binary recommendation at
the outset of the game. More precisely, in our model, we assume that the principal observesconclusive bad news with some probability in each period, but in reality, such conclusive
news is often prohibitively costly or simply infeasible to attain; in most cases, the principal
must evaluate various pieces of evidence with varying degrees of precision to form a belief.
The information-design problem of our analysis can be regarded as a reduced form of thiscomplicated deliberation process. If it is costly to acquire more accurate information, it may
be inevitable to admit some statistical errors in the decision-making process. The relevant
questions to ask are then what type of statistical error—type I (false positive) or type II (false
negative)—should be condoned and to what extent.
Formally, consider a decision-making process in which the principal designs the information
structure that determines how stochastic signals are generated and a decision rule that maps
the observed signal into a binary recommendation. Given that decision rules that involve
randomization are hard to implement in practice, we here only consider pure (deterministic)
decision rules. Below, we build on this framework to explore ways in which to achieve the(second-best) optimal outcome and highlight two practical implications of our analysis.14
False positives versus false negatives. To illustrate what type of statistical error should be moretolerated, consider the two-period example with an augmented information structure. Suppose
the signal distribution is now given by∗∗

P[s1 = b | ω, v1∗ ] = εI e−λω(1−F (v1 )) + (1 − εII )(1 − e−λω(1−F (v1 )) ).
The efficiency of the information structure is measured by (εI , εII ), where εI is the probabilityof a type I error and εII is interpreted loosely as representing the probability of a type II error.
14

In the following argument, we abstract away from the cost of making the information structure moreaccurate to illustrate the key insights. If it is costly to raise the precision of the information structure, theprincipal must trade off the benefit of information design against the cost, but even then, the same insights
should follow.

21

The information structure in the baseline model corresponds to the one with εI = εII = 0.
Under this specification, the probability of observing no signal, denoted by q̂(v1∗ ), is∗∗q̂(v1∗ ) = 1 − m[εI e−λ(1−F (v1 )) + (1 − εII )(1 − e−λ(1−F (v1 )) )] − (1 − m)εI∗

= (1 − εI )q(v1∗ , 1) + εII m(1 − e−λ(1−F (v1 )) ).
Similarly, define r̂(v1∗ ) such that∗∗r̂(v1∗ ) = m[1 − εI e−λ(1−F (v1 )) − (1 − εII )(1 − e−λ(1−F (v1 )) )]∗= (1 − εI )r(v1∗ , 1) + εII m(1 − e−λ(1−F (v1 )) ).
Since there are only two signals, the only non-trivial (pure) decision rule is to continue
after s1 = ∅ and terminate after s1 = b. Given this decision rule, the principal designs theinformation structure by choosing (εI , εII ) at the outset of the game. Observe that the IC
constraint becomesv1∗ − mc = δ(q̂(v1∗ )v1∗ − r̂(v1∗ )c).
for a given (εI , εII ). We argue that the principal can implement any outcome that is achievable
by a binary-message mechanism. To this end, let v ∗ (β) be the threshold (in the two-period
model) when β2 = β, i.e., the threshold that satisfiesv ∗ (β) − mc = δβ(q(v ∗ (β), 1)v ∗ (β) − r(v ∗ (β), 1)c),
The principal’s goal is to implement v1∗ = v ∗ (β). To achieve this outcome, we must haver̂(v ∗ (β))
r(v ∗ (β),1)∗∗q̂(v ∗ (β)) = βq(v ∗ (β), 1) and q̂(v∗ (β)) = µ(v (β)) = q(v ∗ (β),1) , which together imply r̂(v (β)) =βr̂(v ∗ (β)). Note that these conditions can be satisfied if and only if εI = 1 − β and εII = 0.
This argument suggests that false positives (termination in the good state) are more acceptable
than false negatives (continuation in the bad state) in that the optimal information structure
must occasionally admit false positives but no false negatives.
Virtue of overly cautious mechanisms. We extend the same idea further by introducing asecond signal that is not fully informative. Suppose s1 ∈ {b, w, ∅}, and consider the following

22signal distribution:
P[s1 | ω, v ∗ ] =(1 − ε̃)(1 − e−λω(1−F (v∗ )) )if s1 = b,ε e−λω(1−F (v∗ )) + ε̃(1 − e−λω(1−F (v∗ )) ) if s = w,
I
1

where we suppose for illustration that ε̃ is fixed due to some exogenous factors. In this specification, the strong signal s1 = b is still conclusive, corresponding, e.g., to a catastrophic eventthat unambiguously reveals that the state is bad and leads to immediate termination. By
contrast, the weak signal s1 = w represents the ambiguous state that requires the principal’s
judgment.15 Let µw (v ∗ ) be the principal’s belief associated with the weak signal, which is givenby∗∗m[εI e−λ(1−F (v )) + ε̃(1 − e−λ(1−F (v )) )]µw (v ) =.
m[εI e−λ(1−F (v∗ )) + ε̃(1 − e−λ(1−F (v∗ )) )] + (1 − m)εI∗

In this setting, the principal can achieve the optimal outcome by setting εI at 1 − β (whileε̃ ∈ (0, 1) can be arbitrary) and terminating after s1 ∈ {b, w}.
Note that µw (v ∗ ) may take any value between µ(v ∗ ) and 1 and is strictly decreasing in ε̃.
As ε̃ gets smaller, therefore, the weak signal becomes almost equivalent to no news. Under
the optimal decision rule, however, the principal is still instructed to terminate the adoption
process upon observing s1 = w. The decision rule considered above is thus overly cautious(when ε̃ is relatively small) in the sense that the principal “overreacts” to insignificant events
that are hardly informative. Alternatively, this argument points to a possibility that when
it is not feasible to commit to a decision rule ex ante, delegating decision-making authority
to a cautious leader can work as a compromised solution; this can be done by appointing
someone who is known to be cautious or by institutional design to punish failures severely andsomewhat excessively. In either case, our analysis suggests a virtue of mechanisms that may
appear overly cautious, even though we assume that all the concerned parties are risk-neutral.

6

Extensions

In the main body of the analysis, we deliberately consider the simplest setting to illustrate
the key insights in a transparent manner. As it turns out, though, our main characterization
result, Theorem 1, is robust to various alterations to the underlying setup and hence holds
more generally. Below, we discuss two possible extensions—one on the principal’s objectives
15

As before εI corresponds to type I error when the decision rule is to terminate after any signal.

23

and the other on the information-generating process—to ensure that our framework can beapplied widely to a range of situations of practical importance.

6.1

More general objective function

So far, we have considered a benevolent principal whose goal is to maximize the sum of thepayoffs of all agents. In practice, however, the principal’s objective can be more diverse andinclude benefits that are not fully internalized by the agents. For instance, in the case of vaccine
uptake, one major social goal is to achieve herd immunity, the benefit of which can be sharedequally among all citizens irrespective of their vaccination decisions. The vaccination process
also involves third parties, such as pharmaceutical companies, and the government may havesome interest in their well-being as well.
To capture this situation, we now suppose that the principal’s objective function includes
an additional term that depends on past thresholds. Specifically, suppose that the principal’s
payoff in period t is now more generally given by
Z vt−1∗vt∗∗(v − µt c)dF (v) + αH(vt−1, vt∗ ),∗, vt∗ ) measures the external benefit that iswhere H : [0, 1]2 → R+ . In this specification, H(vt−1
contingent on the current and previous thresholds, and α ≥ 0 is the welfare weight given to theexternal benefit with our baseline model corresponding to a special case with α = 0. Note that∗− vt∗ represents
1 − vt∗ represents the stock of agents who have adopted up to period t while vt−1
the flow of agents who adopt in period t. This extended specification of the objective function
can thus accommodate a range of goals that depend on the stock and the current flow in a
flexible manner: for instance, the level of herd immunity should be positively correlated withthe stock of citizens who have been vaccinated, whereas a pharmaceutical firm’s profit in agiven period should be positively correlated with the flow of citizens who get vaccinated inthat period. In what follows, we make the following assumption on the form of the externalbenefit.

Assumption 2 The following conditions are satisfied:∗∗(a) H2 (vt−1, vt∗ ) ≤ 0 for any (vt−1, vt∗ );∗∗∗∗(b) H2 (vt−1, vt∗ ) + H1 (vt∗ , vt+1
) ≤ 0 for any (vt−1, vt∗ , vt+1).

24

In any period t, an increase in the current threshold vt∗ implies a decrease in the stock andthe flow. It is therefore reasonable to assume H2 ≤ 0. By contrast, since an increase in the∗previous threshold vt−1
implies an increase in the flow, it is a priori difficult to determine thesign of H1 . Note that the previous threshold itself was a current threshold in the previous
period. Taking this into account, in part (b) of Assumption 2, we assume that the overall
impact of an increase in vt∗ is weakly negative. An implication of this assumption is that forany two sequences (vs∗ , . . . , vs∗′ ) and (v̂s∗ , . . . , v̂s∗′ ) such that vs∗ = v̂s∗ , vs∗′ = v̂s∗′ , and vt∗ ≥ v̂t∗ forall t = s + 1 . . . s′ − 1,
′ −1
sX

δ

t−s

∗
H(v̂t∗ , v̂t+1
)≥

t=s

′ −1
sX

∗
δ t−s H(vt∗ , vt+1),

t=si.e., if we have two sequences that start from and end at the same thresholds, the one that is
consistently lower (i.e. the one that has faster adoption rate) yields a weakly higher surplus.16
For instance, Assumption 2 is satisfied if the external benefit is given by∗∗H(vt−1, vt∗ ) = η(vt−1− vt∗ ) + h(vt∗ ),where η > 0 is a weight given to the flow and h : [0, 1] → R+ is some decreasing function that
captures the impact of the stock. We can show that our main characterization result holds inthis extended setup.
Theorem 3 Theorem 1 holds for any α > 0 under Assumption 2.

6.2

Good-news caseIn the baseline model, we focus on the case where the principal may observe bad news (a“breakdown”) when the state is bad. Here, we consider the opposite case where the principal
may observe good news (a “breakthrough”) when the state is good. Specifically, let st ∈ {∅, g},and assume
P[st = g | ω, nt ] = 1 − e−λ(1−ω)nt
where nt is the measure of agents who adopt in period t. The principal’s belief jumps down to∗∗) = 0 for any vt−1
∈ [0, 1]. In the absence of good
0 once she observes good news so that µ(vt−1
To see this, if there is an infinitesimal increase in vt∗ (while fixing all other thresholds constant), the change∗∗
in the surplus is H2 (vt−1, vt∗ ) + δH1 (vt∗ , vt+1). Since the first term is positive by assumption, the change must
be nonnegative for any δ ∈ [0, 1].
16

25

news, the principal’s belief is given by∗µpt = µ(vt−1
) :=

m
∗ ,
m + (1 − m)e−λ(1−F (vt ))which is the highest possible belief that can be induced in period t. The consistency restriction∗is then defined in the same way, except that µt must now be bounded between 0 and µ(vt−1).
Note that in the good-news case, the optimistic state is when the principal has observed
good news while the pessimistic state is when she has observed no news. Given this, thecaution property is still the same, where a binary-message mechanism terminates surely in thepessimistic state, whereas it randomizes and continues with some probability in the optimistic
state. With abuse of notation, redefine∗∗∗))(1 − e−λ(1−F (vt )) )q(vt∗ , vt−1
) := (1 − µ(vt−1

as the probability of observing good news in period t, wherem if v ∗ = 1,t−1∗µ(vt−1 ) =0 if v ∗ < 1.
t−1

With this modified definition, in a binary-message mechanism, the probability that the game∗∗) as in the bad-news case.17, vt−2stays in the adoption phase is still βt q(vt−1
Before we state the result, it is worth noting that the good-news case is not the mirror
image of the bad-news case. In the proof of Theorem 1, we consider a mean-preserving spreadthat splits an interior belief into a slightly lower belief and the highest possible one. As notedin Section 4.3, while this mean-preserving spread provides more accurate information andbenefits the agents who adopt now, it could be detrimental to the remaining agents who might
adopt in the future because it necessarily lowers the continuation probability. In the bad-news
case, however, the highest possible belief is always 1, in which case the state is bad for sure.
Therefore, the decrease in the continuation probability, induced by a mean-preserving spread,entails no loss of surplus, and we can take a more direct route to show that such an operation
benefits all agents. This same argument cannot be applied to the good-news case because thehighest possible belief is always bounded away from 1. In this situation, terminating in thepessimistic state (after no news) entails a loss of potential surplus and yields the aforementioned
trade-off, which forces us to explore a different route to establish the characterization result.
17

As we will see below, the game always ends in two periods, so that βt for t ≥ 3 is actually irrelevant.

26

As it turns out, though, the gain from alleviating the free-rider incentive always dominates
the loss of surplus due to the decrease in the continuation probability, such that the same
principles—termination and caution—still hold in this alternative scenario.
Theorem 4 Theorem 1 holds in the good-news case.
The theorem states that the caution property continues to hold in the good-news case. As av∗
consequence, the period-2 belief is either µ2 = 0 (the optimistic state) or µ2 = µ̃(v1∗ , 1; β2 ) > c1
(the pessimistic state). If µ2 = 0, the state is good for sure, and all remaining agents adopt
in period 2. If not, no agent will adopt after period 2 by the termination property. In either
case, therefore, social learning ceases in two periods. This feature of our model draws a clear
contrast with Che and Hörner (2018) who also look at the good-news case in their baseline
specification. In their model, the optimal mechanism always recommends adoption after goodnews and also with some probability after no news. The strategy of mixing good news withno news can extend the learning process even after no news and is valuable in their setting.
In our framework where agents are forward-looking, however, the same strategy would notwork because that would intensify the incentive to wait for news. Although terminating the
adoption process after no news implies a loss of surplus because the principal’s belief is stillbounded away from 1, it is nonetheless part of the optimal mechanism to deter information
free-riding, and the principal cannot fare any better than this.
The fact that social learning cannot extend beyond the second period also suggests that
there is a natural upper bound of social learning in the good-news case. In the best case,the true state can be discovered with a breakthrough, but this occurs only with probability∗(1 − m)(1 − e−λ(1−F (v1 )) ), which is strictly bounded away from 1. In case no news arrives,
the principal’s belief goes down slightly, but she must still terminate the adoption process
altogether, and there is no feasible way to incentivize exploration beyond this point, no matter
how large T becomes. This asymptotic property of the good-news case is clearly different fromthat of the bad-news case where there is no such limit, and increasingly more information canbe acquired as T gets larger. The crucial difference is that news that leads to termination(i.e., bad news) takes a form of conclusive bad news in the bad-news case, so that the true
state is precisely revealed when the adoption process terminates under full disclosure. In the
good-news case, by contrast, the adoption process terminates after inconclusive news (i.e., nonews), making it inherently infeasible to fully identify the true state even asymptotically.
Finally, it is also worth noting that although there is an upper limit of social learning, thefree-rider problem tends to be less severe in the good-news case because of the way informationis generated in this case. In the bad-news case, the arrival of news indicates that the state is27bad for sure. Since no one has an incentive to adopt when the state is bad, this information
is valuable for all agents regardless of their valuations. This is not necessarily the case in the
good-news case where the principal’s belief is bounded away from 1 in the pessimistic state(no news). In this case, additional information may not be informative enough for agentswith sufficiently high valuations for whom there is no free-rider incentive. Although we do not
analyze this aspect in depth as it is outside the scope of this paper, it is of some interest, atleast theoretically, to explore the difference between the bad-news and good-news cases fromthis perspective.

7

Conclusion

Social learning is often hampered by citizens’ desire to wait and see others’ experiences. In thispaper, we explore optimal ways to resolve this issue via information design and characterize
the optimal feedback mechanism to deter information free-riding. The optimal mechanism
is cautious in the sense that it certainly terminates when the principal is pessimistic andterminates with some probability even when she is optimistic. The key driving force of thismechanism is the observation that the value of information varies across agents with different
valuations: more precise information at the low end of the belief distribution is beneficial
for agents with lower valuations but irrelevant for those with higher valuations. As such,a mechanism that is consistently more precise at the low end but obscure at the high end
can induce agents with higher valuations to adopt earlier, thereby alleviating the free-rider
problem, while still enabling those with lower valuations to make more informed decisions. We
show that the optimal mechanism takes a simple form, which effectively reduces the principal’s
problem to an optimal stopping problem. We also show that the key principles identified inthis paper are shown to be robust to various alterations in the underlying setup and can beapplied to a range of social situations.

Appendix A: Proofs∗
Proof of Lemma 1. (i) We consider the incentive of the threshold agent vt−1
, given that∗A∗
) ∈ Mt−1(µt−2 ). First, suppose on the contrary that µt = µ(vt−1) ∈ MtN (µt−1 ).
µt−1 = µ(vt−2∗Then, by part (b) of the definition, the belief stays at µ(vt−1) for all future periods, meaningthat there are no gains from waiting. Therefore, if the threshold agent deviates and choosesv∗∗
to wait in period t − 1, he must choose to adopt in period t given that t−1≥ µ(vt−2
) >
c

28

∗
∗
µT = µ(vt−1). However, this is a contradiction because µt = µ(vt−1) ∈ MtN (µt−1 ). Given this,∗∗we next show µ̃(vt−1, vt−2; βt ) ∈ MtN (µt−1 ). Suppose otherwise. Then, the threshold agent
must adopt in period t with probability 1 if he deviates. This is a contradiction because thethreshold agent would strictly prefer to adopt in period t − 1.
∗
. If µt−1 ∈ MtN (µt−2 ), we(ii) We again consider the incentive to the threshold agent vt−1∗∗
have vt−1= vt−2by definition. Also, by part (b), the belief stays constant for all future periods.
Therefore, if it is not optimal for the threshold agent to adopt in period t − 1, it is not optimal
for him to adopt in period t.

Proof of Theorem 1. The proof consists of two properties, which we label as terminationand caution. These two properties together imply the optimality of binary-message termination
mechanisms.
∗
Termination. We show that if there is no adoption in period t (vt∗ = vt−1), then there will be no∗∗
adoption in any future period as well (vs = vt with probability 1 for all s > t). Consider somemechanism P that pauses the adoption process for one period and then resumes after that.18∗
Fix some history µt−1 and let vt−1
denote the threshold following µt−1 . Also, let vt∗ (µt ) and∗
vt+1(µt , µt+1 ) be the thresholds following (µt−1 , µt ) and (µt−1 , µt , µt+1 ), respectively. We then∗∗.
(µ′t , µ′t+1 ) < vt∗ (µ′t ) = vt−1
have some µ′t ∈ Mt (µt−1 ) and µ′t+1 ∈ Mt+1 (µt−1 , µ′t ) such that vt+1
We now propose an alternative mechanism P̂ that modifies P following µt = µ′t (while keeping
everything else the same). Specifically, suppose the modified mechanism splits µ′t into allµt ∈ Mt+1 (µt−1 , µ′t ) and µ′t such that P̂t (µt | µt−1 ) = δPt (µ′t | µt−1 )Pt+1 (µt | µt−1 , µ′t ) forall µt ∈ Mt+1 (µt−1 , µ′t ) and P̂t (µ′t | µt−1 ) = (1 − δ)Pt (µ′t | µt−1 ). Moreover, (i) following
any µt ∈ Mt+1 (µt−1 , µ′t ) in period t, P̂ implements the same allocation as P and induces noadoption in period T ; (ii) following µ′t in period t, P̂ induces no adoption for all subsequent
periods. Note that this modified mechanism keeps the continuation payoffs of all remaining
agents, and hence the continuation surplus, unchanged (while the allocation following µt = µ′t
is clearly suboptimal). It is therefore without loss of generality to front-load all the information
by setting Pt (µ | µt−1 ) = 0.
This argument suggests that there is no need to consider temporary suspension of the
adoption process, as we can always split this into continuation and permanent termination in∗a profitable way. Alternatively, if vt∗ = vt−1
for some t, that must mean permanent termination∗∗(vs = vt with probability 1 for all s > t).

Caution. We show that we can increase the total surplus by stretching out any interior belief
18

The same argument applies for a mechanism that pauses for multiple periods.

29

v∗

∗
µt ∈ (µ(vt−1), t−1) into two outer points. Alternatively, this suggests that it is not optimal tocassign a positive probability to any interior belief. We establish this result via induction.
We begin with the period-T surplus which can be written as

WT (µT , µT c, vT∗ −1 ) =Z vT∗ −1(v − µT c)dF (v).
µT cTaking derivative of WT with respect to µT then yields −c(F (v ∗ )−F (µT c)), which is increasingin µT , i.e., the period-T surplus is convex in µT . This ensures the caution property for period
T . In the following, we show that if this property holds for all periods s > t, it also holds forperiod t.
v∗∗), t−1), whereConsider some arbitrary history µt−1 and an interior belief µt = µ′t ∈ (µ(vt−1c∗is determined by µt−1 . Let vt∗ = v ∗ denote the threshold following (µt−1 , µ′t ). Under thevt−1
induction hypothesis, the IC constraint is given byv ∗ − µ′t c = δPt+1 (µ(v ∗ ) | µt−1 , µ′t )(v ∗ − µ(v ∗ )c).
The Bayes plausibility, along with the consistency restriction, implies that there is an upperbound P t+1 (µ(vt∗ ) | µt ) of the probability that can be assigned to the lowest possible belief. To
obtain the upper bound, let ν be the probability that the principal is uninformed for a givenµt , which must satisfy∗) + 1 − ν = µt .
νµ(vt−1
1−µt. Since the principal observes no news in period t with
Solving this then yields ν = 1−µ(v
∗
t−1 )
∗ ∗
probability q(vt , vt−1 ), the upper bound is obtained as∗
P t+1 (µ(vt−1) | µt−1 , µ′t ) =1 − µ′tq(v ∗ , v ∗ )∗1 − µ(vt−1) t t−1∗= (1 − µ′t )∗∗∗µ(vt−1)e−λ(F (vt−1 )−F (vt )) + 1 − µ(vt−1)∗
1 − µ(vt−1 )me= (1 − µ′t )−λ(1−F (vt∗ ))

+1−m
,
1−m

(9)

∗
which is determined independently of vt−1.
We now consider a modified mechanism P̂ that applies a mean-preserving spread to µ′t asin the two-period example. In the general case, however, we need to adopt a different strategy

30because the way we split a belief may affect the continuation probability due to the upper∗bound constraint. Specifically, we let P̂ split µ′t into µ′t − ε and µ′′t > vc with probabilities
′
µ′′
t −µt
and µ′′ −µε ′ +ε , respectively, where ε > 0 is some arbitrarily small number, and moreover′µ′′t −µt +εtt
mix µ′′t with the highest belief on the support of Pt (· | µt−1 ). Denote the higher belief by µ̂.
Since the resulting mechanism P̂ is arbitrarily close to P , all the past thresholds up to periodt − 2 (if t ≥ 3) are unaffected by this modification. For t > 1, this modification increases∗∗∗
the continuation payoff of agent vt−1
, raising the threshold slightly. Let vt−1
= v̂t−1be the∗′∗threshold after the modification and vt = v̂ the threshold following µt = µt − ε.
To make the situation directly comparable, it is convenient to combine two contingencies(µt = µ′t − ε and µt = µ̂) to represent the IC constraint. Since agent v̂ ∗ adopts after µt = µ′t − εbut not after µt = µ̂, given the induction hypothesis, the IC constraint combining the twocontingencies can be written asµ′′t − µ′t[v̂ ∗ − (µ′t − ε)c] = δ R̂t+1 (v̂ ∗ − µ(v̂t∗ )c).
µ′′t − µ′t + ε

(10)

where
R̂t+1 :=µ′′t − µ′tP̂t+1 (µ(v̂t∗ ) | µt−1 , µ′t − ε).
µ′′t − µ′t + ε

From (9), the upper bound for the modified mechanism is given by∗P̂t+1 (µ(v̂ ) | µt−1me, µ′t − ε) ≤ (µ′′t − µ′t + ε)−λ(1−F (v̂ ∗ ))

+1−m
,
1−m

from which the upper bound for R̂t+1 is obtained as∗R̂t+1 ≤∗−λ(1−F (v̂ ))µ′′t − µ′tme−λ(1−F (v̂ )) + 1 − m
+1−m
′′
′
′′
′ me
(µ
−
µ
+
ε)
=
(µ
−
µ
)
.
t
t
t
t
′′
′
µt − µt + ε1−m1−mNote that the upper bound depends on and is increasing in µ′′t . To ensure maximum flexibility,
we set µ′′t = 1, so that the upper bound is given bymeR̂t+1 ≤ (1 − µ′t )−λ(1−F (v ∗ ))+1−m
= P t+1 (µ(v ∗ ) | µt−1 , µ′t ),1−m31

when v̂ ∗ = v ∗ . Note also that since
1 − µ′t[v ∗ − (µ′t − ε)c] > v ∗ − µ′t c,
1 − µ′t + ε
R̂t+1 must be set higher than Pt+1 (µ(v ∗ ) | µt−1 , µ′t ) to implement the same threshold as in P .
There are two cases to consider, depending on whether the upper bound is binding. We startwith the case where Pt+1 (µ(v ∗ ) | µt−1 , µ′t ) < P t+1 (µ(v ∗ ) | µt−1 , µ′t ), so that the upper bound is
not binding. In this case, we can find R̂t+1 ∈ (Pt+1 (µ(v ∗ ) | µt−1 , µ′t ), P t+1 (µ(v ∗ ) | µt−1 , µ′t )) that
can implement the same threshold as P . With the same threshold, the modified mechanismP̂ can implement the same allocation as P for all future periods following µt+1 = µ(v ∗ ). The
only difference is that µt+1 = µ(v ∗ ) is now realized with a higher probability so that agentsv ∈ [0, v ∗ ) are better off. In addition, agents v ∈ [v ∗ , v̂ ∗ ) also benefit from the modification.
Next, consider the case where Pt+1 (µ(v ∗ ) | µt−1 , µ′t ) = P t+1 (µ(v ∗ ) | µt−1 , µ′t ), so that theupper bound is binding. In this case, since R̂t+1 cannot be raised any further, we have a lower
threshold v̂ ∗ < v ∗ such that
1 − µ′t[v̂ ∗ − (µ′t − ε)c] = δ R̂t+1 (v̂ ∗ − µ(v̂ ∗ )).
′1 − µt + εWith a lower threshold, we can induce a belief lower than µ(v ∗ ). This means that we canimplement the same allocation following µt = µ′t − ε by inducing µt+1 = µ(v ∗ ) (instead of thelowest possible belief). Since this modification, though suboptimal, gives the same payoff toall agents v < v̂ ∗ following µt+1 = µ(v ∗ ), it is weakly beneficial for those agents. Moreover,since µt+1 = µ(v ∗ ) is still realized with probability P t+1 (µ(v ∗ ) | µt−1 , µ′t ), we have1 − µ′t[v − (µ′t − ε)c] > δP t+1 (µ(v ∗ ) | µt−1 , µ′t )(v − µ(v ∗ )c),
1 − µ′t + ε
for agents v ∈ [v̂ ∗ , v ∗ ), meaning that they are also better off. Finally, it is easy to see that∗the payoffs of agents v ∈ [v ∗ , v̂t−1 ) increase while the payoffs of agents v ∈ [v̂t−1, 1] remainunchanged. This argument shows that the proposed modification is Pareto improving andhence increases the total surplus.
Proof of Theorem 2. Since we have already established part (i) in the main text, here wefocus on part (ii). We first show βT < 1 as δ → 1. To this end, it is important to observethat if βT = 1, vT∗ −1 converges to vT∗ −2 as δ → 1, i.e., no agents adopt in period T − 1, andagents v ∈ [µT −1 c, vT∗ −2 ) adopt in period T . However, the same allocation can be implemented32

a period earlier by setting βT = 0, in which case agents v ∈ [µT −1 c, vT∗ −2 ) adopt in period T − 1(while no agents adopt in period T ). As δ → 1, these two allocations are equivalent and yield
the same continuation surplus. The existence of an interior solution (βT < 1) is then impliedby part (i) which states that it is never optimal to set βT = 0.
This argument suggests that vT∗ −1 is bounded away from vT∗ −2 as δ → 1. Let vT∗ −1 = va∗ andvT∗ = vb∗ be the optimal thresholds in the limit where vT∗ −2 > va∗ > vb∗ . Now consider periodT − 1 and suppose βT −1 = 1 in the optimal mechanism as δ → 1. Then, in the limit, vT∗ −2
converges to vT∗ −3 (where v0∗ = 1 if T = 3), and hence the period-T − 2 surplus converges to 0.
However, the principal can instead implement the same allocation a period earlier by inducing
vT∗ −2 = va∗ and vT∗ −1 = vT∗ = vb∗ . Observe that the implemented allocation is suboptimal because we must have vT∗ < vT∗ −1 in the optimal mechanism. We thus conclude βT −1 < 1 in the
optimal mechanism if δ is sufficiently close to 1. We can then apply this argument repeatedly
to establish the proposition.
Proof of Theorem 3. Recall that we establish the termination property by showing that if
the game is in the no-adoption phase in one period and moves back to the adoption phase in
the next, we can always implement the same allocation a period earlier by a modified mechanism. Since the modified sequence is entirely lower than the original sequence, Assumption
2 ensures that the principal’s payoff is still weakly higher in the modified mechanism, and thetermination property continues to hold. Note also that the caution property holds for a fixedsequence of thresholds if the upper bound defined in the proof of Theorem 1 is not binding. If
it is binding, we consider a modified mechanism (a mean-preserving spread) that implements athreshold slightly lower than the original one in period t and the same allocation after periodt. Again, Assumption 2 ensures that the principal’s payoff is weakly higher in the modified
mechanism, and the caution property continues to hold. This proves Theorem 1 for any α > 0.

Proof of Theorem 4. Observe that the termination property is independent of the underlying information-generating process and holds as it is. Below, we thus focus on establishing∗
the caution property, i.e., MtA (µt−1 ) contains only the lowest possible belief µ(vt−1) = 0 when
v∗
A
∗
t−1
the game is in the adoption phase. To this end, we first note that Mt (µ ) = [µ(vt−1), t−1)c∗v∗, µ(vt−1)] in any optimal mechanism satisfying the termination property.
and MtN (µt−1 ) = [ t−1cv∗∗∗
Alternatively, we need to show that vt∗ < vt−1
if and only if µt ∈ [µ(vt−1), t−1). The necessitycv∗∗
is obvious because it is strictly better to wait for all agents v ∈ [0, vt−1
) if t−1≤ µt . To
c
∗
vt−1
∗
∗
show the sufficiency, suppose on the contrary that vt = vt−1 even if c > µt . Then, by the33termination property, the belief stays constant in all future periods. In period T , agents whose∗
valuations are arbitrarily close to vt−1
have an incentive to adopt because vc > µT = µt for v
v∗∗∗
arbitrarily close to vt−1
. As this is a contradiction, we must have MtA (µt−1 ) = [µ(vt−1), t−1
)
c
∗
vt−1
t−1
N
∗
and Mt (µ ) = [ c , µ(vt−1 )] in an optimal mechanism.
Given this, we next establish the following result.
∗)} ∈ MtN (µt−1 ) or both.
Lemma 2 In the optimal mechanism, either {0} ∈ MtA (µt−1 ), {µ(vt−1∗
Proof. We show that if MtA (µt−1 ) ̸= {0} and MtN (µt−1 ) ̸= {µ(vt−1)} in some mechanism P , wecan construct a modified mechanism P̂ that can improve upon P for a given history µt−1 . First,v∗∗)] into a single one without affecting, µ(vt−1
observe that we can merge all the beliefs in [ t−1cthe thresholds. Therefore, it is without loss of generality to focus on mechanisms that have∗
only one element, say µH , in MtN (µt−1 ). Pick some µL ∈ MtA (µt−1 ) \ {µ(vt−1)} and apply amean-preserving spread to µH , so that µH is split into µL and µH + ε for some arbitrarily small∗
ε > 0. This operation raises the payoff of agent vt−1
and thus (slightly) raises the threshold in∗∗∗∗
period t−1 to some v̂t−1> vt−1. Note that with a higher threshold v̂t−1, µ(v̂t−1) is smaller thanv̂ ∗∗µ(vt−1 ) but is still higher than µH + ε. In period t, some agents adopt if µt ∈ (0, t−1) while no
c
∗
v̂t−1
t−1
∗
∗
∗
agents adopt if µt ∈ [ c , µ(v̂t−1 )). Note that v̂t−1 > vt−1 , P̂t (µL | µ ) > Pt (µL | µt−1 ), andv̂ ∗P̂t (µH + ε | µt−1 ) < Pt (µH | µt−1 ). Therefore, the probability that µt lies in (0, t−1) becomescweakly higher, leading to an increase in the continuation probability. This means that thepayoffs of agents adopting in period t and after increase while the payoffs of those adopting
before period t remain the same, suggesting that the modification is Pareto-improving.

Given the lemma, we now show MtA (µt−1 ) = {0}, which allows us to establish the theorem.
To this end, suppose there is some interior belief µt > 0 in MtA (µt−1 ) for some µt−1 and t ≥ 2.
We then claim that for any history that follows µt−1 , there must be some period s > t suchthat MsA (µs−1 ) = {0} (where s is the earliest period that has this feature). Suppose otherwise.
We then have some history µT −1 such that MTA (µT −1 ) includes some interior belief µT > 0 or
is empty. If MtA (µT −1 ) includes an interior belief, since this is the last period, we can splitv∗
this into µ(vT∗ −1 ) and Tc−1 and improve the payoffs of all types. In period T , therefore, weeither have MTA (µT −1 ) = {µ(vT∗ −1 )} or MTA (µT −1 ) = ∅. If there is a history µT −1 such that
MTA (µT −1 ) = ∅, then period T − 1 is effectively the last period, and we can apply the same
argument to show that MTA−1 (µT −2 ) = {µ(vT∗ −2 )} or MTA−1 (µT −2 ) = ∅. We can apply thisargument repeatedly to provide the claim.

34

Given this, in period s − 1, we have∗∗vs−1
− µs−1 c = δPs (0 | µs−1 )vs−1,

from which we obtain
δPs (0 | µs−1 ) =∗− µs−1 c
vs−1
.
∗
vs−1

∗
) is
Note that the continuation payoff of agent v ∈ (0, vs−1

δPs (0 | µs−1∗− µs−1 c
vs−1
)v =
v.
∗
vs−1
v∗

c
Now consider a mean-preserving spread that splits µs−1 into s−2
with probability µvs−1
and
∗
c
s−2
0 with the remaining probability. Note that this mean-preserving spread does not affect the∗∗∗payoff of agent vs−2
but raises the payoffs of agents v ∈ [vs−1, vs−2]. Also, with this modifiv ∗ −µc∗cation, agents v ∈ (0, vs−1 ) adopt in period s − 1 when the belief is 0 and obtain s−2v∗ s−1 v,
s−2∗∗. Therefore,> vs−1
which is larger than the payoff in the original mechanism because vs−2
MtA (µt−1 ) = {0} for all t as long as the game is in the adoption phase, which proves that
Theorem 1 holds in the good-news case.

Appendix B: Free-rider incentive in collective experimentation
In this appendix, we show that the amount of adoption under full disclosure is generally
insufficient compared to the first best for any (continuous) type distribution and any discountfactor δ ∈ (0, 1]. Consider a two-period model in which each agent’s valuation v ∈ [0, 1] isdistributed according to some distribution function F with full support. The payoff to agent
v of adopting is v − ωc, where c ≥ 1 is some constant.
Let vF∗ be the threshold under full disclosure that solves
vF∗ − mc = δ(q(vF∗ )vF∗ − r(vF∗ )c).

(11)

Here, we use q(v ∗ ) := m(1 − p(1 − v ∗ )) + 1 − m and r(v ∗ ) := m(1 − p(1 − v ∗ )) to save notation
where p : [0, 1] → [0, 1] is a strictly increasing function. The total surplus for a given threshold35

v ∗ in the two-period case can be written as
∗

Z v∗

Z 1
(v − mc)dF (v) + δ

W1 (m, v , 1) =v∗

(q(v ∗ )v − r(v ∗ )c)dF (v),µ(v ∗ )c∗r(v )where µ(v ∗ ) = q(v∗ ) is the principal’s belief when she is uninformed. Now consider a social
planner who can unilaterally impose v ∗ to maximize the total surplus. The first-order conditionis then given by∂
W1 (m, v ∗ , 1) = −[v ∗ − mc − δ(q(v ∗ )v ∗ − r(v ∗ )c)]f (v ∗ ) + δmp′ (1 − v ∗ )
∗
∂v

Z v∗
(v − c)dF (v).
µ(v ∗ )c

(12)
Observe that the first term is simply the IC constraint that must equal 0 at vF∗ . Evaluating(12) at v ∗ = vF∗ thus yields∂
W1 (m, vF∗ , 1) = δmp′ (1 − vF∗ )
∂v ∗

Z vF∗
(v − c)dF (v).
∗ )c
µ(vF

1
The threshold under full disclosure is too high if ∂W< 0. A sufficient condition for this is∂v ∗∗vF ≤ c, which always holds because it is optimal for any agent to adopt in period 1 if v ≥ c.

36

References
Au, Pak Hung, “Dynamic Information Disclosure,” Rand Journal of Economics, 2015, 46(4), 791–823.
Ball, Ian, “Dynamic Information Provision: Rewarding the Past and Guiding the Future,”
Econometrica, 2023, 91 (4), 1363–1391.
Bikhchandani, Sushil, David Hirshleifer, and Ivo Welch, “A Theory of Fads, Fashion,
Custom, and Cultural Change as Informational Cascades,” Journal of Political Economy,
1992, 100 (5), 992–1026.
Bolton, Patrick and Christopher Harris, “Strategic Experimentation,” Econometrica,
1999, 67 (2), 349–374.
Bonatti, Alessandro and Johannes Hörner, “Collaborating,” American Economic Review, 2011, 101, 632–663.
Che, Yeon-Koo and Johannes Hörner, “Recommender Systems as Mechanisms for Social
Learning,” The Quarterly Journal of Economics, 2018, 133 (2), 871–925.
and Konrad Mierendorff, “Optiaml Dynamic Allocation of Attention,” American Economic Review, 2019, 109 (8), 2993–3029.
Correia da Silva, Joao and Takuro Yamashita, “Information Design in Repeated Interaction,” mimeo, 2024.
Ely, Jeffrey C., “Beeps,” American Economic Review, 2017, 107 (1), 31–53.
and Martin Szydlowski, “Moving the Goalposts,” Journal of Political Economy, 2020,
128 (2), 393–782.
Halac, Marina, Navin Kartik, and Qingmin Liu, “Optimal Contracts for Experimentation,” Review of Economic Studies, 2016, 83 (3), 1040–1091.
, , and , “Contests for Experimentation,” Journal of Political Economy, 2017, 125 (5),1523–1569.
Hamel, Liz, Lunna Lopes, and Mollyann Brodie, “KFF COVID-19 Vaccine Monitor:
What Do We Know About Those Who Want to “Wait and See” Before Getting a COVID-19
Vaccine?,” KFF Poll, 2021.
37

Kamenica, Emir and Matthew Gentzkow, “Bayesian Persuasion,” American Economic
Review, 2011, 101 (6), 2590–2615.
Keller, Godfrey and Sven Rady, “Breakdowns,” Theoretical Economics, 2015, 10 (1),
175–202.
, , and Martin Cripps, “Strategic Experimentation with Exponential Bandits,” Econometrica, 2005, 73 (1), 39–68.
Kremer, Ilan, Yishay Mansour, and Motty Perry, “Implementing the “Wisdom of the
Crowd”,” Journal of Political Economy, 2014, 122 (5), 988–1012.
Manso, Gustavo, “Motivating Innovation,” Journal of Finance, 2011, 66 (5), 1823–1860.
Nikandrova, Arina and Romans Pancs, “Dynamic Project Selection,” Theoretical Economics, 2018, 13 (1), 115–143.
Orlov, Dmitry, Andrzej Skrzypacz, and Pavel Zryumov, “Persuading the Principal toWait,” Journal of Political Economy, 2020, 128 (7), 2473–2837.

38