ISSN (Print) 0473-453X

Discussion Paper No. 1060

ISSN (Online) 2435-0982

REPUTATION CONCERNSIN RISKY EXPERIMENTATION

Chia-Hui Chen
Junichiro Ishida
Wing Suen

Revised August 2020
July 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, JapanReputation Concerns in Risky Experimentation∗
CHIA-HUI CHEN
Kyoto University

JUNICHIRO ISHIDA
Osaka University

WING SUEN
University of Hong Kong
August 11, 2020

Abstract. High-ability agents are more likely to achieve early success in risky
experimentation, but learn faster that their project is not promising. These
counteracting effects give rise to a signaling model with double-crossing property. This property tends to induce homogenization of quitting times between
types, which in turn leads to some pooling in equilibrium. Low-ability agents
may hold out to continue their project for the prospect of pooling with the hightype, despite having a negative instantaneous net payoff. A war-of-attrition
mechanism causes low-ability agents to quit only gradually over time, and tostop quitting for a period immediately before all agents exit.
Keywords. double-crossing property; dynamic signaling; venture startups
JEL Classification. D82; D83; O31∗

Part of this research was conducted when Suen visited the Institute of Social and Economic Research at
Osaka University; he likes to thank the Institute for its hospitality. The first author acknowledges financial
support from Japan Society for the Promotion of Sciences (17H13699). The second author acknowledges
financial support from the Joint Usage/Research Center at ISER, International Joint Research Promotion
Program (Osaka University) and Japan Society for the Promotion of Sciences (JP18H03640, JP19H01468
JP15H05728).

1.

Introduction

Reputation concerns are important in many facets of our life, and they are even more
so in the context of exploratory activities undertaken by experts, such as entrepreneurs,
politicians, and scientists, whose reputation is an indispensable asset for advancing theircareers. Reputation is established by both successes and failures, but in ways that are notas straightforward as they seem. Consider, for example, the decision to abandon a risky
venture. Does a better entrepreneur persist longer with his project, because he is better
able to implement a good idea? Or does he quit a risky venture earlier, because he isquicker to recognize its futility? Even success is not always an unblemished boost to one’s
reputation—success that arrives very late may be taken as a sign of mediocrity.
This paper aims to develop a unified framework to analyze the role of reputation concerns in risky experimentation. The setup is a standard bandit problem, where an agent
engages in a project of unknown quality while at the same time attempts to signal his
type to the market. If the project is good, the hazard rate of success (also called a “breakthrough” in this paper) is greater for the high type agent than for the low type. If the
project is bad, neither type ever succeeds with the project. We identify two effects thatare relevant for signaling in this context. The ability effect suggests that the high type has
more incentive to persist longer with a project, because he is better at successfully implementing a good project. But a counteracting learning effect suggests the opposite: the hightype becomes pessimistic about the project quality more quickly if success does not arrive,and therefore has an incentive to abandon it earlier.
The key to our analysis is not which effect outweighs the other. Rather, we show thatthe ability effect dominates in early stages of experimentation (when differences in beliefsof the two types are still small), while the learning effect dominates in later stages (whendifferences in conditional hazard rates are less relevant as the project is quite likely to bebad). As a result, the standard single-crossing property in signaling models does not hold.
The high type initially benefits more from persisting with the project than does the low
type, but eventually benefits less from persisting as he becomes increasingly pessimistic
relative to the low type, generating a double-crossing property in signaling incentives. In
this setting, therefore, perseverance is a sign of strength in early stages of experimentation
but a sign of weakness in later stages.

1

1.1.

Applications

Our model can be applied to a broad range of circumstances in which an agent embarks
on an activity of exploratory nature to acquire new knowledge in the presence of reputation concerns. In each of the cases we discuss below, there is a tradeoff between quitting
early and late, which stems from the aforementioned double-crossing property; as a consequence, it is hardly clear in which direction reputation concerns would influence theagent’s quitting decisions. As we will argue in detail later, the insights we learn from standard models, which are deeply rooted in the single-crossing property, cannot be extended
straightforwardly to these environments.
Venture startups. A leading example of our framework is venture startups. First, any
business startup necessarily involves a process of experimentation in that success is uncertain and its timing is stochastic. In addition, there is ample evidence to suggest that reputation matters in entrepreneurship. Many recent studies report that serial entrepreneurs
outperform first-time entrepreneurs (Chen, 2013; Rocha et al., 2015; Lafontaine and Shaw,
2016). Gompers et al. (2010) find that entrepreneurs with a track record of success are
more likely to succeed than are first-time entrepreneurs or those who have failed previously. They also find that entrepreneurs exhibit performance persistence, and argue that
entrepreneurship is a skill rather than luck. These findings point to the importance of
the “success breeds success” mechanism, where the perception of performance persistence
induces real performance persistence, suggesting that reputation is a crucial input for developing a new business.1 In this type of environment, the entrepreneur needs to protect
his reputation and hence has an incentive to strategically choose when to retreat from thebusiness when it does not appear to take off anytime soon. At a glance, the problem facedby the entrepreneur looks deceptively similar to that of standard reputation models: hechooses an observable action, which is his quitting time, so as to look good in the eyesof the market. A closer look at it reveals, however, that there is no obvious way for theentrepreneur to establish (or salvage) his reputation, begging the type of questions wenoted at the beginning.
Academia. Research and experimentation are almost synonym to each other, as researchers must take an unexplored path to discover new knowledge. There is little doubtthat reputation is a crucial asset for researchers as their career opportunities, such as inter1

Gompers et al. (2010) argue that this is the case if suppliers and customers are more likely to commit
resources to firms perceived to be more likely to succeed.

2nal or external promotion, grants, and potential employment in industry, depend heavily
on the perception of their expected future productivity. Consider an (early-career) economic theorist who tries to prove a conjecture which may or may not be true. A morecapable theorist can prove it more quickly if the conjecture is indeed true, or else it ishighly likely that the conjecture is false. With no sign of success, should the theorist, in
need of signaling his analytical prowess to coauthors, colleagues and the entire academic
community, quit earlier or persist longer with the project?
Political economy. Politicians often need to take actions whose benefits, if any, taketime to materialize. Electoral accountability obviously feeds reputation concerns for officeseeking politicians when they conduct policy experimentation. For a wide range of issues
for which politicians are held accountable, therefore, both experimentation and reputation
formation play a crucial role. Consider, for instance, a politician who chooses to implement
a new policy instead of following his predecessor’s old policy. What if, after a while, things
do not head in the right direction? The politician will face a difficult strategic decision ofwhether to persist with the current policy or to switch back to the old status quo. If heabandons the experimentation, what kind of message does it send to his constituents?
1.2.

Main results

To the best of our knowledge, Matthews and Moore (1987) are the first to consider the
double-crossing property in a multidimensional screening model.2 We exploit this property
to obtain a complete characterization of D1 equilibria in our setup. Equilibrium is unique,but it differs from the least-cost separating equilibrium. This result suggests that the way
reputation concerns work in the context of risky experimentation is qualitatively different
from that in other economic contexts characterized by single-crossing preferences.
A general lesson from our analysis is that the double-crossing property tends to induce
homogenization of quitting times between types, compared to the “standard setup” characterized by the single-crossing property. This in turn leads to some pooling in equilibrium,making inferences based on quitting times less precise. Consider a situation in which separation requires the high type to quit the project inefficiently late (assuming the ability
effect dominates in the relevant parameter region). Suppose reputation concerns become
stronger in the sense that the difference in exit payoffs for the two perceived types becomeslarger. Then separation would require the high type to quit later, but beyond a point the2

Daley and Green (2014) show that the double-crossing property may emerge in an education signaling
model with grades as an additional source of information.

3

learning effect would give the high type more incentive to deviate by stopping earlier, making separation impossible under the D1 refinement. In effect the double-crossing property
imposes an endogenous constraint on how late the high type can quit. But if the hightype cannot quit too late, it becomes cheaper for the low type to mimic the high type, andpooling or semi-pooling is the result. In our model, the low type may “hold out” for theprospect of pooling with the high type, despite the fact that his instantaneous expected
payoff from continuing with the project is strictly worse than the flow payoff from exiting,which in turn forces the high type to conform to avoid adverse inference. This is in contrast to the “standard setup,” where the low type never chooses a signaling action higher
than the full-information optimal level and the high type can always go far enough to separate. It is also worth noting that the pooling equilibrium so determined is sensitive tothe distribution of types, which allows us to discuss the role of prior reputation in risky
experimentation.
Another general lesson is that signaling in risky experimentation is inherently a dynamic process, in which signaling incentives evolve over time. We distinguish between
two cases in this paper, and discuss them separately in Sections 3 and 4. In the case ofexit signaling, the exit payoff to an agent who abandons a project depends on his reputation while the reward to success does not. For example, a research scientist may be
paid a fixed bonus for making a discovery but has to find alternative employment when
he abandons his project. His exit payoff obviously will depend on the market’s perceptionof his ability. In the case of breakthrough signaling, reputation also matters for how large
the reward to a breakthrough is. For a startup entrepreneur, the initial discovery or development of a prototype product from risky experimentation is only a first step toward
ultimate success. How likely the initial discovery will turn into a profitable business isaffected by whether the entrepreneur can attract financial capital and talents to work withhim, which obviously will depend on his reputation. Exit signaling can be analyzed as if
it were a static problem, in that the equilibrium quitting times are deterministic. On theother hand, breakthrough signaling is inherently dynamic, in that the equilibrium quitting
times can be random, with separation of types occurring gradually through time.
To see this, suppose there is efficient separation in which each type stops at his fullinformation optimal timing (with the low type quitting first) if the reward to success does
not depend on reputation. Introduce breakthrough signaling by making the reward tosuccess much larger if the market perceives the agent to be a high type. Efficient separation can no longer be maintained in equilibrium, because the reward to success would
4

rise discontinuously after the low type is supposed to have quit, making it profitable forthe low type to deviate by staying on longer. As low-type agents abandon the project, thereputation for those who remain and succeed becomes higher, therefore inducing these
low-type agents not to abandon so soon. This war-of-attrition feature of the model implies
that equilibrium will entail continuous randomization by low-type agents, which further
exacerbates their incentive to hold out as it slows down the dynamic separation of types.
If this effect is strong enough, the low type stops randomizing to “hold out for exit” as time
draws close to the equilibrium quitting time of the high type. In this case equilibrium will
exhibit a continuous randomization phase which gradually winnows out low-type agents,
followed by a hold-out phase in which the remaining low-type agents get negative instantaneous payoffs, and finally leading to a mass exit by both types for the outside option.
After characterizing equilibria for the two cases, Sections 5.1 and 5.2 study comparative statics with respect to the magnitude of the reputation concern and to the prior reputation of the agent. Section 5.3 extends the model to allow for type-dependent project
quality, and considers the implications of different types of ability (the ability to identify
a good project versus the ability to successfully implement a good project) for organizational design. Section 5.4 extends the baseline model to the case with a continuum oftypes to show robustness of our findings. Finally, in Section 6, we provide a brief illustration of the implications of the analysis for venture startups—our leading example ofrisky experimentation—by offering potential ways to cope with the distortions caused by
reputation concerns.
1.3.

Related literature

Our model falls broadly into the growing literature on strategic experimentation wheremultiple parties are involved in the experimentation process. Since the seminal work of
Keller et al. (2005), much of the literature builds on exponential bandits with constanthazard rates because of their simplicity and tractability. Our analysis suggests that in thesignaling context this assumption can be substantially relaxed while still admitting a clear
characterization of equilibria, thereby offering predictions that are robust to a range of
specifications.3
Some recent works introduce reputation concerns into experimentation models as we3

Our model assumes that the conditional hazard rate is a general non-increasing function of time. See
also Boyarchenko (2017) for a model of strategic experimentation in which the unconditional hazard is ahumped-shaped function of time.

5do. In Halac and Kremer (2020) agents privately observe good or bad news about theirprojects and want to maintain reputation for project quality. In a different vein, Bonattiand Hörner (2017) analyze a career concerns model with exponential learning which incorporates a moral hazard component. Bobtcheff and Levy (2017) and Thomas (2019)consider an exponential bandit model in which a decision maker chooses when to stop theexperimentation, with the reputation payoff determined at the time of termination as inours. An important distinction is that agents in those models are heterogeneous only in
one particular dimension: they differ only in the ability to identify good projects (Thomas,
2019) or in the speed of learning (Bobtcheff and Levy, 2017). Our model encompasses
and integrates these two distinct notions of ability and clarifies the differences they make.4
When agents differ in the speed of learning, the single-crossing property may break
down due to the learning effect.5 In this sense, Bobtcheff and Levy (2017) share an important commonality with our model. In their model, however, the agent privately observes a
conclusive signal which reveals that the project is bad, with the timing of investment serving as the sole signaling device. In contrast the timing of success in our model provides aninconclusive public signal (on top of the timing of project abandonment), so that the agent
is subject to dynamic reputation concerns which further distort the quitting decision.
We can also relate our work to dynamic signaling models which incorporate additionalsources of (noisy) information (Bar-Isaac, 2003; Daley and Green, 2012; Gul and Pesendorfer, 2012; Lee and Liu, 2013). Among them, our model is more closely related to Daleyand Green (2012), who consider an environment where there are two types of seller, either
high or low, and each type decides when to trade. Aside from technical differences (theirinformation arrives via a diffusion process while ours arrives via a jump process), the key
difference is the element of experimentation. In their model, each seller knows his own
type (asset value), so that high-type sellers are always more confident to receive good news
and hence more willing to wait than low-type sellers.6 In our model, the project quality
is initially not known to anyone and needs to be uncovered via experimentation. Whilehigh-ability agents are more likely to achieve early success, they also learn more quickly
that their project is not promising. Therefore, the marginal benefit of experimentation is
4

Prendergast and Stole (1996) is one of the earliest works to explore this point, although in a very
different context.
5
Halac et al. (2016) also make this observation under the exponential bandit framework.
6
In Daley and Green (2012), there is a lower belief threshold at which low-type sellers choose to separate
from high-type sellers by accepting a low price. Meanwhile high-type sellers, who are more confident thatthe belief will recover in future, always wait until the belief becomes sufficiently favorable.

6initially higher for high-ability agents, but this relationship flips as the game progresses,
which generates the double-crossing property.
Finally, our model predicts pooling equilibria under a wide range of parameters, which
can be interpreted as a form of conformity or herding. There is now a very diverse literature which explores this possibility in various ways. For instance, Bernheim (1994)
considers a situation where agents care about status as well as intrinsic utility. He showsthat agents with moderate preferences converge to a homogeneous standard of behavior
to avoid an inference that they have undesirable extreme preferences. The logic of ourmodel is different because agents attempt to signal competence, not preferences, and conform in order to avoid an inference that they are incompetent. In this sense, our model
has a closer connection to Scharfstein and Stein (1990), who show that an agent mimics
the behavior of predecessors in the presence of reputation concerns about his competence.
Their focus is on informational conformity where agents take actions sequentially, whileours rests on stigma attached to off-path deviations.

2.

The Model

2.1.

Setup

An agent undertakes a risky project with an uncertain project quality. If the project quality
is good, it will generate success, or a “breakthrough,” at some random time τ, provided
the agent has not abandoned the project by that time. If the project quality is bad, it will
never generate a breakthrough no matter how much time the agent spends working on it.
The flow cost of working on the risky project for a small time interval of length dt is c dt.
The agent can be of two types, depending on his ability in implementing the risky
project. We capture the difference between types by specifying different distributions of
the stochastic time of success τ, given that the project quality is good. Specifically, letf H (τ) be the density function of τ for the high type and f L (τ) be the density function
for the low type, conditional on good project quality (and let FH and F L represent thecorresponding cumulative distribution functions). We assume that the conditional hazard
rate given good project quality, f i (·)/(1 − Fi (·)), is non-increasing for i ∈ {H, L} (hightype or low type). Moreover, the distributions for the two types are ordered by monotone
likelihood ratio property, in the sense that f H (·)/ f L (·) is strictly decreasing. For example,
in the exponential bandit model, a higher Poisson arrival rate of success implies that thelikelihood ratio, f H (τ)/ f L (τ) = (λH /λ L )e−(λH −λ L )τ , decreases in τ for λH > λ L . While we7

focus on the two-type case for most of our analysis, in Section 5.4, we will consider thecase with a continuum of types to show robustness of our findings.
The state of nature in our model is therefore two-dimensional, defined over the project
quality (good or bad) and the ability type (high or low). Neither the agent nor anyone
else knows the quality of the project. The common prior belief that the project quality isgood is p0 . We will later allow the prior to depend on agent type in Section 5.3; for now
assume that it is type-independent. The agent knows his own ability type, but this private
information is unavailable to the market. The market’s prior belief that the agent is a hightype is q0 . Generally, we use q t to represent the updated belief at time t that the agent isa high type. We sometimes refer to this belief as the agent’s “reputation.”
Time is continuous. At each point in time, the agent decides whether to continue
working on the risky project or to abandon it. For i ∈ {H, L}, let the agent’s strategy be
represented by σi : [0, ∞) → [0, 1], with σi (t) being the probability that the agent is stillworking on the risky project at time t conditional on no breakthrough having occurred.
Abandoning the project is irreversible; i.e., σi (t) is non-increasing in t. If the agent adopts
a pure strategy, there is a unique stopping time si such that σi (t) = 1 for all t < si andσi (t) = 0 thereafter. We sometimes abuse notation by saying that the agent’s strategy issi . (The agent will stop before this time if he has already achieved success in the risky
project.) The agent’s decision at each point in time is publicly observable. The arrival of
success is also publicly observable, representing an additional source of information.
2.2.

Payoffs

The game ends either when the risky project generates a breakthrough, or when the agent
abandons the project. We assume that there is a competitive market that pays an agent
commensurate with his expected output whenever the game ends. Let Wi denote theproductivity (in present value) of type i after he achieves a breakthrough and w i the productivity after he abandons the project. Therefore, if the project generates a breakthroughat time τ, the agent’s payoff is W (τ) = qτ WH + (1 − qτ )WL . If the agent’s reputation
when he abandons the project at time t is q t , his outside opportunity from quitting is
w(t) = q t w H + (1 − q t )w L . As explained below, we take a reduced-form approach andregard W (τ) and w(t) as representing the continuation payoffs of success and failure, respectively. Although we assume that the continuation payoffs depend only on the agent’s
reputation with no type-specific payoffs, adding a type-specific component does not change

8the substance of our analysis.7
We make several assumptions regarding the productivities. First, we generally assume
w H > w L , so that the agent’s reputation always matters in case of failure. In case of
success, we assume WH ≥ WL and consider two cases. If WH = WL , then all reward tosuccess is non-contingent on the agent’s reputation at the time of success. If WH > WL ,part of the reward will depend on the agent’s reputation at the time of success. As we willsee, the difference between the two cases proves to be critical, and we analyze these cases
separately. Finally, we also assume that achieving a breakthrough is always better than theoutside opportunity of working in the labor market. Specifically,
WL > w H .

(1)Assumption (1) implies that W (t) > w(t) at any time t and for any market belief, ensuringthat the net value of success is always positive.8
The agent discounts future payoffs at rate ρ. For i ∈ {H, L}, the expected payoff to anagent of type i if he plans to abandon the project at time s isZs
Ui =e−ρτ p0 f i (τ) [W (τ) − C(τ)] dτ + e−ρs (1 − p0 Fi (s)) [−C(s) + w(s)] ,(2)0where C(t) = c(eρt − 1)/ρ is the accumulated cost of working with the risky project for aperiod of length t.
2.3.

Interpretation of the model

To provide some suitable interpretation of our theoretical framework, consider the caseof a venture startup. A breakthrough in this context refers to reaching a milestone overthe course of the project, such as successfully developing a prototype product, establishing
distribution channels, gaining market recognition, and so on. If the entrepreneur achieves
a breakthrough, he may build on this success and take the next step, the expected payoff ofwhich is summarized by W (τ). If he abandons the project, he will be forced to take an exit
option, such as starting a new business all over again or switching to the labor market as aworker. In either case, the expected payoff of this contingency depends on his reputationand is summarized by w(t).
78

See the Online Appendix for the analysis of this case.
The role of this assumption is purely technical, as the same economic intuition applies even if it does

not hold.

9As noted above, whether the reward to success is contingent on the agent’s reputation
or not is crucial in our model, and we make a clear distinction between WH = WL (which
we label as the case of exit signaling) and WH > WL (the case of breakthrough signaling).
Exit signaling is primarily motivated by the difference in exit payoffs (w H versus w L ) for
agents who fail the experimentation, while the difference in reward to success (WH versus
WL ) also matters for the case of breakthrough signaling. The difference between the twocases reflects, among other things, where in the experimentation process the agent stands.
For instance, in the case of a venture startup, successfully developing a prototype productis only a step toward a bigger goal. Reputation still plays an essential role as it facilitatesaccess to complementary inputs and makes the entrepreneur even more likely to succeedin the future, the extent of which is captured by WH − WL . On the other hand, gaining
global market recognition is valuable in and of itself. Once this level of success is achieved,
reputation matters much less, and WH − WL is regarded as relatively small.
2.4.

Equilibrium selection

A pair of strategies (ŝH , ŝ L ) and the beliefs {q t } constitute an equilibrium if ŝi maximizes
Ui given {q t } for i ∈ {H, L} and if the beliefs {q t } are consistent with Bayes’ rule and thestrategies whenever applicable. As is typical in signaling models, these requirements do
not pin down a unique equilibrium, because we can construct an arbitrary equilibriumby manipulating off-equilibrium beliefs. Consider, for instance, a pooling equilibrium inwhich both type quits at some t p . For a range of values of t p , we can construct such anequilibrium by assigning off-equilibrium belief that an agent is a low type if he quits at any
t 6= t p . To obtain sharper and more reasonable predictions, therefore, we adopt the D1
criterion (Cho and Kreps, 1987) for equilibrium refinement. While there are many specificcases we need to examine, we first note a result that generally holds in our setup (theproof will be provided when we discuss each specific case).
Proposition 1. There always exists a unique equilibrium which survives the D1 criterion.
A crucial feature of our model, which distinguishes it from a standard signaling model,
is that preferences do not satisfy the single-crossing property. Instead, indifference curves
for the two types cross twice in the relevant space. We call this a double-crossing property. It
turns out that the D1 criterion does not always select the least-cost separating equilibriumin this environment.9 Since the problem becomes substantially more complicated when9

It is interesting to note that D1 and the Intuitive Criterion offer slightly different predictions in our

10WH > WL , we first consider the case of exit signaling with WH = WL in Section 3 to focus
only on the issue arising from the double-crossing property. Then, in Section 4, we let
WH > WL to deal with the inherently dynamic issues arising from the case of breakthroughsignaling.

3.

Exit Signaling

3.1.

Reputation concerns

Throughout Section 3 we assume that WH = WL = W > w H , so that the reward to successin the risky project is simply W . In this case, the agent’s reputation matters only when heabandons the project and takes up his exit option, the timing of which can be controlled
perfectly by him. This feature simplifies the problem to a great extent, and allows us toclarify the key mechanism of the double-crossing property.
The updated belief about an agent’s type at the time of project abandonment dependson (i) inferences based on the agent’s choice and its consistency with the equilibrium
strategies of the two types; and (ii) observation about the timing of success τ. We useq̂ = Pr[ high type | σ L , σH , stops at t]
to denote the interim belief based on equilibrium inference alone, and user(t; q̂) = Pr[ high type | σ L , σH , stops at t, τ > t]=q̂(1 − p0 FH (t))
q̂(1 − p0 FH (t)) + (1 − q̂)(1 − p0 F L (t))

to denote the belief based on both (i) and (ii). If the interim belief for an agent who
stops at time t is q̂, the final reputation r(t; q̂) of this agent incorporates the informationfrom the event that he stops without a breakthrough, i.e., τ > t. Monotone likelihood
ratio property implies that FH (t) ≥ F L (t), and hence r(t; q̂) ≤ q̂. Failure to achieve a
breakthrough is bad news for the agent’s ability.
It is straightforward to verify that r(t; q̂) decreases in t if and only if g H (t) > g L (t),
where




f i (t)p0 (1 − Fi (t))
p0 f i (t)g i (t) ==1 − p0 Fi (t)1 − Fi (t)p0 (1 − Fi (t)) + 1 − p0(3)model, even though our model has only two types. All equilibria we identify survive the Intuitive Criterion,
but equilibrium is no longer unique under it. The Intuitive Criterion cannot rule out the least-cost separating
equilibrium when there is a D1 pooling equilibrium.

11

is the unconditional hazard rate of success in the risky project for type i ∈ {H, L}. The firstterm is the conditional hazard function for type i given that the project quality is good.
The second term is the posterior belief that the project quality is good given that an agentof type i fails to obtain success by time t. Monotone likelihood ratio property implies thatthe conditional hazard rate for the high type is always greater than that for the low type.
We call this the ability effect. However, the same property also implies that the posterior
belief about project quality upon failure to obtain success is smaller for the high type thanthat for the low type. We call this the learning effect, because more able agents learn more
quickly that they are likely to be working on a bad project than do less able ones if successhas not been already observed. Note that the learning effect disappears when p0 = 1,
pointing to the essential role of experimentation in our setup.
The following result is crucial for our subsequent analysis.
Lemma 1. The hazard rate g i (·) is strictly decreasing for i ∈ {H, L}. There exists a unique t̂such that g H (t) > g L (t) if and only if t < t̂, and g H (t) < g L (t) if and only if t > t̂.
Proof. The first term of equation (3) is non-increasing by assumption, and the second
term is strictly decreasing. Hence g i (·) is strictly decreasing. At t = 0, g H (0)/g L (0) =f H (0)/ f L (0) > 1. As t approaches infinity, g H (t)/g L (t) approaches lim t→∞ f H (t)/ f L (t),which is less than 1. Therefore, there exists t̂ such that g H ( t̂)/g L ( t̂)−1 = 0. The derivativeof g H (t)/g L (t) at t = t̂ has the same sign as the derivative of f H (t)/ f L (t) at t = t̂, which
is negative. This shows that g H (t)/g L (t) − 1 is single-crossing from above.
The posterior belief about project quality is decreasing in t. Thus, differences in ability
to implement a good project becomes less relevant over time, and Lemma 1 shows thatthe learning effect dominates the ability effect beyond time t̂. If a high-ability agent has
been working on a project for a long time without achieving a breakthrough, he shouldhave recognized that the project is not promising and quit. As noted in the introduction,in this environment, perseverance is a sign of strength in early stages of experimentation,
but a sign of weakness in later stages.

12

3.2.

Double-crossing property

The objective function (2) for type i ∈ {H, L} in the case WH = WL = W reduces to
Zs
Ui (s, q̂) =e−ρτ p0 f i (τ) [W − C(τ)] dτ0+ e−ρs (1 − p0 Fi (s)) [−C(s) + w L + r(s; q̂)(w H − w L )] .

(4)

The marginal rate of substitution between stopping time s and interim belief q̂, denoted
M RSi (s, q̂), is given byg i (s)[W − w L − r(s; q̂)(w H − w L )] − ρ(w L + r(s; q̂)(w H − w L )) − c + (∂ r/∂ s)(w H − w L ).
(∂ r/∂ q̂)(w H − w L )Observe that the marginal rate of substitution depends on agent type only through thehazard rate g i (s). By assumption (1), the term in square brackets is positive. Lemma 1 thenimplies that M RSH (s, q̂) > M RS L (s, q̂) if and only if s < t̂, and M RSH (s, q̂) < M RS L (s, q̂) if
and only if s > t̂. Thus, indifference curves of the two types cross twice: before time t̂ theopportunity cost of quitting is larger for the high type than for the low type; the opposite
is true beyond t̂.
Because g H (t) − g L (t) is single-crossing from above, the slope of the indifference curvechanges more rapidly with respect to the quitting time s for the high type than for the low
type. At t̂, the slopes of the two indifference curves are equal; moreover the indifference
curve of the high type is “more convex” than that of the low type. See Figure 1 for illustration. An implication of this greater convexity is that, for any interim belief q̂, whenever
the high type is indifferent between some outcome (s0 , q̂0 ) and the outcome ( t̂, q̂), the low
type strictly prefers the former to the latter. Formally,
UH (s0 , q̂0 ) = UH ( t̂, q̂) =⇒ U L (s0 , q̂0 ) > U L ( t̂, q̂).

(5)

If there is an equilibrium in which both types quit at t̂, the set of interim beliefs thatwould support deviation to s0 6= t̂ by the high type is strictly contained in the corresponding
set for the low type. By the D1 criterion, the market should assign off-equilibrium belief
that an agent is a low type if he quits at s0 .
3.3.

Equilibrium with exit signaling

To set the stage for equilibrium analysis, we first consider the full-information outcome. Letsi∗ represent the stopping time chosen by type i ∈ {H, L} if his type is known. The optimal
13

(a)

(b)
∗
𝑠"

1

∗
𝑠"

1
𝑈"

𝑈"--

𝑈"-

𝑈#--

𝑈#--

𝑞,

𝑞,

𝑈"-

𝑈#-

0

𝑠

𝑞.

𝑈#-

𝑈"

𝑠#∗

𝑈#

𝑈#

𝑠̅ 𝑡̂

0

𝑠

𝑡̂

𝑠

𝑠#∗
𝑠

𝑠̅

Figure 1. The indifference curves U L and UH are tangent to each other at t̂, with UH “more convex”∗
than U L at that point. The point (sH, 1) is the “bliss point” for the high type. In panel (a), indifference

curves are single-crossing to the left of t̂. The only equilibrium is a separating equilibrium: the low
type quits at s∗L and the high type quits at s. In panel (b), the indifference curve U L0 crosses UH twice—first from below to the left of t̂, then from above to the right of t̂. The only equilibrium is a pooling
equilibrium with both types quitting at t̂.

stopping rule can be obtained by value-matching and smooth-pasting (or, equivalently, by
the first-order condition for maximizing the objective function (4) for the relevant type),
which givesg i (si∗ )(W − w i ) − (ρw i + c) = 0.

(6)

The first term on the left-hand-side is the expected capital gain from extending the risky
project for a small interval of time. The second term is the opportunity cost of doing so.
Note that s∗L can be greater than or less than sH∗ .
In any signaling equilibrium, the low type cannot do worse than choosing s∗L and revealing himself to be a low type, which gives him a utility of U L (s∗L , 0). It is useful to define
two thresholds, s < s, such that
U L (s, 1) = U L (s∗L , 0) = U L (s, 1).
The low type never wants to stop before s or after s, even if by doing so he could successfully
mimic the high type.10 Therefore, if sH∗ and s∗L are sufficiently far apart, then neither type10

It is possible that U L (0, 1) > U L (s∗L , 0), in which case s is defined to be equal to 0. For ease of exposition,

we assume that W is sufficiently large that s is positive.

14

wants to deviate from his optimal stopping time, and the full-information outcome is theunique equilibrium outcome in the model. To focus on the interesting case, we assume in
the following that the incentive compatibility constraint for the low type is binding, in thesense that U L (s∗L , 0) < U L (sH∗ , 1), so that the full-information outcome is not achievable.
This is equivalent to sH∗ ∈ (s, s).
Consider first the case of separating equilibrium.11
Proposition 2. In the case of exit signaling, suppose the full-information outcome is not
feasible.
(a) If t̂ ≤ s, the equilibrium is separating, with the high type quitting at s and the low type
quitting at s∗L .
(b) If t̂ ≥ s, the equilibrium is separating, with the high type quitting at s and the low type
quitting at s∗L .
Proof. When t̂ ≤ s, we have M RSH (s, q̂) ≤ M RS L (s, q̂) for all s ∈ [s, s]. Because the singlecrossing property is satisfied, with the high type having a greater incentive to quit earlier than the low type does. It follows from a standard refinement argument (Cho andKreps, 1987) that the least-cost separating equilibrium (corresponding to the stopping
times (s, s∗L ) for high type and low type, respectively) is the only equilibrium that satisfies
the D1 criterion. When t̂ ≥ s, we have M RSH (s; q̂) ≥ M RS L (s; q̂) for all s ∈ [s, s]. The
single-crossing property is again satisfied, but with the high type having less incentive toquit earlier than the low type does. The least-cost separating equilibrium in this case is forthe high type to quit at s, and for the low type to quit at s∗L , and this is the only equilibriumthat satisfies D1.
The value of t̂ is determined entirely by the statistical properties of the hazard functions
and is independent of the payoff parameters. If t̂ is very low, the learning effect dominates
throughout the relevant region. Because the high type has more incentive to quit early inthis case, the equilibrium outcome is determined by the condition that the high type quits
prematurely at s to just deter the low type from mimicking. If t̂ is very high, the ability
effect dominates throughout the relevant region. Because the high type has more incentive
to stay longer with the project, the equilibrium outcome is determined by the condition
that the high type stays inefficiently long until s to just deter the low type from mimicking.
11

Throughout the analysis, when we say separating equilibrium, it means full separation between the two

types where the quitting time fully reveals the agent’s type.

15

Although equilibrium is separating in both cases of Proposition 2, the direction of how thehigh type separates from the low type differs. In this regard, our result is different fromthe standard setup. Panel (a) of Figure 1 illustrate the least-cost separating equilibrium(with the high type quitting later) corresponding to case (b) of Proposition 2.
The values of s and s needed to deter the low type from mimicking the high type become
more extreme as the signaling incentive (measured by w H − w L ) becomes stronger, whilethe value of t̂ remains unchanged. When the signaling incentive is strong enough, we will
have t̂ ∈ (s, s). In this case the ability effect and the learning effect dominate in different
stages of experimentation, and our model predicts pooling (or semi-pooling) between the
different types.
Proposition 3. In the case of exit signaling, suppose the full-information outcome is not
feasible, and suppose t̂ ∈ (s, s).
(a) If U L ( t̂, q0 ) ≥ U L (s∗L , 0), the equilibrium is full pooling, with both types quitting at t̂.
(b) If U L ( t̂, q0 ) < U L (s∗L , 0), the equilibrium is semi-pooling, with the high type quitting att̂ and the low type randomizing between quitting at t̂ and s∗L .
Proof. We first show that there cannot be a separating equilibrium. Suppose otherwise,
and let the high type quit at some time t in this equilibrium. If t ∈ (s, s), the low type
could profitably deviate by stopping at t. If t ≤ s, the high type could profitably deviate bystopping later at t + ε for some small positive ε, because according to the D1 criterion theoff-equilibrium belief associated with such a deviation is that it comes from a high type.
Similarly, if t ≥ s, the high type could profitably deviate by stopping a bit earlier.
Next, if the two types pool (or partially pool) by both stopping at the same time t withpositive probability, then we must have t = t̂. Otherwise, by stopping a little later (if t < t̂)or a little earlier (if t > t̂), an agent could obtain a discrete improvement in the market’s
belief of his type from some q̂ < 1 to 1.
Finally, condition (5) shows that if there is a semi-pooling equilibrium in which thehigh type randomizes between quitting at t̂ and quitting at some other time t 0 , the low
type strictly prefers to quit at t 0 . This contradicts our earlier conclusion that the two types
cannot partially pool by both stopping at t 0 6= t̂. Hence, in equilibrium, the high type quitsat t̂ with probability 1. Given that the high type quits only at t̂, by D1 the market assigns
interim belief q̂ = 0 to an agent who quits at t 6= t̂. For such interim belief, if a semipooling equilibrium exists, the low type must quit at s∗L . This leaves us with two possible
16

types of equilibrium: (i) full pooling in which both types quit at t̂; or (ii) semi-pooling inwhich the low type quits at both t̂ and s∗L with positive probability, and the high type quitsat t̂.
If U L ( t̂, q0 ) ≥ U L (s∗L , 0), semi-pooling is not an equilibrium. In a semi-pooling equilibrium, because the high type quits with probability 1 while the low type quits with probability strictly less than 1 at t̂, the interim belief q̂ t̂ based on equilibrium inference is strictly
higher than the initial belief q0 . Thus, U L ( t̂, q0 ) ≥ U L (s∗L , 0) implies U L ( t̂, q̂ t̂ ) > U L (s∗L , 0),meaning that the low type strictly prefers quitting at t̂ to quitting at s∗L , a contradiction.
Therefore, it is a unique equilibrium for both types to quit at t̂, and the market assigns
an interim belief q0 upon observing an agent quitting at t̂. Neither type could profitably
deviate because quitting at another time would be interpreted as deviation by a low type.
If U L ( t̂, q0 ) < U L (s∗L , 0), full pooling is not an equilibrium, because the low type could
profitably deviate by quitting at s∗L instead. Given that there exists a unique q ∈ (q0 , 1)such that U L ( t̂, q) = U L (s∗L , 0), the semi-pooling equilibrium is unique, with the high type
quitting at t̂ with probability 1, and the low type doing the same with some positive probability so that the interim belief about an agent who quits at t̂ is exactly q. The remaining
low types quit at s∗L . By construction, the low type is indifferent between quitting at t̂ ands∗L . The high type strictly prefers quitting at t̂ to quitting at another time, because suchdeviation would be interpreted as made by a low type.
When indifference curves of the two types exhibit the double-crossing property, Proposition 3 shows that the D1 refinement does not yield the least-cost separating equilibriumas the unique equilibrium outcome. Instead, equilibrium entails pooling at t̂ (i.e., the point
where the indifference curves of the two types are tangent to one another), supported by
the belief that an agent who abandons the project at any time other than t̂ is a low type.
See panel (b) of Figure 1 for an illustration of a full pooling equilibrium corresponding tocase (a) of Proposition 3. It is interesting that the equilibrium time t̂ for both types to quit
depends only on the distributions of the timing of success (i.e., on p0 , F L and FH ), but noton the costs and benefits of risky experimentation.

4.

Breakthrough Signaling

Beginning from this section, we drop the assumption that WH = WL and assume WH > WL
instead. Because the reward to success, W (τ), is a function of the reputation of the agent
who is staying to work with the risky project at the time of success τ, this introduces a17

dynamic element into the signaling model that is absent in the case discussed in Section 3:
the agent’s reputation now matters even when he achieves a breakthrough, the timing of
which arrives stochastically (as opposed to the timing of quitting, which is deterministic)
and is beyond his control. Since the timing of achieving a breakthrough now has signaling
value,12 we refer to this case as “breakthrough signaling.”
Given the equilibrium strategies σ L and σH of the two types, we useq̃ = Pr[ high type | σ L , σH , has not stopped by t]
to represent the interim belief about an agent who has not abandoned the project by time
t based on equilibrium inference alone, and let
R(t; q̃) = Pr[ high type | σ L , σH , has not stopped by t, τ = t]=q̃ f H (t)q̃ f H (t) + (1 − q̃) f L (t)represent the final belief that incorporates the information from the event that a breakthrough occurs at time t (i.e., τ = t). Because f H (t)/ f L (t) is decreasing, R(t; q̃) is decreasing in t.
It is worth noting that the reputation upon success may be higher or lower than the
reputation upon failure to obtain success. In particular, it is straightforward to show that,for any interim belief q, R(t; q) ≥ r(t; q) if and only if t ≤ t̂, and R(t; q) ≤ r(t; q) if andonly if t ≥ t̂. In other words, success that comes too late may be worse for an agent’s
reputation than no success. This is because late success is evidence for low-ability type,while no success can be partly attributable to bad project quality.
For the case of breakthrough signaling, we continue to have the following result. The
logic of this result is the same as that for the case of exit signaling; and the proof is relegatedto the Appendix.
Lemma 2. If both types of agent abandon the risky project at some time t with positive
probability in equilibrium, then t = t̂.
Lemma 2 implies that a full pooling equilibrium must have both types quit at t̂. Nevertheless, the possibility of breakthrough signaling makes a non-trivial difference to our
12

By this we mean that the timing of achieving a breakthrough reveals payoff-relevant information. It
also reveals some information about the agent’s type in the case of exit signaling, but this information is not
payoff-relevant because the reward to success is fixed and independent of the agent’s reputation.

18

analysis whenever equilibrium entails some separation between the two types. To see this,note that the instantaneous benefit from working with the risky project for a small timeinterval of length dt for an agent of type i ∈ {H, L} is
g i (t) [WL + R(t; q̃)(WH − WL )] dt,
which depends on the value of q̃ when WH > WL . But the interim belief q̃ about an agent
who stays evolves over time as different types of agents quit at different times to separate
from one another. If the high type quits before the low type, then q̃ falls as the high type
quits, and this can reduce the incentive of the remaining agents to continue working withthe risky project. The opposite is true if the low type quits before the high type.
For i ∈ {H, L}, let si∗ (q̃) represent the solution to the following equation:g i (si ) [WL + R(si ; q̃)(WH − WL ) − w i ] − (ρw i + c) = 0.

(7)

Note that s∗L defined in equation (6) is the same as s∗L (0); and sH∗ is the same as sH∗ (1).
When WH > WL , equation (7) gives the optimal stopping rule for an agent of type i, if bycontinuing the market belief about his type would be q̃ and by quitting he would reveal
his true type. Because the left-hand-side of (7) decreases in si and increases in q̃, si∗ (q̃) is
increasing in q̃. A higher interim reputation for stayers raises the reward to success andtends to delay quitting.
The analysis of breakthrough signaling depends crucially on which type quits first. It
turns out that if s∗L (q0 ) > t̂, the high type quits first. The incentive for the low type to stayfalls when the high type quits because the former can no longer pool with the latter. Thus,
the low type may quit before s∗L (q0 ). If s∗L (q0 ) < t̂, the low type quits first. But when the
low type quits, the interim belief about stayers improves, which makes quitting by the low
type self-defeating. Equilibrium in this case generally involves continuous randomizationby the low type, who ends up quitting after s∗L (q0 ). We discuss these two cases in turn.
4.1.

High type quits first

In this subsection, we consider the case where s∗L (q0 ) ≥ t̂. This case obtains, for example,
when the outside option for the low type w L or the flow cost of working with the project c
is small. It can be shown that s∗L (q0 ) ≥ t̂ implies s∗L (q0 ) > sH∗ (q0 ),13 so the high type tends
to quit before the low type.
From equation (7), w L < w H implies g L (s∗L (q0 )) < g H (sH∗ (q0 )). If s∗L (q0 )) ≥ t̂, this implies g H (s∗L (q0 )) <∗g H (sH (q0 )) and therefore s∗L (q0 ) > sH∗ (q0 ).
13

19

For t ≤ s∗L (0), define the following function:v(t; s∗L (0)) =Z s∗L (0)p0 f L (τ)[WL − C(τ − t)] dτ
1 − p0 F L (t)t
1 − p0 F L (s∗L (0)) ∗+ e−ρ(s L (0)−t)−C(s∗L (0) − t) + w L .
1 − p0 F L (t)e−ρ(τ−t)Conditional on the project not having achieved a breakthrough by time t, the functionv(t; s) gives the payoff (from the perspective of time t) from continuing with it until times, given that the agent is known to be a low type. Recall that s∗L (0) is the optimal stopping
time when an agent is known to be a low type, so v(t; s) ≤ v(t; s∗L (0)) for any s ≥ t.
In the following, we simply write v(t) for v(t; s∗L (0)) when there is no confusion. It is
straightforward to verify that v(t) strictly decreases in t for t < s∗L (0), with v 0 (s∗L (0)) = 0
and v(s∗L (0)) = w L .
Define t such that v(t) = w H . (If no such t exists, we let t = 0.) When sH∗ (q0 ) ≤ t,
we have v(sH∗ (q0 )) ≥ w H . The incentive compatibility constraint for the low type is notbinding, because the low type prefers staying with the risky project until time s∗L (0) toquitting at sH∗ (q0 ) to obtain an outside market wage of w H . In this case, it is an equilibrium
for the high type to quit at sH∗ (q0 ) and the low type to quit at s∗L (0).
The following proposition characterizes the equilibrium when the incentive compatibility constraint is binding. The proof is relegated to the Appendix.
Proposition 4. In the case of breakthrough signaling, suppose s∗L (q0 ) ≥ t̂ and sH∗ (q0 ) > t.
(a) If t̂ ≤ t, the equilibrium is separating. The high type quits at t and the low type quitsat s∗L (0).
(b) If t̂ ∈ (t, s∗L (0)) and v( t̂) > w L + r( t̂; q0 )(w H − w L ), the equilibrium is semi-pooling.
The high type quits at t̂ and the low type randomizes between quitting at t̂ and s∗L (0).
(c) If t̂ ∈ (t, s∗L (0)) and v( t̂) ≤ w L + r( t̂; q0 )(w H − w L ), or if t̂ ∈ [s∗L (0), s∗L (q0 )], theequilibrium is full pooling. Both types quit at t̂.
The logic of Proposition 4 is very similar to that described in the case of exit signaling.
Case (a) is a least-cost separating equilibrium in which the high type quits first, because
the learning effect dominates in the relevant region when both s L (q0 ) and sH∗ (q0 ) exceedt̂. In cases (b) and (c), the double-crossing property induces pooling in equilibrium under
the D1 criterion. In case (b), there is an interim belief q̂ > q0 such that if a fraction of low
20

types quit at t̂, then the remaining low types are indifferent between quitting at that timeand continuing with the risky project until s∗L (0). So the equilibrium is semi-pooling. In
case (c), the low type prefers to quit at t̂ and pool with the high type in the labor market
than to continue with the project until s∗L (0). So there is a full pooling equilibrium.
4.2.

Low type quits first

We now consider the case where s∗L (q0 ) < t̂. This case obtains when w L or c is large
relative to the reward to success. Note also that s∗L (q0 ) < t̂ implies sH∗ (q0 ) < t̂.14 Sinceboth s∗L (q0 ) and sH∗ (q0 ) are in the region where the ability effect dominates, the low typehas an incentive to quit before the high type. Nevertheless we will show that any separationin this case must occur gradually over time.
There cannot be an equilibrium in which the high type separates from the low typeby quitting before s∗L (q0 ); otherwise the low type would profitably mimic the high type.15
Lemma 2 also establishes that if the two types pool, then it must occur at t̂ > s∗L (q0 ). Theseobservations imply that, by the time the game reaches time s∗L (q0 ), the high type has notabandoned the project yet. At this time, the low type would prefer to stop if the reputationof stayers were fixed at q0 . However, if the low type stops with positive probability, theinterim belief q̃ about those who stays at time s∗L (q0 ) would jump up, which means that a
low type could profitably deviate by staying a little bit longer instead of quitting at s∗L (q0 ).
The only way to eliminate this deviation incentive is to have the low type exit continuously
at some atomless rate (i.e., σ̇ L (t) < 0) when t ≥ s∗L (q0 ).
The low-type’s payoff is pinned down by the outside option w L when σ̇ L (t) < 0. Ifq̃(t) is the interim belief about an agent who is still staying at time t, the low type must
be indifferent between staying and quitting whenever σ̇ L (t) < 0. This condition can be
written asg L (t) [WL + R(t; q̃(t))(WH − WL ) − w L ] − (ρw L + c) = 0.

(8)

The interim belief q̃(t) satisfiesq̃(t) =

q0.
q0 + (1 − q0 )σ L (t)(9)

As t increases from s∗L (q0 ) to s∗L (1), q̃(t) must rise continuously from q0 to 1 to maintainthe indifference condition (8), and σ L (t) must fall continuously from 1 to 0 according toequation (9).
1415

If s∗L (q0 ) < t̂ and sH∗ (q0 ) ≥ t̂, we would have g L (s∗L (q0 )) > g H (sH∗ (q0 )), which contradicts equation (7).
See Fact 1 in the proof of Proposition 5 for a detailed argument behind this claim.

21

For t ∈ [s∗L (q0 ), t̂], we define the following function:
V (t; t̂) =Z t̂

p0 f L (τ)[WL + R(τ; q̃(t))(WH − WL ) − C(τ − t)] dτ
1 − p0 F L (t)t
1 − p0 F L ( t̂) + e−ρ( t̂−t)−C( t̂ − t) + w L + r( t̂; q̃(t))(w H − w L ) ,1 − p0 F L (t)e−ρ(τ−t)where q̃(t) follows equation (8) for t ≤ s∗L (1) and is equal to 1 for t > s∗L (1). The function
V (t; t̂) gives the payoff to a low type if he stays with the risky project between time t and t̂,and then pools with the high type to quit at t̂. Whether there will be some separation or not
will depend on whether V (s∗L (q0 ); t̂) is less than or greater than w L . If V (s∗L (q0 ); t̂) ≥ w L ,the low type prefers to wait and pool with high types at time t̂ than to quit at s∗L (q0 ).
There will be full pooling. If V (s∗L (q0 ); t̂) < w L , some separation will occur gradually inequilibrium.
Proposition 5. In the case of breakthrough signaling, suppose s∗L (q0 ) < t̂.
(a) If V (s∗L (q0 ); t̂) ≥ w L , the equilibrium is full pooling. Both types quit at t̂.
(b) If V (s∗L (q0 ); t̂) < w L , there exists a unique t 0 ∈ (s∗L (q0 ), t̂) such that V (t 0 ; t̂) = w L .
(i) If t 0 < s∗L (1), the equilibrium is semi-pooling. The high type quits at t̂ with probability 1. The low type’s behavior has four phases: 1) stays (i.e., σ L (t) = 1) fort < s∗L (q0 ); 2) randomizes between quitting and staying (with σ L (t) determinedby equations (8) and (9) and decreasing continuously) for t ∈ [s∗L (q0 ), t 0 ]; 3)stays (with σ L (t) = σ L (t 0 ) > 0) for t ∈ (t 0 , t̂); and 4) quits with probability 1(i.e., σ L (t) = 0) at t = t̂.
(ii) If t 0 ≥ s∗L (1), the equilibrium is separating. There exists a unique t 1 ∈ (s∗L (1), t̂)such that V (s∗L (1); t 1 ) = w L and the high type quits at max{t 1 , sH∗ (1)}. The low
type’s behavior has three phases: 1) stays for t < s∗L (q0 ); 2) randomizes between
quitting and staying for t ∈ [s∗L (q0 ), s∗L (1)]; and 3) quits with probability 1 at
t = s∗L (1).
The strategy of the low type described in Proposition 5 is qualitatively different fromthat described in the earlier propositions. In particular the low type always quits at or
before the optimal time s∗L (q0 ) in other cases, but Proposition 5 shows that it is possible forthe low type to stay beyond s∗L (q0 ). We leave the details of the proof of Proposition 5 to the
Appendix. Intuitively, if V (s∗L (q0 ); t̂) < w L , the low type would prefer to quit than to stayat t = s∗L (q0 ). Because the reward to success would rise as the low type quits, continuous
22

randomization is needed to sustain equilibrium. During the phase of randomization, thepayoff to the low type is pinned down by the exit option w L . As time t gets closer to t̂,the prospect of pooling with the high type by quitting at t̂ becomes increasingly attractive;
so the randomization phase switches to a “hold out” phase in which the low type stays
with probability 1 for t ∈ (t 0 , t̂). Because g(·) and R(·; q̃(t 0 )) are strictly decreasing; theindifference condition (8) at time t = t 0 implies that, for t ∈ (t 0 , t̂),g L (t) [WL + R(t; q̃(t 0 ))(WH − WL ) − w L ] − c < ρw L .
In other words the instantaneous expected payoff from continuing with the project isstrictly worse than the flow payoff from the exit option. Nevertheless the low type strictlyprefers to stay because he is “holding out” in order to pool with the high type to obtain abetter exit payoff at a later date.
If t̂ is large, the t 0 that satisfies V (t 0 ; t̂) = w L will also be large. For t 0 ≥ s∗L (1), therate of exit determined by equations (8) and (9) implies that all low types would have quit
by time s∗L (1). Moreover, since V (s∗L (1); ·) is decreasing on [s∗L (1), t̂] with V (s∗L (1); t 1 ) =w L , the low type cannot profitably deviate to any t 0 ≥ t 1 . Because the high type canalways benefit from moving towards max{t 1 , sH∗ (1)}, the only separating equilibrium thatcan survive the D1 criterion is the one in which the high type quits at max{t 1 , sH∗ (1)}.
4.3.

Dynamic distortions

In our model, the dynamic separation of types is incomplete because we have a public information source which imperfectly reveals the agent’s type—the arrival of a breakthroughis a noisy signal that can come from either type. The presence of public news gives rise to
dynamic reputation concerns which further distort the timing of project abandonment. Although the equilibrium characterization with breakthrough signaling is more complicated,the basic insight is relatively clear: the high type quits too early, and the low type quits toolate (relative to the full-information benchmark).
If the high type quits before the low type, Proposition 4 states that the high type quits
at max{t, t̂}, which is earlier than sH∗ (q0 ). Since sH∗ (q0 ) < sH∗ (1), the high type quits prematurely compared to the full-information benchmark. This is because the “reputational
value of success” is necessarily lower when there are more low-type agents around, which
reduces the continuation payoff of risky experimentation and forces the high type to abandon the project too early.
The timing of project abandonment is even more distorted when the low type quits
23

before the high type. In the case of exit signaling the low type quits once and for all ats∗L (0). In breakthrough signaling, the low type has an incentive to inefficiently wait untils∗L (q0 ) because the “reputational value of success” is higher with more high-type agentsaround. Moreover, when a low-type agent quits, it raises the interim belief about theability type of stayers, and hence the continuation payoff. The game thus resembles awar of attrition in that each low-type agent is waiting for others to drop out. Because ofthis, the low type must randomize over time, causing the separation of types to occur only
gradually, and even later than s∗L (q0 ).
It is worth emphasizing that as a consequence of these forces, there exists no efficient(full-information) equilibrium under breakthrough signaling, i.e., the one in which the
low type quits at s∗L (0) and the high type at sH∗ (1), no matter how far apart they are fromeach other. This is a stark difference from the case under exit signaling, where efficient
separation is feasible as long as s∗L and sH∗ are sufficiently far apart from each other.

5.

Discussion

5.1.

The role of reputation concerns

One of the most important predictions of our model is the presence of pooling equilibria
in which high-type agents quit at the same time as (some of the) low-type agents. But ouranalysis also predicts homogenization of quitting times across types, in comparison to thestandard setup. Below, we will illustrate this point and its efficiency implications.
The extent of reputation concerns in our model can be measured by w H −w L , where theagent possesses stronger reputation concerns when the productivity difference between the
two types is larger. Here, we examine how equilibrium varies with changes in the extent
of reputation concerns by increasing w H for a given w L (while holding WH = WL ).16 There
are two cases, depending on whether s∗L is larger than t̂ or not. See Figure 2 for illustration.
Note that in both cases, the space of w H is divided into four regions.17
16

The effect of an increase in WH for a given WL also has a similar effect and induces more pooling, but
its effect is more complicated and less clear. In this section, therefore, we focus on the impact of w H − w L
while holding WH = WL fixed.
17
It is easy to see that: (1) s∗L is independent of w H while sH∗ decreases in w H (higher exit payoff induces
the high type to quit earlier); (2) s decreases and s increases in w H ; (3) both sH∗ and s decrease and tend to
0 as w H increases. We can also show that s crosses t̂ before sH∗ crosses it in panel (a), while sH∗ and s nevercross each other in panel (b).

24

(a)

(b)

𝑠

𝑠
𝑈𝐿 (𝑡̂, 𝑞0 ) = 𝑈𝐿 (𝑠𝐿∗ , 0)

𝑈𝐿 (𝑡̂, 𝑞0 ) = 𝑈𝐿 (𝑠𝐿∗ , 0)

𝑠𝐿∗

𝑡̂

𝑠𝐿∗

𝑡̂

𝑠𝐻∗

𝑠

full-info

separatingsemi-pooling

Proposition 2(a)Proposition 3(b)full pooling𝑠𝐻∗𝑠𝑤𝐻 − 𝑤𝐿

full-info

Proposition 3(a)separatingsemi-poolingfull pooling

Proposition 2(b)Proposition 3(b)Proposition 3(a)𝑤𝐻 − 𝑤𝐿

Figure 2. Equilibrium as a function of w H for a given w L with WH = WL . Panel (a) depicts thecase of s∗L > t̂. Panel (b) depicts the case of s∗L < t̂. In both cases, in the range where the incentive∗
compatibility is binding (sH∈ (s, s)), the equilibrium changes from separating to semi-pooling and to

full pooling as w H increases.

The key objects to look at in this figure are s and s, which indicate the quitting times
needed to prevent the low type from mimicking the high type to obtain the exit payoff w H
instead of w L . In the standard setup, these represent the expected points of separation:
the high type either quits excessively early at s or excessively late at s to separate from thelow type while the low type always quits at the full-information optimal point s∗L . This is
in fact what happens in equilibrium of our model when w H − w L is relatively small andhence reputation concerns are relatively weak.
As w H − w L becomes large, the gap between s and s widens to contain t̂, at which
point full separation is no longer feasible. The high type cannot quit too early or too late
for the sake of separation, because any deviation towards t̂ is attributed to the high typeunder D1. There is hence a gravitating force towards t̂ which tends to homogenize orcompress quitting times between types and consequently breaks down any separation thatspans over t̂. Since an increase in w H makes pooling a more attractive option, the low
type gradually switches from s∗L to t̂ as w H increases, and full pooling is realized when w H
becomes high enough.
This exercise raises an important observation on the role of reputation concerns inrisky experimentation: the difference in quitting times is compressed and bounded fromabove. This homogenizing force yields both positive and negative efficiency effects. To
see this, note that in the standard setup, the inefficiency in quitting time arises solely from25

the high type as the low type always quits at the efficient time. For illustration, supposet̂ > s∗L (the case of panel (b)). In this case, under single-crossing preferences, the high type
quits later and later to separate from the low type as w H increases. As a consequence, theequilibrium quitting time diverts away from sH∗ without bounds, which leads to a serious
loss of efficiency. In the current setting which generates the double-crossing property, on
the other hand, there is compression of quitting times towards t̂, which constrains howlate the high type can quit. Note, however, that this comes with a cost: knowing that thehigh type cannot persist for too long, the low type now has an incentive to inefficiently
hold out to pool with the high type.
Although the above observation is more about immediate or short-run consequences of
reputation concerns, there are also long-run consequences. In the context of risky experimentation, separation is harder to achieve due to the homogenizing force of the doublecrossing property, and less information about agent type will be revealed as a result. Our
exercise in particular suggests that we cannot learn much from observing that an agent
abandons his risky project. This means that separation of types is less complete and takemore time in environments where success is rare, as is often the case in venture startups.
5.2.

The role of prior reputation

In the standard setup, the D1 criterion always selects the least-cost separating equilibrium, or the Riley outcome, in which the low type chooses his (full-information) optimal
investment level and the high type invests just enough to separate from the low type.
This prediction is somewhat disturbing because the equilibrium allocation in the least-cost
separating equilibrium is independent of the prior belief, implying that the agent’s priorreputation has no real consequences for signaling. In contrast, in our model, the priorbelief q0 plays a crucial role in shaping the equilibrium outcome.
There are basically two ways in which the prior belief affects the equilibrium allocationin our setup. First, an increase in q0 directly raises the value of pooling for the low type,
U L ( t̂, q0 ), relative to the value of quitting at s∗L (0), and hence favors a pooling equilibrium.
This in turn forces the high type to also quit at t̂ in order to avoid adverse inference. Thiseffect is present even in the pure exit signaling case.
With breakthrough signaling, there arises an additional effect of the prior belief whena separating equilibrium prevails. This effect can work either positively or negatively,
depending on which type quits first. In a separating equilibrium in which the high type26

quits first, the high type quits at sH∗ (q0 ), which is earlier than the full-information optimal
time sH∗ (1). The extent of inefficiency thus diminishes as q0 increases. In a separating
equilibrium in which the low type quits first, on the other hand, the low type starts quitting
at s∗L (q0 ), which is later than s∗L (0). An increase in q0 thus raises the extent of inefficiency.
To sum up, the equilibrium selected by D1 is sensitive to the agent’s prior reputation.
The effect of an increase in q0 is hardly straightforward, and can often be negative as
it provides a stronger incentive for the low type to stay and mimic the high type. This
points to the difficulty in predicting the outcome of risky experimentation from publicly
observable traits when reputation matters, because the timing of project abandonment
could be related in some complicated and non-monotonic way to the experimenter’s priorreputation. On top of the inherent randomness of risky experimentation, this fact may
explain why it is so hard to predict success of startup businesses (Kerr et al., 2014).
5.3.

Implementation ability versus identification ability

We now consider an extension of our baseline model to allow for type-dependent project
quality. Suppose that there is a project-selection stage before the game begins, where theagent chooses which project to work on among possible alternatives. Let p0i denote theprior probability that a project handled by type i is of good quality. Here, we assume
that 1 > p0H > p0L > 0; i.e., the high type is better at discovering ideas or identifying
promising projects.18 Also, throughout this subsection, we focus on the exponential bandit
specification where f i (τ) = λi e−λi τ . It is easy to verify that the double-crossing property
still holds, and we can follow the same procedure to characterize equilibria.
With this modification, high-type and low-type agents are different along two dimensions: the ability to implement a project (λi ), and the ability to identify a good project(p0i ). Which one is more important depends on the context. One obvious factor is who
has the right to choose projects: if the agent has discretion over which project to workon, the prior quality of the project p0i most likely will depend on the agent’s type; if theagent has no such discretion and simply works on the project assigned, the prior should
not differ much between the two types. We also argue that identification ability mattersmore in areas where exploration of new ideas is required, and implementation ability matters more in areas where exploitation of existing ideas is sufficient. For any p0H ≥ p0L , theFrom the technical point of view, it is straightforward to extend the analysis to the case where p0L > p0H(while λH > λ L ). The double-crossing property is preserved and our analysis carries through as long as
p0H λH > p0L λ L . If this condition fails, the model then reduces to the standard setup.
1827

dimension concerning the implementation ability becomes relatively more important as
λH becomes farther apart from λ L . Below, we conduct comparative statics with respect to
λH , to show that the equilibrium allocation depends crucially on what is captured by theagent’s “reputation.”
The equilibrium outcome of our model is determined largely by which type quits first,
or alternatively whether t̂ is larger or smaller than s∗L (q0 ). Since s∗L (q0 ) is independent of
λH , we only need to look at how t̂ varies with λH . For clarity, define t̂(λH ) explicitly as afunction of λH , which solves
λH p0H e−λH t̂
1 − p0H + p0H e−λH t̂

=

λ L p0L e−λ L t̂
1 − p0L + p0L e−λ L t̂.

In the proof of the proposition below (provided in the Appendix), we show that t̂(λH )strictly decreases in λH , with limλH →λ L t̂(λH ) = ∞ and limλH →∞ t̂(λH ) = 0 for any p0H > p0L .
Thus t̂ is relatively small when the reputation reflects the implementation ability more (ahigh λH ), and increases as the identification ability gains more importance (a low λH ).
Since s∗L (q0 ) is independent of λH , this result, along with Propositions 4 and 5, immediately
leads to the following statement.
Proposition 6. For any λ L , there exist λ and λ (with λ > λ > λ L ) such that:(a) if λH < λ, the equilibrium entails some separation, and the low type quits first, startingfrom s∗L (q0 );(b) if λH ∈ [λ, λ], the equilibrium entails full pooling, and both types quit at t̂;(c) if λH > λ, the equilibrium entails some separation, and the high type quits first.
Moreover, if w H − w L is sufficiently large, λ → ∞ so that only cases (a) and (b) are relevant.

In existing reputation models of experimentation, agents are assumed to differ only in
one dimension, either in implementation ability (Bobtcheff and Levy, 2017) or in identification ability (Thomas, 2019). Our analysis suggests that this difference could cruciallyaffect the outcome of experimentation. There must be some separation if λH is sufficiently
close to λ L (identification ability is relatively more important). However, if w H − w L islarge, the incentive for the low type to pool with the high type is strong. A larger gap
w H − w L expands the range of parameters which support a pooling equilibrium. For sufficiently large w H − w L , Proposition 6 provides a simple prediction: there exists some λ28such that the equilibrium is full pooling if and only if λH > λ. In this case pooling occurs
when the difference in implementation abilities of the two types is large (λH > λ), whilesome separation occurs when the difference in implementation abilities is small (λH < λ).
Whether equilibrium entails some separation or not offers important implications for
who should retain the right to abandon the project. To put this idea in context, consider aprincipal who decides whether to delegate or not at the outset of the game. Suppose thatthe principal earns a payoff of WP when the agent succeeds and incurs a flow cost c P aslong as the project is implemented. The payoff in case of failure is normalized to 0. The
expected payoff to the principal if the agent is of type i and the project is abandoned attime s is
UiP (s) =Zse−ρτ p0i f i (τ) [WP − C P (τ)] dτ − e−ρs (1 − p0i Fi (s))C P (s),0where C P (t) = c P (eρt − 1)/ρ. When the principal retains the authority, she chooses s to
maximize q0 UHP (s) + (1 − q0 )U LP (s). Let s∗P denote the optimal stopping time for the principal. The costs and benefits of centralization are clear in this setup: by centralization,
the principal must abandon the project without any information about the agent’s type,but can choose to stop at her optimal timing s∗P . This suggests that delegation is of value
only if the agent uses his private information to decide the timing of project abandonment.
Alternatively, for a range of parameters under which the equilibrium is full pooling, theprincipal cannot be worse off by retaining the authority (centralization), even if she acquires no additional information of her own along the way.19 Proposition 6 clarifies whencentralization is more effective relative to delegation. Given that w H −w L is relatively large,
centralization is unambiguously the better choice when implementation ability is more important. As identification ability gains more importance, however, the equilibrium entails
separation, which tends to raise the value of delegation.
5.4.

Model with a continuum of types

As another extension, we may consider a case with more than two types, which would
allow us to check robustness of our main findings. Of particular interest from the analyticalpoint of view is the case with a continuum of types. Consider an agent who is characterizedby his type θ ∈ [θ , θ ], where we assume that the distribution of types is continuous with
19

In many cases, the principal often has means to evaluate the agent’s productivity over time. When theprincipal has access to an additional information source, centralization performs even better.

29

full support. Suppose further that the distribution of arrival times conditional on a goodquality project is exponential with parameter θ .
As in the baseline model, the marginal rate of substitution depends on agent type only
through the conditional hazard rate, which is given by g(t; θ ) = p0 θ e−θ t /(1 − p0 e−θ t ). For
any two types, θ 0 and θ 00 , g(t; θ 0 ) − g(t; θ 00 ) has the same sign ase−(θ 0 −θ 00 )t 1 − p0 e

−θ 00 t1 − p0 e−−θ 0 tθ 00,
θ0which is single-crossing from above in t for any θ 0 > θ 00 . Therefore, there exists t̂(θ 0 , θ 00 )such that g(t; θ 0 ) > g(t; θ 00 ) for t < t̂(θ 0 , θ 00 ) and g(t; θ 0 ) < g(t; θ 00 ) for t > t̂(θ 0 , θ 00 ).
Moreover,p0 e−θ t (1 − p0 e−θ t − θ t)
∂ g(t; θ )=∂θ
(1 − p0 e−θ t )2
is single-crossing from above in θ , meaning that the marginal rate of substitution is nonmonotone in θ . These properties ensure that the double-crossing property holds globally:
between any two types θ 0 > θ 00 , indifference curves of the two types cross twice, with thatof type θ 0 being more convex.
In a companion paper (Chen et al., 2020), we provide a general analysis of signaling
under double-crossing preferences with a continuum of types which includes this extended
model as a special case. We show that equilibrium under double-crossing preferences
exhibits a particular form of pooling, which can be viewed as a generalized version of LSHP
(Low types Separate High types Pool) equilibrium introduced by Kartik (2009): separation
is feasible only at the lower end of types, whereas some forms of pooling emerge at thehigher end. The equilibrium quitting time, as a function of type θ , jumps at most once
and is continuous elsewhere. As such, there is a threshold type below which types are fully
separated and above which they are clustered together in some ways.20
The general analysis raises two important points. First, it shows that pooling is a robust
prediction of signaling under double-crossing preferences. Second, it is also important torecall that equilibrium in the current setting is characterized by the low type’s indifference
condition, where some of low-type agents choose to separate and some others choose
to pool with high-type agents in a semi-pooling equilibrium. This structure has a precise
20

Unlike in Kartik (2009), the model with double-crossing preferences admits various forms of poolingabove the threshold, though with some regularities. See Chen et al. (2020) for a more precise characterization.

30

counterpart in the continuous-type model: equilibrium in this case is also characterized by
the indifference condition of the threshold type, where types below the threshold choose to
fully separate and those above choose to pool with higher types. Types are thus divided into
two segments in a manner similar to the two-type case. As a consequence, the continuoustype model exhibits similar equilibrium properties, i.e., homogenization of quitting times
and some pooling in equilibrium, implying that our predictions are robust in a qualitative
sense and carry over to more general settings.

6.

Policy implications

Our model illustrates how reputation concerns distort the timing of project abandonment.
There are many potential remedies for those distortions, with centralization of decisionmaking rights being one of them. The problem is that centralization is a second-best solution in that the equilibrium quitting time does not reflect the agent’s private information.
The principal can do better if she can commit to and enforce a more sophisticated incentive
scheme. In this section, we explore this possibility and discuss some policy implications
when the principal is equipped with more tools to manipulate the underlying payoff structure. Since the case of venture startups provides a leading example of our model, we focus
on two remedies that are particularly relevant for this industry.
The valley of death. Consider a situation where the principal can raise the cost of continuing the project at some predetermined point in time. Specifically, suppose she can set
a deadline and charge a “fee” when the agent continues the project past the deadline. We
argue that this type of intervention can be quite effective in regulating the dynamic distortion when sH∗ (1) > s∗L (0), or alternatively when identification ability matters more. In
this case, the low type persists for too long in holding out to partially pool with the hightype. If the principal can credibly enforce this scheme, the optimal solution is conceptually straightforward: set a deadline at s∗L (0) and charge a fee high enough to make thecontinuation payoff for the low type nonpositive. Under this scheme, it is indeed optimalfor the low type to quit at s∗L (0), which in turn allows the high type to separate and quit
at the optimal timing sH∗ (1). The optimal scheme can thus restore efficient separation andachieve the full-information outcome.21
21

The effect of the valley of death is most pronounced when there are only two types as we assume here, as

it can achieve efficient separation. When we have more types, it is in general not feasible to achieve efficient
separation by a simple scheme like the valley of death, but the basic ideas remain valid. In particular, when
low-type agents tend to over-experiment, it is always beneficial to let them exit earlier. Moreover, by doing31

This type of midterm screening is often observed in venture financing, where many
venture startups face difficulty raising follow-on (“series A”) funding after initial (seed)funding. In many cases, it is relatively easy to obtain seed funding from public or angelsources; past this stage, many startups struggle to raise follow-on funding. This funding
gap, which inevitably raises the operational cost of running a business, is often referred toas the “valley of death,” as most startups cannot survive past this phase. However, the reason why such funding gap exists is not immediately obvious, especially given the fact thatseed funding typically comes from angel and public sources that are not driven by profit
maximization. The presence of the valley of death thus indicates that the expected returnof a startup business, conditional on its survival, follows a U-shaped path and stagnates in
the middle, so much so that even those funding sources are reluctant to continue.
Our analysis provides a mechanism through which the valley of death naturally emerges,
and offers an important efficiency rationale of this funding gap.22 In the presence of reputation concerns, relatively less productive entrepreneurs tend to “hold out” for too longeven though their projects are in hopeless shape, which could create an interval where theexpected return of a startup business stagnates and dips below the efficient level. Given
this, the valley of death can be efficiency-enhancing as it can work as a screening device
to differentiate entrepreneurs with different degrees of vision and confidence. It prevents
less efficient entrepreneurs from over-experimenting out of reputation concerns. It also
raises the expected return for the remaining, more efficient, entrepreneurs, which is important when the participation constraint is binding for some of them. Although there
is now a heated debate over how to bridge this gap, with some calling for active public
interventions (Murphy and Edwards, 2003; Butler, 2008), our analysis suggests a positiverole of the valley of death as a screening device, which is particularly effective in areas
where exploration of new ideas is crucial for success.
Startup subsidies. When implementation ability matters more, we have s∗L (0) > sH∗ (1),in which case the high type quits too early because the reputational value of success is
not sufficient. In contrast to our previous discussion about the “valley of death,” the issue
here is to induce high-type agents to persist longer to fully explore the true worth of theirprojects. In this case, startup subsidies which lower the operational cost of continuing aso, it raises the reputational value of success for the remaining types, which induces those more efficient
types to enter the game.
22
In a different framework, Chen and Ishida (2018) also discuss a positive role of the valley of death toscreen out less confident entrepreneurs. In their model, there are no reputation concerns in that the payofffrom success or failure is fixed independently of the market belief.

32

startup business can be productive as they allow high-type agents to continue up to theiroptimal timing.
More precisely, consider a startup subsidy which lowers the flow cost of experimentation from c to c̃, possibly for some duration of time. Assume t̂ < t and let v(t; c̃) to denoteits dependence on c̃. If there exists a c̃ such that v(sH∗ (1); c̃) = w H , we can restore efficiency
by providing subsidies for t ∈ [0, s∗L (0)], with the cost reverting back to c after s∗L (0). Under this scheme, the low type is indifferent between quitting at sH∗ (1) and s∗L (0), and thehigh type has no incentive to quit before sH∗ (1). We can thus achieve efficient separationin which the high type quits at sH∗ (1) and the low type quits at s∗L (0).23
Startup subsidies or grants are ubiquitous in both developed and developing economies.
In the United States, for instance, there are several federally funded programs aiming at
getting small startup businesses off the ground. In addition to those public funding sources,
angel investors also play an essential role in early stages of venture financing, accounting
for a substantial fraction of seed money supplied to the venture market. The primary rationale for startup subsidies is to relax credit constraints which small startup firms may face
due to market imperfections. Empirical support for this channel is not strong, however, as
recent evidence suggests that credit constraints may not play as important a role as waspreviously believed (Kerr and Nanda, 2011). Our analysis provides an alternative rationale for startup subsidies, which is to allow more promising projects to persist longer andlive up to their full potential. Note that our argument differs from the conventional one
in an important way, as it stems from dynamic reputation concerns and holds irrespectiveof whether there are credit market imperfections. This implies that our argument can beapplied to a range of situations, outside of venture financing, where credit constraints are
not a crucial factor.

7.

Conclusion

This paper provides a framework to analyze the role of reputation concerns in risky experimentation. We develop a general approach which encompasses a broad class of learning
processes and model specifications and obtain a complete characterization of unique D1equilibrium. Our analysis suggests that signaling incentives in the context of risky exper23

While we focus on the simplest scheme here for the sake of illustration, there are many other schemesthat can equally achieve efficient separation. In particular, if there is a positive transfer cost and the government would like to minimize the amount of subsidies, the scheme must be time-contingent: letting c̃ t denotethe flow cost at time t, the cost-minimizing scheme must satisfy v(t; c̃ t ) = w H for all t ∈ [sH∗ (1), s∗L (0)].

33imentation are qualitatively different from those in other more standard contexts characterized by the single-crossing property, with reputation concerns inducing homogenizationof quitting times between types. We study the implications of these inefficient termination
decisions for organizational design, and offer potential remedies to correct the distortions.
The implications obtained here are far-reaching and can be applied broadly to many situations of interest, such as an entrepreneur experimenting with a business startup, a politician with a policy reform, an engineer with a new product design, and a researcher witha scientific hypothesis.
As a final remark, we would like to offer a broader perspective for the scope of our analysis. The insights obtained from our analysis rely crucially on the double-crossing property
of indifference curves. This property naturally arises in the context of risky experimentation as long as the time horizon is long enough. We work with an infinite-horizon modelin our analysis, but there is sometimes an exogenously imposed deadline for experimentation. The political-economy example is a case in point, where politicians are often subject
to a time frame defined by election cycle. In those instances, the speed of learning becomes an additional factor to determine the form of equilibrium. For some policy issues,
it takes a long time, sometimes decades, to observe outcomes of policy experimentation;
many environmental or foreign policies supposedly fall into this category. In such a case,the deadline (or the election date) binds, and the single-crossing property dictates equilibrium behavior as in the standard setup.24 For some other issues, on the other hand, we
can often expect to learn quickly which policy measures work and which do not, so thatthe time horizon is less likely to be an issue; an example may be domestic economic policy
for which there are many readily available indicators. The insights from our analysis are
more applicable to this type of environment.

24

If the deadline arrives before t̂, preferences are single-crossing for all feasible choices of t. The model
then becomes a standard signaling model with a constrained action space.

34

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Appendix
Proof of Lemma 2. Since both types quit with positive probability at time t, the interim
belief assigned to an agent who quits at that time satisfies q̂ ∈ (0, 1).
Suppose t < t̂. Pick a small ε > 0. There are two cases: (a) σ L (t 0 ) is constant on
t 0 ∈ [t, t + ε). By D1, the market assigns interim belief 1 to an agent who quits at t 0 ∈(t, t + ε). Because the gain in interim belief is discrete while the change in payoff fromdelaying to quit is infinitesimal, such a deviation would be profitable. (b) σ L (t 0 ) is strictly
decreasing on t 0 ∈ [t, t + ε). Since t < t̂, if the low type is indifferent between continuing
and quitting, the high type strictly prefers to continue. This implies that σH (t 0 ) is constanton [t, t + ε). The interim belief assigned to one who quits at t 0 ∈ (t, t + ε) must be 0.
But then this cannot be optimal for the low type to quit at t 0 because he would gain bydeviating to quit at t and obtain an interim belief of q̂ instead of 0.
Suppose t > t̂. There are two cases: (a) σ L (t 0 ) is constant on t 0 ∈ (t − ε, t). By D1,
the market assigns interim belief 1 to an agent who quits at t 0 ∈ (t − ε, t). It would pay
for an agent to deviate by quitting slightly earlier at t 0 instead of t. (b) σ L (t 0 ) is strictly
decreasing on t 0 ∈ (t − ε, t). Since t > t̂, if the low type is indifferent between quitting at
t 0 and quitting at t, the high type strictly prefers quitting at t 0 . Hence, the interim belief
assigned to one who quits at t 0 ∈ (t − ε, t) must be 0. But then the low type could gain bydeviating to quit at t instead of t 0 and obtain an interim belief of q̂ instead of 0.
Proof of Proposition 4. We show that the posited strategy profiles constitute an equilibrium that satisfies D1 in each case. The proof that equilibrium is unique follows the same
logic as that for Propositions 2 and 3 and is omitted for brevity.
(a) Separating equilibrium. The equilibrium strategy profile is that the high type quits
at t and the low type quits at s∗L (0). At time t, the low type has no strict incentive to deviate
by quitting immediately to get w H , because by construction v(t; s∗L (0)) = w H . Moreover,when the low type is indifferent between quitting at t and quitting at s∗L (0), the high typestrictly prefers quitting at the earlier time t, because both t and s∗L (0) are greater thant̂. Now, consider deviating to s0 < t. Because t is lower than sH∗ (q0 ), quitting at such s0
is equilibrium dominated by quitting at t for the high type. This means we can assign
an off-equilibrium belief of 0 to such deviation. Given this off-equilibrium belief, no typehas an incentive to deviate to s0 < t. If s0 > t, then in this case we have s0 > t̂, and D1
requires that deviations to s0 be assigned an off-equilibrium belief of 0. Thus, a low type38

that deviates to s0 > t will get a payoff of v(t; s0 ) from the perspective of time t. Because
it is optimal to quit at s∗L (0) when the belief is 0, we have v(t; s0 ) < v(t; s∗L (0)), meaningthat deviating to s0 is unprofitable for the low type. Further, since g H (s0 ) < g L (s0 ) for anys0 > t ≥ t̂, the payoff from deviating to s0 > t for the high type is strictly less than v(t; s0 ),which in turn is strictly less than v(t; s∗L (0)) = w H . This means that it is not profitable forthe high type to deviate to s0 either.
(b) Semi-pooling equilibrium. The strategy profile is that the high type quits at t̂ andthe low type randomizes between t̂ and s∗L (0). The belief associated with quitting at t̂ issome q̂ > q0 such that v( t̂; s∗L (0)) = w L + r( t̂; q̂)(w H − w L ), and the belief associated withquitting at s∗L (0) is 0. By construction, q̂ is chosen such that the low type is indifferent
between the two equilibrium quitting times. Further, because s∗L (0) > t̂, indifference ofthe low type implies that the high type strictly prefers to quit at the earlier time t̂. Now, we
consider off-equilibrium deviations. Because both types quit at t̂ with positive probability,
condition (5) and D1 require that any deviation s0 be assigned an off-equilibrium belief of
0. Given such belief, for any quitting time after t̂, the one that is optimal for the low type
is by definition s∗L (0). Thus any deviation s0 > t̂ is unprofitable for the low type. By thesame reasoning as in part (a), this implies that such deviation is unprofitable for the hightype as well. Similarly, both types strictly prefer quitting at t̂ to deviating to quit at s0 < t̂.
(c) Full pooling equilibrium. The strategy profile is that both types quit at t̂. The belief
associated with quitting at t̂ is q0 , and the off-equilibrium belief associated with quitting
at any other time is 0. The low type does not want to deviate to s∗L (0), because the deviation payoff v( t̂; s∗L (0)) would be weakly lower than the equilibrium payoff from quitting
immediately. If the low type does not want to deviate to s∗L (0), he does not want to deviate
to any other s0 6= t̂. The same argument as before implies that the high type does not want
to deviate to any s0 6= t̂ either.
Proof of Proposition 5. We begin by showing that V (·; t̂) − w L is single-crossing from below. To see this, note that the derivative of V (t; t̂) with respect to t is
∂ V (t; t̂)= ρV (t; t̂) + c − g L (t) WL + R(t; q̃(t))(WH − WL ) − V (t; t̂)∂t
Z t̂
p0 f L (τ)∂ R(τ; q̃(t))+(WH − WL )dτe−ρ(τ−t)
1
−
p
F
(t)
∂
q̃
0
L
t

∂ r( t̂; q̃(t)) dq̃(t)−ρ( t̂−t) 1 − p0 F L ( t̂)+e(w H − w L ).
1 − p0 F L (t)
∂ q̃
dt
39

If V (t; t̂) = w L at t < s∗L (1), the first term is 0 and the second term is positive. If V (t; t̂) =w L at t > s∗L (1), the first term is positive and the second term is 0. This shows that V (·; t̂)−w L is single-crossing from below. Hence, V (s∗L (q0 ); t̂) < w L and V ( t̂; t̂) > w L imply thatt 0 ∈ (s∗L (q0 ), t̂) exists and is unique, with V (t; t̂) > w L for all t ∈ (t 0 , t̂].
We next show that V (s∗L (1); ·) is decreasing on [s∗L (1), t̂]. The derivative of V (s∗L (1); s)with respect to s is
∂ V (s∗L (1); s)∂s

1 − p0 F L (s)(g L (s) [WH − w H ] − ρw H − c)
1 − p0 F L (s∗L (1))
1 − p0 F L (s)∗(g L (s) [WH − w L ] − ρw L − c) ,< e−ρ(s−s L (1))
1 − p0 F L (s∗L (1))∗= e−ρ(s−s L (1))

which is non-positive for s ∈ [s∗L (1), t̂]. Because V (s∗L (1); s∗L (1)) > w L , and t 0 > s∗L (1)implies V (s∗L (1); t̂) < w L , we can conclude that t 1 ∈ (s∗L (1), t̂) exists and is unique, with
V (s∗L (1); t 0 ) ≤ w L for any t 0 ≥ t 1 .
Equilibrium. We first show that the strategies described in cases (a), (b)(i), and (b)(ii)of the proposition constitute an equilibrium of the corresponding cases.
(a) Let Ji (t) be the value function for type i ∈ {H, L} corresponding to the strategy
profile of full pooling (i.e., both types do not quit until time t̂). By the Bellman equation,
Ji (t) = −c dt + g i (t) dt [WL + R(t; q0 )(WH − WL )] + (1 − g i (t) dt)e−ρ dt Ji (t + dt).
From this, we obtain the differential equation:
Ji0 (t) = ρJi (t) + c − g i (t) [WL + R(t; q0 )(WH − WL ) − Ji (t)] ,(A1)

with terminal condition Ji ( t̂) = w L + r( t̂; q0 )(w H − w L ). By construction, J L (s∗L (q0 )) =V (s∗L (q0 ); t̂) ≥ w L for the low type. Moreover, we can write
J L0 (t) = (ρ + g L (t))(J L (t) − w L ) + (ρw L + c − g L (t) [WL + R(t; q0 )(WH − WL ) − w L ]) .
From the definition of s∗L (q0 ), the second term is equal to zero at t = s∗L (q0 ), and it ispositive when t > s∗L (q0 ) and negative when t < s∗L (q0 ). For t ∈ (s∗L (q0 ), t̂], the second
term is positive, and so J L (t) − w L is single-crossing from below. But J L (t) − w L is nonnegative at t = s∗L (q0 ) and is positive at t = t̂. We therefore must have J L (t) > w L for
all t ∈ (s∗L (q0 ), t̂]. For t ∈ [0, s∗L (q0 )), the second term is negative, and so J L (t) − w L issingle-crossing from above. But since J L (t) − w L is non-negative at t = s∗L (q0 ), we must40

have J L (t) > w L for all t ∈ [0, s∗L (q0 )). Since J L (t) ≥ w L for all t ≤ t̂, it is indeed optimalfor the low type not to quit until t̂. Since g H (t) > g L (t) for all t < t̂, we also have
JH (t) > J L (t) ≥ w L . Thus, the high type also has no incentive to quit until time t̂.
(b)(i) For i ∈ {H, L}, let Ji (t) be the solution to the differential equation (A1), with
with q0 in (A1) and in the terminal condition replaced by q̃(t 0 ). Since the high type neverquits until time t̂, JH (t) is the value function for the high type.
For the low type, let J̃ L (t) be the solution to the differential equation (A1), with terminal condition J̃ L (s∗L (q0 )) = w L . Then, the value function for the low type is given by∗J̃ L (t) if t ∈ [0, s L (q0 )),J L∗ (t) =wLif t ∈ [s∗L (q0 ), t 0 ],J (t) if t ∈ (t , t̂].
L
0

By the same argument as in part (a), we have J L∗ (t) > w L for t ∈ [0, s∗L (q0 )) and fort ∈ (t 0 , t̂]. Thus, it is optimal for the low type not to quit for such t. For t ∈ [s∗L (q0 ), t 0 ],equation (8) ensures that the low type is indifferent between quitting and staying. Thus,
the strategy of the low type is indeed a best response. Furthermore, JH (t) > J L∗ (t) ≥ w L
for all t < t̂. Thus, the high type has no incentive to quit until time t̂.
(b)(ii) Fix any t 0 ∈ [max{t 1 , sH∗ (1)}, t̂]. We have already shown that t 1 < t̂. Moreover,s∗L (1) ≤ t 0 < t̂ implies sH∗ (1) < t̂. Thus, the interval is non-empty.
For t ≤ t 0 , the value function for the high type is given by the solution to the differential
equation:
JH0 (t) = ρJH (t) + c − g H (t) [WH − JH (t)] ,with terminal condition JH (t 0 ) = w H . For the low type, the value function is given by∗J̃ L (t) if t ∈ [0, s L (q0 )),J L∗ (t) =

wL



w

H

if t ∈ [s∗L (q0 ), t 0 ),if t = t 0 ;where J̃ L (t) is as defined in part (b)(i). Note that no agent quits at time t ∈ (s∗L (1), t 0 ).
We assign off-equilibrium belief q̂(t) = 0 for an agent who quits at such time, which isconsistent with the D1 criterion because t < t̂. Because J L∗ (t) > w L for t < s∗L (q0 ), the low
type strictly prefers continuing with the risky project than quitting. For t ∈ [s∗L (q0 ), s∗L (1)],
we have J L∗ (t) = w L . Therefore the strategy σ L (t) that satisfies equations (8) and (9) is41

indeed a best response. At t = s∗L (1), the gain for a low type from deviating to quit at t 0
instead is V (s∗L (1), t 0 ) − w L ≤ V (s∗L (1), t 1 ) − w L = 0. Therefore, the low type cannot gainfrom deviating to wait until t 0 to quit.
Uniqueness. We now show that the equilibria described in cases (a), (b)(i), and (b)(ii)of the proposition are the only candidates for equilibrium for the corresponding cases. To
this end, let t i and t i be the earliest and latest possible time, respectively, for type i to quit
on the equilibrium path. We first establish the following facts.
Fact 1. t H ≥ s∗L (q0 ).
Proof. Suppose t H < s∗L (q0 ). From the proof of Lemma 2, we know that there cannot be
any pooling before time t̂. Given s∗L (q0 ) < t̂, this means that we cannot have t H = t L < t̂.
Therefore, t H 6= t L . Moreover, in this case, it is not possible to have t L < t H , for otherwiseq̂ t L = 0 < q̂ t H , q̃ t L = q0 < q̃ t H , and the low type could profitably deviate by quitting at t H .

This means that we only need to look at the case where t H < t L .

With some abuse of notation, let Vi (t H ; t L ) denote the expected payoff if a type i agent
continues to work with the risky project from t H to t L on the equilibrium path. We obtain
Vi (t H ; t L ) = e−ρ(t L −t H )+Z tL

1 − p0 Fi (t L ) 
1 − p0 Fi (t H )e−ρ(τ−t H )tH
Ji (t L ) − C(t L − t H )p0 f i (τ) 
WL + R(τ; q̃τ )(WH − WL ) − C(τ − t H ) dτ,
1 − p0 Fi (t H )where Ji (t L ) is the equilibrium payoff to type i agent at time t L . Since an agent earns
w H by quitting at t H , the high type quitting at t H and the low type quitting at t L imply
VL (t H ; t L ) ≥ w H ≥ VH (t H ; t L ).
We now show that this condition cannot be satisfied. To this end, observe first thatt L ≤ t̂. Suppose otherwise. Then, since q̃ t is weakly decreasing on [ t̂, t L ) and sH∗ (q̃ t̂ ) < t̂,
there cannot exist t 0 and t 00 in [ t̂, t L ) such that the high type is indifferent between quitting
at t 0 and t 00 . Therefore, there must exist some interval (t L − ε, t L ) such that no type quits,
and as t L > s∗L (q0 ), the low type can profitably deviate by quitting at t ∈ (t L − ε, t L ).
We note that, for t ≤ t L , Vi (t; t L ) solves the differential equation,
Vi0 (t) = ρVi (t) + c − g i (t) [WL + R(t; q̃ t )(WH − WL ) − Vi (t)] ,with terminal condition Vi (t L ) = Ji (t L ). Note also that t L ≤ t̂ implies that g L (t) < g H (t) fort ∈ [t H , t L ). Hence, for any t ∈ [t H , t L ), the right-hand-side of the above is strictly higher
42for i = L than for i = H . Moreover, because it is feasible for the high type to quit at t L , wehave JH (t L ) ≥ w H + r(t L ; q̂ t L )(w H − w L ) = J L (t L ). Therefore, by the comparison theorem
for differential equations, we have VL (t H ; t L ) < VH (t H ; t L ). This contradicts VL (t H ; t L ) ≥w H ≥ VH (t H ; t L ).



Fact 2. Either t L = t H = t̂, or t L = s∗L (q0 ) < t H .
Proof. Suppose t H < t L . Then, q̂ t L < q̂ t H = 1 and q̃τ < q0 for τ ∈ (t H , t L ). Moreover,

we know that t H ≥ s∗L (q0 ) from Fact 1. Given that t L > t H ≥ s∗L (q0 ) > s∗L (q̃ t L ), and alsothat q̃ t is weakly decreasing (as only the high type may quit before t L ), the low type could
receive a higher payoff by quitting at t H than by quitting at t L , a contradiction. Therefore,t L ≤ t H . If t L = t H , we must have t L = t H = t̂ by Lemma 2. If t L < t H , then it is optimalfor the low type to start quitting at s∗L (q0 ).

Fact 3. Either t L = t H = t̂, or t L < t H .
Proof. If t H < t L , we have q̃ t L = 0. Then, since t H ≥ t H > s∗L (q0 ), the low type could
profitably deviate by quitting at t ∈ (t H , t L ). Therefore, t L ≤ t H . Lemma 2 implies that if
t L = t H , then both are equal to t̂.

Fact 4. t H = t H .
Proof. Suppose t H < t H . Suppose further that both types quit at t̂ with positive probability. This means that the high type is indifferent among quitting at t H , t̂ ∈ [t H , t H ], andt H , but condition (5) then implies that the low type must strictly prefer quitting either att H or t H to quitting at t̂, a contradiction. This rules out pooling or partial pooling withtH < tH.
Now suppose that t H < t H and the equilibrium is separating, in which case we have
either (1) t L = s∗L (q0 ) < t L < t H < t H ; or (2) t L = s∗L (q0 ) < t H < t L < t H .
In case (1), we have t L = s∗L (1), q̂ t L = 0, q̂ t H = 1, q̃ t = 1 for t ∈ (t L , t H ), and the low

type weakly prefers quitting at t L to quitting at t H . This implies that for t ∈ (t H , t H ), no
belief q̂ t can give the low type a higher payoff than the equilibrium payoff. Moreover, since
the high type is indifferent between quitting at t H and t H , the expected payoff must go up
first and then go down for t ∈ (t H , t H ) (with q̂ t = q̃ t = 1 over this interval), suggestingthat the set of beliefs q̂ t that give the high type a higher payoff than the equilibrium payoff
is not empty for this interval. Therefore, q̂ t = 1 for t ∈ (t H , t H ) by D1, and the high type
could profitably deviate by quitting at any t in this interval.
In case (2), the proof of Fact 1 shows that if the two types quit separately before t̂, the43low type must quit before the high type. By a similar argument, we can also show that if
the two types quit separately after t̂, the high type must quit before the low type. These
facts imply that given t L < t H < t L , we must have t L < t̂ and t L > t̂. If t L > t̂, however,
we cannot have t L < t H , which implies that t H < t H cannot occur in equilibrium.



Now define t H = t H = t H . The above facts show that there can only be three types of
equilibria: (1) full pooling in which t L = t H = t̂; (2) semi-pooling in which t L = s∗L (q0 ) <t L = t H = t̂; or (3) separating in which t L = s∗L (q0 ) < t L = s∗L (1) < t H . Drawing on thisfact, we show that the equilibrium is unique in each of the cases.
(a) We start with the case where V (s∗L (q0 ); t̂) ≥ w L . In this case, since V (·, t̂) − w L
is single-crossing from below, V (s∗L (q0 ); t̂) ≥ w L implies that V (t; t̂) > w L for all t ∈(s∗L (q0 ), t̂]. Suppose first that t L = s∗L (q0 ) < t L = t H = t̂, so that the equilibrium is semipooling. Then, since V (t; t̂) > w L for all t ∈ (s∗L (q0 ), t̂], the low type strictly prefers quitting
at t̂ to quitting at t L = s∗L (q0 ). Next suppose that t L = s∗L (1) < t H , so that the equilibrium
is separating. Since s∗L (1) ≤ t̂ implies sH∗ (1) < t̂ and s∗L (1) > t̂ implies sH∗ (1) < s∗L (1), we
have sH∗ (1) < max{ t̂, s∗L (1)}. Moreover, t H > max{ t̂, s∗L (1)}, because V (s∗L (1); t̂) > w L ifs∗L (1) < t̂, and thus t H must be greater than t̂ in order to prevent the low type from deviating. By D1, q̂ t = 1 for t ∈ (max{ t̂, s∗L (1)}, t H ), but then the high type could profitably
deviate by quitting at t ∈ (max{ t̂, s∗L (1)}, t H ). This shows that the only possible equilibriumin this case is full pooling.
(b) If V (s∗L (q0 ); t̂) < w L , the full pooling equilibrium cannot exist, because the low
type could profitably deviate by quitting at s∗L (q0 ). We then need to consider two cases,
depending on whether t 0 is larger or smaller than s∗L (1).
(b)(i) Suppose t 0 < s∗L (1). Suppose further that a separating equilibrium exists, i.e.,t L = s∗L (1). The high type must then quit at t H > t̂ such that V (s∗L (1), t H ) = w L . Recall
that sH∗ (1) < max{ t̂, s∗L (1)}, and thus the equilibrium cannot survive the D1 criterion. Thisshows that the only possible equilibrium in this case is semi-pooling.
(b)(ii) If t 0 ≥ s∗L (1), the semi-pooling equilibrium cannot exist, since the low type must
prefer quitting at s∗L (q0 ) to quitting at t̂. Therefore, only the separating equilibrium isfeasible in this case, although there is still a continuum of separating equilibria that arefeasible. To further reduce the set of equilibria, note that since t L = s∗L (1), t H must be
weakly greater than t 1 so that the low type would not deviate. Moreover, since V (s∗L (1); ·)is decreasing on [s∗L (1), t̂], the low type strictly prefers quitting at t L to quitting at anyt ∈ (t 1 , t̂] even if q̂ t = 1. If t H ∈ (t 1 , max{t 1 , sH∗ (1)}) (which implies that sH∗ (1) > t 1 ), by44D1, we assign off-equilibrium belief q̂ t 0 = 1 to an agent who quits at any t 0 ∈ (t H , sH∗ (1)),
since compared to the equilibrium payoff, only the high type could possibly benefit fromsuch a deviation. Then the high type could profitably deviate by quitting at t 0 . If t H ∈(max{t 1 , sH∗ (1)}, t̂), by D1, we also assign off-equilibrium belief q̂ t 00 = 1 to an agent whoquits at any t 00 ∈ (max{t 1 , sH∗ (1)}, t H ), and thus the high type could profitably deviate by
quitting at t 00 . Therefore, we must have t H = max{t 1 , sH∗ (1)}, which uniquely pins down
the equilibrium.
Proof of Proposition 6. We first show t̂(λH ) strictly decreases in λH , with limλH →λ L t̂(λH ) =∞ and limλH →∞ t̂(λH ) = 0.
To show limλH →λ L t̂(λH ) = ∞, suppose that t̂ is bounded from above when λH approaches λ L . Then, in the limit, we must havep0H
1 − p0H + p0H e−λ L t̂=p0L
1 − p0L + p0L e−λ L t̂.

For any p0H > p0L , this is a contradiction because the equality cannot be satisfied for any t̂.
Similarly, to show that limλH →∞ t̂(λH ) = 0, suppose that there exists some ε > 0 such that
limλH →∞ t̂(λH ) > ε. For any such ε, however, we have limλH →∞ λH e−λH ε = 0 by l’Hopital’srule. This implies thatλH p0H e−λH εlim

λH →∞ 1 − p H + p H e −λH ε00= 0 < lim

λ L p0L e−λ L ε

λH →∞ 1 − p L + p L e −λ L ε00,which is a contradiction.
We next show that t̂(λH ) is decreasing. Observe that g H ( t̂(λH )) = g L ( t̂(λH )). Taking
derivative of both sides with respect to λH , we obtain∂ g L ∂ g H d t̂
∂ gH
−
=
.
∂t
∂ t dλH∂ λH
We know that, evaluated at t = t̂(λH ), ∂ g L /∂ t > ∂ g H /∂ t. This means that d t̂/dλH has
the same sign as ∂ g H /∂ λH (evaluated at t = t̂(λH )). Therefore, d t̂/dλH has the same signas(1 − p0H + p0H e−λH t̂(λH ) ) − λH t̂(λH )(1 − p0H ).
This shows that d t̂/dλH is single-crossing from above as λH increases, because the above
expression is decreasing in λH if d t̂/dλH = 0. As λH approaches λ L , we have shown that45t̂(λH ) approaches infinity, and hence the sign of d t̂/dλH is negative. Together with thefact that d t̂(λH )/dλH is single-crossing from above, the fact that limλH →λ L d t̂(λH )/dλH < 0
implies d t̂(λH )/dλH < 0 for all λH > λ L .
We consider how equilibrium varies as the value of t̂ increases. Suppose first that t̂ issufficiently small and t̂ ≤ s∗L (q0 ). Moreover, suppose s∗L (0) > 0 and t > 0 (both of which
are independent of λH ). By Proposition 4, the equilibrium is full pooling if t̂ is sufficiently
close to s∗L (0).
Now suppose t̂ > s∗L (q0 ). First, it is clear that the equilibrium is full pooling if t̂ issufficiently close to s∗L (0), because
lim V (s∗L (q0 ); t̂) = w L + r(s∗L (0); q0 )(w H − w L ) > w L .

t̂→s∗L (q0 )Note also that the derivative of V (s∗L (q0 ); t̂) with respect to t̂ has the same sign asg L (s) WL + R( t̂; q0 )(WH − WL ) − w L − r( t̂; q0 )(w H − w L ) − ρ w L + r( t̂; q0 )(w H − w L ) − c,which is negative for any t̂ > s∗L (q0 ) by definition. Moreover, sincelim V (s∗L (q0 ); t̂) =
t̂→∞

Z∞

p0 f L (τ)[WL + R(τ; q0 )(WH − WL )1 − p0 F L (s∗L (q0 ))s∗L (q0 )−C(τ − s∗L (q0 )) dτ < w L ,∗e−ρ(τ−s L (q0 ))there exists a unique t̂ 0 such that V (s∗L (q0 ); t̂ 0 ) = w L . At this point, the equilibrium must be
semi-pooling. Finally, to satisfy V (t 0 ; t̂) = w L , the benefit of pooling evaluated at t 0 must
be strictly positive, implying that e−ρ( t̂−t 0 ) must be bounded away from 0. This means thatt 0 → ∞ as t̂ → ∞. It then follows that the equilibrium must be separating when t̂ issufficiently large.
We have shown that, as t̂ increases, the equilibrium changes from that described in case(c) to that described in case (b) and then to case (a). Since t̂ decreases in λH , therefore
the proposition follows.
To show that λ can be arbitrarily large, observe that, for any fixed w L , t = 0 if w H issufficiently large. Moreover, for w H sufficiently large, v( t̂) ≤ w L + r( t̂; q0 )(w H − w L ). Thus,
case (c) of Proposition 4 applies and the equilibrium must be full pooling.

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Online Appendix to “Reputation Concerns in Risky Experimentation”
Suppose that the agent’s payoff depends partly on his actual type. To illustrate how itchanges our analysis clearly, we focus on the case of exit signaling (WH = WL = W ).
For i ∈ {H, L}, let w i (s; q̂) denote the agent’s payoff when he quits at time s and when
the interim belief is q̂. This is now type-contingent and is given byw i (s; t̂) = αw(s; q̂) + (1 − α)w i = α[w L + r(s; q̂)(w H − w L )] + (1 − α)w i ,where the parameter α ∈ (0, 1) measures the importance of reputation. The objectivefunction becomesZs
Ui (s, q̂) =e−ρτ p0 f i (τ) [W − C(τ)] dτ + e−ρs (1 − p0 Fi (s)) [−C(s) + w i (s; q̂)] .

0

From this, the marginal rate of substitution between stopping time s and interim belief q̂
is obtained as
g i (s)[W − w i (s; q̂)] − ρw i (s; q̂) − c + α(∂ r/∂ s)(w H − w L ).
α(∂ r/∂ q̂)(w H − w L )The difference in the marginal rates of substitution between the high type and the low
type has the same sign asD(s; q̂) =α[g H (s) − g L (s)][W − w(s; q̂)] + g H (s)(W − w H ) − g L (s)(W − w L ) − ρ(w H − w L ).
1−αObserve that D(s; q̂) may no longer be single-crossing from above when it crosses 0. Definet̃(q̂), if it exists, such thatM RSH ( t̃(q̂), q̂) = M RS L ( t̃(q̂), q̂),for a given α. Observe that t̃(q̂) depends not only on the statistical properties of theunderlying experimentation process but also on other primitives such as W , w i and ρ.
Recall that g H (s) − g L (s) = 0 at s = t̂. Therefore, we have D( t̂; q̂) < 0, and hencet̃(q̂) > t̂ for any α. Clearly, as α → 1, t̃(q̂) → t̂, suggesting that all of our results holdin a qualitative sense for α close to 1. If α is small or ρ is large, on the other hand, themarginal rates may never cross. In this case, the indifference curves are single-crossing,and there can only exist a separating equilibrium (when α = 0, the agent does not careabout his reputation and simply stops at the full-information optimal point).
iWhen α is in an intermediate range, it is possible to have multiple tangency points
along a single indifference curve. Let t̃ j , j = 1, 2, . . . , k, be the tangency point along theequilibrium indifference curve for the high type, where t̃ j < t̃ j+1 . If j is even, the indifference curve for the high type stays below that for the low type, and hence it does notconstitute a D1 equilibrium. If j is odd, the indifference curve stays above, so it survives
D1 for small deviations. Among them, however, only the one that gives the low type thelowest payoff would survive D1 globally. Let this point be denoted by t̃ j ∗ . Now consider jwhich is odd and j 6= j ∗ . This does not survive D1 because the indifference curves that go
through this point cross somewhere between t̂ j and t̂ j ∗ , giving the high type an incentive
to deviate. As such, there is generically a unique pooling equilibrium even if α is in anintermediate range.

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