Essays on Learning and Strategic Investment · Peter Achim Wagner Doctor of Philosophy Graduate Department of Economics University of Toronto 2013 The ﬁrst chapter studies the strategic

Essays on Learning and Strategic Investment

by

Peter Achim Wagner

A thesis submitted in conformity with the requirementsfor the degree of Doctor of PhilosophyGraduate Department of Economics

University of Toronto

Copyright c© 2013 by Peter Achim Wagner

Abstract

Essays on Learning and Strategic Investment

Peter Achim WagnerDoctor of Philosophy

Graduate Department of EconomicsUniversity of Toronto

2013

The first chapter studies the strategic timing of irreversible investments when returns

depend on an uncertain state of the world. Agents learn about the state through privately

observed signals, as well as from each other’s actions and experience. In this environment

there is the possibility of learning feedback in which an agent’s present action affects

how much she can learn from the other agent’s experience in the future. I characterize

symmetric mixed-strategy equilibria, and show that private information mitigates free-

riding and increases efficiency if the prior belief about the state is not too low, but that

it may lead to inefficient over-investment otherwise.

The second chapter examines the effect of trade opportunities on a seller’s incentive to

acquire information through experimentation. I characterize the unique equilibrium out-

come, and discuss the effects of variations in the information structure on the probability

of trade. The main result is that more accurate information for the buyer can reduce so-

cial welfare. Efficiency requires that the buyer offers a price that the seller always accepts

and that the seller experiments when it is socially optimal to do so. When the buyer

receives an informative signal about positive experimentation outcomes, the absence of

such a signal can induce the buyer to purchase the good with low but known quality

at a low price. If the buyer receives an informative signal about negative experimenta-

tion outcomes, the seller might not experiment so as to avoid the risk of generating an

outcome that could trigger the buyer to reduce her offer.

The third chapter analyzes the contracting problem of a principal who delegates research

to two independently experimenting agents. The features of the optimal contract depend

on the principal’s preferences over the agents’ successes. If successes are substitutes for

the principal, the first agent to produce a success receives the greatest reward. The

competition for the first success benefits the principal because it reduces the agents’ in-

ii

centive to delay their effort. In contrast, when successes are complements, the reward for

the second success is greater which results in a second mover advantage that encourages

agents to delay effort.

iii

Contents

1 Learning from Strangers 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.1 Basic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3.2 Histories and strategies . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.3 Equilibrium concept and refinements . . . . . . . . . . . . . . . . 9

1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4.1 Public signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4.2 Public and private signal . . . . . . . . . . . . . . . . . . . . . . . 16

1.4.3 Private signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.4 Welfare analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.A Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2 Experimentation and Trade 47

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.2.1 Optimal experimentation without trade . . . . . . . . . . . . . . . 52

iv

2.2.2 Partial Observability . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.2.3 Comparative statics . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.A Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3 Delegated Problem Solving 69

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Optimal contracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.1 Benchmark: observable effort . . . . . . . . . . . . . . . . . . . . 75

3.3.2 Evolution of rewards . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.3.3 Optimal rewards . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.3.4 Optimal deadlines . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.4.1 Collaboration versus competition . . . . . . . . . . . . . . . . . . 93

3.4.2 Resource-constrained agents . . . . . . . . . . . . . . . . . . . . . 94

3.4.3 Unobservable successes . . . . . . . . . . . . . . . . . . . . . . . . 95

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

3.A Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 103

v

Chapter 1

Learning from Strangers

1.1 Introduction

The economics literature typically assumes that agents learn exclusively from the behav-ior of others, or exclusively from their experiences. In reality, learning through observa-tion often involves both channels of learning. Take for instance the farmer who ascertainsthe value of a new type of crop from the fact that a neighboring farmer uses it, as well asby observing the neighbor’s yield the following year. A doctor may likewise learn aboutthe effectiveness of a new drug by observing that her colleagues prescribe it, and fromthe health outcomes of their patients.

This chapter investigates the strategic timing of investments under uncertainty whenagents learn about the uncertain return from each other’s actions and experience. In thepresence of both channels of learning, each agent can influence the other agent’s beliefs,and hence future experimentation, through her present choice of actions. The questionsaddressed in this chapter are: How does private information about the uncertain stateinfluence the agents’ incentives to invest and how are these incentives affected by thestrength of their private beliefs? Under which conditions is private information revealedin equilibrium and is it revealed instantly or over time? Does private information increaseor decrease efficiency relative to the scenario in which all information is made public?

The analysis is based on a model of strategic experimentation in which agents decidewhen to invest in a risky project. Once an agent has invested, her project yields anuncertain payoff that depends on an unknown state of the world which is either “good”or “bad”. In each state, the project yields a constant positive flow-return. If the state

1

Chapter 1. Learning from Strangers 2

is bad, however, active projects generate costly failures at random times, resulting in anegative expected payoff. Failure times are uncorrelated, and each agent incurs only thecost of her own project. At the outset, agents observe a private signal that conveys eithergood or bad news about the state. Over time, agents learn from observing each other’sactions and from the success or failure of each active project.

The key result in this paper is that if good signals are sufficiently more informative thanbad signals, equilibria are ex-ante more efficient when agents have private informationthan when this information is made public. Intuitively, one might expect that in theabsence of payoff externalities, more information generates better social outcomes, as itreduces uncertainty and allows agents to make better decisions. I show that under fairlygeneral conditions, the social value of disclosing private information is negative.

Inefficiencies arise in the presence of learning externalities, because agents wait too longbefore they invest in the hope that the other provides free information. Here, excessivedelay arises in two forms. There is delay between investments because of free-riding.After the first agent has invested, the other agent trades off the forgone returns frominvesting with the benefit of waiting for more information, but she does not take intoaccount the social value of her own experimentation. Excessive delay before investmentsis the result of leadership-aversion. Both agents prefer to obtain free information fromthe other, so that each has an incentive to delay their investment and wait for the otherto invest first.

When agents have private information, then delaying their investments has drawbacks.One downside is the possibility of negative learning feedback. The first agent can onlybenefit from waiting to invest if the other agent invests first. However, one agent’s delaymay signal bad information to the other, who, in response, may then be less willing toinvest, thereby reducing the first agent’s incentive to delay. As a result, an agent whoreceived a bad signal may have the incentive to invest without delay to encourage theother agent to invest earlier.

The second drawback results from each agent’s uncertainty about the other agent’s be-havior. For example, an agent who received a good signal may wait in the hope that theother agent invests first. The other agent, on the other hand, may have received a badsignal and therefore never invest. Another possibility is that an agent who may, despitehaving received a good signal, delay her investment to learn whether or not the otheragent invests. However, by delaying her investment she discourages future investment bythe other agent, even if the other has received a good signal.


prior belief that state is good

high low

precisionof signals

high An agent with a good signal invests immediately or“waits”; an agent with a bad signal “waits”.

An agent with a good signal invests with randomdelay at decreasing flow-rate; an agent with a badsignal “waits”.

lowAn agent with a good signal invest immediately;an agent with a bad signal invests immediately or“waits”.

An agent with a good signal invests with randomdelay at decreasing flow-rate; an agent with a badsignal “waits”.

1

Table 1.1: Characterization of symmetric equilibria based on the strength of good and bad signals for amoderate prior belief.

These negative effects of delay under private information can induce agents to investearlier, and thereby reduce free-riding and leadership aversion. The reduction increasesex-ante social welfare as long as it does not result in inefficient over-investment. Forexample, when bad signals are very informative, an agent with a good signal may investnot knowing the other agent’s signal, but regret investing if she learns that the otheragent’s signal is bad. In cases like these, it might be socially desirable to make privateinformation public.

The concrete mechanism that induces agents to invest earlier depends on the structureof the equilibrium which varies with the prior belief. There are four distinct types ofsymmetric equilibria. Each depends on the agents’ common prior belief about the stateand the informativeness of good and bad signals. Assuming a fixed prior belief aboutthe state that is “moderate” in the sense that free-riding occurs in equilibrium even whensignals are uninformative, these four different cases are summarized in Table 1.1.

When good signals are informative relative to bad signals, then agents who receive agood signal invest immediately. Agents who receive a bad signal invest immediately withsome probability that depends on the strength of their signal, and with the remainingprobability they “wait”: they do not invest unless the other agent invests first and herproject proves successful for some period of time. Intuitively, when good signals areinformative relative to bad ones, then agents who receive good signals are confident thatthe state is good. Agents with bad signals may then benefit from investing immediatelyto signal good information to the other agent.

When both good and bad signals are informative, then in the symmetric equilibrium,agents who receive a bad signal “wait”. Agents who receive a good signal invest imme-diately with some probability and wait with the remaining probability. The intuition isthat when both good and bad signals are informative, then agents with good signals have


no incentive to delay their investment given that the other agent’s signal is good, butat the same time they are hesitant to invest because an informative bad signal from theother agent would render their investment unprofitable.

When good and bad signals are uninformative, then agents who receive good signals preferto delay their investment, so that in the symmetric equilibrium they play a waiting game,randomly delaying their investment in the hope that the other agent invests first. Agentswho receive bad signals “wait” indefinitely. Therefore, over time both agents becomeincreasingly convinced that the other agent’s signal is bad if neither of them invests.Because bad signals are uninformative, agents with a good signal want to invest even ifthe other agent’s signal is bad, so that the result is a positive feedback loop: as eachagent becomes more pessimistic about the other agent’s signal, her incentive to delay herinvestment decreases, so that in equilibrium the other agent with a good signal has toincrease her rate of investment to keep the first agent indifferent. These effects reinforceeach other, eventually leading to an explosion of flow-rates of investment, ending thewaiting game with certainty before a finite time.

Finally, when bad signals are informative relative to good ones, then in the symmetricequilibrium, agents with bad signals “wait” and agents with good signals play a waitinggame similar to the previous case. The difference is that agents with good signals prefernot to invest if the other agent’s signal is bad. As a result, the equilibrium now exhibits anegative feedback loop: as each agent becomes more pessimistic about the other agent’ssignal, her incentive to delay her investment increases, so that in equilibrium the otheragent with a good signal has to decrease her rate of investment to keep the first agentindifferent. This leads to a dampening of investment rates that fade out over time.

A general result is that in the previously described equilibria, agents invest earlier. Onthe other hand, when agents have good signals, the delays in equilibrium are longer thanin the case in which their signals are publicly disclosed. These effects are the result ofnegative learning feedback and uncertainty about the other agent’s future investmentbehavior. Relative to the case in which signals are all made public, the different dis-tributions of delay under uncertainty lead to increased social welfare. Social gain fromshortened delay when agents have bad signals outweighs the social loss from lengtheneddelay when agents have good signals. The improvement follows from the fact that anagent who is pessimistic about the state of the world benefits more from any additionalpiece of information than an agent who is already certain that the state is good.


1.2 Literature Review

The paper relates to the literature on the strategic delay of investments. In Chamley andGale (1994) and Murto and Välimäki (2010) information is dispersed throughout society,and agents decide when to make an irreversible investment. Information is inefficientlyaggregated because investors have an incentive to delay their decision to acquire moreinformation by observing the behavior of others. In contrast to the current paper, inthese papers no new information is generated over time.

Learning from the experience of others is the main subject of the strategic experimen-tation literature (Bolton and Harris, 1999; Keller, Rady, and Cripps, 2005; Keller andRady, 2010; Klein and Rady, 2011). In these models, a number of agents experimentwith a “risky alternative” which has a payoff distribution that depends on some commonpayoff parameter that is unknown to all agents. The central result in these papers isthat there is too little experimentation in equilibrium, because agents prefer to wait andfree-ride on the information that is provided to them through the experimentation ofothers.

There is a number of papers that consider models of strategic experimentation in whichagents have private information. A closely related paper is Décamps and Mariotti (2004).The authors study a duopoly model in which two firms decide when to invest in a riskyproject, where the project’s value depends on a common state variable. Firms are pri-vately informed about their investment cost and after a firm invests the success of itsproject is publicly observable. Their paper differs from the present paper in that firmshave private information only about their private cost and not about the state variable.The welfare gains described here result from the fact that the agents are simultaneouslyuncertain about the state and each other’s private information. In the model of Décampsand Mariotti (2004), the result is reversed because each firm has in fact an incentive todelay its investment further to convince the other that its cost of investment is higher.

Murto and Välimäki (2011) consider a model of exit in which over time agents privatelylearn from their own experiences about the optimal time to exit. The authors find thatinformation is aggregated in randomly occurring bursts of exit. In contrast to the currentpaper, agents cannot learn from the experience of others, and exit is irreversible so thatthere is no incentive for them to behave so as to influence the belief of others. Rosenberg,Solan, and Vieille (2007) also consider strategic experimentation with private informationand irreversible exit, but they are primarily concerned with characterizing equilibrium


strategies in a general class of games, and not with the informational issues that are thefocus here. Another paper that considers strategic experimentation with private payoffsis Heidhues, Rady, and Strack (2010). Their paper investigates the effects of cheap-talk.The authors show that there exists an equilibrium in which free-riding disappears. Whiletheir main result is somewhat similar to the main finding in this paper, the underlyingmechanisms are very different. In particular, the authors do not limit their attention toMarkov strategies, so that it is possible to construct a punishment scheme that detersagents from free-riding on the other agent’s experimentation.

There are also several papers that investigate learning in R&D competition betweenfirms. Acemoglu, Bimpikis, and Ozdaglar (2011) consider a model in which firms receivea private signal about different projects. The results are much in the spirit of the currentpaper with regards to the structure of the equilibria. The main differences lie in thepresence of payoff externalities and in that experimentation is instantaneously perfectlyrevealing in their paper. The latter eliminates the possibility that an agent times herinvestment in order to encourage others to invest. Moscarini and Squintani (2010) con-sider a model of a winner-takes-all R&D competition in which firms observe a privatesignal about the unknown type of a research project. Over time firms learn about theproject’s type from the actions of their competitor and past payoffs, and they decidewhen to exit irreversibility. Because of the winner-takes-all environment, agents do notbenefit from signaling good news, so that signaling does not play the same role as in thecurrent paper.

The welfare improving effect of private information in the present paper is related tothe “encouragement effect” discussed in Bolton and Harris (1999), and more recentlyin Rosenberg, Salomon, and Vieille (2010). Bolton and Harris (1999) show that whenagents can learn from each other’s experiments, they may experiment more to induce ormove forward the time at which other agents initiate experimentation. The key differenceis that in their paper, agents encourage others by generating more publicly observableinformation, leading to an unambiguous welfare improvement. Here, in contrast, anagent who has bad information encourages the other by mimicking the behavior of anagent with good information, preventing the aggregation of information, so that theex-ante welfare effect is not immediately obvious. In general, both effects may coexist:Rosenberg, Salomon, and Vieille (2010) show that the encouragement effect always arisesin symmetric equilibria of “bad news” models such as the one in this paper.

The welfare improvement is related to the “smoothing effect of uncertainty” (Morris andShin, 2002). Teoh (1997) demonstrates this effect in a model of public goods provision,


in which agents contribute to a joint project. The marginal return to their investment isdetermined by an uncertain state of the world. The author shows that non-disclosure ofinformation may increase ex-ante welfare when the investment has marginally diminishingreturns, because the loss resulting from a reduction in investment after the release of badnews outweighs the benefits from increased investment when the information is favorable.This is the same mechanism that drives the main result in the current paper: when badnews is publicly disclosed, then free-riding and leadership-aversion increase, leading toan over-proportional reduction in the expected value of investment. Contrary to Teoh(1997), however, the welfare improvement from non-disclosure in this paper is purely theresult of learning externalities, and does not require payoff externalities.

1.3 Model

1.3.1 Basic framework

There are two agents, indexed i = 1, 2. Time t ∈ R+ is continuous with infinite horizonand future payoffs are discounted with common discount rate r. Each agent decideswhen to operate a risky project. The profitability of each agent’s project depends on theunknown state θ ∈ {G,B} which is either good (θ = G) or bad (θ = B). To initiate theproject, each agent i must make an investment I > 0. The investment is irreversible, inthe sense that once an agent has initiated her project, the investment cost is sunk andcannot be recovered. An inactive project yields no return. An active project yields thecertain flow return y, but it may fail and generate a lump-sum loss c > 0 if the state isbad. The probability that an active project fails over a time period of length t > 0 is

Fθ(t) =

1 − e−γt if θ = B

0 if θ = G,

where γ > 0 is the time-independent failure rate. The investment that is required toinitiate the project is assumed to be lower than the present value of operating a projectin a good state, i.e., I < y/r. I assume that active projects must be continued indefinitelyunless a failure occurs. To rule out the uninteresting case in which both agents prefer toinvest immediately, it is further assumed that the average flow return from operating aproject in a bad state is negative, i.e., y < γc, given that the mean time a bad projectlasts is 1/γ.


At the outset, each agent i = 1, 2 observes a private binary signal si ∈ {g, b} that providesinformation about the realization of the state variable θ. Define Ω = {G,B} × {g, b}2

and let P denote the common prior belief on Ω. The probability of observing a goodsignal (si = g) is greater than the probability of observing a bad signal (si = b) ifthe state is good, and the probability of observing a bad signal in a bad state is thesame as the probability of observing a good signal in a good state, so that P(g|G) >P(b|G), P(b|B) > P(g|B) and P(G|g, b) = P(G). Note that Bayes’ rule then implies thatP(G|g) > P(G) > P(G|b). I assume that the signals are conditionally independent andthat the probability of observing either signal is positive for each θ ∈ {G,B}. Agents areex-ante identical, i.e., P(s1 = g|G) = P(s2 = g|G) and P(s1 = b|B) = P(s2 = b|B).

I model the continuous-time environment as a multi-stage game based on a technique byMurto and Välimäki (2011). At the beginning of each stage, an agent decides how longto wait before initiating or terminating her project, given the other agent does not makea move and no failure occurs. A stage ends with the first move of an agent or with theoccurrence of a failure. Time is counted from the beginning of each stage, so that actualtime, which accumulates from the beginning of the game, is recorded as the sum of thelengths of all preceding stages. In this framework, it is possible to model the environmentas a fully dynamic game in continuous time1 in which mixed strategies are well-definedand have a natural interpretation as distributions over transition times.

Formally, denote the status of agent i’s project in stage k = 1, 2, . . . by aik ∈ {!, 0, 1},where aik = 1 represents an active project, aik = 0 an inactive project and aik = ! a failedproject. The initial status of project i is ai1 = 0 for each i. An action for agent i in stagek ≥ 1 is a transition time τ ik ∈ R+ ∪ {∞}. If no failure occurs, then stage k ends attk = min{τ 1k , τ 2k}. The flow payoff for agent i in this case is

∫ tk0 e

−rsaiky ds. If τ ik = tk,then the status of agent i’s project in stage k + 1 is aik+1 = 1 − aik. Time is reset to 0,and the next stage proceeds in the same way as the previous one. If a failure occurs att < tk, the game ends. The flow return for each agent i is

∫ t0 e

−rsaiky ds and the agentwhose project generated the failure incurs a loss c. If project i failed in stage k, its finalstatus is aik+1 = ! .

1as opposed to a stopping game in which an agent chooses a single stopping time


1.3.2 Histories and strategies

A non-terminal history in stage k is a list hk = (t1, a2, t2, a3, . . . , tk−1, ak) that consists ofstage lengths tl ∈ R+, 1 ≤ l < k, previous status profiles al = (a1l , a2l ) ∈ {0, 1}2, 1 < l ≤ k,and a current status profile ak ∈ {0, 1}2. A terminal history is a list (hk−1, tk−1, ak), wherehk−1 is a non-terminal history and ak ∈ {!, 0, 1}2 with aik = ! for one agent i = 1, 2,or an infinite list (t1, a2, t2, a3, . . .) with ak ∈ {0, 1}2 for all k ≥ 2. For a given terminalhistory h ending in stage K, the continuation payoff for agent i in stage 0 ≤ k < K is

U ik(h)=K−1∑

l=k

[e−r(Tl−Tk)

∫ tl

0

e−rsail y ds−ail(1−ail−1)e−r(Tl−Tk)I]−1{aiK=!}(h)e

−r(TK−1−Tk)c,

where Tl =∑l

j=1 tj records the actual time that accumulates from the beginning of thegame and 1{aiK=!}(h) is an indicator function that is equal to 1 if agent i’s final status is! and equal to 0 otherwise. Let the set of all non-terminal histories be denoted by H.A behavioral strategy for type si ∈ {g, b} of agent i is a function

σ̂i(si) : H → ∆(R+)

that assigns to every non-terminal history a distribution over transition times. There arecombinations of strategy pairs that are not well-behaved in the sense that they do notgenerate proper paths in time. Suppose for example that each agent’s strategy has theproperty that at time 0 she chooses to start the project immediately if the other agent’sproject is inactive, and to terminate the project immediately if the other agent’s projectis active. Such a pair of strategies generates an infinite history in which agents oscillatebetween starting and stopping their projects at time 0, without progressing in time. Thisdoes not create any difficulties in terms of the formal analysis. If we allow payoffs tobe negative infinity, then the payoff at histories with infinitely many switches is minusinfinity for one of the agents, so that any pair of strategies generating such historiescannot be an equilibrium.

1.3.3 Equilibrium concept and refinements

A profile of strategies σ̂ generates a distribution Pσ̂ over terminal histories. For a givencommon prior belief P, the agents’ beliefs about each other’s signals and the state aredetermined by Bayes’ rule at every non-terminal history hk for which the set of terminalhistories that have hk as a sub-history lies in the support of Pσ. From these beliefs we


can then calculate the distribution Pσ̂|hk which is the distribution over terminal historiesconditional on reaching history hk. If we denote the associated expectation operator byEσ̂|hk , then Eσ̂|hk [U

ik|si] is the expected continuation payoff for type si of agent i at history

hk under the strategy profile σ̂. A strategy profile σ̂ is a perfect Bayesian equilibriumfor the common prior belief if the agents’ strategies are sequentially rational given theirbeliefs, and their beliefs are derived via Bayes’ rule from σ̂ whenever possible.

Because I am interested in the informational aspects of strategic behavior, I focus theanalysis on perfect Bayesian equilibria in Markov strategies. Agent i’s strategy is aMarkov strategy if the distribution over transition times at every history depends onlyon the payoff relevant information that is available to i at that history. The payoffrelevant information that is available to agent i at each history can be characterized bya profile of belief systems π = (πi(g),πi(b))i=1,2, where πi(si) : H → ∆({G,B}× {g, b})assigns to each non-terminal history a distribution over combinations of states and types.A Markov strategy for type si of agent i is then a function

σi(si) : (∆({G,B}× {g, b}))4 × {0, 1}2 → ∆(R+)

and a Markov perfect Bayesian equilibrium (henceforth MPBE) is a pair (σ,π) that hasthe property that σi(si) is sequentially rational at every non-terminal history for eachtype si of each agent i given the agents’ beliefs at that history, and beliefs are derivedvia Bayes’ rule from σ whenever possible.

It should be emphasized here that Markov strategies do not depend on another agent’sprivate beliefs. The notation is convenient, because agent i’s strategy can be expresseddirectly as a function of the belief of each type of the other agent. Equivalently, I couldwrite agent i’s Markov strategies in the conventional way as a function of her private andthe common public belief, and then use Bayes’ rule to back out agent i’s belief about theother agent’s private beliefs. With the above notation I omit this additional step.

Since for perfect Bayesian equilibrium there is no restriction on what agents ought tobelieve after they observe an unexpected move, it is necessary to impose additionalrestrictions on belief systems to rule out a number of implausible equilibria.

Definition 1 (Reasonable beliefs). An MPBE (σ,π) has reasonable beliefs, if at everynon-terminal history hk+1 = (hk, tk, ak+1) ∈ H for which tk is not in the support ofσ̂i(si)(π(hk)) for either si ∈ {g, b} the following conditions hold.

1. For each type sj of agent j *= i, the belief about the state, conditional on each type


si of agent i, is consistent with Bayes’ rule:

πj(sj)(hk+1)(G|si) =πj(sj)(hk)(G|si)

πj(sj)(hk)(G|si) + πj(sj)(hk)(B|si)(1 − Fθ(tk))a1k+a

2k

.

2. For each sj ∈ {g, b} we have

πj(sj)(hk+1)(si = g) =

1 if ajk+1 − a

jk = 1,

0 if ajk+1 − ajk = −1.

The first condition says that when an agent makes an unexpected move, then this shouldnot affect the way an agent updates her belief about the state of the world. The secondcondition states that each agent should associate unexpected investments with good typesand unexpected terminations of projects with bad types. The second condition is neededto rule out a class of artificial equilibria in which agents are deterred from deviatingbecause of the other agent’s off-equilibrium belief after their deviation. For example, itis straightforward to construct an equilibrium in which the good type of each agent doesnot invest because the other agent would then be convinced that the investing agent’stype is bad, and would therefore delay her own investment excessively, so that it is nolonger profitable for good types to invest.

1.4 Results

I begin the analysis with a restricted version of the model in which both agents arerequired to continue an active project indefinitely. Restricting the model is motivatedby theoretical as well as practical considerations. First, I am mainly interested in anenvironment in which the cost of initiating a project is large relative to the possible in-formational gain derived from observing the success or failure of the other agent’s project.If the stakes are sufficiently high, then agents have a strong incentive to strategically de-lay their initial investment, but it is unprofitable to frequently stop and restart a project.Second, restricting the agents’ decision to the timing of their initial investment elimi-nates all coordination problems that may result in “switching” behavior (see Keller et al.(2005)), and it allows me to focus the discussion on the signalling problem of privatelyinformed agents. Finally, assuming that agents never stop projects tremendously sim-plifies the mathematical analysis, without affecting the validity of the results in a moregeneral environment.


The remainder of this paper is structured as follows. I first discuss a benchmark model inwhich each agent’s signal is publicly observable, and I show that strategic conflicts ariseonly when agents are relatively uncertain about the state. I then consider a scenario inwhich only one agent’s signal is private information and present two asymmetric equilibriathat highlight some of the features that are relevant for the subsequent analysis of thesymmetric equilibrium with two privately informed agents. I continue with a welfareanalysis, and I demonstrate that it may be socially preferable if agents are uncertainabout each other’s signals. In the last subsection I discuss to what extent the results canbe generalized to the scenario in which projects may be terminated and restarted.

1.4.1 Public signals

As benchmark scenario, consider the case in which both signals are publicly observable.Beliefs can then be represented by the probability p ∈ [0, 1] each agent assigns to the statebeing good, so that a Markov strategy for each agent i is a function σi : [0, 1]×{0, 1}2 →∆(R+). Since projects cannot be terminated, we can ignore the case in which bothprojects are active, that is, a = (1, 1). I therefore begin by deriving the best response foran agent who has not yet invested, given that the other agent’s project is active.

The flow-payoff of an agent who does not operate a project is 0. For an agent whooperates a project, the expected flow-payoff depends on the belief p at the beginningof the period, whether or not the other agent is operating a project, and on the timeτ at which the other agent invests. Denote the number of currently active projects byα = a1 + a2. I omit subscripts for notational convenience. If no failure occurs beforeτ , then the flow-payoff for an agent who operates a project is

∫ τ0 e

−rsy ds. If a failureoccurs at time t < τ , then an agent with an active project receives

∫ t0 e

−rsy ds, and withprobability 1/α also incurs the loss e−rtc. The probability that a failure does not occuris equal to 1 in a good state and equal to (1− FB(τ))α = e−αγτ in a bad state. A failuredoes occur with the remaining probability 1− e−αγτ , and the random time of the failurein this case has density αγe−αγt on [0, τ). The expected flow-payoff for an agent with anactive project is therefore

uα(p, τ) = p

∫ τ

0

e−rty dt

+ (1 − p)[∫ τ

0

αγe−αγt(∫ t

0

e−rsy ds−e−rtc/α)

dt + e−αγτ∫ τ

0

e−rty dt

].


Solving the integrals and simplifying the resulting expression gives

ruα(p, τ) = p(1 − e−rτ )y + (1 − p)(1 − e−(r+αγ)τ )λα(y − γc),

where the constant λα = r/(r + αγ) is the marginal value of receiving a stream of aconstant flow-payoff up to the first time a project fails, given that α ∈ {1, 2} projects areoperated simultaneously.

1.4.1.1 Leader and follower

Once an agent has started her project, the second agent faces a simple decision problem.She must decide how long to wait before making the investment, trading off the benefitof obtaining additional information about θ with the loss in revenue from delaying herinvestment. In a situation like this, the agent operating a project shall be called the leaderand the other agent shall be referred to as the follower. Given that the follower delaysher investment by τ , the leader receives the expected flow-payoff u1(p, τ) in the currentperiod. The game ends if a failure occurs before the follower makes the investment, andthe payoff for the leader is 0. If a failure does not occur before the follower invests, thenthe continuation value for the leader is u2(p,∞). Given the leader’s belief p about θ, sheexpects that a failure does not occur with probability p + (1 − p)e−γτ . Her value beforemaking the investment is therefore

vl(p, τ) = r[u1(p, τ) +

(p + (1 − p)e−γτ

)e−rτu2(p

′(τ),∞) − I],

wherep′(τ) =

p

p + (1 − p)e−γτ

denotes the updated belief in the next period, derived from Bayes’ rule. After substitutingthe formulas for u1, u2 and p′(τ), the leader’s value becomes

vl(p, τ) = py + (1 − p)(λ1 + (λ2 − λ1)e−(r+γ)τ )(y − γc) − rI. (1.1)

Given the follower delays her investment by τ , her payoff is 0 if the leader’s project failsbefore τ . It does not fail with probability p + (1 − p)e−γτ , in which case the follower’scontinuation value is u2(p′(τ),∞)− I. Using the formulas for u2 and p′(τ), the expected


leader follower

10 20 30 40 50

3

2

1

1

2

vl(p, ·)

p = 0.3

p = 0.5

p = 0.7

τ

10 20 30 40 50t

1

1

2

3vf (p, ·)

p = 0.7

p = 0.5

p = 0.3

Figure 1.1: The value for the leader and follower as functions of the follower’s delay τ for parametervalues r = 0.02, γ = 0.1, y = 6, c = 500, and I = 10.

present value for the follower is

vf (p, τ) = e−rτp(y − rI) + e−(r+γ)τ (1 − p)(λ2(y − cγ) − rI). (1.2)

The following lemma reports basic properties of the functions vl and vf .

Lemma 1. The function vl is linearly increasing in p, convex and decreasing in τ forevery p ∈ (0, 1) and supermodular in (p, τ). The function vf is linearly increasing in pand it has a single peak in τ at

τ ∗(p) =

(φ(p∗f ) − φ(p)

)/γ if p < p∗f

0 if p ≥ p∗f(1.3)

for every p ∈ (0, 1), where φ(p) = log(p) − log(1 − p) is the log-likelihood ratio of p and

p∗f =rI + γI + λ2(γc − y)/λ1y + γI + λ2(γc − y)/λ1

. (1.4)

Moreover, vf (p, τ ∗(p)) > vl(p, τ ∗(p)) if p < p∗f and vf (p, τ ∗(p)) = vl(p, τ ∗(p)) if p ≥ p∗f .

All proofs are in the appendix. I write v∗f (p) = vf (p, τ ∗(p)) and v∗l (p) = vl(p, τ ∗(p)) forthe values of the leader and the follower given the follower uses the optimal delay. Notethat since τ ∗ is weakly decreasing in p, and vl and vf are strictly increasing in p as well asdecreasing in τ , it follows that v∗l and v∗f are strictly increasing functions in p. Moreover,v∗l is positive if p = 1, negative if p = 0 and continuous. Hence, it has a unique root on(0, 1), which I denote by p∗l . Denote further by p∗J the threshold at which the value of


investing jointly without delay is positive. More specifically, p∗J is the lowest value atwhich vf (p, 0) = 0 for all p ≥ p∗J . Using (1.2), we can calculate the threshold explicitly:

p∗J =rI + λ2(γc − y)y + λ2(γc − y)

.

It follows from Lemma 1, that in any equilibrium both agents invest immediately ifP(G|s1, s2) > p∗f . If the prior belief exceed this threshold, then it is optimal for thefollower not to delay investing, so that it is optimal for each agent to invest immediately.The following lemma describes the ordering of these thresholds.

Lemma 2. 0 < p∗J < p∗l < p∗f < 1.

The lemma confirms that it may be profitable to be a leader when the follower delaysher investment.

1.4.1.2 Equilibria when signals are publicly observable

When one agent invests at public belief p, then the second agent becomes the followerand optimally delays her investment by τ ∗(p). If the prior belief is above the thresholdat which delaying investment is optimal, P(G|(s1, s2)) > p∗f , then there is a uniqueequilibrium in which both agents invest immediately. One the other hand, if the priorbelief is below the leader threshold, P(G|(s1, s2)) < p∗l , then being the leader is notprofitable for either agent, so that both agents never invest.

Theorem 1. If P(G|s1, s2) > p∗f then in every MPBE both agents invest immediately. IfP(G|s1, s2) < p∗l then in every MPBE neither agent makes an investment.

If the prior belief lies between p∗l and p∗f , then in equilibrium investments must occursequentially, because each agent prefers to wait if she expects that the other agent in-vests, and she prefers to invest if she expects that the other agent waits. Equilibria inpure strategies are therefore necessarily asymmetric. These equilibria have an obviousstructure: in the first period, one agent invests immediately, and the other agent waitsindefinitely for the first agent to make a move, and in the second period the followermakes her investment with optimal delay. There are of course two equilibria of this kind,one in which agent 1 is the leader and agent 2 is the follower, and one in which theirroles are reversed.


Theorem 2 (Asymmetric equilibria). If p∗l ≤ P(G|s1, s2) < p∗f , there exist two asym-metric simple equilibria, one for each i = 1, 2. In the first period, along the equilibriumpath, agent i starts the project immediately, and agent j *= i waits indefinitely. In thesecond period, agent j *= i delays her investment by τ ∗(P(G|s1, s2)).

There also is a symmetric equilibrium in mixed strategies, in which each agent randomizesover the times of her initial investment. In the mixed strategy equilibrium each agentinvests at a rate that renders the other agent indifferent between investing immediatelyand delaying her investment by any length of time. Note that as long as neither agent hasmade the investment no new information becomes available, so that the agents’ flow rateof investment β is constant over time. We can then immediately calculate the equilibriuminvestment rate, using the fact that each agent must be indifferent between making theinvestment immediately and never making the investment, i.e.,

v∗l (p) =

∫ ∞

0

βe−βte−rtv∗f (p)dt.

Solving the equation for β gives the equilibrium starting rate as a function of the commonbelief p :

β∗(p) =rv∗l (p)

v∗f (p) − v∗l (p). (1.5)

Observe that the flow-rate of investment is positive whenever the value of becoming theleader is greater than 0, because it follows from Lemma 1 that the denominator of β∗ isalways non-negative. Moreover, the difference between the value of the follower and thevalue of the leader converges to 0 as p approaches the follower threshold p∗f , to that theequilibrium flow-rate of investment approaches infinity.

Theorem 3 (Symmetric equilibrium). If p∗l ≤ P(G|s1, s2) < p∗f , then there exists aunique symmetric equilibrium. In the first period of this equilibrium, along the equilibriumpath, each agent makes the investment at flow rate β∗(P(G|s1, s2)). In the second period,the follower starts the project with delay τ ∗(P(G|s1, s2)).

1.4.2 Public and private signal

The purpose of this section is to study the signalling problem of an agent with privateinformation in an environment, in which her investment decision is not confounded by heruncertainty about the other agent’s private information. I show that in an equilibrium


in which the informed agent moves first, bad types invest only if good types invest, andif good types invest the degree to which bad types choose to separate and reveal theirinformation is weakly decreasing in their private belief about the state.

If type g of the informed agent invests without delay in the first period, then type b eitherimitates a good type by investing immediately, or she never invests. For simplicity, Ishall assume that the informed agent is agent 1, and the uninformed agent is agent2. Then, if type b of agent 1 was to delay her investment by some finite time τ > 0,then in an equilibrium with reasonable beliefs, agent 2 believes that agent 1’s type isbad if she delays her investment. Hence the bad type’s value of investing with delayτ is e−rτv∗l (P(G|b)). On the other hand, if she invests immediately, then her value isvl(P(G|b), τ ∗(p)), where p denotes agent 2’s belief that the state is good after observingthat agent 1 invested immediately in the first period. It follows from Bayes’ rule thatp > P(G|b), and therefore τ ∗(P(G|b)) > τ ∗(p), so that

vl(P(G|b), τ ∗(p)) > v∗l (P(G|b)) > e−rτv∗l (P(G|b)).

Hence, type b can profitably deviate so that there cannot exist an equilibrium in whichagent 1 moves first and type b of agent 1 invests with positive and finite delay, so thattype b either invests immediately, or waits indefinitely.

Therefore, I am lead to consider an equilibrium in which type g of agent 1 invests im-mediately, and type b of agent 1 invests immediately with probability ξ and delays herinvestment indefinitely with probability 1 − ξ. Agent 2’s belief after observing agent 1invested immediately in the first period is given by Bayes’ rule,

pξ =P(G, g) + P(G, b)ξ

P(g) + P(b)ξ.

The optimal delay for agent 2 in the second period is then τ ∗(pξ), so that the expectedvalue of investing immediately is vl(P(G|s1), τ ∗(p)) for type s1 of agent 1.

If type b invests immediately with probability ξ = 1, then p = P(G), and thereforeit is sequentially rational for type b to invest immediately if vl(P(G|b), τ ∗(P(G))) ≥ 0.Hence, if vl(P(G|b), τ ∗(P(G))) ≥ 0, then an equilibrium in which the informed agentmoves first is a pooling equilibrium. If type b invests with probability ξ = 0, thenp = P(G|g), and therefore it is sequentially rational for type b to wait indefinitely ifvl(P(G|b), τ ∗(P(G|g))) < 0. Thus, if vl(P(G|b), τ ∗(P(G|g))) < 0 < v∗l (P(G|g)), then anequilibrium in which the informed agent moves first is a fully separating equilibrium. If,


one the other hand, v∗l (P(G|g)) < 0, then the value of investing is negative for type g evenif agent 2 knows her type, and therefore the only equilibrium is a pooling equilibrium inwhich neither agent invests.

If vl(P(G|b), τ ∗(P(G))) < 0 < vl(P(G|b), τ ∗(P(G|g))), then the equilibrium is partiallyseparating. In particular, it has the property that type b of agent 1 is indifferent betweeninvesting immediately and delaying her investment indefinitely given agent 2’s equilibriumbelief pξ that the state is good in the second period, and pξ∗ is consistent with theprobability ξ∗ with which type b invests immediately in equilibrium:

vl(P(G|b), τ ∗(pξ∗)) = 0 and pξ∗ =P(G, g) + P(G, b)ξ∗

P(g) + P(b)ξ∗. (1.6)

To solve for pξ∗ and ξ∗, use equation (1.1) to write the value of the leader as convexcombination vl(p, τ) = (1−e−(r+γ)τ )u1(p,∞)+e−(r+γ)τu2(p,∞). Substituting the formulafor τ ∗ in (1.3), we can rewrite the condition vl(P(G|b), τ ∗(pξ∗)) = 0 and solve for theposterior belief,

pξ∗ =p∗f

p∗f + (1 − p∗f )[1 − u2(P(G|b),∞)/u1(P(G|b),∞)]γ/(r+γ). (1.7)

From (1.6) we then obtain the equilibrium starting probability for agent 1:

ξ∗(P) =P(g)pξ∗ − P(G, g)P(G, b) − P(b)pξ∗

. (1.8)

The following result summarizes these findings.

Theorem 4 (Informed leader). Suppose P(G|g) > p∗l . Then there exists a Markov perfectBayesian equilibrium with the following properties.

1. (Pooling equilibrium) If vl(P(G|b), τ ∗(P(G))) ≥ 0 then both types of agent 1 investimmediately with probability 1 in the first period. In the first period, agent 2 waitsindefinitely. In the second period, agent 2 delays her investment by τ ∗(P(G)).

2. (Semi-Separating) If vl(P(G|b), τ ∗(P(G))) < 0 < vl(P(G|b), τ ∗(P(G|g))) then, inthe first period, type g invests immediately with probability 1 and type b investsimmediately with probability ξ∗(P), given by (1.8), and waits indefinitely with prob-ability 1−ξ∗(P). In the first period, agent 2 waits indefinitely. In the second period,agent 2 delays her investment by τ ∗(pξ∗), where pξ∗ is given by (1.7).


0.0 0.2 0.4 0.6 0.80.0

0.2

0.4

0.6

0.8

1.0ξ∗(P)

P(G|b)

Figure 1.2: The equilibrium probability of immediate investment for a bad type. The parameter valuesr = 0.02,γ = 0.1, y = 6, c = 500, and I = 0. The private belief of a good type is given by P (G|g) = 0.9,and the graphs are plotted for P (g) = 0.7, 0.5 and 0.05 (top to bottom).

3. (Separating) If vl(P(G|b), τ ∗(P(G|g))) < 0 then type g of agent 1 invests immedi-ately with probability 1, and type b never invests. In the first period, agent 2 waitsindefinitely. In the second period, agent 2 delays her investment by τ ∗(P(G|g)).

The probability that type b invests in equilibrium as a function of her private belief aboutthe state is shown in figure 1.2. That the starting probability may be discontinuous inthe private belief of a bad type follows from the fact that a bad type may be too certainthat the state is bad to make it profitable for her to invest, even if the other agent wouldinvest without delay. To be more specific, recall that the lowest private belief at whichthe bad type of agent 1 is willing is to invest is p∗J , the threshold at which operatingboth projects simultaneously is profitable for both agents. Hence, P(G|b) < p∗J , and thenthe value of investing is negative for type b, regardless of the length of the delay of thefollower.

1.4.3 Private signals

I now turn to symmetric equilibria in the case in which each agent observes her signalprivately. In this case, the symmetric equilibria can be characterized by the equilibriumbehavior of good types. Figure 1.3 illustrates the structure of equilibria schematically.The relation of P(G|(g, b)) to the leader threshold p∗l determines whether good types delaytheir investment or invest immediately. If the leader threshold p∗l is below P(G|(b, g)),then in a symmetric equilibrium good types invest with probability one. Intuitively,


(0, 0)

p∗l

1

1

p∗f

P(G|(g, b))

P(G|(g, g))

P(G|(b, b)) P(G|b) P(G|(g, b)) P(G|(g, g))

pooling

semi-sep.

separatingor

semi-sep.separating

semi-sep.or

no investment

noinvestment

type g invests immediately

type g delaysinvestment

type g investswith probability 1

type g investswith probabilityless than 1

Figure 1.3: Schematic characterization of Markov perfect Bayesian equilibria.

good types never “regret” investing if P(G|(g, b)) ≥ p∗l , because the value of being theleader is non-negative even if the other agent’s type turns out to be bad. Conversely, ifp∗l > P(G|(g, b)), then good types do not want to invest if the other agent’s type is bad,and therefore wait to be able to observe the other agent’s investment decision. Hence, inequilibrium, good types invest with a probability less than 1.

The relation of P(G|(g, g)) to the follower threshold p∗f determines the timing with whichprivate information is revealed. When the follower threshold p∗f lies below the P(G|(g, g)),then all private information is revealed immediately. Intuitively, good types prefer notto delay their investment if the other agent’s type is good. Even when good types arenot perfectly revealed because of pooling, the cost for good types of signaling bad newsto non-investing bad types outweighs the potential benefit from learning from bad typeswho do invest. On the other hand, when the follower threshold lies above P(G|(g, g)),then good types can profitably deviate by waiting if the good type of the other agentinvests with probability one. In equilibrium, therefore, good types play a waiting gamerandomly delaying their project hoping for the other agent to invest first.

Theorem 5 characterizes equilibria in which good types invest when p∗f < P(G|(g, g) andp∗l < P(G|(g, b)). In figure 1.3 these equilibria correspond to the three area in the bottomleft. The shaded area represents a set of symmetric equilibria that are either separating


or semi-separating. The distinction depends on parameter values in a more complicatedway than can be represented in the diagram.

Theorem 6 characterizes equilibria in which good types enter immediately with positiveprobability. The equilibria correspond to the left-hand side of the upper shaded area.Equilibria in this area are either semi-separating, when agents of type g invest withpositive probability, or pooling equilibria, when agent of type g do not invest. Again, thedistinction between these equilibria is not possible in the diagram.

Theorem 7 characterizes equilibria with random delayed entry for p∗f > P(G|(g, g)) whichcorrespond to the right hand column in figure 1.3. The lower rectangle represents sepa-rating equilibria for P(G|(g, b)) > p∗l which have increasing investment rates. The areaabove it represents semi-separating equilibria, when good types invest at decreasing rate,and pooling equilibria without investment. Note that there exists no equilibrium in whichgood types invest immediately when p∗f > P(G|(g, g)), because they can profitably devi-ate by waiting for the other agent to invest first, and condition her investment decision onwhether the first agent invested or not. Likewise, good types do no delay their investmentwhen P(G|(g, g)) > p∗f . In an equilibrium in which good type delay their investment,they reveal their types when they make an investment, so that when one agent invests,the good type of the other agent invests without delay. Hence, for good types, there isno value in delaying her investment.

1.4.3.1 Immediate entry

Consider first the case in which P(G|(g, g)) > p∗f , and P(G|(b, g)) > p∗l . Suppose, thattype g of each agent invests immediately in the first period, and type b invests immedi-ately with probability η ∈ [0, 1] and with probability 1 − η she “waits”. Formally, withprobability 1 − η type b randomizes over starting times, with a flow rate of investmentgiven by β∗(P(G|(b, b)) from equation (1.5). This is the equilibrium flow rate of invest-ment in the symmetric equilibrium with observable signals, when the realization of eachsignal is bad. The strategy for a bad type is then as follows: in the first period sheinvests immediately with probability η and with probability 1 − η she invests with flowrate β∗(P(G|(b, b)). If she waits and the other agent invests immediately, then in thesecond period she invests with delay τ ∗(pbη). If she waits and the other agent invests firstwith positive delay, then she invests with delay τ ∗(P(G|(b, b)).

Define Pg = P(·|g) and Pb = P(·|b) to be the interim belief of agent i, i.e., agent i’s belief


after observing the realization of her signal and before the beginning of the game. A badtype’s updated belief that the state is good after observing that the other agent investedis by Bayes’ rule

pbη =Pb(G, g) + Pb(G, b)η

Pb(g) + Pb(b)η. (1.9)

Note that a higher η implies that the posterior belief is closer to the bad type’s privatebelief. If a bad type observes that the other agent invested, her optimal delay in thesecond period is τ ∗(pbη). Type b of the other agent, anticipating her response, assigns toimmediate investment the expected value

V Pinvest(η) = (Pb(g) + Pb(b)η)vl(p

bη, 0) + P

b(b)(1 − η)vl(Pb(G|b), τ ∗(pbη)).

With probability Pb(g)+Pb(b)η the other agent invests immediately, and then both agentscontinue indefinitely. With probability Pb(b)(1− η) the bad type of the other agent doesnot invest, and then delays her investment by τ ∗(pbµ). The value of waiting on the otherhand is

V Pwait(η) = (Pb(g) + Pb(b)η)vf (p

bη, τ

∗(pbη)) + Pb(b)(1 − η) max{0, vl(Pb(G|b), τ ∗(Pb(G|b))}.

With probability Pb(g) + Pb(b)η the other agent invests, and then the value of waiting isequal to the value of being follower at belief pbη. With probability P

b(b)(1 − η) the otheragent does not invest, and then the agents play the symmetric equilibrium of the gamein which they each know that both signals are bad. The equilibrium value for each agentis equal to the value of the leader, if that value is positive, and zero otherwise.

If Pb(G) ≥ p∗f , then there is a pooling equilibrium in which both agents invests immedi-ately. To see why, note that if η = 1 then the posterior belief of a bad type is pbη = P

b(G).Since Pb(G) > p∗f it is optimal for a bad type to invest immediately in the second periodif she does not invest in the first period. It then follows that V Pinvest(1) = V Pwait(1) so thatthere is no benefit for a bad type in delaying her investment.

If Pb(G|g) ≥ p∗f and p∗l > Pb(G|b), there exists a separating equilibrium in which goodtypes invest immediately and bad types invest with probability 0 in the first period andinvest in the second period immediately if the other agent invests. If a bad type observesthat the other agent invests in the first period, then her updated belief is Pb(G|g) ≥ p∗fso that it is optimal to invest without delay. On the other hand, each agent’s type is badand neither invests, then their updated belief is Pb(G|b) < p∗l , so that it is optimal for


them not to invest. There is a set of prior beliefs for which these pooling and separatingequilibria coexist. However, only the separating equilibrium is robust in the sense that ifa bad type expects that the bad type of the other agent does not invest with an arbitrarilysmall but positive probability, then she prefers not to delay her investment.

If p∗f > Pb(G), then bad types invest with a probability less than one. Note that if

bad types invest with certainty, their posterior belief in equilibrium after observing thatthe other agent invests is Pb(g). Since p∗f > P

b(g), the optimal delay following theother agent’s investment is positive, i.e., τ ∗(pbη) > 0. The value of waiting for a badtype is therefore V Pwait(1) = v∗f (P

b(g)) which is now greater than the value of investingimmediately, V Pinvest(1) = vl(P

b(g), 0). In the symmetric equilibrium, the equilibriumprobability η∗ that the bad type of each agent invests immediately must have the propertythat the expected value of investing immediately is the same as the expected value ofwaiting:

V Pinvest(η∗) = V Pwait(η

∗). (1.10)

The equilibrium starting probabilities and resulting posterior beliefs do not have an an-alytical solution. To verify the existence of a solution, note that v∗f (P

b(g)) > vl(Pb(g), 0)

implies V Pwait(1) > V Pinvest(1). If V Pwait(0) < V Pinvest(0), then by the intermediate value the-orem, there exists a point η∗ ∈ (0, 1) such that V Pinvest(η∗) = V Pwait(η∗). If, on the otherhand, V Pwait(0) > V Pinvest(0) then bad types invest with probability 0.

Theorem 5 (Immediate entry). Let P(G|(g, b)) ≥ p∗l and P(G|(g, g)) ≥ p∗f . Then thereexists a symmetric Markov perfect Bayesian equilibrium in which type g of each agentinvests immediately.

1. (Pooling equilibrium) If P(G|b) ≥ p∗f then type b of each agent invests immediatelywith probability 1 in the first period.

2. (Partially Separating) If V Pinvest(0) ≥ V Pwait(0) and p∗f < P(G|b), then type b of eachagent invest with probability η∗ solving (1.10), and with probability 1−η∗ she investsat the constant rate β∗(Pb(G|b)) in the first period. In the second period, type b ofeach agent invests with delay τ ∗(pbη∗), where pbη∗ is given by (1.9), if the other agentinvested immediately.

3. (Separating) If V Pinvest(0) < V Pwait(0) and p∗f < P(G|b), then type b invests withprobability 0 in the first period, and invests with delay τ ∗(P(G)) in the secondperiod.


1.4.3.2 Random immediate entry

Good types invest immediately if Pg(G|b) < p∗l , that is, if their value of investing ispositive even if they learn that other agent’s type is bad. If this constraint is violated,then it is no longer optimal for good types to invest immediately, since good types preferto wait to see whether the other agent invests or not. The symmetric mixed-strategyequilibrium in the case has the property that good types randomize between investingimmediately and delaying their investment indefinitely and bad types “wait”. If a goodtypes expects that the good type of the other agent invests with probability ν then sheis indifferent between investing and delaying her investment if

Pg(g)vl(Pg(G|g), 0) + Pg(b)v∗l (Pg(G|b)) = ν Pg(g)vf (Pg(G|g), 0).

If type g invests immediately, then the other agent invests without delay if her signal isgood. If the other agent’s signal is bad, then she delays her investment by τ ∗(Pb(G|g)).On the other hand, if type g does not invest, then she invests if the other agent investsimmediately, and never invests otherwise. We see immediately from the indifferencecondition that g types are indifferent between investing and waiting if

ν∗ = 1 +

(Pg(b)

Pg(g)

)(v∗l (P

g(G|b))vf (P

g(G|g), 0)

)(1.11)

Note that Pg(G|b) < p∗l implies v∗l (Pg(G|b)) < 0 so that ν∗ < 1. We can thereforesummarize this result as follows.

Theorem 6 (Immediate random entry). Let Pg(G|g) ≥ p∗f and p∗l > Pg(G|b) and supposeE[v∗l (P

g(G|si)] ≥ 0. Then there exists a symmetric MPBE in which type g of each agentinvests immediately with probability ν∗ given by (1.11) and with probability 1 − ν∗ waitsindefinitely. In the second period, type g of an agent who waited in the first period investsimmediately. Bad types never invest in the first period and invest with delay τ ∗(Pb(G|g))in the second period if the other agent invests.

1.4.3.3 Randomly delayed entry

When the good type’s belief is below the follower threshold, p∗f > Pg(G|g), they prefer

to delay their investment when the other agent invests immediately. In the symmetricequilibrium therefore, good types choose their starting time at random and bad typesstrictly prefer to wait, as their belief that the state is good and thus their opportunity


cost of waiting is lower. Then, over time, when agents observe that the other agentcontinues to wait, they become increasingly convinced that the other agent’s type isbad. The increasing pessimism then affects their rate of investment, as the value of theirinvestment depends on the other agent’s information. In equilibrium, changes in the rateof investment determines the speed at which the agent’s learn about each other’s types.The result is a feedback loop in which changes in beliefs affect changes in investmentrates, and changes in investment rates affect the rates of learning. To determine theequilibrium trajectories of flow-rates of investment and beliefs, I derive the flow rate ofagent 1 and the corresponding beliefs for agent 2. The solution is independent of theagent’s identities, by symmetry.

Let πt denote the profile of private beliefs in the first period along the equilibrium pathwhen time t is reached and neither agent has made an investment. Let v2(πt, a) be thevalue function of type g of agent 2 at (πt, a) in the symmetric equilibrium, where a is thecurrent status profile. Let ρ1t be the flow-rate of investment of agent 1 at time t. Agent2 is willing to randomize if she is indifferent between starting her project immediatelyand delaying it by another instant. Hence, the value function for type g of agent 2 mustsatisfy the indifference condition

v2(πt, (1, 0))−rI = ρ1t q2t v2(πt, (0, 1))dt + (1−rdt−ρ1t q2t dt)(v2(πt+dt, (1, 0)) − rI), (1.12)

where q2t = π2t (g)(g) denotes the probability type g of agent 2 assigns to type g of agent1. If we apply Bayes’ rule, then agent 2’s updated private belief about agent 1’s type inthe next period is

q2t+dt =q2t (1 − ρ1t dt)1 − q2t ρtdt

.

The differential change in belief is therefore dqt/dt ≡ qt+dt−qt = −ρtqt(1−qt). Note thatwhile the belief about agent 1’s type changes over time, the conditional belief about thestate remains unchanged, i.e., π2t (g)(G|s1) = P(G|g, s1) for each s1 ∈ {g, b} and all t ≥ 0.Hence, a first-order Taylor approximation of the indifference condition (1.12) yields

r(v2(π, (0, 1))−rI) = ρ1q2v2(π, (1, 0))−ρ1q2(v2(π, (0, 1))−rI)−ρ1q2(1−q2)(dv2(π, (0, 1))/dq2),

where subscripts are omitted for notational convenience. If agent 2 makes the investmentfirst, then type s1 of agent 1 delays by τ ∗(P(G|s1, g)) in any equilibrium. The expectedvalue of becoming the leader for agent 2 is therefore

v2(π, (0, 1)) − rI = q2v∗l (Pg(G|g)) + (1 − q2)v∗l (Pg(G|b)).


The derivative of agent 2’s value function with respect to q2 is thereforedv2(π, (0, 1))/dq2 = v∗l (P

g(G|g)) − v∗l (Pg(G|b)). Hence, the indifference condition (1.12)is, in first-order approximation, equal to

r[q2v∗l (P

g(G|g)) + (1 − q2)v∗l (Pg(G|b))]

=

ρ1q2 v∗f (Pg(G|g)) − ρ1q2

[q2v∗l (P

g(G|g)) + (1 − q2)v∗l (Pg(G|b))]

− ρ1q2(1 − q2) [v∗l (Pg(G|g)) − v∗l (Pg(G|b))] .

Simplifying and solving the equation for the investment rate yields the solution

ρ∗(q2) =r [q2v∗l (P

g(G|g)) + (1 − q2)v∗l (Pg(G|b))]q2

[v∗f (P

g(G|g)) − v∗l (Pg(G|g))

] . (1.13)

The function ρ∗(q2) is the equilibrium flow-rate of investment for type g of agent 1 as afunction of agent 2’s belief about agent 1’s type. If q2t is an arbitrary trajectory of agent2’s private belief, then agent 2 is indifferent between making the investment immediatelyand delaying her investment by an arbitrary time period if she expects that agent 1 startsthe project at flow-rate ρ∗(q2t ) at all t ≥ 0. In equilibrium, the evolution of qt must beconsistent with agent 1’s investment rate. If type g of agent 1 invests at rate ρ∗(q2t ) forall t ≥ 0, then by Bayes’ rule the law of evolution must solve the differential equationdq2t /dt = −ρ∗(q2t )q2t (1 − q2t ). Substituting ρ∗(q2t ) yields

dq2t = −rq2t (1 − q2t )(

q2t v∗l (P

g(G|g)) + (1 − q2t )v∗l (Pg(G|b))q2t

[v∗f (P

g(G|g)) − v∗l (Pg(G|g))

])

dt.

The initial belief type g of agent 2 assigns to agent 1’s type being good is Pg(g). Hencewe impose the initial value condition q20 = P

g(g). This initial value problem has theunique solution

q∗t (Pg) =

e−tβ∗(Pg(G|g))E[v∗l (P

g(G|s1))] − Pg(b)v∗l (Pg(G|b))Pg(b) [v∗l (P

g(G|g)) − v∗l (Pg(G|b))] + e−tβ∗(Pg(G|g))E[v∗l (P

g(G|s1))] . (1.14)

If we substitute q∗t into the starting rate (1.13) and simplify, we obtain the desiredequilibrium flow-rate of investment

ρ∗t (Pg) =

e−tβ∗(Pg(G|g))E[v∗l (P

g(G|s1))]β∗(Pg(G|g))e−tβ∗(P

g(G|g))E[v∗l (Pg(G|s1))] − Pg(b)v∗l (P

g(G|b)) . (1.15)

The flow-rates of investment for different prior beliefs of agent 2 about agent 1’s type


0 10 20 30 40 50 60 700.00

0.05

0.10

0.15

0.20

0.25

0.30Pg(g) = 0.5

Pg(g) = 0.472

Pg(g) = 0.4

ρ∗t (P)

t

Figure 1.4: The equilibrium flow-rate of investment for parameter values r = 0.02, γ = 0.1, y = 6,c = 500, and I = 10. The prior beliefs are given by Pg(G|g) = 0.7, Pg(G|b) = 0.4.

are shown in figure 1.4. Note that the evolution of the flow-rate of investment hingeson v∗l (P

g(G|b)). If v∗l (Pg(G|b)) > 0 then the denominator converges to 0 as t → t∗(Pg)where

t∗(Pg) = log

(1 +

Pg(g)v∗l (Pg(G|g))

Pg(b)v∗l (Pg(G|b))

)β∗(Pg(G|g)), (1.16)

so that the flow-rate of investment approaches infinity and the belief about agent 1’s typeapproaches 0. Hence, in equilibrium, the good type of each agent makes the investmentwith probability 1 before time t∗ is reached. The intuition is that if v∗l (P

g(G|b)) > 0, thenthe good types’ value of becoming the leader is positive so that delaying is valuable onlyif there is a chance that the other agent will start first. Since bad types never start, goodtypes become more convinced that the other agent’s type is bad which increases theirincentive to invest. As a result the other agent has to increase her flow-rate of investmentover time to keep the first agent indifferent. The increased rates of investment acceleratethe learning process, so that good types become convinced that the other agent’s typeis bad at an increasing rate which again causes both agents to increase their rates ofinvestment. The result is a spiraling effect in which investment rates eventually explode,and beliefs converge to zero in finite time, conditional on no investment.

When v∗l (Pg(G|b)) < 0, the opposite effect occurs. Good types then become less eager

to invest when they become increasingly convinced that the other agent’s type is bad.This causes investment rates to fall which in turn diminish the rate of of learning. The


result is a tapering of investment rates and with beliefs about types converging to thethreshold

q∗∞ =−v∗l (Pg(G|b))

v∗l (Pg(G|g)) − v∗l (P

g(G|b)) ,

at which a good type’s value of becoming the leader is 0. Since v∗l (Pg(G|b)) < 0, it

follows that q∗∞ > 0 which implies that with positive probability good types never startthe project. More specifically, denote the probability that a good type never invests byPr(τ = ∞|s1). Then Bayes’ rule implies q∗∞ = Pr(τ = ∞|g) P(g)/ Pr(τ = ∞). Since typeb waits indefinitely, it follows that Pr(τ = ∞, b) = P(b). Hence, the probability that agood type never invests is

Pr(τ = ∞|g) = q∗∞

1 − q∗∞Pg(b)

Pg(g)= −v

∗l (P

b(g))

v∗l (Pg(g))

Pg(b)

Pg(g).

In summary, we have the following result.

Theorem 7 (Symmetric equilibrium with delayed entry). Suppose E[v∗l (P(G|si, g)] ≥ 0and p∗f > P

g(G|g). Then there exists a symmetric Markov perfect Bayesian equilibrium inwhich type g of each agent invests at flow-rate ρ∗t (P) at time t in period 1, and type b delaysinvestment until t∗(Pg) and invests at constant rate β∗(P(G|b, b)) after. If agent i investsin the first period, then type sj of agent j *= i delays her investment by τ ∗(P (G|sj, g)) inthe second period.

1.4.3.4 No entry

In the previous case, we assumed either implicitly or explicitly that for good types theexpected payoff of becoming the leader, E[v∗l (P(G|si, g)], is positive. If that is not thecase, then the only equilibrium is one in which neither agent invests.

Theorem 8 (Symmetric equilibrium without entry). When E[v∗l (P(G|si, g)] < 0, thenno type of either agent invests.

1.4.4 Welfare analysis

This section examines the efficiency of learning and the social value of public informa-tion. When information is public, then all equilibria are ex-post efficient when the priorbelief that the state is good is either very high or very low. Equilibria are inefficient


for intermediate values, when agents are uncertain about the realization of θ. The in-efficiencies manifest in three different ways. The first source of inefficiency is excessivedelay of the follower, who waits too long before making her investment because she doesnot take into account the positive externality of her experimentation on the value of theleader. This is an expression of the standard free-riding effect that is well known fromthe strategic experimentation literature. The second source of inefficiency is the delayprior to the first investment. Agents prefer being the follower and therefore wait if thereis a sufficiently high probability that the other agent invests first. The initial delay is theresult of the irreversibility of investment which forces the leader to continue her projectindefinitely. Free-riding aggravates its effect, because the excessive delay of the followermakes it less profitable to become the leader, and reduces the rate at which the agents arewilling to invest. The third type of inefficiency is the possible absence of any investmentin equilibrium when investing would be socially optimal. When signals are private thereis also the possibility that an agent makes an investments even though investing is notex-post socially optimal.

I begin the analysis by deriving the ex-post socially optimal timing of investments. Con-sider a social planner whose objective it is to maximize the sum of the agents’ expectedpresent values and who can observe the signal of each agent. The planner’s belief canthen be summarized by a single variable p representing the probability assigned to thestate being good. Because both projects are identical, the social value depends only onthe number of active projects, α, and not on the projects’ identities. Therefore, we candefine a strategy for the social planner to be a pair (τ, n) that assigns to every pair (p,α)a switching time τ(p,α) ∈ R+ and the number of active projects n(p,α) ∈ {0, 1 , 2} inthe following period. For α ∈ {0, 1} define

wα(p, τ) = vl(p, τ) + vf (p, τ) + αrI.

Note that vl is the leader’s value before making the investment. The last term adjustsfor the cost of investment once it is sunk. Using (1.1) and (1.2), we can write w1(p, τ)explicitly:

w1(p, τ) = (1+e−rτ )py+(1−p)

[λ1 + (2λ2 − λ1)e−(r+γ)τ

](y−γc)−e−rτ

[p + e−γτ (1 − p)

]rI

It follows from the definition λα = r/(r + αγ) that 2λ2 − λ1 = λ1λ2. Therefore, the


marginal value of delaying the second investment is

∂w1(p, τ)

∂τ= re−rτ [−p(y − rI) + (1 − p)(λ2(γc − y) + (r + γ)I)e−γτ ]. (1.17)

The expression in brackets is strictly decreasing in τ . This implies that dw1(p, τ)/dτ < 0for all τ ≥ 0 whenever p > psf , where

psf =rI + γI + λ2(γc − y)y + γI + λ2(γc − y)

, (1.18)

in which case it is socially optimal to make the second investment immediately. If p < psf ,then the socially optimal delay solves the first-order condition dw1(p, τ)/dτ = 0. Insummary, the optimal strategy for the planner at α = 1 is τ(p, 1) = τ s(p), where

τ s(p) =

[φ(psf ) − φ(p)]/γ if p < psf ,

0 if p ≥ psf .

Define ws1(p) = w1(p, τ s(p)) and ws0(p) = w0(p, τ s(p)). From the envelope theorem, itfollows that dws0(p)/dp = ∂w0(p, τ s(p))/∂p. By Lemma 1, the functions vl and vp areincreasing in p which implies that ws0 is an increasing function. Since ws0 is continuouswith ws0(0) < 0 and ws0(1) > 0, it follows that there exists a unique threshold psl thatsolves ws0(psl ) = 0. The socially optimal timing of investments is summarized in thefollowing theorem.

Theorem 9 (Planner’s solution). There exist thresholds psl and psf > psl so thatit is socially optimal for both agents to invest immediately if P(G|s1, s2) > psf . Ifpsl ≤ P(G|s1, s2) < psf , then it is socially optimal for one agent to invest immediately, andfor the second to invest with delay τ s(P(G|s1, s2)) = [φ(psf ) − φ(P(G|s1, s2))]/γ, whereφ(p) denotes the log-likelihood ratio of p. If P(G|s1, s2) < psl then it is socially optimalnot to invest for either agent.

What is noteworthy here is that staggered investment is socially optimal. This contrastswith other models that are based on a similar framework without switching cost, in whichthe optimal policy is to let all agents experiment simultaneously (Keller et al., 2005). Inthe absence of switching costs, each additional project provides additional informationwithout increasing the loss when a failure occurs. In contrast, if starting a project iscostly, the value additional information obtained from an additional project must betraded off with the cost of initiating it.


That equilibrium outcomes are efficient for extreme belief and inefficient for intermediatebeliefs can be seen by comparing Theorem 1 with the socially optimal timing of invest-ments characterized in Theorem 9 and by noticing that psf < p∗f and psl < p∗l . The firstinequality implies that the socially optimal delay is shorter than the equilibrium delaywhenever the equilibrium delay is positive. The second inequality implies that there arebelief at which no agent invests but investing would be socially optimal.

Theorem 10. Suppose signals s1 and s2 are publicly observable. If P(G|s1, s2) ≥ p∗for P(G|s1, s2) < psl then all equilibria are efficient. If psl < P(G|s1, s2) < p∗f , then allequilibria are inefficient.

Since for intermediate values of belief equilibrium outcomes are inefficient one may askwhat the social value is of disclosing private information. As motivating example, con-sider a social planner who can force agents to verifiably disclose their private information.Is it ever socially beneficial not to exercise this option? Naïve logic may lead to the con-clusion that, in the absence of any payoff externalities, preventing complete aggregationof information will unambiguously lead to worse outcomes. This view, however, disre-gards the fact that agents wait inefficiently long for others to provide that informationfor free when they are uncertain about the state which negatively effects the efficiencyof learning. The correct view is that there is a tradeoff between aggregating informationto prevent agents from making bad choices and limiting public access to information inorder to reduce the agents’ incentive to free-ride.

To see that non-disclosure of information may improve social outcomes, consider thecase in which agent 1’s signal is private information and agent 2’s signal is made public.Suppose that the prior belief satisfies the conditions P(G) > p∗f and p∗f > P(G|b) > psf .It follows from Theorem 9 that it is socially optimal in this case for both agents to investimmediately. However, from Theorem 10 we know that if the informed agent’s signal isbad and made public, then any equilibrium is inefficient. Therefore, irrespective of theequilibrium that is played after each signal, the equilibrium outcome is ex-ante inefficient.

On the other hand, when agent 1’s signal is private, there is an equilibrium that gener-ates the efficient outcome. To see that, suppose that both types of the informed agentinvest immediately, independent of the uninformed agent’s strategy. Then agent 2 in-vests immediately after agent 1 has made the investment, thus implementing the ex-postefficient outcome. The welfare improvement is the result of the signalling effect of agent1’s actions. If agent 1 does not invest she sends a bad signal to agent 2 about the state,and then agent 2 prefers to wait before making her investment, thereby withholding


information.

These examples are somewhat contrived, as the positive welfare effects rely on specificproperties of the prior belief and they may be sensitive to equilibrium selection. How-ever, if restrain the comparison to symmetric equilibria, the conditions in which welfareimprovement are possible become more transparent. Denote by W ∗publ(P(G|s1, s2)) theexpected social surplus generated in the symmetric equilibrium when each agent’s signalis made public, and let W ∗priv(P) denote the ex-ante expected social surplus generatedin a symmetric equilibrium in which each agent’s signal is private information. We thenhave the following result.

Theorem 11. If P(G|g, b) ≥ p∗l and P(G|b, b) > p∗J , then W ∗priv(P) ≥E[W ∗publ(P(G|s1, s2))].

The theorem says that if the prior belief that the state is good is not too low, then dis-closure of private information has no social value. The second condition is only requiredbecause for high prior beliefs that the state is good, there is a pooling equilibrium inwhich agents invest immediately, even though it would be socially optimal for them notto invest at all. The result is inefficient over investment. As mentioned in the previ-ous section, if P(G|b, b) > p∗J , there also exists a separating equilibrium in which thisinefficiency does not arise.

There are two parts to the argument that explains this result. In an symmetric equi-librium with delayed entry, the welfare improvement results from the spiraling effect, inwhich increasing pessimism about each other’s types and increasing eagerness to investresult in ex-ante earlier investment. In a symmetric equilibrium with immediate entryof good types, the reason is a slightly different one. When bad types with some proba-bility between 0 and 1, then the good types are worse off compared to the case of publicinformation, because the delay of the follower is longer. However, it turns out that thesupermodularity of the leader’s value, vl, implies that the loss of good types from ad-ditional delay is lower than the gain for bad types from shorter delays. From a socialwelfare perspective therefore, it is better to keep private information secret, because thisinduces bad type agents to invest earlier, and thus decreases delay when it causes a higherloss.

It is natural to ask whether Theorem 11 extends to asymmetric equilibria. Unfortunately,Theorem 11 does not extend to asymmetric equilibria in general. One problem is thatthere is multiplicity among asymmetric equilibria, so that the welfare comparison hingeson the equilibrium selection. But even for reasonable selection criteria, the theorem does


not extend. Consider for example the asymmetric equilibria in which one designatedagent invests immediately, if her expected payoff from doing so is positive, and considerthe prior belief with P(G|b, b) = p∗l and P(G|g, g) = p∗f . The conditions in Theorem 11are satisfied, but the social welfare in the equilibrium with public signals in higher thanwith private signals. To see why, suppose that agent 1 is the designated leader, and agent2 the designated follower and let the signal of the follower agent 2 be fixed. When signalsare public, then the expected social welfare is E[v∗f (P(G|s1, s2)) + v∗l (P(G|s1, s2))|s2] .When signals are private, then the expected social welfare is

E[vf (P(G|s1, s2), τ ∗(E[P(G|s1, s2)|s2])) + vl(P(G|s1, s2), τ ∗(E[P(G|s1, s2)|s2]))

= vf (P(G|s2), τ ∗(P(G|s2)])) + vl(P(G|s2), τ ∗(P(G|s2])).

Now note that both v∗f (p) as well as v∗l (p) are convex on (p∗l , p∗f ) which implies that

E[v∗f (P(G|s1, ŝ2)) + v∗l (P(G|s1, ŝ2))]

> vf (P(G|ŝ2), τ ∗(P(G|ŝ2)])) + vl(P(G|ŝ2), τ ∗(P(G|ŝ2]))

for each s2 ∈ {g, b}. Hence the social welfare under public information is greater thanunder private information.

1.5 Conclusion

The goal of this paper is to investigate how private information affects agents’ incentivesto make irreversible investments to engage in experimentation. Agents benefit fromthe experimentation of others because it provides free information, but delaying theirinvestment can signal bad news and discourage experimentation of others. I showed thatwhen using symmetric equilibria as a benchmark, private information increases socialwelfare when good signals are strong relative to bad ones.

One aspect I did not discuss in this paper, is how the results would change when onewould allow the agents to communicate prior at the beginning of the game. However, it isrelatively easy to anticipate the effects on communication in the present model. Cheap-talk between agents does not change the outcome, because if it did, each agent wouldalways send the most positive signal to induce the other to invest early. Communicationwith verifiable messages will lead to unraveling: agents prefer to send communicate theirsignal only if it is good, but then each agent can infer the other agent’s signal perfectly.


1.A Proofs

Proof of Lemma 1. (i) That vl is linear in p is obvious from its definition in (1.1) andvl is increasing in p because it follows from γc > y > 0 that the second term in (1.1) isnegative. To see that vl is decreasing in τ , note that λ2 < λ1, and hence

d

dτvl(p, τ) = −(r + γ)(1 − p)(λ1 − λ2)e−(r+γ)τ (γc − y) < 0,

for all p ∈ (0, 1) and τ ≥ 0. That vl is convex for all p ∈ (0, 1) follows from

d2

d2τvl(p, τ) = (r + γ)

2(1 − p)(λ1 − λ2)e−(r+γ)τ (γc − y) > 0.

Finally, supermodularity holds because

d2

dpdτvl(p, τ) = (r + γ)(λ1 − λ2)e−

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Essays on Learning and Strategic Investment · Peter Achim Wagner Doctor of Philosophy Graduate Department of Economics University of Toronto 2013 The ﬁrst chapter studies the strategic