Essays on Learning and Strategic Investment. Peter Achim Wagner

Transcription

1 Essays on Learning and Strategic Investment by Peter Achim Wagner A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Economics University of Toronto Copyright c 2013 by Peter Achim Wagner

2 Abstract 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 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 freeriding 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 outcome, 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 social 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 experimentation 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 inii

3 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

4 Contents 1 Learning from Strangers Introduction Literature Review Model Basic framework Histories and strategies Equilibrium concept and refinements Results Public signals Public and private signal Private signals Welfare analysis Conclusion A Proofs Experimentation and Trade Introduction Model Optimal experimentation without trade iv

5 2.2.2 Partial Observability Comparative statics Conclusion A Proofs Delegated Problem Solving Introduction Model Optimal contracts Benchmark: observable effort Evolution of rewards Optimal rewards Optimal deadlines Discussion Collaboration versus competition Resource-constrained agents Unobservable successes Conclusion A Proofs Bibliography 103 v

6 Chapter 1 Learning from Strangers 1.1 Introduction The economics literature typically assumes that agents learn exclusively from the behavior of others, or exclusively from their experiences. In reality, learning through observation often involves both channels of learning. Take for instance the farmer who ascertains the value of a new type of crop from the fact that a neighboring farmer uses it, as well as by observing the neighbor s yield the following year. A doctor may likewise learn about the effectiveness of a new drug by observing that her colleagues prescribe it, and from the health outcomes of their patients. This chapter investigates the strategic timing of investments under uncertainty when agents learn about the uncertain return from each other s actions and experience. In the presence 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 questions addressed in this chapter are: How does private information about the uncertain state influence the agents incentives to invest and how are these incentives affected by the strength of their private beliefs? Under which conditions is private information revealed in equilibrium and is it revealed instantly or over time? Does private information increase or 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 decide when to invest in a risky project. Once an agent has invested, her project yields an uncertain 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

7 Chapter 1. Learning from Strangers 2 is bad, however, active projects generate costly failures at random times, resulting in a negative expected payoff. Failure times are uncorrelated, and each agent incurs only the cost of her own project. At the outset, agents observe a private signal that conveys either good or bad news about the state. Over time, agents learn from observing each other s actions 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 than bad signals, equilibria are ex-ante more efficient when agents have private information than when this information is made public. Intuitively, one might expect that in the absence of payoff externalities, more information generates better social outcomes, as it reduces uncertainty and allows agents to make better decisions. I show that under fairly general conditions, the social value of disclosing private information is negative. Inefficiencies arise in the presence of learning externalities, because agents wait too long before they invest in the hope that the other provides free information. Here, excessive delay 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 from investing with the benefit of waiting for more information, but she does not take into account the social value of her own experimentation. Excessive delay before investments is the result of leadership-aversion. Both agents prefer to obtain free information from the other, so that each has an incentive to delay their investment and wait for the other to 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 only benefit from waiting to invest if the other agent invests first. However, one agent s delay may signal bad information to the other, who, in response, may then be less willing to invest, thereby reducing the first agent s incentive to delay. As a result, an agent who received a bad signal may have the incentive to invest without delay to encourage the other agent to invest earlier. The second drawback results from each agent s uncertainty about the other agent s behavior. For example, an agent who received a good signal may wait in the hope that the other agent invests first. The other agent, on the other hand, may have received a bad signal and therefore never invest. Another possibility is that an agent who may, despite having received a good signal, delay her investment to learn whether or not the other agent invests. However, by delaying her investment she discourages future investment by the other agent, even if the other has received a good signal.

8 Chapter 1. Learning from Strangers 3 prior belief that state is good high low precision of signals high low An agent with a good signal invests immediately or waits ; an agent with a bad signal waits. An 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 random delay at decreasing flow-rate; an agent with a bad signal waits. An agent with a good signal invests with random delay at decreasing flow-rate; an agent with a bad signal waits. Table 1.1: Characterization of symmetric equilibria based on the strength of good and bad signals for a moderate prior belief. These negative effects of delay under private information can induce agents to invest earlier, and thereby reduce free-riding and leadership aversion. The reduction increases ex-ante social welfare as long as it does not result in inefficient over-investment. For example, when bad signals are very informative, an agent with a good signal may invest not knowing the other agent s signal, but regret investing if she learns that the other agent s signal is bad. In cases like these, it might be socially desirable to make private information public. The concrete mechanism that induces agents to invest earlier depends on the structure of the equilibrium which varies with the prior belief. There are four distinct types of symmetric equilibria. Each depends on the agents common prior belief about the state and the informativeness of good and bad signals. Assuming a fixed prior belief about the state that is moderate in the sense that free-riding occurs in equilibrium even when signals 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 a good signal invest immediately. Agents who receive a bad signal invest immediately with some probability that depends on the strength of their signal, and with the remaining probability they wait : they do not invest unless the other agent invests first and her project proves successful for some period of time. Intuitively, when good signals are informative relative to bad ones, then agents who receive good signals are confident that the state is good. Agents with bad signals may then benefit from investing immediately to 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 immediately with some probability and wait with the remaining probability. The intuition is that when both good and bad signals are informative, then agents with good signals have

9 Chapter 1. Learning from Strangers 4 no incentive to delay their investment given that the other agent s signal is good, but at the same time they are hesitant to invest because an informative bad signal from the other agent would render their investment unprofitable. When good and bad signals are uninformative, then agents who receive good signals prefer to 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. Agents who receive bad signals wait indefinitely. Therefore, over time both agents become increasingly 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 if the other agent s signal is bad, so that the result is a positive feedback loop: as each agent becomes more pessimistic about the other agent s signal, her incentive to delay her investment decreases, so that in equilibrium the other agent with a good signal has to increase her rate of investment to keep the first agent indifferent. These effects reinforce each other, eventually leading to an explosion of flow-rates of investment, ending the waiting game with certainty before a finite time. Finally, when bad signals are informative relative to good ones, then in the symmetric equilibrium, agents with bad signals wait and agents with good signals play a waiting game similar to the previous case. The difference is that agents with good signals prefer not to invest if the other agent s signal is bad. As a result, the equilibrium now exhibits a negative feedback loop: as each agent becomes more pessimistic about the other agent s signal, her incentive to delay her investment increases, so that in equilibrium the other agent with a good signal has to decrease her rate of investment to keep the first agent indifferent. 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. On the other hand, when agents have good signals, the delays in equilibrium are longer than in the case in which their signals are publicly disclosed. These effects are the result of negative learning feedback and uncertainty about the other agent s future investment behavior. Relative to the case in which signals are all made public, the different distributions of delay under uncertainty lead to increased social welfare. Social gain from shortened delay when agents have bad signals outweighs the social loss from lengthened delay when agents have good signals. The improvement follows from the fact that an agent who is pessimistic about the state of the world benefits more from any additional piece of information than an agent who is already certain that the state is good.

10 Chapter 1. Learning from Strangers Literature Review The paper relates to the literature on the strategic delay of investments. In Chamley and Gale (1994) andmurto and Välimäki (2010) informationisdispersedthroughoutsociety, and agents decide when to make an irreversible investment. Information is inefficiently aggregated because investors have an incentive to delay their decision to acquire more information by observing the behavior of others. In contrast to the current paper, in these papers no new information is generated over time. Learning from the experience of others is the main subject of the strategic experimentation literature (Bolton and Harris, 1999; Keller, Rady, and Cripps, 2005; Keller and Rady, 2010; Klein and Rady, 2011). In these models, a number of agents experiment with a risky alternative which has a payoff distribution that depends on some common payoff parameter that is unknown to all agents. The central result in these papers is that there is too little experimentation in equilibrium, because agents prefer to wait and free-ride on the information that is provided to them through the experimentation of others. There is a number of papers that consider models of strategic experimentation in which agents 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 risky project, where the project s value depends on a common state variable. Firms are privately informed about their investment cost and after a firm invests the success of its project is publicly observable. Their paper differs from the present paper in that firms have 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 simultaneously uncertain about the state and each other s private information. In the model of Décamps and Mariotti (2004), the result is reversed because each firm has in fact an incentive to delay 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 privately learn from their own experiences about the optimal time to exit. The authors find that information is aggregated in randomly occurring bursts of exit. In contrast to the current paper, agents cannot learn from the experience of others, and exit is irreversible so that there 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 information and irreversible exit, but they are primarily concerned with characterizing equilibrium

11 Chapter 1. Learning from Strangers 6 strategies in a general class of games, and not with the informational issues that are the focus here. Another paper that considers strategic experimentation with private payoffs is 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. While their main result is somewhat similar to the main finding in this paper, the underlying mechanisms are very different. In particular, the authors do not limit their attention to Markov strategies, so that it is possible to construct a punishment scheme that deters agents from free-riding on the other agent s experimentation. There are also several papers that investigate learning in R&D competition between firms. Acemoglu, Bimpikis, and Ozdaglar (2011) consider a model in which firms receive a private signal about different projects. The results are much in the spirit of the current paper with regards to the structure of the equilibria. The main differences lie in the presence of payoff externalities and in that experimentation is instantaneously perfectly revealing in their paper. The latter eliminates the possibility that an agent times her investment in order to encourage others to invest. Moscarini and Squintani (2010) consider a model of a winner-takes-all R&D competition in which firms observe a private signal about the unknown type of a research project. Over time firms learn about the project s type from the actions of their competitor and past payoffs, and they decide when to exit irreversibility. Because of the winner-takes-all environment, agents do not benefit from signaling good news, so that signaling does not play the same role as in the current paper. The welfare improving effect of private information in the present paper is related to the encouragement effect discussed in Bolton and Harris (1999), and more recently in Rosenberg, Salomon, and Vieille (2010). Bolton and Harris (1999) showthatwhen agents can learn from each other s experiments, they may experiment more to induce or move forward the time at which other agents initiate experimentation. The key difference is that in their paper, agents encourage others by generating more publicly observable information, leading to an unambiguous welfare improvement. Here, in contrast, an agent who has bad information encourages the other by mimicking the behavior of an agent with good information, preventing the aggregation of information, so that the ex-ante welfare effect is not immediately obvious. In general, both effects may coexist: Rosenberg, Salomon, and Vieille (2010) show that the encouragement effect always arises in 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 and Shin, 2002). Teoh (1997) demonstratesthiseffectinamodelofpublicgoodsprovision,

12 Chapter 1. Learning from Strangers 7 in which agents contribute to a joint project. The marginal return to their investment is determined by an uncertain state of the world. The author shows that non-disclosure of information may increase ex-ante welfare when the investment has marginally diminishing returns, because the loss resulting from a reduction in investment after the release of bad news 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 bad news is publicly disclosed, then free-riding and leadership-aversion increase, leading to an 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 the result of learning externalities, and does not require payoff externalities. 1.3 Model Basic framework There are two agents, indexed i =1, 2. Timet R + is continuous with infinite horizon and future payoffs are discounted with common discount rate r. Each agent decides when to operate a risky project. The profitability of each agent s project depends on the unknown state θ {G, B} which is either good (θ = G) orbad(θ = B). To initiate the project, each agent i must make an investment I > 0. The investment is irreversible, in the sense that once an agent has initiated her project, the investment cost is sunk and cannot be recovered. An inactive project yields no return. An active project yields the certain flow return y, but it may fail and generate a lump-sum loss c>0 if the state is bad. The probability that an active project fails over a time period of length t>0 is 1 e γt if θ = B F θ (t) = 0 if θ = G, where γ>0isthetime-independent failure rate. The investment that is required to initiate the project is assumed to be lower than the present value of operating a project in a good state, i.e., I<y/r. I assume that active projects must be continued indefinitely unless a failure occurs. To rule out the uninteresting case in which both agents prefer to invest immediately, it is further assumed that the average flow return from operating a project in a bad state is negative, i.e., y<γc,giventhatthemeantimeabadproject lasts is 1/γ.

13 Chapter 1. Learning from Strangers 8 At the outset, each agent i =1, 2 observes a private binary signal s i {g, b} that provides information 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 good signal (s i = g) is greater than the probability of observing a bad signal (s i = b) if the state is good, and the probability of observing a bad signal in a bad state is the same 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 that P(G g) > P(G) > P(G b). I assume that the signals are conditionally independent and that the probability of observing either signal is positive for each θ {G, B}. Agentsare ex-ante identical, i.e., P(s 1 = g G) =P(s 2 = g G) and P(s 1 = b B) =P(s 2 = b B). I model the continuous-time environment as a multi-stage game based on a technique by Murto and Välimäki (2011). At the beginning of each stage, an agent decides how long to wait before initiating or terminating her project, given the other agent does not make a move and no failure occurs. A stage ends with the first move of an agent or with the occurrence of a failure. Time is counted from the beginning of each stage, so that actual time, which accumulates from the beginning of the game, is recorded as the sum of the lengths of all preceding stages. In this framework, it is possible to model the environment as a fully dynamic game in continuous time 1 in which mixed strategies are well-defined and have a natural interpretation as distributions over transition times. Formally, denote the status of agent i s project in stage k =1, 2,... by a i k {, 0, 1}, where a i k =1represents an active project, ai k =0an inactive project and ai k = afailed project. The initial status of project i is a i 1 =0for each i. Anactionforagenti in stage k 1 is a transition time τ i k R + { }. If no failure occurs, then stage k ends at t k =min{τk 1,τ2 k }. The flow payoff for agent i in this case is t k 0 e rs a i k yds. If τ k i = t k, then the status of agent i s project in stage k +1is a i k+1 =1 ai k. Time is reset to 0, and the next stage proceeds in the same way as the previous one. If a failure occurs at t<t k, the game ends. The flow return for each agent i is t 0 e rs a i kyds and the agent whose project generated the failure incurs a loss c. Ifprojecti failed in stage k, itsfinal status is a i k+1 =. 1 as opposed to a stopping game in which an agent chooses a single stopping time

14 Chapter 1. Learning from Strangers Histories and strategies A non-terminal history in stage k is a list h k =(t 1,a 2,t 2,a 3,...,t k 1,a k ) that consists of stage lengths t l R +, 1 l<k, previous status profiles a l =(a 1 l,a2 l ) {0, 1}2, 1 <l k, and a current status profile a k {0, 1} 2.Aterminalhistoryisalist(h k 1,t k 1,a k ), where h k 1 is a non-terminal history and a k {, 0, 1} 2 with a i k = for one agent i =1, 2, or an infinite list (t 1,a 2,t 2,a 3,...) with a k {0, 1} 2 for all k 2. For a given terminal history h ending in stage K, thecontinuationpayoffforagenti in stage 0 k<kis U i k(h)= K 1 l=k [ tl e r(t l T k ) 0 ] e rs a i l yds a i l(1 a i l 1)e r(t l T k ) I 1 {a i K = }(h)e r(t K 1 T k ) c, where T l = l j=1 t j records the actual time that accumulates from the beginning of the game and 1 {a i K = }(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 s i {g, b} of agent i is a function ˆσ i (s i ):H (R + ) that assigns to every non-terminal history a distribution over transition times. There are combinations of strategy pairs that are not well-behaved in the sense that they do not generate proper paths in time. Suppose for example that each agent s strategy has the property that at time 0 she chooses to start the project immediately if the other agent s project is inactive, and to terminate the project immediately if the other agent s project is active. Such a pair of strategies generates an infinite history in which agents oscillate between starting and stopping their projects at time 0, without progressing in time. This does not create any difficulties in terms of the formal analysis. If we allow payoffs to be negative infinity, then the payoff at histories with infinitely many switches is minus infinity for one of the agents, so that any pair of strategies generating such histories cannot be an equilibrium Equilibrium concept and refinements A profile of strategies ˆσ generates a distribution Pˆσ over terminal histories. For a given common prior belief P, the agents beliefs about each other s signals and the state are determined by Bayes rule at every non-terminal history h k for which the set of terminal histories that have h k as a sub-history lies in the support of P σ. From these beliefs we

15 Chapter 1. Learning from Strangers 10 can then calculate the distribution Pˆσ hk which is the distribution over terminal histories conditional on reaching history h k. If we denote the associated expectation operator by Eˆσ hk,theneˆσ hk [Uk i si ] is the expected continuation payoff for type s i of agent i at history h k under the strategy profile ˆσ. A strategy profile ˆσ is a perfect Bayesian equilibrium for the common prior belief if the agents strategies are sequentially rational given their beliefs, and their beliefs are derived via Bayes rule from ˆσ whenever possible. Because I am interested in the informational aspects of strategic behavior, I focus the analysis on perfect Bayesian equilibria in Markov strategies. Agent i s strategy is a Markov strategy if the distribution over transition times at every history depends only on the payoff relevant information that is available to i at that history. The payoff relevant information that is available to agent i at each history can be characterized by a profile of belief systems π =(π i (g),π i (b)) i=1,2, where π i (s i ):H ({G, B} {g, b}) assigns to each non-terminal history a distribution over combinations of states and types. A Markov strategy for type s i of agent i is then a function σ i (s i ):( ({G, B} {g, b})) 4 {0, 1} 2 (R + ) and a Markov perfect Bayesian equilibrium (henceforth MPBE) is a pair (σ, π) that has the property that σ i (s i ) is sequentially rational at every non-terminal history for each type s i of each agent i given the agents beliefs at that history, and beliefs are derived via Bayes rule from σ whenever possible. It should be emphasized here that Markov strategies do not depend on another agent s private beliefs. The notation is convenient, because agent i s strategy can be expressed directly as a function of the belief of each type of the other agent. Equivalently, I could write agent i s Markov strategies in the conventional way as a function of her private and the common public belief, and then use Bayes rule to back out agent i s belief about the other 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 to believe after they observe an unexpected move, it is necessary to impose additional restrictions on belief systems to rule out a number of implausible equilibria. Definition 1 (Reasonable beliefs). An MPBE (σ, π) has reasonable beliefs, if at every non-terminal history h k+1 = (h k,t k,a k+1 ) H for which t k is not in the support of ˆσ i (s i )(π(h k )) for either s i {g, b} the following conditions hold. 1. For each type s j of agent j i, thebeliefaboutthestate,conditionaloneachtype

16 Chapter 1. Learning from Strangers 11 s i of agent i, isconsistentwithbayes rule: π j (s j )(h k+1 )(G s i )= π j (s j )(h k )(G s i ). π j (s j )(h k )(G s i )+π j (s j )(h k )(B s i )(1 F θ (t k )) a1 k +a2 k 2. For each s j {g, b} we have 1 if a j π j (s j )(h k+1 )(s i k+1 = g) = aj k =1, 0 if a j k+1 aj k = 1. The first condition says that when an agent makes an unexpected move, then this should not affect the way an agent updates her belief about the state of the world. The second condition states that each agent should associate unexpected investments with good types and unexpected terminations of projects with bad types. The second condition is needed to rule out a class of artificial equilibria in which agents are deterred from deviating because of the other agent s off-equilibrium belief after their deviation. For example, it is straightforward to construct an equilibrium in which the good type of each agent does not invest because the other agent would then be convinced that the investing agent s type is bad, and would therefore delay her own investment excessively, so that it is no longer profitable for good types to invest. 1.4 Results I begin the analysis with a restricted version of the model in which both agents are required to continue an active project indefinitely. Restricting the model is motivated by theoretical as well as practical considerations. First, I am mainly interested in an environment in which the cost of initiating a project is large relative to the possible informational 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 delay 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 eliminates 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 privately informed agents. Finally, assuming that agents never stop projects tremendously simplifies the mathematical analysis, without affecting the validity of the results in a more general environment.

17 Chapter 1. Learning from Strangers 12 The remainder of this paper is structured as follows. I first discuss a benchmark model in which each agent s signal is publicly observable, and I show that strategic conflicts arise only when agents are relatively uncertain about the state. I then consider a scenario in which only one agent s signal is private information and present two asymmetric equilibria that highlight some of the features that are relevant for the subsequent analysis of the symmetric equilibrium with two privately informed agents. I continue with a welfare analysis, and I demonstrate that it may be socially preferable if agents are uncertain about each other s signals. In the last subsection I discuss to what extent the results can be generalized to the scenario in which projects may be terminated and restarted 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 state being good, so that a Markov strategy for each agent i is a function σ i :[0, 1] {0, 1} 2 (R + ). Since projects cannot be terminated, we can ignore the case in which both projects are active, that is, a =(1, 1). I therefore begin by deriving the best response for an 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 who operates a project, the expected flow-payoff depends on the belief p at the beginning of the period, whether or not the other agent is operating a project, and on the time τ at which the other agent invests. Denote the number of currently active projects by α = a 1 + a 2. I omit subscripts for notational convenience. If no failure occurs before τ, then the flow-payoff for an agent who operates a project is τ 0 e rs yds. If a failure occurs at time t<τ,thenanagentwithanactiveprojectreceives t 0 e rs yds,andwith probability 1/α also incurs the loss e rt c.theprobabilitythatafailuredoesnotoccur is equal to 1 in a good state and equal to (1 F B (τ)) α = e αγτ in a bad state. A failure does occur with the remaining probability 1 e αγτ,andtherandomtimeofthefailure in this case has density αγe αγt on [0,τ). Theexpectedflow-payoffforanagentwithan active project is therefore τ u α (p, τ) =p e rt ydt 0 [ τ ( t +(1 p) αγe αγt 0 0 ) τ ] e rs yds e rt c/α dt + e αγτ e rt ydt. 0

18 Chapter 1. Learning from Strangers 13 Solving the integrals and simplifying the resulting expression gives ru α (p, τ) =p(1 e rτ )y +(1 p)(1 e (r+αγ)τ )λ α (y γc), where the constant λ α = r/(r + αγ) is the marginal value of receiving a stream of a constant flow-payoff up to the first time a project fails, given that α {1, 2} projects are operated simultaneously 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 benefit of obtaining additional information about θ with the loss in revenue from delaying her investment. In a situation like this, the agent operating a project shall be called the leader and the other agent shall be referred to as the follower. Given that the follower delays her investment by τ, theleaderreceivestheexpectedflow-payoffu 1 (p, τ) in the current period. The game ends if a failure occurs before the follower makes the investment, and the payoff for the leader is 0. If a failure does not occur before the follower invests, then the continuation value for the leader is u 2 (p, ). Given the leader s belief p about θ, she expects that a failure does not occur with probability p +(1 p)e γτ. Her value before making the investment is therefore v l (p, τ) =r [ u 1 (p, τ)+ ( p +(1 p)e γτ) e rτ u 2 (p (τ), ) I ], where p (τ) = p p +(1 p)e γτ denotes the updated belief in the next period, derived from Bayes rule. After substituting the formulas for u 1, u 2 and p (τ), the leader s value becomes v l (p, τ) =py +(1 p)(λ 1 +(λ 2 λ 1 )e (r+γ)τ )(y γc) ri. (1.1) Given the follower delays her investment by τ, herpayoffis0iftheleader sprojectfails before τ. It does not fail with probability p +(1 p)e γτ,inwhichcasethefollower s continuation value is u 2 (p (τ), ) I. Usingtheformulasforu 2 and p (τ), theexpected

19 Chapter 1. Learning from Strangers 14 leader v l (p, ) p = p = τ v f (p, ) follower p =0.7 p =0.5 p = p = t 3 1 Figure 1.1: The value for the leader and follower as functions of the follower s delay τ for parameter values r =0.02, γ =0.1, y =6, c = 500, andi = 10. present value for the follower is v f (p, τ) =e rτ p(y ri)+e (r+γ)τ (1 p)(λ 2 (y cγ) ri). (1.2) The following lemma reports basic properties of the functions v l and v f. Lemma 1. The function v l is linearly increasing in p, convexanddecreasinginτ for every p (0, 1) and supermodular in (p, τ). The function v f is linearly increasing in p and it has a single peak in τ at ( φ(p τ f ) φ(p) ) /γ if p<p f (p) = (1.3) 0 if p p f for every p (0, 1), whereφ(p) =log(p) log(1 p) is the log-likelihood ratio of p and p f = ri + γi + λ 2(γc y)/λ 1 y + γi + λ 2 (γc y)/λ 1. (1.4) Moreover, v f (p, τ (p)) >v l (p, τ (p)) if p<p f and v f(p, τ (p)) = v l (p, τ (p)) if p p f. All proofs are in the appendix. I write vf (p) =v f(p, τ (p)) and vl (p) =v l(p, τ (p)) for the values of the leader and the follower given the follower uses the optimal delay. Note that since τ is weakly decreasing in p, andv l and v f are strictly increasing in p as well as decreasing in τ, itfollowsthatvl and vf are strictly increasing functions in p. Moreover, is positive if p =1, negative if p =0and continuous. Hence, it has a unique root on v l (0, 1), whichidenotebyp l. Denote further by p J the threshold at which the value of

20 Chapter 1. Learning from Strangers 15 investing jointly without delay is positive. More specifically, p J is the lowest value at which v f (p, 0) = 0 for all p p J.Using(1.2), we can calculate the threshold explicitly: p J = ri + λ 2(γc y) y + λ 2 (γc y). It follows from Lemma 1, that in any equilibrium both agents invest immediately if P(G s 1,s 2 ) >p f. If the prior belief exceed this threshold, then it is optimal for the follower 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 delays her investment Equilibria when signals are publicly observable When one agent invests at public belief p, then the second agent becomes the follower and optimally delays her investment by τ (p). If the prior belief is above the threshold at which delaying investment is optimal, P(G (s 1,s 2 )) > p f, then there is a unique equilibrium in which both agents invest immediately. One the other hand, if the prior belief is below the leader threshold, P(G (s 1,s 2 )) < p l,thenbeingtheleaderisnot profitable for either agent, so that both agents never invest. Theorem 1. If P(G s 1,s 2 ) >p f then in every MPBE both agents invest immediately. If P(G s 1,s 2 ) <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 occur sequentially, because each agent prefers to wait if she expects that the other agent invests, and she prefers to invest if she expects that the other agent waits. Equilibria in pure strategies are therefore necessarily asymmetric. These equilibria have an obvious structure: in the first period, one agent invests immediately, and the other agent waits indefinitely for the first agent to make a move, and in the second period the follower makes 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 their roles are reversed.

21 Chapter 1. Learning from Strangers 16 Theorem 2 (Asymmetric equilibria). If p l P(G s 1,s 2 ) <p f,thereexisttwoasymmetric simple equilibria, one for each i =1, 2. In the first period, along the equilibrium path, agent i starts the project immediately, and agent j i waits indefinitely. In the second period, agent j i delays her investment by τ (P(G s 1,s 2 )). There also is a symmetric equilibrium in mixed strategies, in which each agent randomizes over the times of her initial investment. In the mixed strategy equilibrium each agent invests at a rate that renders the other agent indifferent between investing immediately and delaying her investment by any length of time. Note that as long as neither agent has made the investment no new information becomes available, so that the agents flow rate of investment β is constant over time. We can then immediately calculate the equilibrium investment rate, using the fact that each agent must be indifferent between making the investment immediately and never making the investment, i.e., v l (p) = 0 βe βt e rt v f(p)dt. Solving the equation for β gives the equilibrium starting rate as a function of the common belief p : β (p) = rvl (p) vf (p) (1.5) v l (p). Observe that the flow-rate of investment is positive whenever the value of becoming the leader is greater than 0, because it follows from Lemma 1 that the denominator of β is always non-negative. Moreover, the difference between the value of the follower and the value of the leader converges to 0 as p approaches the follower threshold p f,tothatthe equilibrium flow-rate of investment approaches infinity. Theorem 3 (Symmetric equilibrium). If p l P(G s 1,s 2 ) < p f, then there exists a unique symmetric equilibrium. In the first period of this equilibrium, along the equilibrium path, each agent makes the investment at flow rate β (P(G s 1,s 2 )). Inthesecondperiod, the follower starts the project with delay τ (P(G s 1,s 2 )) Public and private signal The purpose of this section is to study the signalling problem of an agent with private information in an environment, in which her investment decision is not confounded by her uncertainty about the other agent s private information. I show that in an equilibrium

22 Chapter 1. Learning from Strangers 17 in which the informed agent moves first, bad types invest only if good types invest, and if good types invest the degree to which bad types choose to separate and reveal their information 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 either imitates a good type by investing immediately, or she never invests. For simplicity, I shall assume that the informed agent is agent 1, and the uninformed agent is agent 2. 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 is bad if she delays her investment. Hence the bad type s value of investing with delay τ is e rτ vl (P(G b)). On the other hand, if she invests immediately, then her value is v l (P(G b),τ (p)), where p denotes agent 2 s belief that the state is good after observing that agent 1 invested immediately in the first period. It follows from Bayes rule that p>p(g b), and therefore τ (P(G b)) >τ (p), sothat v l (P(G b),τ (p)) >vl (P(G b)) >e rτ vl (P(G b)). Hence, type b can profitably deviate so that there cannot exist an equilibrium in which agent 1 moves first and type b of agent 1 invests with positive and finite delay, so that type b either invests immediately, or waits indefinitely. Therefore, I am lead to consider an equilibrium in which type g of agent 1 invests immediately, and type b of agent 1 invests immediately with probability ξ and delays her investment indefinitely with probability 1 ξ. Agent 2 s belief after observing agent 1 invested immediately in the first period is given by Bayes rule, p ξ = 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 (p ξ ),sothattheexpected value of investing immediately is v l (P(G s 1 ),τ (p)) for type s 1 of agent 1. If type b invests immediately with probability ξ =1, thenp =P(G), and therefore it is sequentially rational for type b to invest immediately if v l (P(G b),τ (P(G))) 0. Hence, if v l (P(G b),τ (P(G))) 0, thenanequilibriuminwhichtheinformedagent moves first is a pooling equilibrium. If type b invests with probability ξ = 0, then p =P(G g), and therefore it is sequentially rational for type b to wait indefinitely if v l (P(G b),τ (P(G g))) < 0. Thus, if v l (P(G b),τ (P(G g))) < 0 <vl (P(G g)), thenan equilibrium in which the informed agent moves first is a fully separating equilibrium. If,

23 Chapter 1. Learning from Strangers 18 one the other hand, vl (P(G g)) < 0, then the value of investing is negative for type g even if agent 2 knows her type, and therefore the only equilibrium is a pooling equilibrium in which neither agent invests. If v l (P(G b),τ (P(G))) < 0 <v l (P(G b),τ (P(G g))), thentheequilibriumispartially separating. In particular, it has the property that type b of agent 1 is indifferent between investing immediately and delaying her investment indefinitely given agent 2 s equilibrium belief p ξ that the state is good in the second period, and p ξ is consistent with the probability ξ with which type b invests immediately in equilibrium: v l (P(G b),τ (p ξ )) = 0 and p ξ = P(G, g)+p(g, b)ξ P(g)+P(b)ξ. (1.6) To solve for p ξ and ξ,useequation(1.1) towritethevalueoftheleaderasconvex combination v l (p, τ) =(1 e (r+γ)τ )u 1 (p, )+e (r+γ)τ u 2 (p, ). Substitutingtheformula for τ in (1.3), we can rewrite the condition v l (P(G b),τ (p ξ )) = 0 and solve for the posterior belief, p ξ = p f p f +(1 p f )[1 u. (1.7) 2(P(G b), )/u 1 (P(G b), )] γ/(r+γ) From (1.6) we then obtain the equilibrium starting probability for agent 1: The following result summarizes these findings. ξ (P) = P(g)p ξ P(G, g). (1.8) P(G, b) P(b)p ξ Theorem 4 (Informed leader). Suppose P(G g) >p l.thenthereexistsamarkovperfect Bayesian equilibrium with the following properties. 1. (Pooling equilibrium) If v l (P(G b),τ (P(G))) 0 then both types of agent 1 invest immediately with probability 1 in the first period. In the first period, agent 2 waits indefinitely. In the second period, agent 2 delays her investment by τ (P(G)). 2. (Semi-Separating) If v l (P(G b),τ (P(G))) < 0 <v l (P(G b),τ (P(G g))) then, in the first period, type g invests immediately with probability 1 and type b invests immediately with probability ξ (P), givenby(1.8), andwaitsindefinitelywithprobability 1 ξ (P). Inthefirstperiod,agent2waitsindefinitely.Inthesecondperiod, agent 2 delays her investment by τ (p ξ ), wherep ξ is given by (1.7).

24 Chapter 1. Learning from Strangers 19 ξ (P) P(G b) Figure 1.2: The equilibrium probability of immediate investment for a bad type. The parameter values r =0.02,γ =0.1, y =6, c = 500, andi =0.TheprivatebeliefofagoodtypeisgivenbyP (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 v l (P(G b),τ (P(G g))) < 0 then type g of agent 1 invests immediately with probability 1, and type b never invests. In the first period, agent 2 waits indefinitely. 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 about the state is shown in figure 1.2. That the starting probability may be discontinuous in the private belief of a bad type follows from the fact that a bad type may be too certain that the state is bad to make it profitable for her to invest, even if the other agent would invest without delay. To be more specific, recall that the lowest private belief at which the bad type of agent 1 is willing is to invest is p J, the threshold at which operating both projects simultaneously is profitable for both agents. Hence, P(G b) <p J,andthen the value of investing is negative for type b, regardless of the length of the delay of the follower Private signals I now turn to symmetric equilibria in the case in which each agent observes her signal privately. In this case, the symmetric equilibria can be characterized by the equilibrium behavior 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 delay their 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,

25 Chapter 1. Learning from Strangers 20 p l type g invests immediately type g delays investment 1 P(G (g, g)) P(G (g, b)) no investment semi-sep. or no investment type g invests with probability less than 1 semi-sep. separating or semi-sep. separating type g invests with probability 1 pooling (0, 0) P(G (b, b)) P(G b) P(G (g,b)) P(G (g,g)) 1 p f 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 the leader is non-negative even if the other agent s type turns out to be bad. Conversely, if p l > P(G (g, b)), thengoodtypesdonotwanttoinvestiftheotheragent stypeisbad, and therefore wait to be able to observe the other agent s investment decision. Hence, in equilibrium, 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 which private 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 not to delay their investment if the other agent s type is good. Even when good types are not perfectly revealed because of pooling, the cost for good types of signaling bad news to non-investing bad types outweighs the potential benefit from learning from bad types who 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 agent invests with probability one. In equilibrium, therefore, good types play a waiting game randomly 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) and p l < P(G (g, b)). Infigure1.3 these equilibria correspond to the three area in the bottom left. The shaded area represents a set of symmetric equilibria that are either separating