Irreversible Investment and Learning Externalities - idei

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rival will invest first decreases (so that they both become increasingly pessimistic about ... his investment cost, on the initial belief about the value of the project.
Irreversible Investment and Learning Externalities Jean-Paul D´ecamps†



Thomas Mariotti‡

First draft: March 1999 This draft: February 2000

Abstract In this paper, we develop a continuous time duopoly model of irreversible investment under uncertainty, where each player may learn about the profitability of an investment by observing the experience of his rival, as well as some costless background information. In a world where the opportunity costs of investing are private information to the firms, we study how this pure informational externality creates further incentives to delay investment, on top of the standard real option effect generated by the availability of background information. We show that the resulting war of attrition game has a unique, symmetric perfect Bayesian equilibrium. One important feature of this equilibrium is that it is not Markovian with respect to current public beliefs about the quality of the project. This can be interpreted as a form of history-dependence. We determine the impact of changes in the cost distribution and the stochastic environment on the timing of investment. Last, we study how the equilibrium is affected by the introduction of a first-mover advantage. Keywords: Real Options, Duopoly, Incomplete Information, War of Attrition. JEL Classification: C73; D82; D83. ∗

We are grateful to Martin Cripps, Jocelyn Donze, Godfrey Keller, Bart Lambrecht, S´ebastien Pouget, Jean Tirole, St´ephane Villeneuve, Helen Weeds and Lucy White for thoughtful comments and discussions. We would also like to thank seminar participants at CEPR, CIRANO, ESEM, GREMAQ, Institut Henri Poincar´e, LSE, Nuffield College, UCL, and at the 3rd Annual International Conference on Real Options for their comments. Financial support from CNRS and STICERD is gratefully acknowledged by the authors. We remain of course solely responsible for the content of this paper. † GREMAQ-IDEI, Universit´e de Toulouse I, Manufacture des Tabacs, 21 all´ee de Brienne, 31000 Toulouse, France. E-mail: [email protected] ‡ London School of Economics and Political Science, Houghton Street, WC2A 2AE London, United Kingdom. E-mail: [email protected].

1

Introduction

Uncertainty and irreversibility have long been recognized as major determinants of the investment decision (see e.g. Arrow and Fisher (1974), Henry (1974), McDonald and Siegel (1986), Dixit and Pindyck (1994)). In particular, much attention has recently been devoted to the impact of informational spillovers on the timing of investment decisions. The basic insight of this literature is that individual agents have a strategic incentive to delay their investment decision in order to learn from the success or failure of other investors. Rob (1991) and Caplin and Leahy (1993) study models of industry equilibrium in which firms may learn about the size of the market or the profitability of an industry by observing past quantities and prices. In settings with pure informational externalities, Hendricks and Kovenock (1989) and Chamley and Gale (1994) study timing games of investment, in which the common value from exercising an investment option is correlated with the private signals of players. Last, Bolton and Harris (1999) have recently extended the classic two-armed bandit problem to a multi-agent setting in which each player may learn from the current experimentation of his rivals. The purpose of this paper is to study the impact of asymmetric information about private investment costs in a duopoly model where a player’s investment enhance the quality of the signal received by his rival about the return of a common value project. Compared with previous works on pure informational externalities, the environment we study has three distinctive features: Private vs Common Values. Players’ payoffs incorporate both a common and a private value components. The first relates to the return of the investment project, which is assumed to be the same for both investors, independently of the timing of investments. The second relates to the opportunity cost borne by each player when he exercises his option to invest. For simplicity, there is no strategic interaction between players once they both have invested. Asymmetric Information. The common and private components of players’ payoffs are associated to two different kinds of information. Indeed, we consider a model where at each point in time, players have incomplete but symmetric information about the underlying common value of the investment project, but at the same time have asymmetric information about their private investment costs. This differs from the pure common value models of Hendricks and Kovenock (1989) and Chamley and Gale (1994), in which players have private signals about the common value of an investment, but not about privately borne investment costs. 1

Learning. Last, there are two possible sources of public information about the common value of the project. Independently of the players’ investment decisions, an exogenous background signal continuously provides free information about the profitability of the project. Moreover, once a player, acting as a leader, has sunk his investment, an additional signal is generated that may be used by the remaining player to optimally adjust his investment decision. Note that this formulation is closer to the symmetric information strategic multi-armed bandit problem studied by Bolton and Harris (1999) and Cripps, Keller and Rady (1999) than to the model of Chamley and Gale (1994), in which investment by a player does not generate any information besides that revealed by the very fact that investment took place. The basic elements of our model are worthy of some comments. Note first that this formulation allows for both nonstrategic and strategic incentives to delay investment. Indeed, when the underlying value of the investment is unknown, the availability of a background signal generates a standard real option effect. Next, since there is no incentive to preempt one’s rival in the absence of post investment market interaction, there is no strategic interaction between the firms other than informational. The primary goal of our paper is to study the pure learning externality generated by the increase in signal’s quality following the leader’s investment. In that respect, the crucial step consists in analyzing how private information about the opportunity costs of investment is gradually revealed when new public information about the value of the investment becomes available. Intuitively, each player uses strategically his private information about his investment cost in order to convince his rival to invest first. Here, the main difference with a pure common value model is that each player does not care about the information of his rival per se, but only in so far as it measures the likelyhood of investing second and hence of benefiting from a better signal. To address this issue, we focus on perfect Bayesian equilibria of the investment game. The main difficulty with the use of this equilibrium concept in a learning model with asymmetric information is that a priori, there is no well-defined and reasonably simple state variable with respect to which strategies could be defined. Without extrinsic uncertainty, time itself may be such a state variable, as in the literature on the war of attrition with incomplete information (Riley (1980), Fudenberg and Tirole (1986)). In a learning context with symmetric information about the value of the investment, it might seem reasonable to consider strategies contingent on their corresponding current common belief about the quality of the project. For instance, Bolton and Harris (1999) have characterized the Markov perfect equilibria of the strategic two-armed bandit problem 2

in terms of belief triggers. With irreversible investment, the analogue would be that in equilibrium, each player invests as soon as his belief that the project will yield a high return reaches a certain trigger, conditional on his cost. However, for general learning processes (such as the one considered by Bolton and Harris), this approach is inappropriate when players have incomplete information about each others’ costs. Indeed, as opposed to time, the prospects associated to investment may then fluctuate randomly as new information becomes available. This implies that, even if one player is using a trigger strategy, and will therefore sunk his investment as soon as the current belief reaches this trigger, his rival’s best response is not itself a trigger strategy. The reason is that there is not a one-to-one correspondence between what this player knows about the value of the project and what he knows about his rival’s investment cost. Note that, by assumption, this cannot occur when agents have private signals about a pure common value: in that case, there is always a one-to-one correspondence between what a player knows about his rivals and what he can get by investing. To overcome this issue, we restrict ourselves to a class of simple, though economically relevant learning processes. Specifically, the project may be of a low or high value. If the project has a low value, then players eventually will learn this by observing failures, that occur according to a Poisson process. If the project has a high value, failures occur with probability zero. As an illustration, one may think for instance of two countries that consider developing nuclear energy. Before they invest, the only information available to them is the experience of other (unmodeled) countries. If a major nuclear incident occurs in one of these countries, this is a sure sign that the existing nuclear technology is not safe, and that investing in it is not desirable. However, as long as no such accident has occurred, the common belief in the safety of nuclear technology will raise in both countries. The informational externality comes here from the fact that if one country develops nuclear energy then, conditional on the technology being risky, the hazard rate of failures will increase. Overall, the follower will benefit from a more informative signal than the leader. With such a simple learning model, the set of histories that may occur before investment takes place is simple: if investment costs are sufficiently high, players will never invest after a failure has occurred. We show that in this case, each player’s decision to invest at any point in time is only contingent on his cost and on the current public belief about the investment’s value, which necessarily coincides with the historic maximum of past beliefs. In other terms, each perfect Bayesian equilibrium is in trigger strategies. These equilibria are characterized by a double filtering problem, in which each player 3

uses the flow of public information to update both his beliefs about the value of the investment and his rival’s private opportunity cost. As time goes by and new upper values are reached by the stochastic process of public beliefs (so that players become increasingly optimistic about the value of the investment), each player’s belief that his rival will invest first decreases (so that they both become increasingly pessimistic about benefiting from the second-mover advantage). An important qualification of our results is that, while each perfect Bayesian equilibrium of the game is in trigger strategies, it is in general not Markovian with respect to the common belief process. Indeed, the equilibrium trigger of each player depends, besides his investment cost, on the initial belief about the value of the project. The intuition of this result is simple. One can indeed show that in any perfect Bayesian equilibrium of this game, each player’s investment trigger is an increasing function of his investment cost. In particular, if a player’s cost is equal to the lower bound of the support of the cost distribution, then, under the assumption of a symmetric equilibrium, this player knows that with probability one he will invest first in equilibrium. If the initial belief is above the investment trigger he would adopt if he was the only player, his equilibrium trigger is then just equal to the initial belief. Since each player’s equilibrium trigger is an increasing function of his cost, this implies that an increase of the initial belief will lead each player to raise his investment trigger. This dependence of the equilibrium on the initial conditions contrasts with the Markov perfect equilibrium that prevails in the complete information version of the game. This reflects the fact that, when players have incomplete information about each other’s characteristics, public beliefs not only represents the information accumulated about the desirability of the investment, but also summarize each players’ belief about his opponent’s cost. Before proceeding with the analysis, we would like to relate our paper to the growing literature aiming at unifying real option models with strategic considerations. Most papers in this area have focused on the dual case of preemption. In a symmetric information setting, Smets (1993) has analyzed the trade-off between the value of waiting to gain more information and the fear of preemption. This intuition also underlines the work of Boyer, Lasserre, Mariotti and Moreaux (1999), who also discuss the case of multiple capacity unit acquisitions. The paper most related to the present work is Lambrecht and Perraudin (1997). In a standard valuation framework, they analyze how the threat of preemption is limited when players have incomplete information about each others’ costs. Compared to the attrition game studied in this paper, the main difference is that even if the value of investment may fluctuate through time (in their 4

case, according to a geometric Brownian process), there exist perfect Bayesian equilibria in trigger strategies. The reason is that if a player is using a trigger strategy conditional on his cost, his rival is not threaten by preemption as long as the current value is below the maximum historic value. If the current value also happens to be below the rival’s monopolistic trigger, the latter has no incentive to invest and may safely wait until a new upper value is reached. This paper is organized as follows. Section 2 describes the model. In Section 3, we solve the optimal investment problem of a single player when the information available to him is either the background or the follower’s signal, and we briefly analyze the symmetric information version of the game. In Section 4, we describe our strategy and equilibrium concepts, and show that the equilibrium investment strategies are characterized by investment triggers conditional on each player’s cost and on the common belief about the value of the investment. In Section 5, the existence of an unique symmetric perfect Bayesian equilibrium is established. In Section 6, several comparative statics exercises are performed, and some extensions of the results and of the model are discussed. Section 7 concludes.

2

The Model

We first describe the players and their payoffs in the investment game. Next, we describe the information structure of the game. Notation. Throughout the paper, we will use the following notation. For any real numbers x and y, x ∨ y denotes the maximum of x and y, and x ∧ y the minimum of x and y. For any open subset X of an Euclidean space, B(X) is the space of Borel subsets of X, and C 1 (X, R+ ) is the space of continuously differentiable positive realvalued functions on X.

2.1

Players and Preferences

Time is continuous, and labeled by t ∈ [0, ∞). I = {1, 2} is the set of players. Each player i ∈ I can invest at most once in a risky project with unknown value v ∈ {l, h}, where h > l ≥ 0. This investment is irreversible and entails an opportunity cost θi ≥ 0 for player i. A distinctive feature of the model is that the value v of the project is common to both players, and does not depend on the timing of investment. In particular, there is no market interaction between the players, hence no incentive to preempt one’s rival. 5

Hence, assuming that both players are risk neutral and discount future payoffs at rate ρ > 0, player i’s payoff at date 0 from investing at date t is u(v, t, θi ) = e−ρt (v − θi ). Investment decisions are assumed to be perfectly observable.1

2.2

Public Information

Henceforth, all uncertainty about v and any public signal released during the game is represented by a probability space (Ω, F, P ). At date 0, v is drawn from a common knowledge probability distribution on {l, h}, with P (v = h) = p0 ∈ (0, 1). Neither player is informed of the outcome of this lottery. However, players can learn about θ through two types of public signals. The Background Signal. Before any investment takes place, players can learn at no cost about v by observing the outcome of a Poisson process N B (v) = {NtB (v)} on (Ω, F, P ), with intensity λB ∈ (0, ρ) if v = l, and zero if v = h. A jump of the process N B (v) will be referred to as a failure. A failure is common knowledge when it occurs, and it clearly provides an unambiguous signal of the poor quality of the project.2 As in Bolton and Harris (1999), the background signal N B (v) may be interpreted as the information generated by investment analyzes, consumer surveys, or previous experience by other (unmodeled) players. The Follower’s Signal. While real interactions are ruled out by assumption, our model is characterized by a pure learning externality. Specifically, we posit that once a player (the leader) has publicly sunk his investment, the signal observed by remaining player (the follower) is more informative about v than the background signal. This is modeled by assuming that, after the leader’s investment, the signal observed by the follower is a Poisson process N F (v) = {NtF (v)} on (Ω, F, P ), with intensity λF ∈ (λB , ρ) if v = l, and zero if v = h.3 Conditional on v, the follower’s signal is assumed to be independent from the past realizations of the background signal. The increase in the quality of the signal creates a strategic incentive to delay investment, on top of the pure real option effect induced by the availability background signal. 1

Our model can however be easily extended to take preemption into account, see Section 6. In a recent paper, Cripps, Keller and Rady (1999) make use of a similar signal structure in the context of a strategic multi-armed bandit problem. 3 This typically holds if, conditional on v = l, the follower observes, on top of the background signal, an independent Poisson process with intensity λF − λB representing the experience of the leader. We can then rewrite the payoffs h and l as π/ρ and π/(ρ + λF − λB ) for some π > 0. The interpretation is that both type of projects yield π per unit of time, but bad projects have a failure rate of λF − λB . 2

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2.3

Private Information

The investment costs (θi )i∈I incorporate both the value of foregone opportunities in other activities and expenditures at the date of investment. At date 0, they are independently drawn from a common knowledge Borel probability measure µ on Θ = R+ with c.d.f. G and density g with respect to Lebesgue measure. The support of µ is taken to be a compact interval [θ, θ]. Each player is informed of his own cost, but not of the cost of his rival. Throughout the paper, we assume that: A1. [θ, θ] ∩ [l, h] 6= ∅; A2. g ∈ C 1 (Θ, R+ ); A3. g(θ) > 0 ∀θ ∈ (θ, θ). Assumption A1 ensures that the problem is not trivial. (If θ ≥ h, the project would almost never be undertaken, and if θ ≤ l, only a negligible subset of types would have an incentive to delay investment.) Together, A2 and A3 will imply that the players’ equilibrium entry triggers are differentiable function of their costs in any symmetric perfect Bayesian equilibrium. We will also assume that each player’s cost is independent from v and the signal processes. In particular, a player’s own cost does not convey information about the value of the project. It follows that players have incomplete but symmetric information about v at any stage of the game where no failure has still occurred.

3

Preliminaries

A natural state variable for players’ strategies is their common belief about v. The first step of our analysis is therefore to study the players’ filtering problem, i.e. to describe how they incorporate new information about v as time evolves. In a second step, we derive optimal investment policies in two benchmark cases. Last, we briefly analyze a complete information version of the investment game, in which players have no uncertainty about their rival’s cost.

3.1

The Filtering Problem

Prior to investment by any player, the only source of information about v is the background signal. Let {HtB } denote the public information filtration generated on (Ω, F, P ) 7

by N B (v). As v can only take on two values l and h, the players’ posterior beliefs about B v prior to any investment are completely characterized by pB t (ω) = P (v = h | Ht )(ω) for each ω ∈ Ω. Suppose that no one has invested at date t, and let I[0,t] (ω) = 1 if a failure occurs during [0, t], and 0 otherwise. Then we have:  p0   p + (1 − p ) e−λB t if I[0,t] (ω) = 0 0 0 pB . (1) t (ω) =   0 if I[0,t] (ω) = 1 Suppose now that the leader invested at date τ . Then, at any date t ≥ τ , the follower’s information about v is represented by HtF (τ ) = σ({NsB | 0 ≤ s ≤ τ } ∪ {NsF | τ ≤ s ≤ t}), and his belief pFt (ω, τ ) = P (v = h | HtF (τ ))(ω) at date t satisfies:  pB  τ (ω)  if I[τ,t] (ω) = 0  B −λF (t−τ ) pτ (ω) + (1 − pB τ (ω)) e pFt (ω, τ ) = . (2)    0 if I[τ,t] (ω) = 1 Since λF > λB , other things being equal the beliefs of the follower evolves more rapidly F than the beliefs of the leader. By construction, the processes {pB t } and {pt (τ )} are respectively {HtB }- and {HtF (τ )}-martingales. However, since a failure only conveys bad news about the value of the investment project, players’ beliefs that v = h are monotonically increasing as long as no failure occurs.

3.2

Benchmarks

As benchmarks, we derive the optimal investment policies in two polar cases. First, we consider the optimal investment problem of a single player with investment cost θ, when the only information available to him about v is the background signal N B (v). This problem is denoted ΠB (θ). Next, we solve the optimal investment problem of a follower with cost θ, given the information provided by the follower’s signal N F (v). This problem is denoted ΠF (θ). Due to the stationarity of the belief processes {pkt }, the optimal stopping regions in Πk (θ) (k ∈ {B, F }) are (possibly empty) intervals [P k (θ), 1] and the corresponding optimal stopping times are inf{t ≥ 0 | P k (θ) ≤ pkt } for well-chosen investment triggers P k (θ). The following lemma is proved in the Appendix: Lemma 1 Let ∆v = h − l. For each k ∈ {B, F } the optimal investment trigger in

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Πk (θ) is given by:

k

P (θ) =

 0       q     −(ρ − λk ) + (ρ − λk )2 −          

2λk ∞

if θ ≤ l 4ρλk (l−θ) ∆v

if θ ∈ (l, h) .

if θ ≥ h

Moreover, P k is increasing on (l, h) and P B (θ) < P F (θ) on this interval. The value function V k (·, θ) : [0, 1] → R+ associated to Πk (θ) can be explicitly computed. To do so, let: µ ¶ρ/λk p k u (p) = 1−p for each p ∈ [0, 1). With this notation, for each (p, p˜) ∈ (0, 1) × (0, 1) the quantity Lk (p, p˜) = uk (p)/uk (˜ p) ∧ 1 represents the expected discounted value of a claim to a dollar at the first date where {pkt } reaches at least the level p˜ given a current value p. Thus, for each θ ∈ (l, h):  k ¡ ¢ if p ∈ [0, P k (θ)]  L (p, P k (θ)) ∆v P k (θ) + l − θ k V (p, θ) = . (3)  ∆v p + l − θ if p ∈ [P k (θ), 1] These value functions are depicted in Figure 1. Note that V k (·, θ) smoothpastes at the exercise profit line p 7→ ∆v p + l − θ at P k (θ), a property that will be used repeatedly in the equilibrium analysis. It is straightforward to check that for each θ ∈ (l, h), P k (θ) is strictly greater than the Marshallian trigger θ−l at which the net present value ∆v of investing is equal to zero. As in standard real option models (e.g. McDonald and Siegel (1986) or Dixit and Pindyck (1994)), the difference between the optimal and the Marshallian triggers reflects the value of waiting for new information before investing. Similarly, the fact that for each θ ∈ (l, h), P B (θ) < P F (θ), and consequently that V F (p, θ) > V B (p, θ) for each p ∈ [0, P F (θ)) reflects the positive value of information provided by the leader to the follower.

3.3

The Investment Game: Complete Information

To conclude this section, we would like to underline that the non strategic investment triggers P B (θ) and P F (θ) also admit a natural game-theoretic interpretation. To illus9

trate this point, consider a complete information version of the investment game where players know each others’ investment costs. Suppose for simplicity that for each i, θi ∈ (l, h). The following proposition describes the unique subgame perfect equilibrium that arises when there is a (arbitrarily small) cost asymmetry between the players:4 Proposition 1 If θ1 < θ2 , there exists a unique subgame perfect equilibrium, in which for any (t, ω) ∈ [0, ∞) × Ω such that no player has invested at date t: F 2 (i) If pB t (ω) ∈ [P (θ ), 1], both players simultaneously invests; B 1 F 2 (ii) If pB t (ω) ∈ [P (θ ), P (θ )), player 1 invests, and player 2 does not invest; B 1 (iii) If pB t (ω) ∈ [0, P (θ )), no player invest.

For any (t, ω) ∈ [0, ∞) × Ω such that player i invested at time τ < t, player −i invests if and only if pFt (ω, τ ) ≥ P F (θ−i ). The intuition of this result is simple. The leader is endogenously determined as the player with the lowest investment cost, here player 1. Since he knows that he will never benefit from the high-quality signal, he invests as if he were the only player, adopting (on the equilibrium path) the trigger strategy P B (θ1 ) described in Lemma 1. Conversely, since P B (θ2 ) > P B (θ1 ) and P F (θ2 ) > P F (θ1 ), player 2 can credibly threaten player 1 to always invest after or, at the very least, at the same date as him. The proof, much in the spirit of the backward induction argument of Ghemawat and Nalebuff (1985), is presented in the Appendix. In Section 6, we compare the outcomes of the game with and without asymmetric information about costs. Remark. Clearly, the above equilibrium is inefficient. Indeed, when taking his investment decision, the leader does not internalize the positive externality that he exerts on the follower in the form of a more precise information about the value of the project.

4

Strategies and Equilibrium Concept

We first provide a general formalization of the strategies and of the perfect Bayesian equilibrium (PBE) concept. We next show that the equilibrium investment strategies are characterized by investment triggers conditional on the beliefs {pB t }. 4

Note that if θ1 = θ2 , the game also admits mixed strategy equilibria. However, these are not robust to arbitrarily small perturbations of the players’ investment costs.

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4.1

A General Framework

We begin by an observation. Obviously, in any perfect Bayesian equilibrium, the optimal behavior of a follower is described by the Markov policy P F in any continuation game where one player has already invested. We may therefore reduce our attention to the following stopping game: 1. The only signal is the background signal N B (v); 2. The game is over as soon as one player invests; i 3. If player i invests first at time t, he receives ∆v pB t + l − θ , and his opponent −i −i receives V F (pB t , θ ).

Since the only signal available in this auxiliary game is the background signal and the game is over as soon as one player invests, the information generated by public histories up to any time t such that the game is not over at t can be represented by HtB . Accordingly, for each (t, ω), we shall sometimes denote by ht (ω) the history at time t generated by ω and such that no player has invested at date t. A pure strategy for player i specifies whether or not he should invest first at time t, for each t ≥ 0 and each history ht (ω) of the game up to t, conditional on his cost θi . Hence the following definition (1 stands for “invest” and 0 for “do not invest” respectively): Definition 1 A pure strategy for player i ∈ I is a process σ i = {σti } on Ω × Θ with values in {0, 1} such that: (i) σti0 (ω, θi ) ≥ limt↓t0 σti (ω, θi ) ∀(t0 , ω, θi ) ∈ [0, ∞) × Ω × Θ; (ii) σ i is progressively measurable with respect to the filtration {HtB ⊗ B(Θ)}, i.e. {(s, ω, θi ) ∈ [0, t] × Ω × Θ | σsi (ω, θi ) = 1} ∈ B([0, t]) ⊗ HtB ⊗ B(Θ) ∀t ≥ 0. This definition is worthy of some comments. First, our strategy concept is more general than the stopping time formalization of Fudenberg and Tirole (1986). One may for instance have σti0 (ω, θi ) = 1 and σti (ω, θi ) = 0 for some (ω, θi ) and each t ∈ (t0 , t0 + ∆t), corresponding to the case where player i with type θi plans to invest first at date t0 following history ht0 (ω), but does not invest first in an interval of time consecutive to t0 if he deviated at t0 and the history at each date t ∈ (t0 , t0 + ∆t) is ht (ω). As in Dutta and Rustichini (1993), (i) is a regularity condition that ensures that a strategy is intuitively playable, ruling out for instance the type- and history-independent process 11

σti = I(t0 ,+∞) (t), i.e. player i invests first “just after” t0 . (This strategy is clearly progressively measurable since the mapping t 7→ σti is left-continuous.) Last, a pure strategy for player i generates a family of mappings τt (σ i ) : (ω, θi ) 7→ inf{s ≥ t | σsi (ω, θi ) = 1}, t ≥ 0. For each θi , τt (σ i )(θi ) is the first time after t at which player i with type θi will invest provided he has not invested yet at t. τt (σ i ) can be seen as a family of stopping times, parametrized in a measurable way by the type of the player. Indeed, ¡ ¢ {(ω, θi ) ∈ Ω × Θ | τt (σ i )(ω, θi ) ≤ t˜} = proj {(s, ω, θ) ∈ [t, t˜] × Ω × Θ | σsi (ω, θ) = 1} Ω×Θ

i for each t˜ ≥ t, and this set belongs to HtB ˜ ⊗ B(Θ) since σ is progressively measurable. To define a perfect Bayesian equilibrium, we must describe the evolution of players’ beliefs about their opponent’s type. A belief process for player i is essentially a collection of {HtB }-measurable transition probabilities from Ω to ∆(Θ), the space of probability measures on (Θ, B(Θ)). Formally:

Definition 2 A belief process for player i ∈ I is a parametrized collection of mappings µit : Ω × B(Θ) → [0, 1], t ≥ 0, such that: (i) For each (t, ω) ∈ [0, ∞) × Ω, µit (ω, ·) is a probability measure on (Θ, B(Θ)); (ii) For each (t, A) ∈ [0, ∞) × B(Θ), µit (·, A) is HtB -measurable. Intuitively, µit (ω, A) represents player i’s belief that θ−i ∈ A given the information held at date t on the state of nature ω and the fact that player −i has not invested at any date s < t. We are now in a position to define our equilibrium concept: Definition 3 A profile of strategies and belief processes (σ i , µi )i∈I is a perfect Bayesian equilibrium (PBE) if, for each i ∈ I: (i) For each (t, ω) ∈ [0, ∞) × Ω such that µ({θ−i ∈ Θ | τ0 (σ −i )(ω, θ−i ) ≥ t}) > 0, µit (ω, A) =

µ({θ−i ∈ A | τ0 (σ −i )(ω, θ−i ) ≥ t}) µ({θ−i ∈ Θ | τ0 (σ −i )(ω, θ−i ) ≥ t})

∀A ∈ B(Θ);

(ii) For any (t, ω, θi ) ∈ [0, ∞) × Ω × Θ, τt (σ i )(θi ) maximizes: ³ ´ i −i i −rτ B i −i ) T , and ω such that no failures occur before time t. Then the event “player i has not invested yet at time t > T ” has measure zero if no failure has occurred up to time t. However, if the support [θ, θ] of the type distribution does not coincide with the space of possible types Θ = R+ , the zero probability event in the above example can always be rationalized by another zero probability event, namely that θi > h. This seems more reasonable than any alternative interpretation, which would require an infinite number of deviations by player i between times T and t.

4.2

Entry Triggers

Since the only payoff relevant variable for the players, besides their own investment costs, is described by their common belief about the underlying value of the project, a natural question to ask is whether there is a loss of generality in describing the players’ behaviour on the equilibrium path by investment triggers as in the benchmark cases analyzed in Section 3. The following proposition gives a positive answer to this question: Proposition 2 Let (σ i , µi )i∈I be a PBE. Then for each i ∈ I, there exists a Borel measurable function P i : Θ 7→ [p0 , 1] ∪ {∞} such that: i τ0 (σ i )(ω, θi ) = inf{t ≥ 0 | P i (θi ) ≤ pB t (ω)} ∀(ω, θ ) ∈ Ω × Θ.

13

(6)

Proof. For each t ≥ 0, let Ft ∈ HtB be the event “a failure occurs before or at date t”. We first prove that for each θi ∈ Θ, there exists a unique T i (θi ) ∈ [0, ∞] such that:  i i / FT i (θi )  T (θ ) if ω ∈ i i τ0 (σ )(ω, θ ) =  ∞ if ω ∈ FT i (θi ) If θi ≤ l, it is obvious that (6) holds with T i (θi ) = 0, since there is no reason to delay investment in that case. Conversely, if θi ≥ h, one has T i (θi ) = ∞, since investment is ˜ ) ∈ Ω × Ω and never profitable. Consider now θi ∈ (l, h), and suppose there exist (ω, ω (t, t˜) ∈ [0, ∞) × [0, ∞) such that τ0 (σ i )(ω, θi ) = t, τ0 (σ i )(˜ ω , θi ) = t˜ and t < t˜. Notice first that since θi > l and a failure is an unambiguous signal that v = l, necessarily ω∈ / Ft and ω ˜∈ / Ft˜. In particular, the histories ht (ω) and ht (˜ ω ) of the game up to time i i B t under ω and ω ˜ are identical. Hence τ0 (σ )(θ ) is not an {Ht }-adapted stopping time, a contradiction. It follows that necessarily t = t˜ = T i (θi ). Hence, on the equilibrium path player i with type θi invests at T i (θi ) if no failure occurs until T i (θi ) and player −i has not already invested at this date. Conditional on no failure occurring, there is a one-to-one correspondence between time and beliefs. Hence: ½ ¾ p0 i i B τ0 (σ )(ω, θ ) = inf t ≥ 0 | ≤ pt (ω) ∀ω ∈ Ω. p0 + (1 − p0 ) e−λB T i (θi ) Let us define P i by:

P i (θi ) =

      

if

p0

θi ≤ l

p0 if θi ∈ (l, h) . −λB T i (θi ) p + (1 − p ) e  0 0      ∞ if θi ≥ h

(7)

To complete the proof, note that T i : Θ → [0, ∞] is measurable since σ i is measurable with respect to θi , hence P i is measurable as well. ¥ Proposition 2 shows that any PBE induces a pair of trigger function (P i )i∈I . This does not imply that a PBE is Markovian with respect to the current belief about v. Quite on the contrary, it is immediate from (7) that the trigger functions derived from an equilibrium strategy profile depends on the initial belief p0 , or equivalently, on calendar time. The reason for this is that since players have asymmetric information about their investment costs, the true Markovian state variable (in the sense of Maskin and Tirole 14

(1999)) embodies players’ beliefs about their opponent’s type, as well as their beliefs about the value of the project (see Bergin (1992)). We will come back on this point in Section 6 when we discuss the symmetric equilibrium. A related fact is that the trigger functions (P i )i∈I derived in Proposition 2 only describe the players’ behaviour on the equilibrium path, and should not be confused with equilibrium strategies. Note in particular that nothing ensures a priori that the trigger strategies σ ˜ti (ω, θi ) = I{pBt (ω)≥P i (θi )} , i ∈ I induced by (P i )i∈I are sequentially rational off the equilibrium path, given the equilibrium beliefs processes (µi )i∈I .6 Accordingly, Proposition 2 only asserts that in equilibrium, everything happens as if players were playing these trigger strategies. The reason for this latter fact is simple to understand: the Poisson specification of the background process implies that, conditional on no failure and no investment occurring up to time t, all the histories of the game up to this date are identical. Hence in equilibrium each player knows at which time, if ever, he will invest if he invests first. In this sense, the investment game is essentially a game of timing. This last point is best understood by considering the equilibrium belief processes. In equilibrium, player i’s belief at date t about his opponent’s type only depends on P −i and on the maximal value reached by the posterior {pB s } up to time t: µit (ω, A) =

µ({θ−i ∈ A | P −i (θ−i ) ≥ sups∈[0,t] pB s (ω)}) , µ({θ−i ∈ Θ | P −i (θ−i ) ≥ sups∈[0,t] pB s (ω)})

∀A ∈ B(Θ),

(8)

provided that µ({θ−i ∈ Θ | P −i (θ−i ) ≥ sups∈[0,t] pB s (ω)}) > 0. Since failures are an unambiguous signal that v = l, beliefs always keep increasing according to (1) up to the first time where a failure occurs, if ever. Hence, conditionally on not observing a i failure, the belief processes {pB t } and ({µt })i∈I prior to any investment are completely predictable by the players. In particular, if it is optimal at date 0 for player i with i i type θi to invest first at the first date where {pB t } reaches P (θ ), this policy remains optimal at any date s ≤ inf{t ≥ 0 | P i (θi ) ≤ pB t }. This is because at any time where no failure nor investment have occurred, there is a one-to-one correspondence between the expected payoff that a player can get by investing at this time and what he knows about his opponent. This property typically does not hold in a Gaussian model as in Moretto (1996), because the state variable is no longer monotonically increasing. 6

In Section 6, we nevertheless prove that this is the case.

15

5

Equilibrium Analysis: Symmetric Equilibrium

In this section, we prove that the investment game has an essentially unique symmetric PBE. To do so, we characterize the trigger function induced by a symmetric PBE.

5.1

Statement of the Problem

Hereafter, we shall use the shortcut “player i invests at pi ” to mean that, on the equilibrium path, i invests at τpi = inf{t ≥ 0 | pi ≤ pB t } if −i has not invested at this time. i −i i Let V(p0 , p , P , θ , µ) be time zero expected discounted payoff of player i with cost θi ’s if player −i’s trigger function is P −i and i invests at pi ≥ p0 , given his beliefs p0 and µ at date 0 about v and θ−i . Then:7 Z i −i i V(p0 , p , P , θ , µ) = LB (p0 , P −i ) V F (P −i , θi ) dµ i −i −i {p >P (θ )≥p0 } (9) + LB (p0 , pi ) (∆v pi + l − θi ) µ({P −i (θ−i ) ≥ pi }). The first term on the right-hand side of (9) represents i’s expected gain if −i invests before pi is reached by the belief process {pB t }. The second term is i’s expected gain if −i i does not invest before p is reached by the belief process. Note that future payoffs in (9) are appropriately discounted by the stochastic discount factor LB (p0 , p) = E(e−ρτp | H0B ). From Proposition 2, a necessary and sufficient condition for a pair of trigger functions (P i )i∈I to be induced by a PBE is: P i (θi ) =

argmax V(p0 , pi , P −i , θi , µ),

pi ∈[p

0 ,1]∪{∞}

∀(i, θ) ∈ I × Θ.

(10)

Our main objective is to characterize the symmetric solutions of the system (10). Before we proceed with the analysis, some comments might be helpful. An immediate remark is that if (P i )i∈I is a solution of (10), then necessarily P i = p0 on [0, l) and P i = ∞ on (h, ∞). Next, since the behaviour of a negligible subset of each player’s types does not affect the distribution of entry triggers and thus the incentives of his rival, the bulk of the argument consists in characterizing (P i )i∈I on the support [θ, θ] of µ. Last: Lemma 2 Let (P i )i∈I be a solution of (10). Then, for each (i, θi ) ∈ I × Θ : (i) P i (θi ) ≥ p0 ∨ P B (θi ); 7

Henceforth, we systematically simplify the expressions by suppressing unnecessary mentions of bounded variables.

16

(ii) p0 ∨ P F (θi ) ≥ P i (θi ). Though straightforward, this lemma will be very useful. It relies on two revealed preference arguments. First, since there is no strategic interaction besides informational ones, it is obvious that in equilibrium each player is ready to wait at least as much as if he were the only player, hence (i). Symmetrically, unless p0 ≥ P F (θi ), there is no point for player i in investing at a higher trigger than P F (θi ), since he would be ready to invest at such a trigger if he were a follower, hence (ii). An immediate consequence of Lemma 2 is that if P F (θi ) ≤ p0 , then P i (θi ) = p0 . Indeed, in that case player i would be ready to invest at the initial level of belief p0 even if he were a follower at this level of belief. He has therefore no incentive to wait at date 0. Denoting by φk (k ∈ {B, F }) the inverse of P k on [0, 1], we will henceforth assume ¡ ¢ that Θ0 = φF (p0 ) ∨ θ, h ∧ θ 6= ∅ to ensure that the problem is not trivial. As usual with games of incomplete information, the analysis will be mainly in terms of the inverse correspondence φ = P −1 : [p0 , 1] ∪ {∞} → 2Θ . For each p ∈ [p0 , 1] ∪ {∞}, φ(p) is the set of possible costs of a player who invests at p in equilibrium. Thus the equilibrium distribution of each player’s entry trigger is G ◦ φ.

5.2

Separating Properties of Equilibria

The purpose of this subsection is to show that in any (possibly asymmetric) PBE, the equilibrium trigger functions (P i )i∈I are (strictly) increasing functions of each player’s opportunity cost on Θ0 , in other terms, any PBE is strictly separating on this interval.8 As shown in the following lemma, weak monotonicity follows from a standard revealed preference argument: Lemma 3 In any PBE, the equilibrium trigger functions are nondecreasing on Θ0 . Proof. Let (P i )i∈I be equilibrium trigger functions, and let (θi , θ˜i ) ∈ Θ0 × Θ0 such that θi > θ˜i . Denote P i (θi ) = pi and P i (θ˜i ) = p˜i . If P F (θ˜i ) ≤ pi ≤ P F (θi ), the claim is obvious. Assume now that P B (θ˜i ) ≤ pi ≤ P F (θi ) ∧ P F (θ˜i ) = P F (θ˜i ). By (10), V(p0 , pi , P −i , θi , µ) ≥ V(p0 , p˜i , P −i , θi , µ) and V(p0 , p˜i , P −i , θ˜i , µ) ≥ V(p0 , pi , P −i , θ˜i , µ). Summing and rearranging these two inequalities yields: v(pi , θi ) − v(˜ pi , θi ) ≥ v(pi , θ˜i ) − v(˜ pi , θ˜i ), 8

(11)

It is clear that any PBE involves pooling on [0, φF (p0 )] and [h, ∞). It follows that if θ < φF (p0 ) (respectively θ > h), the distribution of entry triggers has an atom at p0 (respectively at ∞).

17

where for each (p, θ) ∈ (p0 , 1) × Θ0 : v(p, θ) =

(∆v p + l − θ) µ({P −i (θ−i ) ≥ p}) uF (P F (θ)) × . ∆v P F (θ) + l − θ uB (p)

From (11), we deduce that: Z

θi

θ˜i

∂θ (v(pi , θ) − v(˜ pi , θ)) dθ ≥ 0.

(12)

¿From (12), it is sufficient to prove that for each θ ∈ [θ˜i , θi ], pi ≥ p˜i if and only if pi , θ)) ≥ 0. More specifically, we prove that p 7→ ∂θ v(p, θ) is nondecreasing ∂θ (v(pi , θ)−v(˜ for all θ ∈ [θ˜i , θi ] and for all p ∈ [P B (θ˜i ), P F (θ˜i )]. By construction of P F (θ): µ ¶ ∆v p + l − θ ∂p (P F (θ)) = 0, uF (p) so that ∆v uF (P F (θ)) = (∆v P F (θ) + l − θ) ∂p uF (P F (θ)). It follows that: µ ¶ uF (P F (θ)) uF (P F (θ)) ∂θ = ∆v P F (θ) + l − θ (∆v P F (θ) + l − θ)2 identically in θ. A direct computation yields then: ∂θ v(p, θ) =

uF (P F (θ)) ∆v (p − P F (θ)) µ({P −i (θ−i ) ≥ p}) × × . ∆v P F (θ) + l − θ ∆v P F (θ) + l − θ uB (p)

(13)

Let a(p) and b(p) be respectively the second and third term in the product on the righthand side of (13). It is straightforward to see that a is nondecreasing in p and that b is nonincreasing in p. Moreover, for each value of p lying between pi and p˜i and for each θ ∈ [θ˜i , θi ], a(p) ≤ 0 since pi ∨ p˜i ≤ P F (θ˜i ). It is then easy to check that for each p > p˜ lying between pi and p˜i , one has a(p)b(˜ p) ≤ a(p)b(p) ≤ 0 and a(˜ p)b(˜ p) ≤ a(p)b(˜ p). Thus ab is nondecreasing on [P B (θ˜i ), P F (θ˜i )] and the proof is complete. ¥ We now prove that in any PBE, the equilibrium distributions of entry triggers have no atoms on (p0 , 1), i.e. µ({P i (θi ) = p}) = 0 for each i and p ∈ (p0 , 1). From Lemma 3 and the continuity of G, this will imply that the equilibrium trigger function P i is increasing on Θ0 , hence that φi is single-valued on P i (Θ0 ). This step of the proof is more involved than in other models of war of attrition (e.g. Riley (1980), Fudenberg and Tirole (1986)) or auctions (Maskin and Riley (1999a)), because it is not a priori clear that if player −i’s trigger distribution has an atom at 18

p ∈ (p0 , 1), all types of player i will be better off waiting that the belief process reaches p before investing. In particular, if −i’s hazard rate of entry in a small interval [p − ε, p] is high, we cannot exclude that some types of player i with an opportunity cost θi arbitrarily close to, but smaller than φF (p) would be ready to wait almost up to the point where p is attained by the belief process before investing. If it were the case, and thus a positive measure of types of player i were investing in any small interval [p − ε, p], the standard argument used to show that the types of player −i investing at p would be better off investing slightly before p breaks down. The following lemma shows that if p is an atom in the distribution of −i’s entry trigger, no type θi of player i such that θi > φF (p) will invest in an interval [p − ε, p] for ε > 0 close enough to zero. Lemma 4 For each i ∈ I, if p ∈ (p0 , 1) is an atom in the equilibrium distribution of player −i’s entry trigger, then: © ª ∃ε > 0 s.t. [p − ε, p] ∩ P i (θi ) | θi > φF (p) = ∅. Proof. The result is obvious if θi > h. Let η ≥ 0 such that p − η > p0 . If player i with type θi > φF (p) invests first at p − η, his expected payoff at the time of the investment is ∆v (p − η) + l − θi . If he invests “just above” p, his expected payoff when the current value of {p0t } is p − η is given by: Z dµ LB (p − η, P −i ) V F (P −i , θi ) −i −i µ({P (θ ) ≥ p − η}) {p>P −i (θ−i )≥p−η} µ + LB (p − η, p)

µ({P −i (θ−i ) = p}) V F (p, θi ) µ({P −i (θ−i ) ≥ p − η})

(14)

¶ µ({P −i (θ−i ) > p}) i + (∆v p + l − θ ) . µ({P −i (θ−i ) ≥ p − η}) The first term of (14) is positive, and the mapping q 7→ µ({P −i (θ−i ) ≥ q}) is leftcontinuous, so that the second term converges as η ↓ 0. To prove our claim, it is sufficient to note that V F (p, θi ) > ∆v p + l − θi since θi > φF (p), so that the limit of the ¥ second term is strictly greater than ∆v p + l − θi . Having characterized the best response of player i with θi > φF (p) to an atom in −i’s trigger distribution at p, we now consider the case where θi ≤ φF (p). Intuitively, if the distribution of player −i’s investment trigger exhibits an atom at p, then player 19

i with cost θi ≤ φF (p) must invest with a positive probability in any interval of the form [p − ε, p], ε > 0, otherwise some types of player −i that invest at p would have an incentive to invest before p. For our purposes, it is sufficient to focus on the marginal player i with cost φF (p). We have: Lemma 5 For each i ∈ I, if p ∈ (p0 , 1) is an atom in the equilibrium distribution of player −i’s entry trigger, then P i (φF (p)) = p. Proof. Suppose that P i (φF (p)) < p. Then, by Lemmas 3 and 4, there exists ε > 0 such that [p − ε, p] ∩ {P i (θi ) | θi ∈ Θ0 } = ∅, and we can suppose that p − ε > p0 . Let J = {θ−i ∈ Θ0 | P −i (θ−i ) = p}. By Lemma 3, J is an interval. Let j0 = inf J. Two cases may happen. First, if j0 ∈ J and P B (j0 ) < p, then player −i with cost j0 would be better off investing at the trigger P B (j0 ) ∨ p − ε < p, since doing so would not affect the probability that i invests first and would increase his own entry payoff, a contradiction. If j0 ∈ J and P B (j0 ) = p, then there exists θ−i ∈ J such that p = P −i (θ−i ) < P B (θ−i ), a contradiction to Lemma 2. Second, if j0 ∈ / J and P B (j0 ) < p, then the same argument as above shows that there will be some type θ−i ∈ J of player −i that would strictly gain by investing at P B (θ−i ) ∨ p − ε < p, a contradiction. Last, if P B (j0 ) = p, one should have j0 ∈ J contrary to the assumption. ¥ We are now ready to prove the main result of this subsection: Proposition 3 In any PBE, the equilibrium trigger functions are increasing on Θ0 . Proof. If P −i is not increasing on Θ0 , there exists p ∈ (p0 , 1) such that µ({P −i (θ−i ) = p} > 0. It follows from Lemma 5 that P i (φF (p)) = p. We will prove that this equality yields to a contradiction. Note first that if the equilibrium distribution of player −i’s trigger admits an atom p, there exists η > 0 such that µ({p > P −i (θ−i ) ≥ p − η}) > 0. Otherwise, player −i would almost never invest in [p − η, p), and player i with cost φF (p) would be better off investing earlier, e.g. at P B (φF (p)) ∨ p − η < P F (φF (p)) = p, a contradiction to Lemma 5. For player i with cost φF (p) to be ready to invest at p, the following must hold for each p˜ ∈ (p0 , p]: V(p0 , p˜, P −i , φF (p), µ) ≤ V(p0 , p, P −i , φF (p), µ).

(15)

As q 7→ LB (p0 , q) V F (q, φF (p)) is strictly decreasing on (p0 , 1), and ∆v p˜ + l − φF (p) > LB (˜ p, p) (∆v p + l − φF (p)) as well as LF (˜ p, p) (∆v p + l − φF (p)) > ∆v p˜ + l − φF (p) for

20

each p˜ ∈ (p0 , p], one can rearrange (15) to obtain: 0
P −i (θ−i ) ≥ p˜}) < . LF (˜ p, p) (∆v p + l − φF (p)) − ∆v p˜ − l + φF (p) µ({P −i (θ−i ) ≥ p˜})

(16)

We prove that the central term in (16) goes to +∞ as p˜ ↑ p, thus obtaining a contradiction since the right-hand side is bounded. On one hand, using L’Hˆopital’s rule and the smooth pasting condition ∆v − ∂1− LF (p, p) (∆v p + l − φF (p)) = 0 for ΠF (φF (p)), one obtains: ∆v p˜ + l − φF (p) − LB (˜ p, p) (∆v p + l − φF (p)) lim p˜↑p p˜ − p = ∆v − ∂1− LB (p, p) (∆v p + l − φF (p)) = (∂1− LF (p, p) − ∂1− LB (p, p)) (∆v p + l − φF (p))

(17)

ρ/λF − ρ/λB = (∆v p + l − φF (p)), p(1 − p) which is strictly negative since λF > λB and the optimal trigger is strictly greater than the Marshallian trigger. On the other hand, using again the smooth pasting condition for ΠF (φF (p)), one gets: lim p˜↑p

LF (˜ p, p) (∆v p + l − φF (p)) − ∆v p˜ − l + φF (p) = 0− . p˜ − p

(18)

Dividing (17) through (18) yields the required conclusion.

¥

The intuition underlying the proof of Proposition 3 is straightforward: whatever the hazard rate of investment of player −i in a left neighborhood of p, there is no gain for player i with opportunity cost φF (p) to wait up to p before investing. Technically, this is because the smooth pasting condition for ΠF (θ) at θ = φF (p) implies that, as p˜ ↑ p, the relative gain of investing immediately at p˜ becomes infinite compared to the marginal gain obtained by delaying investment until p is reached. Since by Lemma 5 P i (φi (p)) = p is a necessary condition for p ∈ (p0 , 1] to be an atom in the equilibrium distribution of player −i’s entry trigger, this implies that no pooling can occur on this interval.

5.3

Continuity and Differentiability of Symmetric Equilibria

The purpose of this subsection is to show that in any symmetric PBE, each player’s equilibrium trigger function P is a continuous function of his cost on Θ0 . Specifically, 21

we shall prove that there are no gaps in the supports of equilibrium distributions of firms’ entry triggers, implying that the probability of investment by any player is strictly positive on any interval of triggers in P(Θ0 ). For any equilibrium trigger functions (P i )i∈I , and for each i and θ ∈ Θ0 , let P i (θ− ) and P i (θ+ ) be respectively the left- and right-hand limits of P i at θ. Those limits are well-defined since, by Proposition 2, equilibrium trigger functions are increasing. The following result will be used repeatedly in the sequel. Lemma 6 For each i ∈ I, if θ ∈ Θ0 is a discontinuity point of player i’s equilibrium trigger function, then P B (θ) < P i (θ+ ). Proof. By Lemma 2, P B (θ) ≤ P i (θ+ ). Suppose the equality holds. Since P i is strictly increasing and discontinuous at θ and P B is continuous, there would exist ε > 0 such that P i (θ − ε) ≤ P i (θ− ) < P B (θ − ε), a contradiction since P B (θ − ε) ≤ P i (θ − ε). ¥ The following lemma shows that if equilibrium trigger functions presents discontinuities, the corresponding gaps in the firms’ distributions of entry triggers cannot overlap. The intuition is straightforward: if any two such intervals were to overlap, then there would exist at least one player i and one discontinuity point θ of P i , such that almost surely no investment by player −i takes place in an interval of triggers U with upper bound P i (θ+ ). By Lemma 6, there is no loss of generality in supposing that the lower bound of U is greater than P B (θ). But then, there is no point for player i with a type just above θ to invest just beyond P i (θ+ ) rather than at the lower bound of U . By doing so, he only marginally increases the probability of entering second, while foregoing current benefits since P B (θ) < P i (θ+ ). Lemma 7 For each i ∈ I, if θ ∈ Θ0 is a discontinuity point of player i’s equilibrium trigger function, then for any open interval U ⊂ (p0 , 1) with upper bound P i (θ+ ), µ({P −i (θ−i ) ∈ U }) > 0. Proof. Suppose the contrary. Let {θn } and {pn } be two decreasing sequences in Θ0 and (p0 , 1) converging respectively to θ and P i (θ+ ), and such that P i (θn ) = pn for each n ∈ N. For each n, let p˜n = inf U ∨ P B (θn ). By Lemma 6, P B (θ) < P i (θ+ ). Hence, we can assume that p˜n < P i (θ+ ) for all n ∈ N. Since P i (θn ) = pn , we have: Z dµ LB (˜ pn , P −i ) V F (P −i , θn ) −i −i µ({P (θ ) ≥ P i (θ+ )}) {pn >P −i (θ−i )≥P i (θ+ )}

22

+

µ({P −i (θ−i ) ≥ pn }) pn , pn ) (∆v pn + l − θn ) LB (˜ µ({P −i (θ−i ) ≥ P i (θ+ )}) ≥ ∆v p˜n + l − θn

∀n ∈ N.

pn , q) V F (q, θn ) is strictly decreasing on (p0 , 1), it follows that: Since q 7→ LB (˜ µ({pn > P −i (θ−i ) ≥ P i (θ+ )}) B pn , P i (θ+ )) V F (P i (θ+ ), θn ) L (˜ −i −i i + µ({P (θ ) ≥ P (θ )}) +

µ({P −i (θ−i ) ≥ pn }) pn , pn ) (∆v pn + l − θn ) LB (˜ µ({P −i (θ−i ) ≥ P i (θ+ )}) ≥ ∆v p˜n + l − θn

(19)

∀n ∈ N.

By Proposition 2, the equilibrium distribution of player −i’s entry trigger does not admit an atom at P i (θ+ ). Thus the first term on the left hand side of (19) tends to zero as n → ∞. Therefore, taking the limit: ¡ ¢ LB inf U ∨ P B (θ), P i (θ+ ) (∆v P i (θ+ ) + l − θ) (20) ≥ ∆v inf U ∨ P B (θ) + l − θ. However, inf U ∨ P B (θ) < P i (θ+ ). Since p 7→ (∆v p + l − θ)/uB (p) is strictly decreasing on [P B (θ), P i (θ+ )], this is a contradiction to (20). ¥ The following proposition is an immediate consequence of Lemma 7. Proposition 4 In a symmetric PBE, the common equilibrium trigger function is continuous on Θ0 . Note that this result holds independently of any assumption on the distribution of costs. Unlike the model of Fudenberg and Tirole (1986), however, it is not a priori clear that any PBE exhibits continuous trigger functions. In Section 6, we will prove that under additional conditions on the support of the cost distribution, the converse of Proposition 4 holds, so that any PBE with continuous trigger functions on Θ0 is symmetric. It follows from Proposition 4 that P(Θ0 ) is an (open) interval of (p0 , 1). The following lemma is proved in the Appendix: Lemma 8 In a symmetric PBE, the inverse φ of the common equilibrium trigger function P is differentiable on P(Θ0 ), and: ∂φ(p) = f (p, φ(p)) ∀p ∈ P(Θ0 ), 23

(21)

where, by definition: f (p, θ) =

1 − G(θ) ∂1− LB (p, p) (∆v p + l − θ) − ∆v × g(θ) V F (p, θ) − ∆v p − l + θ

(22)

on the open domain D = {(p, θ) ∈ (0, 1) × (l, h) ∩ (θ, θ) | P B (θ) < p < P F (θ)}. Intuitively, the first order condition (21) states that player i with investment cost φ(p) ∈ Θ0 cannot increase his payoff by investing at p + dp > p instead of p. Heuristically, this formula can be derived as follows. Note first that the probability that player i’s investment trigger lies in [p, p+dp) is ∂φ(p) g(φ(p)) dp. Hence, a marginal change dp > 0 yields an expected gain: ¡ F ¢ V (p, φ(p)) − ∆v p − l + φ(p) ∂φ(p) g(φ(p)) dp + o(dp). (23) This change however costs ∆v p + l − φ(p) − LB (p, p + dp) (∆v (p + dp) + l − φ(p)) if player i’s investment trigger lies in [p + dp, 1] ∪ {∞}, which has probability 1 − G(φ(p + dp)). The expected incremental cost is therefore equal to: ¡ ¢ −∆v − ∂2+ LB (p, p) (∆v p + l − φ(p)) (1 − G(φ(p))) dp + o(dp) (24) Equating (23) and (24), and taking advantage of ∂1− LB (p, p) = −∂2+ LB (p, p) yields (21). Note that the smooth pasting condition for ΠB (φB (p)) implies limθ↑φB (p) f (p, θ) = 0, and the value matching condition for ΠF (φF (p)) implies limθ↓φF (p) f (p, θ) = +∞ for each p in the relevant range. This will imply that the equilibrium trigger function P in a symmetric equilibrium strictly lies between P B and P F . Intuitively, since the identity of the player with the lowest investment cost is not common knowledge ex ante, each player wants to delay his decision in order to convince his rival that the latter has indeed the lowest cost. Hence, the uncertainty about private investment costs creates further inefficient delays on top of those generated by the presence of strategic interactions.

5.4

Existence and Uniqueness of a Symmetric Equilibrium

The main objective of this section is to prove the existence of an essentially unique symmetric PBE. The proof considers the family of solutions to the differential equation (21) and establishes the existence and uniqueness of a solution φ∗ satisfying appropriate equilibrium boundary conditions. It will follow directly from the construction of φ∗ that its inverse is a symmetric PBE trigger function on Θ0 . Second, we describe the players’ beliefs and strategies off the equilibrium path. The following lemma describes the boundary conditions that are satisfied by a symmetric equilibrium trigger function. 24

Lemma 9 In a symmetric PBE, the common equilibrium trigger function P satisfies: (i) If P F (θ) > p0 , then P(θ) = p0 ∨ P B (θ); (ii) If P F (θ) ≤ p0 , then P(φF (p0 )) = p0 . Proof. (i) By Proposition 2, a player with cost θ knows that he will almost surely invest first at equilibrium. Hence, he has no reason to delay investment, except if p0 < P B (θ), in which case he will wait until his belief that v = h reaches P B (θ) to invest. (ii) This is only a re-statement of Lemma 2(ii). ¥ To prove the existence of a solution to (21) satisfying these boundary conditions, we must be distinguish two cases. Either (i) holds, in which case θ is the lower bound of Θ0 so that no bunching occurs on the support [θ, θ] of µ. This case is relatively simpler, since P(θ) < P F (θ) implies that the differential equation (21) satisfies a Lipschitz condition at p0 ∨ P B (θ). Or (ii) holds, in which case there is a strictly positive probability that each player will invest at date 0. The Lipschitz condition no longer holds at p0 , and an ad hoc argument is needed. The following technical result is proved in the Appendix. Lemma 10 For any (π, ϑ) ∈ D, there exists a unique solution φπ,ϑ on D to the ODE (21) passing through (π, ϑ). Moreover, if Iπ,ϑ is the maximal open interval I ⊂ (0, 1) such that for each p ∈ I, (p, φπ,ϑ (p)) ∈ D, then supp∈Iπ,ϑ φπ,ϑ (p) = h ∧ θ. A more intuitive formulation of Lemma 10 is that a solution of the ODE entering the domain D cannot escape from D (except possibly at p = 1 if θ ≥ h). This result is illustrated on Figure 2, which represents the flow of the ODE (21). The central step of the argument is developed in the following lemma: Lemma 11 For each p0 ∈ (0, 1), the ODE (21) has a unique solution φ∗ in D starting at max(p0 , P B (θ)) and satisfying the equilibrium boundary conditions. The proof is technical, and is relegated to the Appendix. Let P ∗ : Θ0 → [0, 1] ∪ {∞} be the inverse function of φ∗ . It is straightforward to check that for each θ ∈ Θ0 , P ∗ (θ) satisfies the second order conditions of (10) with P −i = P ∗ . Indeed, let p ∈ P ∗ (Θ0 ). Using the notation of the proof of Lemma 11, let us rewrite (21) as: g(φ∗ (p)) ∂φ∗ (p) = ϕ(p, φ∗ (p)). ∗ 1 − G(φ (p))

25

For each q ∈ (0, 1), the mapping θ 7→ ϕ(q, θ) is strictly decreasing on (φF (p), φB (p)). Moreover, φ∗ is strictly increasing on P ∗ (Θ0 ). It follows that g(φ∗ (˜ p)) p) ≷ ϕ(˜ p, φ∗ (p)) as p ≷ p˜, ∂φ∗ (˜ 1 − G(φ∗ (˜ p))

(25)

for each p˜ ∈ [P B (φ∗ (p)) ∨ p0 , P F (φ∗ (p))). In other words, a player with cost φ∗ (p) optimally invests at p, since the expected incremental payoff of a marginal delay of investment is everywhere negative between p and P F (φ∗ (p)), and everywhere positive between P B (φ∗ (p)) ∨ p0 and p. It follows that the incentive compatibility conditions are globally satisfied, as required. To complete the existence proof, we proceed in two steps. First, we describe the equilibrium behaviour of a player with cost lying outside of Θ0 . Next, we construct each player’s strategy off the equilibrium path. The following proposition, proved in the Appendix, summarizes our findings: Proposition 5 For each p0 ∈ (0, 1), there exists an essentially unique symmetric PBE. By saying that the symmetric PBE is essentially unique, we mean here that two symmetric PBEs can only differ off the equilibrium path. More precisely, two cases must be distinguished. If θ ≥ h, then for any large enough t, there is a positive probability that the first investment will take place later than t, conditional on no failure occurring up to time t: in the terminology of Fudenberg and Tirole (1986), this is a situation of “perpetual selection”. In that case, beliefs can be computed by Bayes’ rule at any date. If on the contrary θ < h, there is “finite selection”: there exists a time T such that the probability that the first investment will occur later than T is zero, if no failure occurred up to time T . The out of equilibrium beliefs that we impose in that case have a unit mass on [h, ∞), so that each player believes that his opponent will never invest.

6

Comments and Extensions

We first compare our results to the complete information benchmark and the existing literature on attrition games (with and without incomplete information). Next, we provide several comparative statics exercises. We then provide a sufficient condition that ensures the uniqueness of the equilibrium. Last, we study how the introduction of a first-mover advantage affects our results.

26

6.1

Interpretation of the Results

As already mentioned in the discussion of Proposition 2, it would be misleading to describe the symmetric PBE as a Markov equilibrium in which the players’ strategies are only contingent on their current beliefs about v. Indeed, the analysis in the previous section implies that in equilibrium, players’ investment triggers depend on their initial belief p0 about the value of the project, and more precisely on the position of p0 with respect to P F (θ). E.g., if P F (θ) < p0 , the equilibrium distribution of entry triggers has an atom at p0 , since there is a positive measure of each player’s types for which the additional information provided by becoming a follower has no value at this initial level of belief. In that sense, history matters. Specifically, let P ∗ (θ, p0 ) be the symmetric PBE entry trigger of a player with cost θ given initial belief p0 . Then: Corollary 1 Assume that θ > l. Then for each θ ∈ [θ, θ], (i) P ∗ (θ, ·) is continuous on [0, 1]; (ii) P ∗ (θ, ·) is constant on [0, P B (θ)] and strictly increasing on [P B (θ), P F (θ)]. For each p0 ≥ P F (θ), P ∗ (θ, p0 ) = p0 . This result, an immediate consequence of the differential characterization of the equilibrium provided in Subsection 5.4, is illustrated on Figures 3-5.9 The intuition is straightforward. If p0 > P B (θ), a small increase dp0 increases the symmetric PBE entry trigger of type θ by dp0 . Besides, since in this case the symmetric PBE trigger function is increasing on [θ, h ∧ θ], the probability that a player with cost θ in this interval invests second is G(θ). Hence, in order to maintain a constant probability of investing second, the equilibrium entry trigger of each type θ > θ must increase.10 It follows from Corollary 1 that, conditional on his cost, a player tends to invest at higher level of beliefs as the initial belief p0 about the quality of the project raises. Let us now study how the marginal incentives to invest at p vary as a function of p0 . Let 9

In the same vein, it is straightforward to prove that the set of equilibrium entry functions on [θ, θ] obtained as p0 varies is equicontinuous. 10 Another way to state this point is that when players have incomplete information about each other’s costs, an increase in p0 induces the same effect on the players’ behaviour as a decrease in the lower bound θ of the support of the distribution of types, holding p0 constant. The only difference is that in this latter case, each player has an incentive to increase his investment trigger, not because the value itself has increased, but in order to increase the probability that his rival enters before him. Note that if p0 < P B (θ), a small increase in p0 has no impact on type θ, who will continue investing at P B (θ), and a fortiori on types θ > θ.

27

φ∗ (·, p0 ) be the inverse of P ∗ (·, p0 ). By independence of θi and θ−i , one has: µ (θi ∈ [φ∗ (p, p0 ), φ∗ (p + dp, p0 )] | θi, θ−i ≥ φ∗ (p, p0 )) =

g(φ∗ (p, p0 )) ∂φ∗ (p, p0 ) dp + o(dp). 1 − G(φ∗ (p, p0 ))

Hence the hazard rate of investment of each player when the current belief is p > p0 , conditional on no player having invested between p0 and p, is given by: g(φ∗ (p, p0 )) (26) ∂φ∗ (p, p0 ). 1 − G(φ∗ (p, p0 )) ¡ ¢ For each p0 ∈ (0, 1), let Θ0 (p0 ) = φF (p0 ) ∨ θ, h ∧ θ . We then have the following result: h(p, p0 ) =

Corollary 2 Let p˜0 > p0 in (0, 1). Then: p0 ), p˜0 ). h(p, p˜0 ) > h(p, p0 ) ∀p ∈ P ∗ (Θ0 (p0 ), p0 ) ∩ P ∗ (Θ0 (˜ Proof. From (26), one has, using the notation of Lemma 11, h(p, p0 ) = ϕ(p, φ∗ (p, p0 )) and h(p, p˜0 ) = ϕ(p, φ∗ (p, p˜0 )) for each p in the relevant range. From Corollary 1, it follows that φ∗ (p, p0 ) > φ∗ (p, p˜0 ) for each such p, hence the result since θ 7→ ϕ(p, θ) is decreasing with respect to θ for each (p, θ) ∈ D. ¥ The economic interpretation is straightforward. On average, an increase in the initial belief p0 about the quality of the project increases the likelyhood that each player will invest in any interval [p, p+dp] given that no player has invested before p. In other terms each player, unconditionally on his type, is less cautious in his investment decision. This illustrates the fact that a slight perturbation of the initial conditions of the problem may have long-term consequence: ceteris paribus, an increase in the players’ initial confidence level p0 in the good quality of the project leads to an acceleration of the timing of investment at any level of beliefs p > p0 . This result contrasts sharply with those obtained in the complete information version of our model. When players have complete information about each other’s investment costs, we have shown in Proposition 1 that for generic choice of (θi )i∈I in (l, h), there exists an unique subgame perfect equilibrium, which is in pure strategies. In this equilibrium, the player with the lower cost, say player 1, invests at P B (θ1 ), while player 2 credibly threaten to invest at P F (θ2 ) if player 1 has not invested at this date. In particular, this equilibrium is truly Markovian with respect to current beliefs about v, since players’ trigger strategies do not depend on their initial assessment p0 of the quality of the project. 28

A related point is that, as in standard models of war of attrition with complete information (e.g. Ghemawat and Nalebuff (1985), Hendricks, Weiss and Wilson (1988)), players do not actually “fight” in this pure strategy equilibrium. Indeed, the only reason why concession by the low-cost player might not occur at date 0 is that p0 might be below P B (θ1 ). The only source of delay in the complete information model is the real option effect induced by the availability of the background signal. More generally, this reflects the fact that, when players have complete information about each other’s characteristics, public beliefs only represents the information accumulated about the desirability of the investment. In contrast, in the presence of incomplete information, they indirectly perform a dual task, that of summarizing each players’ belief about his opponent’s cost. Remark. While the symmetric PBE of the incomplete information game is not Markovian with respect to the current beliefs about v, it nonetheless shares an important property with the equilibrium of the complete information game: the equilibrium strategy of a player with cost θ in the support [θ, θ] of the cost distribution is a trigger strategy. Formally, this means that his equilibrium entry trigger is sequentially rational off as well as on the equilibrium path, so that the symmetric PBE strategy is given by: σt∗ (ω, θ) = I{pBt (ω)≥P ∗ (θ)}

∀(t, ω, θ) ∈ [0, ∞) × Ω × [θ, θ].

Indeed, suppose without loss of generality that θ > l and that p0 ≤ P B (θ) so that almost surely no player invests at date 0. It is then immediate from the second order condition (25) that if a player with cost θ ∈ Θ0 failed to invest first at P ∗ (θ), and the current belief that v = h is greater than P ∗ (θ), he should invest immediately. Note that this result does not follow from Proposition 2, which only asserts that trigger strategies are optimal on the equilibrium path. As in Fudenberg and Tirole (1986), it is more specifically linked to the non atomicity of the equilibrium distribution of entry triggers on (p0 , 1). A contrario, if there were an atom at some p˜ > p0 , it might be optimal for a player with cost θ ∈ φ∗ (p) < inf φ∗ (˜ p) to wait until the belief p˜ is reached before investing, in case he failed to invest at p < p˜ as he was supposed to.

6.2

Comparative Statics

The characterization of the unique symmetric PBE by the ODE (21) greatly simplifies the comparative statics analysis. We have already examined how the symmetric PBE is affected by a change in p0 . We now consider three types of changes: in the distribution of investment costs G, in the intensities of the background and follower signals λB and λF , and in the payoffs l and h. 29

6.2.1

Changes in the Cost Distribution

Let µ and µ ˜ be two distributions satisfying A1-A3. Assume for simplicity that the supports of µ and µ ˜ have the same lower bound θ, and that p0 < P B (θ). Denote by ˜ = g˜/(1 − G) ˜ the corresponding hazard rates. We want to compare H = g/(1 − G) and H the symmetric PBEs P ∗ and P˜ ∗ of the investment games corresponding respectively to µ ˜ > H on supp µ∩supp µ and µ ˜, under the assumption that H ˜ ∩(θ, h).11 More specifically, we obtain the following corollary: ˜ > H on supp µ ∩ supp µ Corollary 3 If H ˜ ∩ (θ, h), then P˜ ∗ > P ∗ on this interval. Proof. The inverse functions φ∗ and φ˜∗ of P ∗ and P˜ ∗ both satisfy (21), and since θ > φF (p0 ), they both satisfy the same equilibrium boundary conditions at P B (θ). It follows that in a right neighborhood of P B (θ), we have ∂φ∗ > ∂ φ˜∗ , and hence φ∗ (p) > φ˜∗ (p) for each p close enough to P B (θ). If φ∗ and φ˜∗ re-cross at some p > P B (θ), then, at the first such p, ∂φ∗ (p) ≤ ∂ φ˜∗ (p) and φ∗ (p) = φ˜∗ (p). But then, from (21), it follows that ∂φ∗ (p) > ∂ φ˜∗ (p), a contradiction. Thus φ∗ (p) > φ˜∗ (p) for each p ∈ ˜) \ {1, ∞}, which implies the result. ¥ P ∗ (supp µ) ∩ P˜ ∗ (supp µ To interpret this result, one should note that if the hazard rate of µ ˜ majorates the hazard rate of µ, then, for each p in the relevant range, the likelyhood that entry by player i will occur in the interval of beliefs [p, p + dp] is higher, ceteris paribus, for the distribution µ ˜ than for the distribution µ. This gives his opponent −i an additional incentive to delay entry at p when the cost distribution is µ ˜ rather than µ. Conversely, if there is a shift from µ to µ ˜ but player −i does not adjust his investment strategy P ∗ , then player −i will enter at a faster rate, which in turn incites player i to enter at higher values of {pB t }. In particular, a shift in the cost distribution that decreases the likelyhood of high costs may actually incite a player to increase his investment trigger. However, one should bear in mind that this kind of “mixed strategies” effect holds conditionally on a particular value of a player’s cost, but not necessarily when one aggregates across ˜) = P ∗ (supp µ) ∩ P˜ ∗ (supp µ ˜) \ {1, ∞}. Then, under the types. Indeed, let Π∗ (µ, µ 11

This type of comparison is standard in the first price auction literature (see e.g. Maskin and Riley ˜ > H everywhere on their common support and both g and g˜ are strictly (1999b)). Note that if H positive on their respective supports, then necessarily supp µ ˜ ⊂ supp µ, so that µ inevitably puts more weight on high costs than µ ˜. As an illustration, consider a shift from a distribution G(θ) on [θ, θ] to the ˜ distribution G(θ) = αG(θ) for α > 1 and θ ∈ [θ, G−1 (1/α)]. If g˜ is not required to be strictly positive ˜ on supp µ ˜, then µ and µ ˜ may have the same support. For instance, if G(θ) = 1 − (1 − G(θ))α for α > 1, α−1 g(θ) = 0. then g˜(θ) = α(1 − G(θ))

30

the assumptions of Corollary 3, one can compare the equilibrium distributions of entry triggers under µ and µ ˜. ˜ > H on supp µ ∩ supp µ ˜ ◦ φ˜∗ first order stochastically Corollary 4 If H ˜ ∩ (θ, h), then G dominates G ◦ φ∗ on Π∗ (µ, µ ˜). Proof. Since by assumption p0 < P B (θ), it follows that φ(P B (θ)) = φ˜∗ (P B (θ)) = θ. Moreover, both φ∗ and φ˜∗ are integrable from P B (θ) on. For each p ∈ Π∗ (µ, µ ˜), integrating (21) for φ∗ and φ˜∗ on (P B (θ), p) and substracting yields: ¶ Z p ³ µ ´ 1 − G(φ∗ (p)) ϕ(q, φ˜∗ (q)) − ϕ(q, φ∗ (q)) dq. = ln ˜ φ˜∗ (p)) 1 − G( P B (θ) By Corollary 3, φ˜∗ (q) < φ∗ (q) for each q ∈ (P B (θ), p). Hence, since the mapping θ 7→ ϕ(q, θ) is decreasing on (φF (q), φB (q)) for each q ∈ Π∗ (µ, µ ˜), the ratio (1−G(φ∗ (p)))/(1− ˜ φ˜∗ (p))) is greater than one, which implies the result. G( ¥ In other terms, even if each player has, conditionally on each particular value of his cost, greater incentive to wait before investing when the cost distribution puts more weight on low costs, the resulting distribution of entry triggers will be biased towards lower triggers as a result of such a shift in the cost distribution. 6.2.2

Changes in the Quality of Public Signals

We now examine the sensitivity of our equilibrium to changes in the intensity of public signals. As previously, we assume for simplicity that p0 < P B (θ). Changes in λF . We first compare the symmetric PBEs P ∗ and P˜ ∗ of the investment ˜ F of the background process, game corresponding respectively to intensities λF and λ ˜ F > λF . We have: under the assumption that λ ˜ F > λF , then P˜ ∗ > P ∗ on supp µ ∩ (θ, h). Corollary 5 If λ ˜ F ), Proof. A straightforward computation shows that for each (p, θ) ∈ D(λF ) ⊂ D(λ ˜ F ) > V F (p, θ, λF ), and hence f (p, θ, λF ) > f (p, θ, λ ˜ F ). Since the inverse funcV F (p, θ, λ tions φ∗ and φ˜∗ of P ∗ and P˜ ∗ coincide at P B (θ), the remaining of the proof follows the same lines as the proof of Corollary 3. ¥ Changes in λB . We next compare the symmetric PBEs P ∗ and P˜ ∗ of the investment ˜ B of the background process, game corresponding respectively to intensities λB and λ ˜ B > λB . We have: under the assumption that λ 31

˜ B > λB , then P˜ ∗ > P ∗ on supp µ ∩ (θ, h). Corollary 6 If λ Proof. The equilibrium boundary conditions of the inverse functions φ∗ and φ˜∗ of ˜ B )) = θ. Suppose P ∗ and P˜ ∗ are respectively φ∗ (P B (θ, λB )) = θ and φ˜∗ (P B (θ, λ ˜ B ) ∈ P ∗ (supp µ) (otherwise, the result is trivial). Since φ∗ is strictly inthat P B (θ, λ ˜ B )], necessarily φ∗ (P B (θ, λ ˜ B )) > φ˜∗ (P B (θ, λ ˜ B )). Morecreasing on [P B (θ, λB ), P B (θ, λ over, a straightforward computation shows that for each p ∈ (0, 1), ∂1− LB (p, p, λB ) > ˜ B ), hence f (p, θ, λB ) > f (p, θ, λ ˜ B ) for each (p, θ) ∈ D(λ ˜ B ) ⊂ D(λB ). The ∂1− LB (p, p, λ remaining of the proof follows then the same lines as the proof of Corollary 3. ¥ Compared to the previous case of an increase in the quality of the follower’s signal, the impact of an increase in the quality of the background signal is two-fold. First, by reducing the cost of waiting to become a follower, it increases the marginal incentives of each player with cost θ > θ to delay investment: this is a strategic effect. Second, it affects directly the investment decision of a cost θ player, who in any case is doomed to invest first in a symmetric equilibrium: an increase in the background signal’s intensity ˜ B leads him to change his trigger from P B (θ, λB ) to P B (θ, λ ˜ B ) > P B (θ, λB ). from λB to λ Technically, this non-strategic effect modifies the initial condition for the ODE describing the equilibrium trigger function. 6.2.3

Changes in Payoffs

Last, we analyze the sensitivity of our equilibrium to changes in the payoff structure. Changes in l. We first compare the symmetric PBEs P ∗ and P˜ ∗ of the investment game corresponding respectively to minimal values l and ˜l of the project, under the assumption that ˜l > l, holding ∆v constant. This may be interpreted as an investment subsidy of an amount ∆l = ˜l − l, received at the instant of the investment. We will assume that p0 < P B (θ, ˜l), i.e. that the increase in value is not so drastic as to imply immediate investment by a cost θ player. An immediate effect of such a policy is to incite players with costs θ ∈ [h, h + ∆l] to invest, provided they become sufficiently optimistic about the value of v. For the remaining types, we have: Corollary 7 If ˜l > l and p0 < P B (θ, ˜l), then P ∗ > P˜ ∗ on supp µ ∩ (θ, h). Proof. The equilibrium boundary conditions of the inverse functions φ∗ and φ˜∗ of P ∗ and P˜ ∗ are respectively φ∗ (P B (θ, l)) = θ and φ˜∗ (P B (θ, ˜l) = θ. Suppose that P B (θ, l) ∈ P˜ ∗ (supp µ) (otherwise, the result is trivial). Since φ˜∗ is strictly increasing on [P B (θ, ˜l), P B (θ, l)], necessarily φ˜∗ (P B (θ, l)) > φ∗ (P B (θ, l)). The numerator of f 32

is clearly increasing with respect to l. Moreover, by the envelope theorem ∂l V F (p, θ, l) = LF (p, P F (θ)) < 1, so that the denominator of f is decreasing with respect to l. It follows ˜ B ) ∩ D(λB ). The remaining of the proof that f (p, θ, ˜l) > f (p, θ, l) for each (p, θ) ∈ D(λ follows then the same lines as the proof of Corollary 3. ¥ As an increase in the quality of the background signal, a subsidy at the date of the investment has a double impact. First, by reducing the relative cost of becoming a leader, it decreases the marginal incentives of each player with cost θ > θ to delay investment. Second, it has a direct impact on a cost θ player, who will change his trigger to P B (θ, ˜l) < P B (θ, l). Changes in h. Next, we compare the symmetric PBEs P ∗ and P˜ ∗ of the investment ˜ of the project, under the game corresponding respectively to maximal values h and h ˜ > h, holding l constant. Again, we assume p0 < P B (θ, h). ˜ We have: assumption that h ˜ > h and p0 < P B (θ, h), ˜ then P ∗ > P˜ ∗ on supp µ ∩ (θ, h). Corollary 8 If h Proof. A direct calculus shows that the map ∆v 7→ ∂1− LB (p, p) (∆v p + l − θ) − ∆v defined on {∆v > 0 | p > P B (θ, ∆v)} is increasing if and only if p > 1 − ρ/λB . Similarly, the map ∆v 7→ V F (p, θ)−∆v p−l +θ defined on {∆v > 0 | p < P F (θ, ∆v)} is decreasing if and only if p > 1 − ρ/λF . Keeping in mind that λF ∈ (λB , ρ), it follows immediately ˜ > f (p, θ, h) for each (p, θ) ∈ D(h) ˜ ∩ D(h). The remaining of the proof that f (p, θ, h) follows then the same lines as the proof of Corollary 7. ¥ Again, we obtain the double impact enlightened in Corollary 7. A relative change in the payoff differential ∆v, decreases the marginal incentives of each player with cost θ > θ to delay investment. Moreover, the direct impact implies that a player with cost ˜ < P B (θ, h). θ will change his trigger to P B (θ, h)

6.3

Uniqueness of Equilibrium

So far, we have essentially focused on the symmetric PBE of the investment game. In this section, we provide a sufficient condition for the uniqueness of the equilibrium. This condition states that with positive (but arbitrarily small) probability, each player’s cost can be sufficiently large to discourage him to invest whatever his beliefs about the profitability of the investment. Formally: A4. θ ∈ (h, +∞). 33

Assumption A4 plays the same role in our model as Assumption A(ii) in Fudenberg and Tirole (1986). In a duopoly model of exit, they argue that there is no a priori reason to impose a natural monopoly assumption as in the standard war of attrition. In the present context of entry decision, the analogue of this point is that one cannot exclude that players with prohibitively high costs would abstain to invest even if they knew that the value of the project is high. We already know from Proposition 5 that there exists a unique symmetric PBE. The first step of the uniqueness proof consists to show that this equilibrium is the unique continuous equilibrium. Lemma 12 Under Assumption A4, the symmetric equilibrium is the unique PBE such that both players’ equilibrium trigger functions are continuous on Θ0 . Proof. Suppose that there exists a continuous asymmetric PBE. It follows from Proposition 3 that the equilibrium trigger functions (P i )i∈I have continuous and strictly increasing inverse (φi )i∈I defined respectively on (P i (Θ0 ))i∈I . Following the same lines as in Lemma 8, it is easy to check that the mappings (φi )i∈I are differentiable on the common part of their domain of definition P(Θ0 ) = P i (Θ0 ) ∩ P −i (Θ0 ), and that they satisfy the following system of ODEs: ∂φi (p) = f (p, φ−i (p)) ∀(i, p) ∈ I × P(Θ0 ).

(27)

It follows from A4 that sup P(Θ0 ) = 1. Suppose for instance that φi (p) > φ−i (p) for some p ∈ P(Θ0 ). Since φ−i is strictly increasing and continuous on (p, 1), there exists exactly one p ∈ (p, 1) such that φi (p) = φ−i (p). For each (q, θ) ∈ D, let ϕ(q, θ) = f (q, θ) g(θ)/(1 − G(θ)). Integrating (27) respectively for φi and φ−i on (p, p) and (p, p) and substracting yields: Z p³ Z p ´ ¡ ¢ −i i −i i ln R(p, φ , φ ) = ϕ(q, φ (q)) − ϕ(q, φ (q)) dq + ϕ(q, φ−i (q)) dq (28) p

p

for each p ∈ (p, 1), where the ratio R(p, φ−i , φi ) is defined as in the proof of Lemma 11. By Assumption A4, φi (1) = φ−i (1) = h < θ, so that the ratio R(p, φ−i , φi ) should converge to one as p ↑ 1. Note that φi (p) > φ−i (p) implies that φi (q) > φ−i (q) for each q ∈ (p, 1). It is easy to check that the mapping θ 7→ ϕ(q, θ) is decreasing on (φF (q), φB (q)) for each q ∈ (0, 1). Hence, for each p ∈ (p, 1), R(p, φ−i , φi ) > 1, and by (28) it is increasing. R(p, φ−i , φi ) cannot therefore converge to one as p ↑ 1, a contradiction. ¥ 34

Note that if θ ≤ h, one may have P i (θ) 6= P −i (θ), so that the end of the proof would not carry on as above. The role of Assumption A4 is therefore to provide a boundary condition at p = 1 which pins down a unique solution to the system (27). As already mentioned, the continuity of equilibrium trigger functions is not automatically granted in any PBE. However, it turns out that A4 is also sufficient to eliminate all asymmetric equilibria. To prove this result, the following lemma will be helpful: Lemma 13 For each i ∈ I, if θ ∈ Θ0 is a discontinuity point of player −i’s investment trigger, then for each θi ∈ φB ([P −i (θ− ), P −i (θ+ )]), P i (θi ) = P B (θi ). Proof. Let p− = P −i (θ− ), p+ = P −i (θ+ ). To prove the result, note first that, by Lemma 2, player i with cost θi such that P F (θi ) < p− or P B (θi ) > p+ will not invest on [p− , p+ ]. Similarly, if P F (θi ) ∈ [p− , p+ ], then θi ≤ θ, hence P B (θi ) ≤ p− so player i with such a cost θi does not gain from investing on (p− , p+ ] rather than at p− . The only remaining case is when player i has a cost θi ∈ φB ([p− , p+ ]). By Lemma 7, a positive measure of such types of player i must invest on any interval of triggers with upper bound p+ , hence one must have φi (P −i (θ+ )) = φB (P −i (θ+ )) which implies the result since none of these types have an incentive to delay investment on [p− , p+ ], as player −i invests with probability zero on this interval. ¥ The intuition for this result is clear: if the equilibrium distribution of player −i’s investment triggers has a gap, then player i’s types θi who invest within this gap must do so at P B (θi ). Otherwise, there may be overlapping gaps in the distribution of both player’s equilibrium triggers, which never occurs in equilibrium. We can now state the main result of this section: Proposition 6 Under Assumption A4, the symmetric equilibrium is the unique PBE. The proof is given in the Appendix. The idea of the proof consists to show that if player −i’s equilibrium trigger function P −i has a discontinuity at θ, then P −i > P i on [θ, h], and to apply the same boundary condition as in Lemma 12 to get a contradiction.

6.4

Preemption vs War of Attrition

In the previous sections, we have assumed that the value of the project is common to both players and independent of the timing of investment. We now investigate how our results are affected when a preemption motive is introduced besides the pure informational externality. 35

6.4.1

The Model

The assumptions about public and private information remain the same as in the basic model; in particular, the follower still benefits from a more informative signal than the leader. As for payoffs, we first assume without loss of generality that l = 0. Second, to introduce a potential first-mover advantage, we suppose that player i’s payoff at date 0 from investing as a leader at date t is given by u` (v, t, θi ) = e−ρt (`v − θi ) for some ` > 1, while his payoff from investing as a follower remains defined as u(v, t, θi ) = e−ρt (v − θi ). Since ` > 1, the investment game now presents a combination of first- and second-mover advantages. 6.4.2

Benchmarks

Note that in this new framework, the optimal investment trigger P F (θ), hence the value function V F (·, θ) of a follower with investment cost θ are unchanged compared to the pure war of attrition model. By contrast, the investment problem of a single player with investment cost θ is now affected by the value of `. The corresponding optimal trigger P`B (θ) is now given by:  0 if θ ≤ 0       q     −(ρ − λB ) + (ρ − λB )2 + 4ρλ`hB θ if θ ∈ (0, `h) . P`B (θ) = 2λB           ∞ if θ ≥ `h In addition to this trigger, it will be helpful to define the value of the posterior belief about the project’s quality at which a player with cost θ is exactly indifferent between preempting his rival or being preempted by him, i.e. the solution to: V F (p, θ) − `hp + θ = 0.

(29)

Note that, if he is preempted, a player with cost θ ≥ h effectively loses any opportunity to invest, so V F (·, θ) ≡ 0. It follows that for θ ≥ h, the unique solution to (29) is given θ by P`P (θ) = `h , the Marshallian trigger in the single player investment problem. Thus the threat of preemption is extreme for firms with high costs. The solution to (29) for θ < h is characterized in the following lemma. 36

Lemma 14 For any θ ∈ (0, h), there exists a unique solution P`P (θ) to (29) in the θ interval ( `h , P F (θ)). Moreover: (i) There exists ` > 1 such that if ` > `, ∃ε > 0 s.t. P`B (θ) > P`P (θ) ∀θ ∈ (0, ε).

(30)

(ii) The mapping θ 7→ P`P (θ) is increasing and continuously differentiable on (0, `h). Proof. For each θ ∈ (0, h), the mapping p 7→ Ξ` (p, θ) ≡ V F (p, θ) − `hp + θ is θ θ θ , P F (θ)]. Moreover, Ξ` ( `h , θ) = V F ( `h , θ) > 0 and continuously differentiable on [ `h F F F Ξ` (P (θ), θ) = (1 − `)hP (θ) < 0. Last, from the convexity of V (·, θ) and the smooth θ pasting condition at P F (θ), it follows that Ξ` (·, θ) is decreasing on [ `h , P F (θ)]. The existence and uniqueness of P P (θ) follow immediately. To prove (i), observe first that λF ∈ (λB , ρ) implies that: µ `≡

λF λB

¶λF /ρ µ

ρ − λF ρ − λB

¶1−λF /ρ > 1.

Next, a Taylor expansion on a right neighborhood of θ = 0 yields: Ξ` (P`B (θ), θ)

´ λB ³ ρ/λF (`/`) = − 1 θ + o(θ), ρ − λB

which implies the result. Last, (ii) follows from a direct application of the implicit functions theorem. ¥ Note from (i) that, depending on the size of the preemption effect, P`P can be either larger or smaller than P`B in a right neighborhood of θ = 0. Obviously, the larger `, the larger the threat of preemption. 6.4.3

Equilibrium Triggers

The definitions of strategies and equilibria are similar to those provided in Section 4. It is also immediate to check that Proposition 2 still holds, and that any perfect Bayesian equilibrium is characterized by a pair of equilibrium trigger functions. The following is an analogue of Lemma 2: Lemma 15 Let (P i )i∈I be a pair of equilibrium triggers functions. Then, for each (i, θi ) ∈ I × Θ : 37

(i) P i (θi ) ≥ p0 ∨ (P`B (θi ) ∧ P`P (θi )); (ii) p0 ∨ (P`B (θi ) ∨ P`P (θi )) ≥ P i (θi ). Lemma 15 relies on revealed preference arguments. Assume for instance P`B (θi ) > P`P (θi ). Then there is no reason for player i to invest before P`P (θi ) since, below this threshold, he does better as a follower than as a leader and has not reached his optimal investment trigger as a single player. Hence (i). Similarly, unless p0 > P`B (θi ), there is no point for player i in investing after P`B (θi ) since, beyond this threshold, he does better as a leader than as a follower and he is above his optimal investment trigger as a single player. Hence (ii). Similar arguments hold for the case P`B (θi ) ≤ P`P (θi ). 6.4.4

Equilibrium Analysis

We now turn to an heuristic description of a symmetric PBE. Let us suppose for simplicity that p0 < P`B (θ) ∨ P`P (θ), so that a player with cost θ will not invest at time zero. Assume also that θ = h. It follows immediately from Lemma 13 that for each θ ∈ [θ, θ] such that P`B (θ) = P`P (θ), P i (θ) = P`B (θ) = P`P (θ) for any player i in equilibrium. If P`B (θ) 6= P`P (θ), then by continuity of P`B and P`P , there exists a maximal open interval ˜ − P P (θ)) ˜ > 0 for any θ˜ ∈ Iθ . Iθ ⊂ [θ, θ] containing θ such that (P`B (θ) − P`P (θ))(P`B (θ) ` Intuitively, if θ˜ ∈ Iθ , then two players with types θ˜ or θ essentially face the same type of incentives. Two cases must therefore be distinguished. Case 1: P`B (θ) < P`P (θ). In that case, the attrition motive dominates in Iθ ; the analysis parallels that of the pure attrition case, except that P`P (θ) now plays the role of the trigger P F (θ). Following similar lines as in Section 5, one can show that, in a symmetric PBE, the inverse φ of the common equilibrium trigger function P is differentiable on P(Iθ ), and satisfies the differential equation:   ∂φ(p) = f` (p, φ(p)) ∀p ∈ P(Iθ ) (31)  φ(P`B (inf Iθ ) = inf Iθ , where, by definition: 1 − G(θ) ∂1− LB (p, p) (`hp − θ) − `h f` (p, θ) = × g(θ) V F (p, θ) − `hp + θ

(32)

on the domain D` = {(p, θ) ∈ (0, 1) × (θ, θ) | P`P (θ) ∧ P`B (θ) < p < P`P (θ) ∨ P`B (θ)}. The differential equation (31) simply expresses the first-order condition for a layer with 38

cost φ(p). Note in particular that the boundary condition in (31) is an initial condition as in the previous analysis of the pure attrition model. Case 2: P`B (θ) > P`P (θ). In that case, the preemption threat dominates in Iθ . One can again show that, in a symmetric PBE, the inverse φ of the common equilibrium trigger function P is differentiable on P(Iθ ), and satisfies the differential equation:   ∂φ(p) = f` (p, φ(p)) ∀p ∈ P(Iθ ) (33)  P φ(P` (sup Iθ )) = sup Iθ . Contrary to (31), the boundary condition in (33) is a terminal condition. This reflects the P threat of preemption. If in particular θ = h, then when {pB t } tends to P` (θ) = 1/` and no one has invested yet, the hazard rate of being preempted per unit of time explodes to infinity. The terminal condition expresses then the fact that at the upper end of the cost distribution, the option value of waiting disappears, and players’ equilibrium triggers become arbitrarily close to their Marshallian triggers. This is illustrated on Figure 6, which represents the flow of the ODE (31) on the domain D` . Remark. Lambrecht and Perraudin (1997) study a preemption game with incomplete information in which there is only one investment project. Our model is more general than theirs in the sense that (i) a follower with cost θ < h does not lose any opportunity to invest, and (ii) he benefits from a better information than the leader. However, focusing on the pure preemption case allows them to deal with the case where the value {xt } of the project follows a geometric Brownian motion dxt = µxt dt + σxt dZt . The argument is intuitively as follows. Suppose that player −i follows a trigger strategy X −i : Θ → R++ . Then the expected payoff at date t for player i of investing at the first date τxi where the value of the project reaches xi > 0 is given by: ¡ ¢ Vti (xt , xˆt , xi , X −i , θi , µ) = Et e−r(τxi −t) (xi − θi ) µit ({X −i (θ−i ) > xi }) =

³ x ´β t xi

µ ({X −i (θ−i ) > xi }) (x − θ ) , µ ({X −i (θ−i ) > xˆt }) i

i

(34)

where β > 1 is the positive root of the quadratic equation β(β − 1)σ 2 /2 + βµ = r and xˆt = sups∈[0,t] xs is the historic maximum of the value at date t. It is immediate to see from (34) that if xi maximizes Vti (xt , xˆt , xi , X −i , θi , µ), then it also maximizes Vsi (xs , xˆs , xi , X −i , θi , µ) for any s 6= t. In particular, optimal trigger strategies are time consistent, even if the value of the project fluctuates. This is due to the fact that contrary to (9), the expected payoff of a player depends upon the strategy of his opponent (hence upon the path of the state variable {xt }) only through the probability of investing first. 39

7

Conclusion

In this paper, we have built a strategic real option model where the information available to players is partly exogenous (through the background signal), and partly endogenous (through the investment decision of the leader). We have focused on the problem of private provision of information as a public good, and on the impact of cost uncertainty on the incentives of players to provide this information by investing. When the identity of the player with the smallest opportunity cost of investing is unknown ex ante, we have shown that inefficient delays occur in equilibrium because each investor tries to convince his rival that the latter has the lowest investment cost and should therefore bear the whole burden of experimentation. In equilibrium, the first player to invest does so at a time where the marginal value of public information per se is negative. This implies that failing to take into account the potential, or indirect information generated by the players’ investment themselves leads to under-estimate the value of public information. Our analysis has shown that the current common belief about the prospects of the investment is not a sufficient statistics for the equilibrium investment decisions of the players. In particular, an increase in the players’ initial level of optimism has long-lasting consequences in the form of a higher investment rate at any levels of beliefs. For more general learning processes, this form of history-dependence would lead to more complex investment dynamics. This, and related questions, must await for future work.

40

Appendix Proof of Lemma 1. Let k ∈ {B, F }. It is obvious that if θ > h, investment is a strictly dominated strategy, and conversely that delaying is strictly dominated if θ < l. Henceforth, let θ ∈ [l, h] and V k (·, θ) : [0, 1] → R+ be the value function of Πk (θ), i.e. ¡ ¢ V k (p, θ) = sup E u(v, τ k , θ) | pk0 = p , p ∈ [0, 1]. (B.1) τk

In (B.1), the expectation is defined with respect to the law P (· | pk0 = p) for the belief process {pkt } starting at p, and the supremum is taken over stopping times of the filtration {Htk }. From the explicit expression of payoffs, it is immediate that for each p ∈ (0, 1) and h ∈ (0, 1 − p), V k (p + h, θ) ≤ V k (p, h) + h∆v. In particular, if p belongs to the optimal stopping region for (B.1), then V k (p, h) = p ∆v + l − θ, so that V k (p + h, θ) ≤ (p + h) ∆v + l − θ. Since the reverse inequality is always true, this implies that p + h also belongs to the optimal stopping region, which must be therefore of the form [˜ p, 1] for some p˜ ∈ [0, 1]. Letting τp˜k = inf{t ≥ 0 | pkt ≥ p˜}, (B.1) can be rewritten as: ³ ´ −ρτp˜k k k V (p, θ) = sup E e | p0 = p (˜ p ∆v + l − θ), p ∈ [0, 1]. p˜∈[0,1]

³ ´ k The Laplace transform E e−ρτp˜ | pk0 = p is easily computed by noting that: p = p˜ iff p + (1 − p) e−λk t˜

1 t˜ = − k ln λ

µ

p (1 − p˜) p˜ (1 − p)

¶ .

When p ≤ p˜, the resulting expression for the Laplace transform is simply Lk (p, p˜). The remaining of the proof follows then from standard computations. Note in particular that the solution satisfies the smooth-fit principle (Shiryayev (1978)), in the sense that (P k (θ), V k (·, θ)) solves the following free boundary problem:  k λ p (1 − p) ∂p V k (p, θ) = ρ V k (p, θ) ∀p ∈ [0, p˜]         limp↓0 V k (p, θ) = 0  (B.2)  k  V (˜ p, θ) = ∆v p˜ + l − θ        ∂p V k (˜ p, θ) = ∆v. The first equation in (B.2) is the standard Hamilton-Jacobi-Bellman equation, which describes the value of the project before investment takes place. The second equation is 41

the no-bubble condition as p ↓ 0. The third and fourth equations represent respectively the value matching and the smooth pasting conditions at the optimal exercise trigger. A general solution to (B.2) is of the form C k (θ)uk (p). The value matching and the smooth pasting conditions then determine the optimal trigger P k (θ) and the constant C k (θ). ¥ Proof of Proposition 1. The proof of (i) and (iii) is immediate. Suppose now that F 1 F 2 pB t (ω) ∈ [P (θ ), P (θ )) and that no player has invested yet. Then it is a dominant 2 B 2 strategy for player 1 to invest; since V F (pB t (ω), θ ) > ∆v pt (ω) + l − θ , the best reply of player 2 is to abstain. The most pessimistic assumption that player 2 can make about the behavior of player 1 is that he will not invest until P F (θ1 ) is reached (if ever). Now, define a sequence {pn } as follows. First, let p0 = P F (θ1 ). Next, for each n ∈ N: (i) If pn ≤ P B (θ2 ), pn+1 = pn ; (ii) If for each p ∈ [0, pn ], LB (p, pn ) V F (pn , θ2 ) > ∆v p + l − θ2 , pn+1 = 0; (iii) Otherwise, pn+1 = sup{p ∈ [0, pn ] | ∆v p + l − θ2 ≥ LB (p, pn ) V F (pn , θ2 )}. Obviously, pn+1 < pn as p0 < P F (θ2 ). Note that if there exists some n ∈ N such that pn ≤ P B (θ2 ), then player 2’s best reply at any (t, ω) such that pB t (ω) ∈ (0, p0 ) is to B 1 wait until player 1 invests, which therefore occurs at P (θ ). Otherwise, if no player has invested at the time where pn+1 is reached by {pB t } (if ever), player 1 can make no credible threat to wait until pn to invest. Hence, if pn+1 ≥ P B (θ1 ), player 1 should invest immediately, and player 2 therefore abstain. To complete the proof, it is thus sufficient to show that for some n ∈ N, pn ≤ P B (θ2 ). Suppose the contrary. Then, since the sequence {pn } is bounded below by P B (θ2 ) and decreasing, it must converge to a limit p∞ ≥ P B (θ2 ). By continuity, (iii) implies that ∆v p∞ + l − θ2 = V F (p∞ , θ2 ). This contradicts p∞ < P F (θ2 ). ¥ Proof of Lemma 8. It follows from Propositions 3 and 4 that P has a continuous and strictly increasing inverse function φ on P(Θ0 ). Let p ∈ P(Θ0 ), and consider a player with cost φ(p). Then we have: Z ∆v p + l − φ(p) ≥

+

p



LB (p, q) V F (q, φ(p))

dG(φ(q)) 1 − G(φ(p)) (B.3)

1 − G(φ(˜ p)) B p ∈ (p, P F (φ(p))). L (p, pe) (∆v p˜ + l − φ(p)) ∀˜ 1 − G(φ(p)) 42

Taking advantage from the fact that q 7→ LB (p, q) V F (q, φ(p)) is strictly decreasing on (p, P F (φ(p))) and rearranging (B.3), we obtain: (1 − G(φ(˜ p)))

∆v p + l − φ(p) − LB (p, p˜) (∆v p˜ + l − φ(p)) V F (p, φ(p)) − ∆v p − l + φ(p)

(B.4)

≥ G(φ(˜ p)) − G(φ(p)) ≥ 0 ∀˜ p ∈ (p, P F (φ(p))). Dividing both sides of (B.4) by p˜ − p and letting p˜ ↓ p, we conclude that: 0 ≤ ∂ + G(φ(p)) = lim sup pe↓p

G(φ(˜ p)) − G(φ(p)) ≤ g(φ(p)) f (p, φ(p)), p˜ − p

and a similar inequality holds for any q in a neighborhood of p. From the proof of Lemma 2, q < P F (φ(q)) for each such q. It follows that ∂ + G ◦ φ is bounded in a neighborhood of p, and since p is arbitrary, on any closed interval K ⊂ P(Θ0 ). In particular, G ◦ φ is Lipschitzian, hence absolutely continuous on any such K. Since the density g is bounded away from zero, it follows that φ is absolutely continuous, hence almost everywhere differentiable on P(Θ0 ). The ODE (21) follows then from differentiating V(p0 , p, P, θ, µ) with respect to p and requiring that the resulting derivative, when it exists, be equal to zero at P(θ). To conclude the proof, notice that p 7→ f (p, φ(p)) is continuous, and that φ is absolutely continuous. Since φ satisfies (21) almost everywhere, the fundamental theorem of calculus implies that it is in fact everywhere differentiable on P(Θ0 ), and its derivative is given everywhere by (21). ¥ Proof of Lemma 10. It is easy to check that since g is continuously differentiable and bounded away from zero on (θ, θ), f has continuous partial derivatives on D, and therefore f ∈ C 1 (D, R+ ). It follows from the Cauchy-Lipschitz local existence theorem that for any (π, ϑ) ∈ D, (21) has a unique maximal solution φπ,ϑ defined on an open interval Iπ,ϑ ⊂ (0, 1) containing π. Set Iπ,ϑ = (α, β). By the extension theorem for the solution of an ODE, for each compact K ⊂ D, there exists ε > 0 such that for each p ∈ (β − ε, β), (p, φπ,ϑ (p)) 6∈ K. Extending φπ,ϑ at β by continuity, three cases are possible: (i) β ∈ [P B (h ∧ θ), P F (h ∧ θ)] and φπ,ϑ (β) = h ∧ θ, which is our claim; (ii) β < P B (h ∧ θ) and φπ,ϑ (β) = φB (β); (iii) β > P F (h ∧ θ) and φπ,ϑ (β) = φF (β). Consider first case (ii). Since f ∈ C 1 (D, R+ ), it is clear that ∂φπ,ϑ (p) → 0+ as p → β − . Since ∂φB is bounded away from zero on [0, 1], it follows that there exists η > 0 such that ∂φB (p) > ∂φπ,ϑ (p) > 0 for each p ∈ (β − η, β]. Since φπ,ϑ (β) = φB (β), it follows by integration that one would have φB (p) < φπ,ϑ (p) for each p ∈ (β − η, β], a contradiction. 43

Consider now case (iii). In this case, one has ∂φπ,ϑ (p) → +∞ as p → β − . Since ∂φF is bounded on [0, 1], it follows that there exists η > 0 such that ∂φF (p) < ∂φπ,ϑ (p) for each p ∈ [β − η, β). But since φF (β − η) < φπ,ϑ (β − η), one should have φF (β) < φπ,ϑ (β), a contradiction to the assumption. ¥ Proof of Lemma 11. If P F (θ) > p0 , the result follows readily from Lemma 10, since the Cauchy-Lipschitz theorem can be applied in a neighborhood of p0 ∨ P B (θ). Suppose now that P F (θ) ≤ p0 . Let {ϑn } be a decreasing sequence in (φF (p0 ), φB (p0 )) converging to φF (p0 ). By Lemma 10, for each n ∈ N, there exists a unique solution φn = φp0 ,ϑn on D to (21) starting at (p0 , ϑn ), and defined on a maximal interval In = Ip0 ,ϑn . If n ≥ m, then sup In ≥ sup Im and φn (p) ≤ φm (p) for each p ∈ [p0 , sup Im ). Note that if θ < h, the functions {φn } are not defined on the same set. To remedy this, we take p∞ to be the limit of {sup In } as n → +∞, which is well-defined since the sequence is increasing and bounded above by P F (θ). Then, for each n ∈ N, we extend φn by continuity by setting φn (p) = θ for each p ∈ [sup In , p∞ ]. Since the sequence {φn } is bounded below by φF , it admits a pointwise limit φ on [p0 , p∞ ]. We now show that φ is the required solution. Since φ obviously satisfies the boundary condition at p0 , we must only show that φ is the unique solution to (21) starting at (p0 , φF (p0 )). First, we show that for each p ∈ (p0 , p∞ ), φ(p) > φF (p). Suppose the contrary holds for some p. Let δ > 0. Since f (q, θ) goes to +∞ as (q, θ) → (p, φF (p)) in D and ∂φF is bounded, there exists an open ball B ⊂ (0, 1) × (l, h) centered at (p, φF (p)) such that f (q, θ) ≥ ∂φF (q) + δ for each (q, θ) ∈ B ∩ D. If {φn (p)} tends to φF (p) as n → ∞, then for each ε > 0, there exists N ∈ N such that φN (p) − φF (p) < ε/2. Since φN and φF are continuous, and the {φn } form by construction a decreasing sequence of functions, there exists a neighborhood U ⊂ (p0 , p∞ ) of p such that φn (q) − φF (q) ≤ ε for each q ∈ U and n ≥ N . Hence, by choosing ε small enough, we can ensure that for each q ∈ U and n ≥ N , (q, φn (q)) ∈ B and thus ∂φn (q) ≥ ∂φF (q) + δ. Integrating from u0 = inf U , it follows that for each n > N , φn (p) − φn (u0 ) ≥ φF (p) − φF (u0 ) + δ(p − u0 ). As φn (u0 ) > φF (u0 ) for each n ≥ N , it follows that {φn (p)} cannot converge to φF (p), a contradiction. Next, we prove that φ satisfies (21) on (p0 , p∞ ). For each (n, p) ∈ N × (p0 , 1), φn (p) > φ(p) and (p, φ(p)) ∈ D. Since f ∈ C 1 (D, R+ ), for each p ∈ (p0 , p∞ ) there exists a neighborhood V ⊂ (p0 , p∞ ) of p and a real number M > 0 such that for each (n, q) ∈ N × V , ∂φn (q) = f (q, φn (q)) ≤ M and therefore |φn (q) − φn (r)| ≤ M |q − r| for each (n, q, r) ∈ N × V × V . This implies that the sequence {φn } is equicontinuous on V , 44

and thus converges uniformly on V to its pointwise limit φ by Ascoli’s theorem. Since ˜ > 0 such f is locally Lipschitzian with respect to θ on D, there exists a real number M ˜ |φn (p) − φ(p)|, so that the that for each (n, p) ∈ N × V , |f (p, φn (p)) − f (p, φ(p))| ≤ M sequence {f (·, φn (·))} converges uniformly to f (·, φ(·)) on V . Letting v0 = inf V , we can Rq therefore take the limits as n → ∞ on both sides of φn (q) = φn (v0 ) + v0 f (r, φn (r)) dr for each q ∈ V , which yields the desired result. It remains only to show that φ is the unique solution to (21) satisfying the equilibrium boundary conditions. Supposing the contrary, let φ and φ˜ be two solutions to (21) ˜ starting at (p0 , φF (p0 )), and such that φ(p) > φ(p) for some p ∈ (p0 , 1). Since φ˜ is ˜ = increasing and continuous on (p0 , p), there exists exactly one p ∈ (p0 , p) such that φ(p) φ(p). For each (q, θ) ∈ D, let ϕ(q, θ) = f (q, θ) g(θ)/(1 − G(θ)). Consider now the ratio ˜ = (1 − G(φ(p)))/(1 − G(φ(p))). ˜ R(p, φ, φ) Integrating (21) respectively for φ and φ˜ on (p, p) and (p, p) and substracting yields: Z p ³ ´ Z p³ ´ ˜ ˜ ln R(p, φ, φ) = ϕ(q, φ(q)) − ϕ(q, φ(q)) dq + ϕ(q, φ(q)) dq (B.5) p

p

˜ > φ(p) implies that φ(q) ˜ > φ(q) for each q ∈ (p0 , p). for each p ∈ (p0 , p). Note that φ(p) It is easy to check that the mapping θ 7→ ϕ(q, θ) is decreasing on (φF (q), φB (q)) for each ˜ > 1, and by (B.5) it is decreasing. q ∈ (0, 1). Hence, for each p ∈ (p0 , p], R(p, φ, φ) ˜ cannot therefore converge to one as p ↓ p0 , a contradiction. R(p, φ, φ) ¥ Proof of Proposition 5. We first describe the equilibrium entry triggers of a player with cost θ < inf Θ0 . If θ ≤ φF (p0 ) and θ ≤ φF (p0 ), then his equilibrium entry trigger is p0 . If θ > φF (p0 ) and θ ∈ (φF (p0 ), θ), then the equilibrium entry trigger of a player with cost θ maximizes V(p0 , p, P ∗ , θ, µ). Since in that case φ∗ ∈ C 1 (P(Θ0 ), [0, 1]), and notably ∂ + φ∗ is bounded at max(p0 , P B (θ)), this problem has a solution that we denote by P ∗ (θ) as well. Next, we construct the equilibrium entry trigger of a player with cost θ > sup Θ0 . Obviously, if θ > h, then the equilibrium entry trigger of a player with type θ is ∞. If θ < h and θ ∈ (θ, h), then a simple revealed preference argument implies that a player with type θ will not invest before P ∗ (θ) is reached by {pB t }. Let T be the nonrandom date at which this occurs, if ever. Let ν ∈ ∆(Θ) such that supp(ν) ⊂ [h, ∞). Let us define out of equilibrium beliefs as follows: µt (ω, ·) = ν

∀(t, ω) ∈ (T , ∞) × Ω \ Ft .

(B.6)

According to this belief system, if no entry and no failure have occurred up to time t > T , then each player believes that his opponent will never enter because his investment cost 45

is larger than h. It is then clear that the equilibrium entry trigger of a player with type θ ∈ (θ, h) is P ∗ (θ) = max(P ∗ (θ), P B (θ)). Last, we describe our symmetric PBE (σ, µ). For each θ, let T ∗ (θ) = inf{t ≥ 0 | P ∗ (θ) ≤ pB t (ω)} if this quantity is finite and ∞ otherwise. For each (t, ω), σt (ω, θ) = 1 if θ ≤ l and σt (ω, θ) = 0 if θ ≥ h. Consider now θ ∈ (l, h). On the equilibrium path, one has σt (ω, θ) = 0 for each ω and t < T ∗ (θ), and σT ∗ (θ) (ω, θ) = 1 if ω 6∈ FT ∗ (θ) . Suppose now that a player with cost θ has not invested first at T ∗ (θ), despite that no failure has occurred up to this time. If t ≥ T ∗ (θ) ∨ T ∗ (θ) then, given the out of equilibrium beliefs (B.6), σt (ω, θ) = 1 if and only if ω 6∈ Ft . Suppose now that θ < sup Θ0 , and let t ∈ (T ∗ (θ), T ∗ (θ)). Obviously, σt (ω, θ) = 0 if ω ∈ Ft . Moreover, a close inspection of (25) reveals that the equilibrium trigger P ∗ (θ) remains optimal off the equilibrium path. Hence σt (ω, θ) = 1 if ω 6∈ Ft and t ≥ T ∗ (θ). By construction, σ is sequentially rational given the beliefs (8) on the equilibrium path and (B.6) off the equilibrium path. If θ ≥ h, T ∗ (θ) = +∞, so that this is the only symmetric PBE. If θ < h and hence T ∗ (θ) < +∞ any other symmetric PBE differs from the constructed one only off the ¥ equilibrium path (and from date T ∗ (θ) onwards), hence the claim. Proof of Proposition 6. We first prove that for each i ∈ I, if θ ∈ Θ0 is a discontinuity point of player −i’s investment trigger, then P i is continuous and P −i > P i on [θ, h]. Suppose instead that P i has a discontinuity at some θ0 ∈ [θ, h ∧ θ). Let p+ be defined as in the proof of Lemma 13. Note that as P i coincides with P B on φB ([p− , p+ ]), and P B is continuous, one has in fact θ˜ = inf{θ0 | P i has a discontinuity at θ0 } > θ. From Lemma 13 and the fact that P −i cannot have a downward jump at θ˜ by Proposition 3, it follows ˜ = P B (θ). ˜ Note that by construction, that P −i must be continuous at θ˜ and that P −i (θ) ˜ since P i has no discontinuity on this interval. Consider we must have P i < P −i on [θ, θ) now the ratio R(p, φ−i , φi ) as in the proof of Lemma 12. Note that R(·, φ−i , φi ) may ˜ since P −i might be discontinuous on (θ, θ). ˜ be not defined everywhere on (p+ , P B (θ)) Nevertheless, we can extend the function φ−i by requiring it to be constant on any interval of discontinuity. Without risk of confusion, call φ−i the function generated in ˜ R(·, φ−i , φi ) > 1 on [p+ , P B (θ)). ˜ this way. Note that since P i < P −i on [θ, θ), We −i i + B ˜ now prove that R(p, φ , φ ) is increasing on (p , P (θ)), thereby contradicting the fact ˜ since θ˜ < h ∧ θ. Three cases must be that it must converges to one as p ↑ P B (θ) ˜ is an distinguished. Consider first a value of p ∈ [π − , π + ) where [π − , π + ] ⊂ (p+ , P B (θ)) interval of discontinuity of P −i . Then:

46

∂ + R(p, φ−i , φi ) = lim sup p˜↓p

R(˜ p, φ−i , φi ) − R(p, φ−i , φi ) ≥0 p˜ − p

(B.7)

since φi = φB and φ−i is constant on any interval [p, p˜] for p˜ close enough to p. Consider ˜ such that the derivative of φ−i exists and is strictly next a value of p ∈ (p+ , P B (θ)) positive at p. The first order condition (27) is then satisfied by both φi and φ−i at p, from which it follows that: ¡ ¢ ∂R(p, φ−i , φi ) = R(p, φ−i , φi ) ϕ(p, φ−i (p)) − ϕ(p, φi (p)) ≥ 0 (B.8) since the mapping θ 7→ ϕ(q, θ) is decreasing on (φF (q), φB (q)) for each q ∈ (0, 1). Last, consider a value of p such that p is the upper bound of an interval of discontinuity of P −i . From the formula f (x) ∂ + g(x) + g(x) ∂ + f (x) ≥ ∂ + (f g)(x) which is valid for any nonnegative and continuous functions f and g, we get: R(p, φ−i , φi ) ∂ + (1 − G(φi (p))) + (1 − G(φi (p))) ∂ + R(p, φ−i , φi ) ≥ ∂ + (1 − G(φ−i (p))). Since obviously ∂ + (1 − G(φi (p))) ≤ 0 and 1 − G(φi (p)) > 0, a sufficient condition for ∂ + R(p, φ−i , φi ) ≥ 0 is that ∂ + (1 − G(φ−i (p))) = 0. Consider player i with cost φB (p). It follows from Lemma 13 that P i (φB (p)) = p. Proceeding as in the proof of Lemma 8, we have: (1 − G(φ−i (˜ p)))

∆v p + l − φB (p) − LB (p, p˜) (∆v p˜ + l − φB (p)) V F (p, φB (p)) − ∆v p − l + φB (p)

(B.9)

≥ G(φ−i (˜ p)) − G(φ−i (p)) ≥ 0 ∀˜ p ∈ (p, P F (φB (p))). Dividing both sides of (B.9) by p˜ − p and letting p˜ ↓ p, we conclude that: 0 ≤ ∂ + G(φ−i (p)) = lim inf pe↓p

≤ × lim inf pe↓p

G(φ−i (˜ p)) − G(φ−i (p)) p˜ − p

1 − G(φ−i (p)) V F (p, φB (p)) − ∆v p − l + φB (p)

∆v p + l − φB (p) − LB (p, p˜) (∆v p˜ + l − φB (p)) p˜ − p = 0,

where the last equality follows from L’Hˆopital’s rule and the smooth pasting condition ∆v − ∂1− LB (p, p) (∆v p + l − φB (p)) = 0 for ΠB (φB (p)) and the fact that ∂1− LB (p, p) = 47

−∂2+ LB (p, p). Hence ∂ + (1 − G(φ−i (p))) = ∂ + G(φ−i (p)) = 0. Together with (B.7) ˜ and (B.8), this implies that ∂ + R(p, φ−i , φi ) ≥ 0 everywhere on (p+ , P B (θ)), hence −i i that R(·, φ , φ ) is increasing on this interval, which yields the required contradiction. Summing up, P i cannot jump over P −i on [θ, h]. The same argument as above, but this time applied to h instead of θ˜ implies that when A4 holds and therefore G(h) = G(φi (1)) = G(φ−i (1)) < 1, R(p, φ−i , φi ) is increasing and strictly greater than one over [p+ , 1], and hence cannot converge to one as p ↑ 1. ¥

48

V F (·, θ) V B (·, θ)

6

∆vp + l − θ V F (·, θ)

.. .. .. .. .. .. . B P (θ) l−θ

.. .. .. .. .. .. .. .. .. .. .. .. .. . F

P (θ)

-

1

V B (·, θ)

Figure 1 : The value functions V B (., θ ) and V F (., θ)

49

6

PF ∂φ = ∞

® - -... . 6.. 6 ... ?6 .. 6 ... 6 .. 6 .. 6 6 ... 6 .. 6 .. D .. 6 .. B 6 P .. 6 ... 6 6 .. 6 .. 6 ... .. 6 ..- 6 .. ∂φ = 0 .. ... .. 6 .. .. .. .. 6 .

1

l

.. .. .. .. . θ

.. .. .. .. .

θ

-

h

Figure 2 : The flow of the ODE (21) on the domain D

50

θ

6

PF ª

1

6

P∗ 6

B

P (θ) ......................................... .. .. .. .. p0 .. .. .. .. l

θ

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .

]

θ

Figure 3 : Equilibrium in the case p0 < P B (θ)

51

PB

-

h

θ

6

PF ?

1

6

P F (θ) ........................................... .. .. .. .. ........................................... p0 .. . B P (θ) ............................................ .. .. .. .. .. .. .. .. .. l

P∗

θ

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . θ

I

PB

-

h

Figure 4 : Equilibrium in the case P B (θ) < p0 < P F (θ)

52

θ

6

PF À

1

p0 P F (θ)

P B (θ)

......................................... .. .. .. .. . ......................................... .. .. .. .. .. .. .. ......................................... .. .. .. .. .. .. .. .. .. l

6

P∗

θ

.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..

K

θ

Figure 5 : Equilibrium in the case p0 > P F (θ)

53

PB

-

h

θ

6

P`B 1 P



? - 6 6 6 6 N-- 6 ..- 66 .. .. 66 .. 6 .. 6 .. 6 ..6

P`P

.. ... .. .. .. .. .. θ

.. .. ? .. ? ? . ??¾ ¾ ... ? ¾ q ? ... ?¾¾ .. ?¾ .. ¾ .. ¾ .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. h=θ

® K

θ `h

-

`h

Figure 6 : The flow of the ODEs (31) and (33) on D` .

54

θ

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[13] Hendricks, K., and D. Kovenock (1989): “Asymmetric Information, Information Externalities, and Efficiency: The Case of Oil Exploration,” Rand Journal of Economics, 20, 164-182. [14] Henry, C. (1974): “Investment Decisions under Uncertainty: The Irreversibility Effect,” American Economic Review, 64, 89-104. [15] Lambrecht, B., and W. Perraudin (1997): “Real Options and Preemption,” JIMS Working Paper No. 15/97, University of Cambridge. [16] Maskin, E., and J. Riley (1999a): “Equilibrium in Sealed High Bid Auction,” Review of Economic Studies, forthcoming. [17] Maskin, E., and J. Riley (1999b): “Asymmetric Auctions,” Review of Economic Studies, forthcoming. [18] Maskin, E., and J. Tirole (1999): “Markov Perfect Equilibrium, I: Observable Actions,” mimeo GREMAQ, Universit´e des Sciences Sociales, Toulouse. [19] McDonald, R., and D. Siegel (1986): “The Value of Waiting to Invest,” Quarterly Journal of Economics, 101, 707-728. [20] Moretto, M. (1996): “Option Games: Waiting vs. Preemption in the Adoption of New Technology,” Nota di Lavoro 55.96, Fondazione Eni Enrico Mattei, Milan. [21] Riley, J. (1980): “Strong Evolutionary Equilibrium and the War of Attrition,” Journal of Theoretical Biology, 82, 383-400. [22] Rob, R. (1991): “Learning and Capacity Expansion under Demand Uncertainty,” Review of Economic Studies, 58, 655-675. [23] Smets, F. (1993): Essays on Foreign Direct Investment, PhD Thesis, Yale University, New Haven. [24] Shiryayev, A.N. (1978): Optimal Stopping Rules, Berlin, Springer-Verlag.

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