Learning and Collusion in New Markets withUncertain Entry Costs∗

Francis BlochEcole Polytechnique

Simona FabriziMassey University

and

Steffen LippertUniversity of Otago

August 17, 2011

Abstract

This paper analyzes an entry timing game with uncertain entrycosts. Two firms receive costless signals about the cost of a new projectand must decide when to invest. We characterize the equilibrium ofthe investment timing game with private values, private and publicsignals. We show that competition leads the two firms to invest tooearly and analyze collusion schemes whereby one firm prevents theother firm from entering the market. We show that, in the efficientcollusion scheme, the active firm must transfer a large part of thesurplus to the inactive firm in order to limit preemption.

JEL Classification Codes: C63, C71, C72, D82, D83, O32

Keywords: Learning; Preemption; Innovation ; New Markets ;

Entry Costs ; Collusion ; Private Information

∗We are grateful to seminar participants at Erasmus University, Queen Mary, the ParisSchool of Economics for helpful comments. We want to thank particularly Rossella Argen-ziano, Nisvan Erkal, Emeric Henry, Hodaka Morita and Francisco Ruiz Aliseda for theirhelp and support.

1 Introduction

The development of new products and processes, the access to new marketsare characterized by a high degree of uncertainty. Firms engage in longresearch process and accumulate information about the cost and benefitsof the project before actually investing in new products or entering newmarkets. Project selection is often very fierce, and very few projects endup being implemented. For example, in the pharmaceutical industry, only avery small fraction of the molecules which are tested end up being patented.A realistic description of investment in new products and markets must allowfor learning about the cost and benefit of the investment.

When different firms or research teams compete to enter a new market,the dynamics of learning and competition may become very complex. Signalsreceived by competitors may transmit information about the profitability ofthe market, affecting the incentives to acquire information and the speed ofinvestment in the project. A further distinction exists between situationswhere information received by competitors is made public or kept private.In the first case, information becomes a public good, resulting possibly infree-riding and low levels of information acquisition. In the second case,participants in the race must control the diffusion of information that theiractions convey to their opponents, resulting in complex investment dynamics.In particular, private information may lead to preemption and accelerate therace between the competitors.

In this paper, our objective is to better understand the interplay betweenlearning and preemption in entry timing games, and to study simple mecha-nisms of collusion between firms. We contrast the outcome of an entry gameamong firms which cooperate in project selection, and between competitorswith public and private information. We consider a simple compensating pay-ment cooperation mechanism, whereby one of the two firms pays the otherteam to stay out of the market, and show when this compensating paymentenables firms to reach the collusive outcome.

Our analysis is motivated by situations where firms are engaged in inno-vation races to launch a new product, and upon success, one of the two teamsis acquired by the other so that a single firm extracts the monopoly profit ofthe innovation. For example in the late 1990s and early 2000s, the computerindustry saw an acquisition frenzy in which firms were bought out even beforethey had an initial success. After acquiring a Switch maker in 1993 (switcheswere the main competition to their core business, routers), Cisco acquired 69companies between 1995 and 2000 (more than eleven companies per year).Many of these acquisitions were early stage acquisitions that turned out not

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to be viable. Cisco often did not learn about their targets: They bought outpotential competitors early, before knowing whether or not their own R&Dstrategy is better. Subsequently, Cisco switched to an acquisition strategy,in which it acquired fewer, but more mature companies, after learning abouttheir potential.2

Similarly, threatened by looming patent expiry for blockbuster drugsas well as by competition from generics, and facing a declining productpipeline due to low in-house R&D productivity, big pharmaceutical com-panies have recently been acquiring biotechnology companies. For example,in 2009 Hoffmann-La Roche acquired the biotech company Genentech fornearly $47bn. Merck & Co. purchased a follow-on biologics platform fromInsmed in 2010. However, contrary to Cisco in the 1990s and early 2000s,these pharmaceutical companies did these acquisitions in order to access theirtargets’ existing early-stage successes. This seems to suggest that competingteams sometimes wait until there is a success signal in early phases of theR&D process and then purchase their potential competitor.

On the other hand, in the pharmaceutical industry, companies at timesengage in what to outside observers seems wasteful investment: Several phar-maceutical companies target the same biological mechanism with differentmolecules, leading to several drugs with the same therapeutic indications.3

The phenomenon is called “me-too” drugs, and the presumption is that, afterthey see a competitor succeed with a particular mechanism, pharmaceuticalcompanies engage in imitation and invest in finding a different molecule thattarget this mechanism. However, contrary to this commonly held belief,DiMasi and Faden (2011) show that many “me-too” drugs are not the re-sult of one company imitating another, but of parallel research targeting thesame mechanism, with no one knowing which drugs will work until they havecleared their regulatory trials, often in rapid succession.4 In these cases, itseems firms fail to cooperate and instead invest in competition with eachother.

We construct a model of entry where firms initially ignore the fixed cost ofentry. They gradually acquire signals about the entry cost through research

2Between 2001 and 2003, they only acquired 10 companies.3For example, in the U.S., more than six anti-cholesterol drugs on the market, from

Crestor to Zocor, are frequently advertised on television. The market for male sexualenhancements started with Pfizer Inc.s Viagra and now includes two other drugs, Cialismarketed by Eli Lilly and Co. and Levitra, which is sold by a partnership of GlaxoSmithK-line and Bayer Pharmaceuticals Corp.

4See DiMasi and Faden (2011); and Economist article entitled “Me too! Me too!” (Apr17, 2007; http://www.economist.com/blogs/freeexchange/2007/04/me_too_me_too).

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and experimentation. We consider here the case of private values, where theprofitability of entry differs across firms (as would be the case if uncertaintyprimarily involves the private cost of investment) and not the case of commonvalues, where the profitability of innovation is the same for all teams (aswould be the case if uncertainty primarily involved the value of the newmarket).

Our model results in project selection. In the cooperative outcome, onlythe most profitable firm should be allowed to enter. We show that com-petition results in preemption, as the two firms may, for some parametervalues, choose to invest too early, before they learn the entry cost whereasit would have been optimal for them to wait until they learn their cost. Ifsignals on entry costs are public, compensating payments can be used to en-force monopolization of the market. If the signals are private, firms do notobserve whether their opponent has dropped from the race or not. Hence,over time, if they don’t observe investment by their opponent, teams becomemore optimistic as they believe that with high probability their opponenthas decided not to invest (”no news is good news”). This dynamics of beliefswill eventually lead a firm to invest before it has learned the profitability ofthe innovation, aggravating preemption and resulting in an inefficient accel-eration of the entry race. Compensating payments can be designed in orderto mitigate this preemption effect and lead to market monopolization. Wecharacterize a lower and upper bound on the share of the surplus which istransferred to the firm which drops out of the race, and show that the optimalcompensating payment will typically involve a large share of the surplus to betransferred to the trailing firm. However, the compensating payment cannotrestore efficiency in all circumstances, and in fact, there exist states whereno efficient, budget balanced, individually rational and incentive compatiblemechanism can be designed.

Our analysis sheds light on situations of project selection, where twoindependent firms run parallel research programs and a third party can en-force a cooperative scheme to prevent inefficiencies. The third party can forinstance be a venture capitalist or a granting agency running competing re-search projects, the editor of an academic journal or organizer of a scientificconference who discovers that two teams of scientists are working on the sameproblem. Our analysis suggests that selection should neither occur too early(before the profitabilities of the projects are known), nor too late (when thefirms have become very optimistic about their prospects given that the otherfirm has not entered). It also shows that the share of the surplus transferredto the firm which is not selected should neither be too large (in which casethe selected firm may have an incentive to delay the research project) nor too

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small (the higher the payoff transferred to the firm which is not selected, thesmaller the gap between the payoffs of the leading and trailing firms, whichreduces inefficiencies due to excess momentum.)

Our analysis is rooted in the literature on patent races in continuous timepioneered by Reinganum (1982) and Harris and Vickers (1985). The first ex-tensions of patent races allowing for symmetric uncertainty are due to Spattand Sterbenz (1985), Harris and Vickers (1987) and Choi (1991). Modelsof learning in continuous time with public information have been studied byKeller and Rady (1999) and Keller, Cripps and Rady (2005). Rosenberg,Solan and Vieille (2007) and Murto and Valimaki (2010) extend the modelto allow for public signals. The model of preemption we consider is for-mally closely related to Fudenberg and Tirole (1985)’s models of technologyadoption with preemption. Innovation timing games which can result eitherin preemption or in waiting games have been studied by Katz and Shapiro(1987). Hoppe and Lehmann-Grube (2005) propose a general method foranalyzing innovation timing games. Fudenberg and Tirole (1985)’s modelhas been extended by Weeds (2002) and Mason and Weeds (2010) to allowfor stochastic values of the technology. However, none of these models allowsfor private information. The closest papers to ours are the recent papersby Hopenhayn and Squintani (2010) on preemption games with private in-formation and Moscarini and Squintani (2010) on patent races with privateinformation. Moscarini and Squintani (2010) analyze a common values prob-lem, where agents learn about the common arrival rate of the innovation. Ourmodel with common values shares the same characteristics as theirs, albeitin a much simpler setting, and the results are closely related. Our modelwith private values and preemption displays very different results. Hopen-hayn and Squintani (2011)’s model is much more general than ours but onlycovers situations where agents receive positive information over time. In ourmodel, research teams may either receive positive or negative signals aboutthe profitability of the research project, so that the results of Hopenhayn andSquintani (2011) do not directly apply. However, the spirit of the analysis isvery similar, and we outline in the body of the paper the similarities and dif-ferences between their results and ours. Cooperation among research teamswith private information has been studied in a mechanism design contestby Gandal and Scotchmer (1993). In ongoing work, Akcigit and Liu (2011)study cooperation in a patent race with learning and private signals, wherecooperation involves the disclosure of private signals.

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2 The Model

2.1 Firms, new markets and entry costs

We consider two firms which may invest in order to enter a new market,launch a new product or exploit a new process. The monopoly and duopolyprofits obtained after investment are fixed and given by πm and πd respec-tively. We suppose that πm > 2πd. The entry cost to the new market isuncertain, and can either take a high or low value, θi = θ or θi = θ. Weconsider the case of private values where costs are independently distributedamong the two firms. Assuming that the two values of the cost are equiprob-

able, the expected value of the entry cost is θ = θ+θ2

for both firms.During the experimentation phase, each firm receives a costless perfectly

informative signal about its cost according to a Poisson process with intensityµ. Hence, the probability that a firm receives a signal during the interval[0, t] is 1 − e−µt. With probability 1

2the firm learns that it is of high type,

and with probability 12, it learns that it is of low type. We assume that

the Poisson processes generating signals to the two teams are independent.In the private values case, independence furthermore means that the signalsreceived by the two teams are independent. In the common values case, thesignals which are perfectly informative, are not independent.

We assume that high cost firms never have an incentive to invest, evenif they receive monopoly profit. Low cost firms always have an incentiveto invest, even if they receive duopoly profit. When entry cost remainsunknown, research teams have an incentive to invest as monopolists but notas duopolists. Formally,

Assumption 1θ ≤ πd ≤ θ ≤ πm ≤ θ. (1)

2.2 Entry timing and strategies

At any date t = 0,∆, 2∆, ..., both firms can choose whether to enter themarket. (We will analyze situations where the time grid becomes infinitelyfine, and ∆ converges to zero.) If team i enters the market, it pays thefixed cost θi and starts collecting monopoly (or duopoly) profits immediately.Investments to enter the market are immediately observed by the other firm.

Given Assumption 1, it is a dominant strategy for a high cost firm notto invest. Hence, the only relevant choices are choices made by a firm whichlearns that its cost is low, or by a firm which still ignores its entry cost. A

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strategy specifies, after every possible history, a pair of probabilities withwhich the firm invests when it learns that its cost is low and when it ignoresits cost. We consider perfect equilibrium strategies which maximize the firm’sexpected discounted payoff after every possible history.

2.3 Cooperative benchmark

If the two firms cooperate, they either choose to enter immediately, and earnan expected profit of πm− θ, or wait until they identify whether one firm hasa low entry cost, and obtain an expected profit of

VC =µ

2µ+ r(πm − θ)(1 +

µ

2(µ+ r)).

The optimal cooperative choice depends on the values of the parameters.Clearly, the incentive to experiment rather than enter immediately is increas-ing in the difference between the costs θ and θ, and in the intensity of thePoisson process generating signals, µ. It is decreasing in the discount rate rand in the value of the monopoly profit πm. Finally, notice that if a singlefirm is present on the market, it will only be able to experiment with one ofthe two research processes, and obtain an expected payoff by experimentinggiven by

VO =µ

2(µ+ r)(πm − θ).

This expected payoff is lower than the expected payoff obtained by twocooperating firms for two reasons. First, by experimenting in parallel, thetwo firms accelerate the rate at which signals arrive, as the Poisson processgenerating signals now has intensity 2µ instead of µ. Furthermore, in theprivate values case we consider, by experimenting on two projects in parallel,the teams draw two independent signals about the costs. In other words,even if the first team receives a signal that it has a high cost, there is apositive probability as the second team receives a signal that it has a lowcost and implements the project.

2.4 Leader and follower payoffs

Suppose that one firm (the leader) invests first. In the private values model,the second firm (the follower) will only follow suit if it learns that its cost islow. Hence the expected value of the follower is given by

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VF =µ

2(µ+ r)(πd − θ).

The leader thus extracts monopoly profit as long as the other firm has notentered, and we compute the value of the leader after investment as:

VL = πm −µ

2(µ+ r)(πm − πd)

3 Entry timing

3.1 Entry timing with public signals

We suppose that the signals received by the two firms during the experimen-tation phase are public. We first establish that a firm which learns that itscost is low invests immediately. Depending on the parameters, a firm whichhas not yet learned its cost will either choose to preempt at zero or to waituntil it learns that its cost is low. In the preemption case, both firms investwith positive probability at zero, resulting in coordination failures. In thewaiting game, we show that, because the leader’s and follower’s payoffs areindependent of time, the equilibrium strategy is for both firms not to en-ter before they learn the value of their costs. Summarizing, we obtain thefollowing characterization of equilibrium.

Proposition 1 In the entry timing game with public information, a firmswhich learns that its cost is low invests immediately. If VL − θ > VF , pre-emption occurs and (i) a firm which ignores its cost invests with positiveprobability at any date t = 0,∆, ... whenever the other firm has not investedand (ii) a firm invests immediately after it learns that the other firm has a

high cost. If VL− θ < VF , firms do not enter unless they learn that their costis low.

Proposition 1 shows that the entry timing game is either a preemptiongame (when VL − θ > VF ), or a waiting game (when VL − θ < VF ). Noticethat VF − VL = VO − πm < VC − πm. Hence, as compared to the cooperativebenchmark, competition results in excess momentum. In the competitiveentry timing game, firms have an incentive to invest too early, before theylearn the true value of their cost.

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3.2 Entry timing with private signals

When signals are private, teams do not learn whether the other team hasdrawn a bad or good signal about its cost. Each firm holds beliefs γt(θ)about the cost of the other firm. These beliefs evolve over time given thestrategies and the observation of investments. In order to compute the beliefs,we let G(t, τ) denote the probability that a firm which learns that its costis low at date τ invests at t ≥ τ , and g(t, τ) the instantaneous probabilitythat the firm invests at date t. We also let h(t) denote the instantaneousprobability that a firm which ignores its cost invests exactly at date t. UsingBayes’ rule, the beliefs at period t are then given by:

γt(θ) =

∫ t0[1−G(t, τ)]µe−µτdτ

A(t),

γt(θ) =1− e−µt

A(t),

γt(θ) =2[e−µt −

∫ t0e−µτh(τ)]

A(t)

where

A(t) =

∫ t

0

[1−G(t, τ)]µe−µτdτ + 1− e−µt + 2[e−µt −∫ t

0

e−µτh(τ)].

We first establish that a firm which learns that its cost is low has anincentive to invest immediately:

Lemma 1 In the entry timing game with private signals, a firms whichlearns that its cost is low has an incentive to enter immediately.

Lemma 1 enables us to focus attention on situations where beliefs onlyinvolve situations where a firm has learned that its cost is high, or ignoresits cost. In other words, we now consider beliefs:

γt(θ) = 0,

γt(θ) =1− e−µt

1 + e−µt − 2∫ t

0e−µτh(τ)

,

γt(θ) =2[e−µt −

∫ t0e−µτh(τ)]

1 + e−µt − 2∫ t

0e−µτh(τ)

.

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It is easy to check that γ(t)(θ), increases over time. As t increases, theprobability that the competing firm has a high entry cost increases. No newsis good news: as time passes, each firm becomes more pessimistic about thecost of the other firm and thus more optimistic about its own prospects. Thisdynamics of beliefs is the driving force behind the dynamics of the model, asit ensures that firms which ignore their costs will eventually find it profitableto invest because they believe that the other firm has drawn a high cost andwill never invest with high probability.

In order to characterize equilibrium strategies, we compute the expecteddiscounted payoff of the leading team which is the first to invest at time t:

VL(t) = γ(t)(θ) + pimγ(t)(θ)VL = πm − γ(t)(θ)µ

2(µ+ r)(πm − πd).

Because γ(t)(θ) is decreasing over time, the value of the leader is increas-ing over time. As time passes, firms become more optimistic, and the valueof the leader increases from VL = VL(0) to πm = limt∞ VL(t). Notice that thevalue of the follower, VF , remains independent of time. In order to analyzethe equilibrium entry times, we distinguish between three cases: (i) Case 1

when VL − θ > VF ; (ii) Case 2 when πm − θ ≥ VF ≥ VL − θ; and (iii) Case 3

when VF > πm − θ. The three cases are illustrated below:

Cases 1 and 3 correspond to the preemption and waiting cases in thetiming game with public signals. Case 2 exploits the fact that beliefs evolveover time, and describes a new situation which is similar to the preemptiongame studied by Fudenberg and Tirole (1985). The expected payoff of theleading firm is initially lower than the expected payoff of the following firm,but is increasing over time and eventually becomes higher than the payoff ofthe following firm. As in Fudenberg and Tirole (1985), the unique subgameperfect equilibrium results in rent equalization. One firm invests at the firsttime at which the payoff of the leading firm is higher than the payoff of thefollowing team, VL(t)− θ = VF . Formally,

Theorem 1 In the entry timing game with private signals, a firm whichlearns its cost invests immediately. If VL− θ > VF , preemption occurs at thebeginning of the game and both teams invest with positive probability at time0. If πm − θ ≥ VF ≥ VL − θ, in a symmetric equilibrium, rents between theleader and the follower are equalized and each firm invests with probability

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Figure 1: Case 1: VL − θ > VF

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at time t such that: VL(t) − θ = VF . If VF > πm − θ, firms do not enterunless they learn that their cost is low.

Theorem 1 shows that when signals are private, excess momentum dueto preemption is higher than when signals are public. When πm − θ ≥ VF ≥VL− θ, firms do not invest before learning their cost when signals are public,but rush to invest at time t when information about costs are private.Thisincentive to preempt earlier with private signals than with public signals isdifferent from the result of Hopenhayn and Squintani (2011) who show thatpreemption is stronger with public signals than with private signals, whennew information can only lead to improvements. In their model, publicity ofinformation strengthens competition between agents, and results in higherpreemption. In our model, publicity of information reduces competition asfirms learn that the other firm is out of the market, and reduces preemption.

We now focus on Case 2 and analyze how the preemption time t dependson the parameters of the model. The preemption time is implicitly definedas the unique solution to the equation:

πm −µ

2(µ+ r)πd −

µ

µ+ r(πm − πd)

e−µt

1 + e−µt− θ

2− θr

2(µ+ r)= 0. (2)

Implicit differentiation of equation (2) immediately results in the followingcomparative statics:

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Figure 2: Case 2: πm − θ ≥ VF ≥ VL − θ

πm -πd +

θ +θ +r -µ ?

Table 1: Preemption time t – Comparative statics

An increase in the monopoly profit πm increases incentives to preempt andresults in a lower preemption time ; conversely, an increase in the duopolyprofit πd reduces incentives to preempt and lengthens preemption time. Anincrease in entry costs (both θ and θ) makes entry more costly and resultsin longer delays before preemption. When firms become less patient (r in-creases), preemption time decreases. Changes in the Poisson arrival rate µhave ambiguous effects, as an increase in µ simultaneously increases the pay-off of the leader VL(t) and of the follower VF . For small values of µ, (µ closeto zero), the magnitude of the marginal effect of an increase in µ is higheron the follower payoff than on the leader payoff, so that the preemption timeincreases with µ. For large values of µ, (µ close to infinity), the comparisonis reversed, so that the preemption time decreases with µ.

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Figure 3: Case 3: VF > πm − θ

3.3 Efficiency comparison

We now compare the industry profits in the three regimes of cooperation,competitive entry with public signals and competitive entry with privatesignals. We distinguish between four parameter regions, depending on themagnitude of the expected entry cost θ:

1. −θ > VC − πm: immediate entry in the cooperative regime, and pre-emption at zero in both competitive regimes

2. VC − πm > −θ > VF − VL: delayed entry in the cooperative regime,and preemption at zero in both competitive regimes

3. VF − VL > −θ > VF − πm: delayed entry in the cooperative regimeand in the competitive regime with public signals, preemption at finitetime t in the competitive regime with private signals

4. VF − πm > −θ: delayed entry in all regimes.

We define the industry profits when both firms delay their entry untilthey learn that their cost is low as:

VS =µ

2µ+ r[(πm − θ) +

µ

2(µ+ r)(2πd − 2θ)]

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and the industry profits with preemption at finite time t as:

VP = (1− e−(2µ+r)t)[µ

2µ+ r[(πm − θ) +

µ

2(µ+ r)(2πd − 2θ)]]

− e−(µ+r)t(1− e−µt) µ

2(µ+ r)(πm − θ)

+ e−(2µ+r)t2µ

2(µ+ r)(πd − θ)

It is easy to check that VC > VS > VP > 2VF . The following table lists theindustry profits under the three regimes in the four parametric configurations:

parameter region cooperative public private

−θ > VC − πm πm − θ 2VF 2VFVC − πm > −θ > VF − VL VC 2VF 2VFVF − VL > −θ > VF − πm VC VS VP

VF − πm > −θ VC VS VS

Table 2: Efficiency comparisons

Table 2 illustrates the three sources of inefficiencies due to competition inour model of entry. First, by competing on the market, the firms forgo thebenefits of market monopolization, the difference between monopoly profits,πm and the sum of duopoly profits, 2πd. Second, by competing on the market,the firms pay twice the entry cost θ, whereas in the cooperative benchmark,only one firm enters. Finally – and this is the new element of the model wewish to emphasize– competition results in excess momentum, making firmsenter the market before they learn their cost, whereas in the cooperativebenchmark, they should wait until they learn their cost before entering.

4 Market foreclosure and cooperation

In this section, we analyze cooperation mechanisms which would allow thefirms to reach the cooperative outcome. We focus attention on compensatingpayment schemes which are paid by one firm in order to compensate the otherfirm for not entering the market. Compensating payment schemes could beimplemented at three different points in time:

• ex ante: payments are made before the firms learn their entry cost

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• at the interim stage: payments are made by one firm after it learns itscost, when it ignores the cost of the other firm

• ex post: payments are made after the costs of both firms are commonknowledge.

The best timing of compensating payments depends on the specific para-metric configuration. If −θ > VC − πm, compensating payments should bemade ex ante in order to prevent inefficiencies due to market competitionand duplication of entry costs. If VC −πm > −θ, the cooperative benchmarkis reached when one of the two firms learns that its cost is low and we arguethat the best timing for compensating payments is the interim stage. At theex ante stage, if one of the firm pays the other firm to leave the market, it stillneeds to run the other firm’s project in order to select the project with thelowest cost. This requires ongoing cooperation on the part of the managersof the firm which has been bought off, and agency problems will arise, whichmay limit the gains from cooperation. When signals are public, cooperationmay happen at the ex post stage, with compensating payments being paidto the follower firm only when it learns that its cost is low. When signals areprivate and firms cannot credibly convey information about their entry cost,ex post compensating payments can only be made after the follower firm hasentered the market. In that case, the entry cost of the follower firm has beensunk, so that it becomes impossible to alleviate the duplication of entry costswith payments at the ex post stage.

We now concentrate our attention to compensating payments made at theinterim stage when VC − πm > −θ and signals are private. First note thatcompensating payments should satisfy individual rationality and incentivecompatibility. Hence the utility obtained by the leader after the compensat-ing payment is paid at date t, UL(t) and the utility obtained by the followerUF (t) should satisfy the following two conditions:

UL(t) ≥ VL(t), (3)

UF (t) ≥ VF (4)

The first inequality stems from the individual rationality constraint of theleader, who may choose to forgo the compensating payment. The secondinequality results from the follower’s incentive compatibility and individualrationality constraint. As the leader cannot verify information about thefollower’s type, he ignores whether the follower has received a high cost signalor not, and must pay the follower the fixed payment VF corresponding to a

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firm ignoring its cost. As compensating payments are budget balanced, thesum of utilities received by the leader and follower must exactly equal themonopoly profit:

UL(t) + UF (t) = πm (5)

Given inequalities (3), (4) and equation (5), a necessary condition for theexistence of a budget balanced, individually rational and incentive compatibletransfer scheme is

VL(t) + VF ≥ πm.

As VL(t) is increasing, VL(0) < πm − VF and VL(∞) > πm − VF , there existsa unique date t∗, such that no budget balanced individually rational compen-sating payments exist if the first firm enters at date t ≥ t∗. This remarkcaptures the following simple intuition. As time passes, firms become moreoptimistic about their prospects. If a firm enters at a late date, it will expectthe other firm to have left the race and will not be willing to compensatethe other firm at the level VF , which is the minimal level at which a firmwhich ignores its cost is willing to leave the race. Furthermore, this remarkshows that there is no efficient, budget balanced and individually rational co-operation mechanism. To see this, consider a realization of the signals whereno firm has learned its cost before t∗. Either the mechanism prescribes thatone of the team invests before t∗, and the mechanism is inefficient becauseit will result in a high cost team investing with positive probability, or themechanism prescribes to wait until one of the firm has learned it has a lowcost, and the mechanism is inefficient because there is no budget balanced,individually rational compensating payment which prevents the other firmfrom entering the race.

We now consider the following problem: How should compensating pay-ments be designed in order to guarantee that, whenever one firm learns thatit has a low cost before t∗, it is chosen to be the only firm operating on themarket?

Proposition 2 A differentiable compensating payment scheme UF (t) imple-ments the cooperative benchmark when a firm learns that it has a low costbefore t∗ if and only if for all t < t∗,

πm − θ < 2UF (t) <2r + µ

r + µπm +

U ′F (t)

r + µ

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Proposition 2 shows that efficient compensating payment schemes mustbe balanced to satisfy two requirements. First, the payment to the followermust be large enough to prevent early entry by firms which ignore their costs.Second, the payment to the follower should not be too large, in order to giveincentives to a firm which learns that its cost is low to enter immediately.These two requirements provide an upper and a lower bound on the expectedpayoffs of the follower and leader firm and show that the cooperative surplusmust be shared in a balanced way between the two firms.

In order to provide additional intuition, we specialize the model by as-suming that the compensating payment scheme assigns a fixed bargainingpower to the leader and the follower firm, so that

UL(t) = VL(t) + α(πmVL(t)− VF ),

UF (t) = VF + (1− α)(πm − VL(t)− VF ),

We observe that UL(t) is increasing and UF (t) decreasing over time. Thefollowing graph illustrates these mappings for α = 0 and α = 1. It displaysthe maximal time t∗ at which compensating payments can be implemented,shows that payoffs are independent of time if all of the bargaining power isgiven to the leader and that the gap between the payoff of the leader andfollower is increasing in α

The cooperative benchmark can be implemented when a firm learns thatit has a low cost before t∗ if and only if

2VF ≥ πm − θ, (6)

2VF + 2(1− α)(πm − VL − VF ) ≤ 2r + µ

r + µπm −

V ′L(0)

r + µ. (7)

These conditions put a lower bound (but no upper bound) on the shareof the bargaining surplus accruing to the leader. Notice that if the firstcondition (6) fails, it is impossible to guarantee that no early preemptionoccurs before t∗. In that situation, the value of α can be manipulated inorder to increase preemption time. By reducing α, and giving a larger shareof the surplus to the follower, the mechanism designer reduces incentives topreempt and increases the value of t.5 The optimal compensating payment

5The fact that giving a prize to the loser of a contest may be efficient, as it reducesthe gap between the winner and the loser and minimizes wasteful expenditures, has long

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Figure 4: Expected utilities with compensating payments

mechanism is thus given by the lowest value of α for which condition (7)holds.

5 Conclusion

This paper analyzes a model of entry with learning. Two firms contemplatethe entry into a new market, or the development of a new product and grad-ually learn about their private entry costs. We show that when signals arepublic, the model either results in a preemption game or a waiting game, and

been noted in the literature on contests. See Moldovanu and Sela (2008) for a recentformalization of the problem and the references therein.

17

when signals are private, firms which ignore their cost may choose to enter ata finite time, resulting in the same rent equalization phenomenon as in Fu-denberg and Tirole (1985). As opposed to Hopenhayn and Squintani (2011),we find that preemption is greater when signals are private, as firms ignorewhether the other firm has left the race or not. As compared to the collusiveoutcome, the equilibrium of the entry timing game exhibits three sourcesof inefficiencies: dissipation of the monopoly rent, duplication of entry costsand excess momentum. We analyze how compensating payments by one firmto prevent the other firm from entering the market can be implemented. Weobserve that collusion can only be effective if the first firm enters sufficientlyearly, and that compensating payments must allocate a significant share ofthe surplus to the excluded firm.

The model we consider is one instance of models of competition withlearning which have recently attracted considerable attention in economictheory. There are two directions in which we would like to continue the anal-ysis. First, in our model, teams do not choose the intensity of the Poissonprocesses generating signals. We believe that a more general model, whereexperimentation is endogenously chosen by the two firms is worth investigat-ing. At a more abstract level, our model is an example of a situation where adesigner chooses a mechanism without knowing when the agents learn theirtypes. The general problem of mechanism design in a dynamic context whereagents gradually learn their types is the next item in our research agenda.

18

References

[1] Akcigit, U. and Liu, Q., 2011, Mechanism Design for Efficient Compet-itive R & D. Work in progress presented at AEA Meetings in Denver.

[2] Choi, J.P., 1991, Dynamic R&D Competition under ‘Hazard Rate’ Un-certainty. The RAND Journal of Economics 22, 596–610.

[3] Cripps, M., Keller, G. and Rady, S., 2005, Strategic Experimentationwith Exponential Bandits. Econometrica 73(1), 39–68.

[4] DiMasi, J.A., and Faden, L.D., 2011, Competitiveness in follow-on drugR&D: a race or imitation? Nature Reviews Drug Discovery 10, 23–27.

[5] Fudenberg, D., and Tirole, J., 1985, Preemption and Rent Equalizationin the Adoption of New Technology. The Review of Economic Studies52(3), 383-401.

[6] Gandal, N., and Scotchmer, S., 1993, Coordinating Research throughResearch Joint Ventures. Journal of Public Economics 51(2), 173–193.

[7] Harris, C. and Vickers, J., 1985, Perfect Equilibrium in a Model of aRace. Review of Economic Studies 52, 193–209.

[8] Harris, C. and Vickers, J., 1987, Racing with Uncertainty. Review ofEconomic Studies 54, 1–21.

[9] Hopenhayn, H. and Squintani, F., 2011, Preemption Games with PrivateInformation. Forthcoming, Review of Economic Studies.

[10] Hoppe, H.C. and Lehmann Grube, U., 2005, Innovation Timing Games:A General Framework with Applications, Journal of Economic Theory,121, 30-50.

[11] Katz, M. and Shapiro, C., 1987, R &D rivalry with licensing or imitation.American Economic Review 77 (1987), 402420.

[12] Keller, G. and Rady,S., 1999, Optimal Experimentation in a ChangingEnvironment. Review of Economic Studies 66(3), 475–507.

[13] Mason, R., and Weeds, H., 2010, Investment, Uncertainty and Preemp-tion. International Journal of Industrial Organization 28(3), 278–287.

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[14] Moscarini G. and Squintani, F., 2010, Competitive Experimentationwith Private Information: The Survivor’s Curse. Journal of EconomicTheory 145(2), 639–660.

[15] Murto, P. and Valimaki, J., 2010, Learning and Information Aggregationin an Exit Game. Forthcoming Review of Economic Studies.

[16] Reinganum, J.F., 1982, A Dynamic Game of R and D: Patent Protectionand Competitive Behavior. Econometrica 50, 671–688.

[17] Rosenberg, D., Solan, E., and Vieille, N., 2007, Social Learning in OneArmed Bandit Problems. Econometrica 75, 1591–1611.

[18] Spatt, C. and Sterbenz, F., 1985, Learning, Preemption and the Degreeof Rivalry. RAND Journal of Economics 16, 85–92.

[19] Weeds, H., 2002, Delay in a Real Options Model of R&D Competition.The Review of Economic Studies 69(3), 729–747.

20

A Proofs

Proof of Proposition 1: We first note that, if both firms learn that theircost is low, they have no incentive to delay their investment and will bothinvest immediately. Consider then a situation where firm i has learned thatits cost is low and firm j has not learned its cost yet. We will show that it isa dominant strategy for firm i to invest immediately. If firm j invests, firmi obtains πd − θ by investing immediately and (1− r∆)(πd − θ) ,by delayingits investment, and thus prefers to invest immediately. If firm j does notinvest, and chooses to invest with probability p at period t+ ∆, by delayingits investment until t+ ∆, firm i will obtain a payoff:

W (t+ ∆) = (1− r∆)[(1− µ∆)[(1− p)VL + pπd]

+µ∆πd + πm

2− θ]

Now

W (t+ ∆)− (VL − θ) = −r∆(VL − θ)− µ∆(VL −πd + πm

2)

−p(1− r∆)(1− µ∆)(VL − πd) +O(∆2).

Note that

VL −πd + πm

2=

r

2(µ+ r)(πm − πd) > 0,

so that W (t+ ∆)− (VL − θ) < 0, establishing that firms invest immediatelyafter they learn that their cost is low.

Next, it is easy to check that a firms invests immediately after it learnsthat the other firm has high cost if and only if

πm − VO = πm −µ

2(µ+ r)(πm − θ) = VL − VF ≥ θ

Consider the investment game played by the two firms if none of themhas invested up to date t and costs are not known:

invest not invest

invest (πd − θ, πd − θ) (VL − θ, VF )

not invest (VF , VL − θ) (W (t+ ∆),W (t+ ∆)

21

where

W (t+∆) = (1−r∆)[(1−2µ∆)W (t)+2µ∆VL − θ + VF + max[V0, πm − θ]

4+O(∆2).

We first consider a symmetric equilibrium where both firms invest withpositive probability p ∈ (0, 1). In that equilibrium,

W (t) = p(πd) + (1− p)VL − θ

and

W (t) = pVF + (1− p)(W (t) + δ)

Solving this equation, we find

p =VL − θ − VFVL − πd

,

showing that an equilibrium with preemption exists if and only if VL−VF ≥ θ.Next, we consider a symmetric equilibrium in the waiting game when

VL − VF ≤ θ. Notice that, by delaying investment one period, the firmobtains a payoff:

W (t+∆) = (1−r∆)[(1−2µ∆)(VL−θ)+2µ∆VL − θ + VF + πm − θ

4+O(∆2).

Now notice that

VL − θ + VF =πm(µ+ 2r)− θ(2mu+ 2r) + 2muπd

2(µ+ r).

As πm > 2pid,

πm(µ+ 2r) + 2muπd > 4πd(µ+ r).

Hence,

VL − θ + VF > 2[πd − θ] =4VF (µ+ r)

µ.

As VF > VL − θ,

22

µ

2(VL − θ + VF + πm − θ) >

µ

2(VL − θ + VF ),

> 2(µ+ r)VF

> 2(µ+ r)(VL − θ)> (2µ+ r)(VL − θ),

establishing that W (t+ ∆) > VL − θ, so that firms always have an incentiveto wait.

Proof of Lemma 1: Suppose that firm i learns that its cost is low at date t.If firm j invests at t, firm i obtains πd−θ by investing at t and (1−r∆)(πd−θ)by delaying investment. As πd − θ > 0, it has an incentive to invest. If firmj does not invest, and firm i invests at t, it obtains a discounted expectedpayoff:

W (t) = γt(θ)πm + γt(θ)VL + γt(θ)πd − θ.

By delaying investment until time t + ∆, the firm will obtain an expecteddiscounted payoff:

W (t+ ∆) = e−r∆([1−∫ t+∆

t

(

∫ t+∆

0

g(ρ, τ)µ

2e−µτdτ + e−µρh(ρ))dρ]

(γt+∆(θ)πm + γt+∆(θ)VL + γt+∆(θ)πd − θ)

+

∫ t+∆

t

(

∫ t+∆

0

g(ρ, τ)µ

2e−µτdτ + e−µρh(ρ)dρ)(πd − θ)).

For small ∆, we have:

W (t+ ∆) = [1− r∆−∆

∫ t

0

g(t, τ)µ

2e−µτdτ − e−µth(t)]W (t)

+ [πd(∂γt(θ)

dt+ πm

∂γt(θ)

dt+ VL

∂γt(θ)

dt]∆

+ ∆[

∫ t

0

g(t, τ)µ

2e−µτdτ − e−µth(t)](πd − θ).

We compute:

23

∂γt(θ)

dt=

µ2e−µt −

∫ t0g(t, τ)µ

2e−µτdτ

A(t)− A′(t)

A(t)γt(θ),

∂γt(θ)

dt=

µ2e−µt

A(t)− A′(t)

A(t)γt(θ),

∂γt(θ)

dt=−µe−µt − e−µth(t)

A(t)− A′(t)

A(t)γt(θ)

Hence

W (t+ ∆)−W (t) = −r∆W (t)−∆µe−µt

A(t)[VL −

πm + πd2

]

+ ∆

∫ t

0

g(t, τ)µ

2e−µτdτ

W (t)− πdA(t)

+ ∆h(t)e−µtW (t)− πdA(t)

.

Notice that if h(t) = g(t, τ) = 0, W (t + ∆) −W (t) < 0, so it never pays todelay investment if the other firm does not delay investment.

However, if the other firm delays investment (g(t, τ) > 0 or h(t) > 0), afirm may benefit from delaying investment, as delaying will enable it to learnthe type of the other firm. We now prove that in fact firms will never face apositive incentive to delay investment in order to learn the type of the otherfirm. To see this, consider now a firm which has not yet learned its cost andcontemplates delaying its investment between t and t+ ∆. We compute thediscounted expected value of leaving at t as:

Y (t) = γt(θ)πm + γt(θ)VL + γt(θ)πd − θ.

and the discounted expected value of leaving at t+ ∆ as:

24

Y (t+ ∆) = e−r∆([1−∫ t+∆

t

(

∫ t+∆

0

g(ρ, τ)µ

2e−µτdτ) + e−µρh(ρ)dρ]

[e−µ∆(γt+∆(θ)πm + γt+∆(θ)VL + γt+∆(θ)πd − θ)

+1− e−µ∆

2(γt+∆(θ)πm + γt+∆(θ)VL + γt+∆(θ)πd − θ)]

+ (

∫ t+∆

t

(

∫ t+∆

0

g(ρ, τ)µ

2e−µτdτ) + e−µρh(ρ)dρ)

[e−µ∆(πd − θ) +1− e−µ∆

2(πd − θ)]).

We compute:

Y (t+ ∆)− Y (t) = −r∆Y (t)−∆µe−µt

A(t)[VL −

πm + πd2

]

+ ∆

∫ t

0

g(t, τ)µ

2e−µτdτ

W (t)− πdA(t)

+ +∆h(t)e−µtW (t)− πdA(t)

+ ∆θ − Y (t)

2.

Hence,

(Y (t+ ∆)− Y (t))− (W (t+ ∆)−W (t)) = r∆(W (t)− Y (t)) + ∆θ − Y (t)

2.

As Y (t) < W (t) and θ−Y (t) > 0, Y (t+∆)−Y (t) > W (t+∆)−W (t) for allt,∆. Hence, a firm always has a stronger incentive to wait when it ignoresits cost than when it knows that its cost is low. In particular, this impliesthat whenever g(t) > 0 (so that the firm is indifferent between investing att and t + ∆ when it knows that its cost is low), than a firm must prefer towait when it ignores its cost.

Suppose by contradiction that g(t, τ) > 0 for some t, τ < t and let t∗ =min{τ |g(t, τ) > 0 for some τ < t} be the earliest date at which one of thefirms delays its investment. By the previous argument, at t∗, a firm whichignores its cost must prefer to wait so that h(t∗) = 0. Furthermore, by

construction,∫ t∗

0g(t∗, τ)µ

2e−µτdτ = 0. But, as VL − πm+πd

2> 0, this implies

that

25

W (t∗ + ∆)−W (t∗) < 0,

contradicting the fact that a firm which learns its cost at t∗ has an incentiveto delay its investment.

Proof of Theorem 1: As in the proof of Proposition 1, we first note that,if VL − θ ≥ VF , there exists an equilibrium where both firms preempt withpositive probability at time t = 0 and at any time t > 0. Suppose nextthat VL − θ ≤ VF and πm − θ ≥ VF . Then there exists t > 0 such thatVL(t) − θ = VF . By investing at t < t, a firm either obtains πd − θ < VF(if the other firm invests) or VL(t) − θ < VF (if the other firm does notinvest). By investing at time t, the firm obtains VF . Hence it is a dominatedstrategy to invest at any time t < t. At any time t ≥ t, there is a preemptionequilibrium where both firms invest with positive probability p(t) at date t.At t converges tot, the loss due to coordination failures converges to zero, sothat at t = t, as in Fudenberg and Tirole (1985), rent equalization occursand both firms receive an expected payoff of VL(t) = VF .

Finally, suppose that πm − θ < VF . We show that, any time t, the firmhas an incentive to wait. If the firm waits one period before investing it willobtain a payoff of VF > πd − θ if the other team invests. If the other teamdoes not invest, it obtains a payoff of

VL(t)− θ = γt(θ)(πd − θ) + γt(θ)(VL − θ)by investing and

W (t+ ∆) = (1− r∆)W (t) + ∆h(t)e−µt(VF −W (t)) + ∆mu

2(VF + VL(t)− θ)

−3∆mu

2W (t) + ∆γ′t(θ)(πm − VL(t)).

by waiting one period. Now VF > W (t) and γ′t(θ) > 0. Furthermore,

VF + VL(t)− θ > VF + VL − θ >4VF (µ+ r)

µ>

4W (t)(µ+ r)

µ.

Hence,

µ

2(VF + VL(t)− θ) > (2µ+ 2r)W (t) > (

3

2µ+ r)W (t),

showing that W (t+ ∆) > W (t), so that the firm has an incentive to wait.

26

Proof of Proposition 2 In order to implement the cooperative benchmark,two conditions must be satisfied: (i) no firm must be willing to enter themarket at t < t∗ if it ignores its cost and (ii) a firm which learns that ithas a low cost must be willing to enter the market immediately. The firstcondition will hold as long as :

UF (t) > UL(t)− θ

As UL(t) = πm − UF (t), this results in

2UF (t) > πm − θ.

For the second condition to hold, we characterize the conditions under whichan equilibrium where a firm immediately invests after it observes that itscost is low exists. The discounted expected payoff of investing at period twhen the other firm does not invest is:

W (t) = UL(t)− θ,

whereas by waiting one period the firm will obtain a discounted expectedpayoff of

W (t+ ∆) = e−r∆[(1− e−µ∆

2)UF (t+ ∆) +

e−µ∆

2UL(t+ ∆)]

For ∆ small enough and assuming that utilities are differentiable,

W (t+ ∆)−W (t) = ∆[(−2r − µ)UL(t) + µUF (t) + U ′L(t)]

so that the firm has an incentive to enter immediately if and only if:

2UF (t) <2r + µ

r + µπm +

U ′F (t)

r + µ.

27