Strategic investment evaluation

Kirpalani, Rishabh; Madsen, Erik Article Strategic investment evaluation Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Kirpalani, Rishabh; Madsen, Erik (2023) : Strategic investment evaluation, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 18, Iss. 3, pp. 1141-1180, https://doi.org/10.3982/TE4806 This Version is available at: https://hdl.handle.net/10419/296436 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 18 (2023), 1141–1180 1555-7561/20231141 Strategic investment evaluation Rishabh Kirpalani Department of Economics, University of Wisconsin-Madison Erik Madsen Department of Economics, New York University We study the interaction of incentives to free-ride on information acquisition and strategically delay irreversible investment in environments in which multiple firms evaluate an investment opportunity. In our model, two firms decide how quickly to privately obtain information about the profitability of a project and when (if ever) to publicly invest in it. Multiple equilibria exist, differing with respect to how much information firms acquire as well as how quickly they invest. The equilibrium that maximizes aggregate payoffs features asymmetric play with distinct leader and follower roles when firms are patient, but features symmetric play when firms are impatient and information acquisition costs are sufficiently high. Keywords. Social learning, investment timing, strategic information acquisition. JEL classification. C73, D82, D83, G24. 1. Introduction In many economic settings, decision makers may strategically delay irreversible action so as to learn from the actions of others. For instance, oil firms can delay drilling on leased tracts to learn from the drilling decisions of firms on nearby tracts,1and venture capitalists can delay investing in startups to learn from the funding decisions of other investors.2More generally, incentives for strategic delay arise whenever information about payoffs is dispersed, opportunities are nonrival, and decision makers may freely time their actions. Our starting point is the observation that in many applications, a decision maker’s private information is the result of costly information-acquisition activities. For instance, oil firms conduct seismic surveys to estimate the extent of oil deposits on a tract, Rishabh Kirpalani: [email protected] Erik Madsen: [email protected] We are grateful to Nageeb Ali for many stimulating conversations and suggestions. We would also like to thank Alessandro Bonatti, Laura Doval, Marina Halac, Boyan Jovanovic, Emir Kamenica, Laurent Mathevet, Dmitry Orlov, David Pearce, and Andy Skrzypacz for helpful discussions and comments. 1See Hendricks and Kovenock (1989) for a discussion of incentives for social learning in offshore oil drilling and Hendricks and Porter (1996) for empirical evidence of strategic delay in this setting. 2Paul Graham, a prominent entrepreneur and venture capitalist, has discussed the importance of social learning among venture capitalists: “The biggest component in most investors’ opinion of you is the opinion of other investors...When one investor wants to invest in you, that makes other investors want to, which makes others want to, and so on” (Graham (2013)). ©2023 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4806 1142 Kirpalani and Madsen Theoretical Economics 18 (2023) and venture capitalists perform due diligence to gauge the quality of a startup’s product and management team. Strategic incentives then shape both how much information is produced through private effort, as well as how much is aggregated through public actions. A key insight from the literature on experimentation in teams is that when information acquisition is costly, decision makers tend to free-ride by inefficiently reducing their rate of information acquisition. We build on that insight by assuming, in a departure from existing work, that learning is both private and imperfect, and that decision makers reveal what they know only by irreversible action. These features create an incentive for players to strategically delay acting on good news in addition to, or instead of, free-riding on the acquisition of news. Our model provides a tractable framework for studying the equilibrium interplay of incentives for free-riding and strategic delay. In our model, two firms have the opportunity to invest in a nonrival risky project. Firms may dynamically exert variable costly effort, a process we call prospecting, for the chance of receiving a binary signal that is informative about the project’s value. Each firm can acquire at most one signal, and signals are conditionally independent. As a result, aggregating signals from multiple firms yields information about the profitability of investment beyond what any one firm could learn. Any information a firm acquires through prospecting is private but investment is public. We show that there are exactly three perfect Bayesian equilibria of our model. In the unique symmetric equilibrium, each firm prospects as intensively as possible until a cutoff time, after which it abandons prospecting forever if it has not seen investment by the other firm. At any time before the cutoff, if a firm receives a positive signal, it invests without delay. This equilibrium exhibits no free-riding or investment delay. There are also two asymmetric “leader–follower” equilibria. In these equilibria, one firm takes the role of a leader, prospecting until acquiring a signal and investing without delay if the signal is positive. Meanwhile the remaining firm follows the leader by either free-riding on the leader’s prospecting efforts, delaying investment after acquiring a signal, or both. The mix of the two behaviors depends on the cost of prospecting: Investment delay arises when costs are low, free-riding emerges when costs are high, and for intermediate costs, delay is followed by eventual free-riding. In the low-cost regime, not only is there no free-riding, but the follower spends more time prospecting than it would have in the symmetric equilibrium. In contrast to existing models of free-riding and investment delay, neither equilibrium generates unambiguously larger amounts of social learning. In general, the symmetric equilibrium produces more information early on, while the leader–follower equilibrium produces more at later times. As a result, either equilibrium can generate higher total payoffs, depending on model parameters. We show that when firms are patient, the leader–follower equilibrium generates higher aggregate payoffs, while when firms are impatient and prospecting costs are sufficiently high, the symmetric equilibrium is superior. The remainder of the paper is organized as follows. Section 1.1 surveys related literature. Section 2describes the model. Section 3characterizes the set of perfect Bayesian equilibria of the model. Section 4compares payoffs across equilibria. Section 5con- cludes. Theoretical Economics 18 (2023) Strategic investment allocation 1143 1.1 Related literature Our paper is most closely connected to models of collective experimentation, in particular Bonatti and Hörner (2011,2017), Bolton and Harris (1999), Keller, Rady, and Cripps (2005), Keller and Rady (2010,2015), and Dong (2021).3These papers study environments in which effort simultaneously dictates both the production and the aggregation of information. This linkage is a key feature of the canonical bandit experimentation framework, in which players learn by monitoring the returns to incrementally investing effort in a project. Our paper departs from this literature by separating learning from the payoffs generated by a project. This separation allows us to make learning completely private, with information aggregation instead associated with a distinct decision to collect payoffs. Our model builds most directly on the work of Bonatti and Hörner (2011) (hereafter BH), who study strategic experimentation with private effort and learning, in a setting where signals arrive via a Poisson good news process with perfectly informative breakthroughs.4A key dynamic in both their model and ours is a gradual deterioration of each player’s beliefs due to continued inaction by another player, which is taken as a negative signal about their private information. Methodologically, our model differs by modeling negative signals as arriving discretely rather than continuously, simplifying equilibrium characterizations by avoiding belief divergences following deviations from equilibrium effort. Conceptually, it differs by assuming that good news is not perfectly revealing, creating a motive for players to delay investment once they have obtained a positive signal. Several papers pursue related approaches to separating information production and aggregation in collective experimentation. Heidhues, Rady, and Strack (2015) find that in a classic bandit model, outcomes improve when payoffs are private and disclosed with delay via a cheap-talk communication channel. Our paper more severely restricts possibilities for communication, generating distinctive welfare implications from private learning. Guo and Roesler (2018) augment the model of BH with discrete negative signals, which can be signaled by irreversibly dropping out of the project. In their model these signals are perfectly revealing, so that delay in dropping out is driven by free-riding rather than social-learning concerns. Our paper is also related to models of investment timing. One set of papers assumes that players receive exogenous private signals of the state, either at time zero or dynamically. Papers in this tradition include Chamley and Gale (1994), Gul and Lundholm (1995), Chari and Kehoe (2004), Rosenberg, Solan, and Vieille (2007), and Murto and Välimäki (2011,2013). Aghamolla and Hashimoto (2020) endogenize the precision of 3See Hörner and Skrzypacz (2017) for an excellent survey of this literature. 4Formally, in their model breakthroughs are publicly observed and immediately accrue a common payoff to all players. Because breakthroughs are perfectly informative, their results would not change if breakthroughs were private and players publicly invested to collect a state-contingent payoff. 1144 Kirpalani and Madsen Theoretical Economics 18 (2023) a private signal received at the start of the game, but do not allow agents to dynamically acquire information. A second strand of the literature abstracts from private information about the project and instead assumes that investment generates public signals of the project’s profitability. Papers in this tradition include Décamps and Mariotti (2004), Fajgelbaum, Schaal, and Taschereau-Dumouchel (2017), and Frick and Ishii (2020). Klein and Wagner (2022) span the two sets of papers by endowing players with time-zero private information and assuming that investment generates further public information. None of these papers features a trade-off between free-riding and investment delay, a tension that plays a key role in our model. In addition, most of these papers focus on symmetric play. One exception is Gul and Lundholm (1995), who find that asymmetric play reduces delay and raises aggregate payoffs. In contrast, our analysis identifies a nontrivial trade-off between the payoffs generated by symmetric and asymmetric play, which can yield higher aggregate payoffs for either type of equilibrium depending on model parameters. Finally, our paper shares important features with work by Ali (2018)andCampbell, Ederer, and Spinnewijn (2014). Ali (2018) endogenizes information acquisition in a model where players invest in a predetermined sequence. It can, therefore, be viewed as a fixed-move-order analog to our exercise of endogenizing information acquisition when players invest flexibly. Campbell, Ederer, and Spinnewijn (2014) study a team production problem in which production is private and separate from the decision to disclose progress. This separation is analogous to the separation of learning and information aggregation in our setting. 2. The model Two firms have the opportunity to invest one unit of capital in a nonrival risky project of unknown quality. The project has underlying type θand is either good (θ=G)orbad (θ=B). If θ=G, each unit of capital invested in the project generates cash flows with a net present value of R, beginning at the time that unit of capital is invested; if θ=B, the project generates no cash flows. We assume that R>1, so that each unit of capital invested in the project generates positive returns in the good state. Each firm is free to invest in the project at any time t∈R+. Firms are risk-neutral with common discount rate r>0. Capital is indivisible, investment in the project is irreversible, and project outcomes are observed only by players who invest.5 Both firms begin with a common prior belief π0∈(0, 1)that the project is good. Each firm i=1, 2 can exert costly effort to privately search for an informative signal about the project’s quality, an activity we will refer to as prospecting. The goal of prospecting is to uncover a binary signal Si∈{H,L}, i.e., high or low, which is correlated with the state of the project: Pr(Si=H|θ=G)=qHand Pr(Si=L|θ=B)=qL,withqH,qL∈(1/2, 1). Any prospecting that a firm undertakes and any signal that results are observed only by 5A natural interpretation of the private observability of outcomes is that the project’s cash flows are realized far in the future. Theoretical Economics 18 (2023) Strategic investment allocation 1145 the firm conducting the prospecting. Each firm can obtain at most one signal, and firms observe conditionally independent, identically distributed signals. Prospecting is a dynamic process unfolding in continuous time. Over every time interval [t,t+dt], each firm ichooses a prospecting rate λi t≥0, which causes a signal to arrive with probability λi tdt while incurring an effort expense of C(λi t)dt. Following much of the literature on collective experimentation,6we assume a linear cost structure C(λ)=cλ,λ∈[0, λ] ∞,λ∈(λ,∞) for some constant marginal cost c>0 and maximum prospecting rate λ,bothofwhich are symmetric across firms. Conditional on prospecting rates, signal arrival times are independent across firms and independent of the state of the project. Firms cannot observe each other’s signals or prospecting intensities, or observe whether another firm has received a signal or obtained a good outcome from investment. There are also no communication channels between firms. However, all investment decisions are public, introducing a channel for social learning. 2.1 Notation and assumptions We will denote the posterior beliefs induced by one or more signals as follows: π+and π++ are the posteriors induced by one and two high signals, respectively; π−and π−− are the posteriors induced by one and two low signals; π+− is the posterior induced by one high and one low signal. (Exchangeability implies that posterior beliefs are independent of the order of receipt of signals.) Given that high signals are more likely when the state is good, and conversely for low signals when the state is bad, π++ >π +>π 0, π+− >π −>π −−. Note that, in general, π+− = π0, except in the special case when qH=qL. Suppose that a firm receives a signal when its current beliefs that θ=Gare μ∈[0, 1]. Then the total probability that the signal is high is h(μ)≡qHμ+(1−qL)(1−μ), while the corresponding probability that the signal is low is l(μ)≡1−h(μ).Thequantities h(μ)and l(μ)are the transition probabilities that a firm’s posterior belief jumps up or down upon receiving a signal. Following acquisition of a signal, we will write μ+≡qHμ/h(μ)for the firm’s updated belief if the signal is high and μ−≡(1−qH)μ/l(μ) if the signal is low. We impose several bounds on prospecting costs and the payoff of a good project. Assumption 1. The payoff of a good project satisfies 1/π+<R<1/π0. Under this assumption, investment in the project is ex ante unprofitable, but becomes profitable conditional on observation of a high signal.7 6See Keller, Rady, and Cripps (2005) and BH for classic examples of team experimentation models assuming linear experimentation costs. 7The case R<1/π+is uninteresting, as the unique equilibrium involves no prospecting and no investment by either firm. 1146 Kirpalani and Madsen Theoretical Economics 18 (2023) Assumption 2. The payoff of a good project satisfies R<1/π+−. This assumption ensures that learning another firm’s signal is useful even after acquisition of a high signal, since an additional low signal would push beliefs back below the break-even threshold. This assumption in conjunction with Assumption 1rules out perfectly informative good news, as such signals correspond to π+=π+− =1, in which case no Rcan simultaneously satisfy the bounds in both assumptions. These assumptions, therefore, distinguish our setting from classic experimentation models like Keller, Rady, and Cripps (2005) and BH, where a single positive outcome is definitive. Assumption 3. The cost of prospecting satisfies c≤c≡h(π+)(π++R−1)−(π+R−1). This assumption ensures that a second signal is at least potentially profitable to acquire, in the sense that if it could be attained instantaneously, it would provide enough information to be worth the cost. As with Assumption 2, this assumption focuses our analysis on environments in which combining information from multiple signals is strategically relevant. Note that Assumptions 1and 2ensure that c>0. 2.2 Single-player benchmark Consider a single firm prospecting and investing on its own, shutting down the social learning channel of our model. The firm’s initial beliefs that the project is good will be taken to be μ<1/R. We will refer to this benchmark setting as autarky. As long as the firm has acquired no signal, it learns nothing about the project and its beliefs remain fixed at μ. An optimal prospecting strategy is, therefore, stationary. This behavior differs from the cutoff strategies that are optimal when learning from Poisson bandits, for instance as in BH. In Poisson bandit models, lack of arrival of a signal is itself news about the underlying state, leading to belief updating. In our model, in contrast, lack of signal acquisition does not signal anything, positive or negative, about the true project state; no news truly is no news until a signal arrives.8Once the firm has acquired a signal, no further information is available. It then faces a simple static choice of whether or not to invest, which it resolves by comparing its posterior beliefs to the investment threshold 1/R. The optimal prospecting strategy depends on whether the firm’s initial beliefs μlie above a critical threshold, which we will denote πAand refer to as the autarky threshold. It is formally characterized as the unique belief satisfying h(πA)(πA+R−1)=c, which equalizes the marginal flow gains and costs from a unit of prospecting. (Recall our notational convention that πA+=qHπA/h(πA)are the posterior beliefs following receipt of a high signal, when beliefs are πAprior to observing the signal.) If μ<π A,the firm abandons prospecting immediately. On the other hand, if μ>π A, then the firm prospects at the maximum rate λuntil a signal is acquired. Note that πAis increasing in the cost parameter c. 8A similar signal acquisition technology is employed in Akcigit and Liu (2016). Theoretical Economics 18 (2023) Strategic investment allocation 1147 Abandonment of prospecting if beliefs fall below πAoccurs even with multiple firms: If π0lies below πA, then no prospecting or investing takes place in equilibrium, despite the potential for social learning. Going forward we will assume that π0>π Afor all costs below c, which the following lemma establishes is equivalent to assuming Ris sufficiently large. Lemma 1. There exists a unique R0∈(1/π+,1/max{π+−,π0})such that π0>π Afor every c≤cif and only if R>R 0. Assumption 4. The payoff of a good project satisfies R>R 0. Lemma 1ensures that this bound is compatible with the restrictions on Rimposed by Assumptions 1and 2. The bound could be dispensed with, at the cost of a more stringent upper bound on allowed prospecting costs. To streamline our analysis, we maintain Assumption 4going forward. 3. Equilibrium analysis In this section, we characterize the set of perfect Bayesian equilibria of the model. Going forward, we will use the term “equilibrium” without qualification to refer to elements of this set. We find that our model has exactly three equilibria. One equilibrium is symmetric and exhibits no free-riding or investment delay, but leads both firms to eventually abandon prospecting for information about the project. The remaining leader– follower equilibria feature distinct roles for the two firms, with one firm that takes the lead in prospecting and investing while the other firm plays a passive follower role. In general, this equilibrium features either free-riding, investment delay, or both by the follower, with the mix shifting from investment delay toward free-riding as prospecting costs rise. This section is structured as follows. In Section 3.1, we describe each firm’s optimal continuation strategy after observing investment by the other firm. In Sections 3.2 and 3.3, we characterize the symmetric and leader–follower equilibria and provide intuition for their properties. In Section 3.4, we prove that no other equilibria exist. 3.1 Behavior after observing investment In the spirit of backward induction, we first characterize a firm’s optimal continuation strategy after observing the other firm invest. It can be shown that in any equilibrium, the first firm to invest is always in possession of a high signal.9The remaining firm, therefore, finds itself in a stationary single-player environment analogous to the autarky benchmark studied in Section 2.2. If the firm has already acquired a signal, its beliefs are either π++ >1/R or π+− <1/R, and no further information can be acquired. The 9See Appendix Afor a formal derivation of this result. 1148 Kirpalani and Madsen Theoretical Economics 18 (2023) firm, therefore, either invests immediately if its signal is high or abandons the project otherwise. On the other hand, if the firm has not yet acquired a signal, its beliefs are π+>1/R and it has the opportunity to acquire a signal before investing. This additional signal is pivotal, given that π+− <1/R, and would be worth the cost of acquiring if not for time discounting, given c≤c. Whether the signal is, in fact, worth acquiring depends on the comparison between the gains from more information, mediated by R, and the cost and delay of obtaining it, captured by c,r,andλ. As Rrises, the firm becomes less willing to acquire an additional signal, because the downside of a bad project becomes less important relative to the upside of a good one. The firm also becomes less willing to wait if prospecting or delay costs rise, i.e., if cor r increase or λdecreases. Either acquiring an additional signal or investing immediately can be optimal, depending on parameters. The following lemma formally states how the optimal strategy changes with the discount rate r, a comparative static that will be particularly useful for later results.10 (The proof is straightforward and so is omitted for brevity.) Lemma 2. There exists a threshold discount rate r∗≥0such that in any equilibrium, subsequent to investment by some firm, continuation play proceeds as follows: •If r≤r∗, the remaining firm prospects at rate λuntil acquiring a signal and invests immediately if it acquires a high signal. •If r>r ∗, the remaining firm invests immediately if it has not yet acquired a low signal. 3.2 The symmetric equilibrium We now characterize the unique symmetric equilibrium of the model. This equilibrium exhibits no free-riding or investment delay, but does involve eventual abandonment of prospecting by both firms. To state the equilibrium, we define a time threshold at which a firm’s posterior beliefs reach πA, assuming the other firm never delays signal acquisition or investment. Suppose that some firm iprospects at rate λforever and invests immediately whenever it obtains a high signal. Let μλ(t)denote the associated posterior beliefs of firm −i that θ=G, conditional on observing no investment by firm iuntil time t. These beliefs decline over time, converging to π−as t→∞and firm −ibecomes sure that continued lack of investment implies that firm ihas obtained a low signal. Since π+− <1/R, it must be that πA>π −,andsoμλ(t)crosses the autarky threshold πAat some finite time, which we will denote TA≡(μλ)−1(πA). Proposition 1 (The symmetric equilibrium). There exists a symmetric equilibrium in which, whenever no investment has occurred, each firm’s continuation play proceeds as follows: 10An identical result holds with respect to 1/λ. Analogous results could also be stated for Rand c,but with some additional care needed to account for the boundary conditions on these parameters. Theoretical Economics 18 (2023) Strategic investment allocation 1155 investing, depending on model parameters. Once this time is pinned down, it can be shown that the leader’s unique best response is to remain active prior to time  T−i,which uniquely determines the remainder of the equilibrium. 4. Comparing equilibrium payoffs We have seen that our model has exactly two distinct equilibrium structures. In this section we compare individual and aggregate payoffs across equilibria. Let VSbe the expected payoff of each firm in the symmetric equilibrium, and let VL and VFbe the expected payoffs to the leader and follower, respectively, in the leader– follower equilibrium. Aggregate payoffs in the symmetric equilibrium are then 2VS, while in the leader–follower equilibrium they are VL+VF. The following proposition examines how both individual and aggregate payoffs compare across the two equilibria. Proposition 4. Equilibrium payoffs satisfy VF>VS≥VL.Ifris sufficiently small, then VL+VF>2VS.Thereexistsac < c (independent of r)suchthatifc>c,then2VS> VL+VFfor rsufficiently large. The first result of the proposition is that the symmetric equilibrium generates lower payoffs for each firm than the follower’s payoff, but (weakly) higher payoffs than the leader’s payoff. Intuitively, the leader–follower equilibrium generates more information from the leader but less from the follower than each firm would produce in the symmetric equilibrium, and this change in social learning is reflected in the remaining firm’s payoff. Interestingly, the leader is not necessarily strictly worse off than it would be in the symmetric equilibrium. This is because when the discount rate is low, each firm acquires its own signal before investing, even after seeing the other firm invest. (See Lemma 2.) In addition, in the symmetric equilibrium, neither firm waits for the other to invest once they are in possession of a high signal. Social learning, therefore, turns out not to be pivotal for investment in the symmetric equilibrium at low discount rates, and so the additional information generated does not raise payoffs. This second result of the proposition is that when firms are patient, the leader– follower equilibrium yields higher total firm profits than the symmetric one, while when firms are impatient (and costs are not too low), the symmetric equilibrium is superior. If the discount rate is low enough that VL=VS, this result is an immediate consequence of the individual payoff ranking. However, if VL<VS, comparing aggregate payoffs requires a balancing of higher payoffs generated by the symmetric equilibrium early on against higher payoffs generated by the leader–follower equilibrium later. Consequently, the comparison between the two equilibria turns on the discount rate. More precisely, in the symmetric equilibrium, both firms contribute to social learning by actively prospecting and investing until the time TA. By contrast, in the leader– follower equilibrium the leader remains active forever, while the follower remains active only up to some time  TF<TA. Comparing welfare therefore amounts to comparing aggregate social learning in each equilibrium, taking into account time discounting. 1156 Kirpalani and Madsen Theoretical Economics 18 (2023) In the symmetric equilibrium, more social learning occurs during the time interval [ TF,TA]than in the leader–follower equilibrium, while the latter equilibrium features more social learning during the time interval [TA,∞). When firms are patient, the long duration of social learning in the leader–follower equilibrium is the most important factor determining welfare, and so aggregate payoffs are higher in this equilibrium. By contrast, when firms are impatient, the additional social learning generated early on in the symmetric equilibrium becomes important. The subtlety in this argument is that  TFapproaches TAas the discount rate grows large, requiring a careful calculation of total gains in the limit as r→∞.Itturnsoutthat, under an appropriate normalization, the limiting gains from additional social learning early in the symmetric equilibrium are strictly positive and increasing in c. Thus when r and care sufficiently large, these normalized gains outweigh the normalized losses from reduced social learning later on, yielding higher aggregate payoffs than in the leader– follower equilibrium.18 The ambiguity of this payoff comparison stands in contrast to the findings of previous work on investment timing and collective experimentation. A classic example in the investment timing literature is Gul and Lundholm (1995), who find that asymmetric equilibria eliminate the war of attrition inherent in symmetric play and reveal private information more quickly, improving aggregate welfare. By contrast, since information acquisition is endogenous in our model, asymmetric play generates an additional profit loss by reducing the incentives for the second mover to produce and reveal information. The relative performance of symmetric and asymmetric play then becomes a horse race between the war of attrition of the symmetric equilibrium and the free-riding of the asymmetric equilibrium. This finding demonstrates the importance of modeling incentives for information acquisition alongside investment timing when both effects are present in applications. Meanwhile in the collective experimentation literature, BH find that asymmetric play increases aggregate payoffs versus symmetric play. Specifically, in the two-player version of their model, they characterize a continuum of asymmetric equilibria indexed by the time at which the follower stops free-riding and begins exerting effort. They show that aggregate payoffs are increasing in the amount of time the follower spends freeriding. Key to their result is the fact that the more players are actively exerting effort, the less total effort is exerted. By contrast, in our setting asymmetric play has an ambiguous effect on total effort: Effort early on is lower than in the symmetric equilibrium, while effort at later times is higher. Our contrasting welfare results are, therefore, driven by important differences in behavior across the two models in both symmetric and asymmetric equilibria. 5. Conclusion We study a model of strategic investment timing with endogenous information acquisition, with the aim of understanding the interplay of incentives for free-riding and investment delay. We find that the extent and mix of free-riding and investment delay varies 18The cost bound cneed not be very stringent. In particular, it can be shown that c=0 when positive and negative signals are both sufficiently informative. Theoretical Economics 18 (2023) Strategic investment allocation 1157 across equilibria as well as with the cost of acquiring information. We further find that the equilibrium that maximizes aggregate payoffs varies with model parameters, the discount rate in particular. These results are closely linked to our central assumption that positive signals are imperfectly informative, revealing new economic forces that are absent in models of strategic experimentation with observable or perfectly revealing signals. One limitation of our current analysis is its focus on a two-player setting. Extending our work to accommodate many players would bring it closer to applications as well as permit a richer study of the possibilities of asymmetric play. In particular, with many players, there might exist additional asymmetric equilibria featuring multiple active players. Comparing aggregate payoffs across different asymmetric configurations could reveal novel trade-offs that further illuminate when and how asymmetric play boosts payoffs. Our analysis also restricts attention to environments in which investment represents a pure information externality. In some applications, investment may additionally generate payoff externalities. For instance, early-stage startups may exhibit increasing returns to scale and generate higher profits, or a greater probability of success, when they are better funded. In that case, investment by one firm would raise the return on investment by another, strengthening incentives for strategic delay. Extending our model to incorporate increasing returns to scale would enhance its realism in such applications and allow informational and payoff externalities to be compared as sources of strategic delay. Appendix A: Regular strategies In this appendix, we establish that in any perfect Bayesian equilibrium, lack of investment is (weakly) bad news about the state, while investment signals that the investing firm has received a high signal. Definition A.1. A firm’s strategy is regular if the following conditions hold: •Investment never occurs after receipt of a low signal •Investment without a signal occurs only in histories in which the other firm has invested. The following lemma establishes that all firms choose regular strategies in equilibrium. We shall invoke this fact repeatedly in what follows to focus our analysis on a firm’s best response to play of a regular strategy by its rival. Lemma A.1. In any equilibrium, each firm’s strategy is regular. Proof. Fix an equilibrium. First consider a firm that has obtained a low signal. Then regardless of its beliefs about the content of any signal obtained by the other firm, its posterior belief that the state is good cannot be higher than π+−.Asπ+−R−1<0by 1158 Kirpalani and Madsen Theoretical Economics 18 (2023) assumption, investment in such a history is unprofitable. Thus, in any equilibrium, no firm invests in such a history. Now consider a firm ithat has obtained no signal by time t.Ifibelieves that −i,when following its equilibrium strategy, would have invested with probability strictly less than 1bytimet, then this history is on-path. Firm imay then use Bayes’ rule to update its beliefs about firm −i’s signal, and as −idoes not invest when in receipt of a low signal, lack of investment by time tis weakly negative news about S−iand, therefore, about θ. Thus, firm i’s posterior beliefs that θ=Gare no higher than π0, and investment in such a history is unprofitable. Thus, in any equilibrium, no firm invests in such a history. The remaining possibility is that firm ihas obtained no signal and is in a history at which firm −i’s strategy called for investment with probability 1 prior to time t.Such histories are off-path, and firm iis then free to choose its beliefs about S−iarbitrarily. To complete the proof, we argue that such off-path histories cannot arise in any equilibrium. Let ρi(t)be the cumulative time-tprobability that each firm, under its equilibrium strategy, invests prior to time tabsent observing investment by its rival. Each ρiis weakly increasing and left-continuous. Off-path histories correspond to ρi(t)=1. Let t∗≡inf{t:max{ρ1(t),ρ2(t)}=1}, and suppose by way of contradiction that t∗<∞. Prior to time t∗, histories are on-path, and so no firm invests when in possession of no or a low signal. As a result, it must be that ρi(t)≤(1−exp(−λt))h(π0)for each t<t ∗and firm i, since each firm’s prospecting rate is bounded above by λ. Thus, by leftcontinuity, ρ1(t∗),ρ2(t∗)<1. Then also at time t∗, histories involving lack of investment are on-path for both firms, meaning no firm invests when in possession of a low signal. But also by definition there exists a firm ifor which ρi(t)=1 for arbitrarily small t>t ∗, meaning that firm imust invest with strictly positive probability when in possession of no signal at time t∗. This is a contradiction, and so t∗=∞, meaning all histories are on-path. Appendix B: Belief updating identities In this appendix, we derive several useful identities involving posterior beliefs about the state in the event no investment by the other firm has been observed. Fix a firm iand a pure strategy for firm −iwhich is regular (as defined in Appendix A) and invests when in possession of a high signal if and only if t<T −ifor some T−i∈ R+∪{∞}.Wewillletμi(t)denote firm i’s time-tbelief that the state is good, supposing it has obtained no signal and observed no investment. Additionally, we will let νi(t) denote firm i’s time-tbelief that firm −ihas not yet obtained a signal, given that it has not yet invested. Lemma B.1. The posterior belief path μiis absolutely continuous and satisfies ˙μi(t)=−1t<T −iνi(t)λ−i(t)h(π0)π+−μi(t) almost everywhere. Theoretical Economics 18 (2023) Strategic investment allocation 1159 Proof. For all times t>T −i,firmiis in autarky with fixed beliefs, in which case μi(t) is trivially absolutely continuous and satisfies the stated identity. So consider times t≤T−i.Thenfirm−iinvests at variable Poisson rate νi(t)λ−i(t)h(π0),andarrivalof investment causes beliefs to jump from μi(t)to π+. It follows that μi(t)is absolutely continuous and satisfies the Bayes plausibility condition that the average rate of change of beliefs must be zero, i.e., νi(t)λ−i(t)h(π0)π+−μi(t)+˙μi(t)=0, which is the desired identity. Lemma B.2. The posterior belief path μisatisfies ˙μi(t)=−1t<T −iλ−i(t)μi(t)−π− π+−π−π+−μi(t) almost everywhere. Proof. For all times t≥T−i,firmiis in autarky with fixed beliefs, in which case the identity trivially holds. So assume t<T −i. Define −i(t)=exp−t 0 λ−i(s)ds to be the cumulative probability that firm −ihas not obtained a signal by time t.By Bayes’ rule, μi(t)=−i(t)+1−−i(t)1−qHπ0 −i(t)+1−−i(t)l(π0)=−i(t)π0+1−−i(t)l(π0)π− −i(t)+1−−i(t)l(π0). Solving this identity for −i(t)yields −i(t)=l(π0) h(π0) μi(t)−π− π+−μi(t). Taking the log of both sides, differentiating, and using the identity d dt log−i(t)= −λ−i(t)yields the desired relationship. Appendix C: Value functions and the HJB equation In this appendix, we describe properties of a given firm i’s continuation value function in several important classes of histories, supposing that firm −iuses a pure strategy which is regular (as defined in Appendix A) and invests when in possession of a high signal if and only if t<T −ifor some T−i∈R+∪{∞}. We will use the following notation for value functions in different histories: Vi(t)will denote firm i’s time-tcontinuation value function given no signal and no investment by firm −i;Vwill denote i’s continuation value upon seeing firm −iinvest (note that Vis independent of iand t); Vi +(t)will denote firm i’s time-tcontinuation value function given 1160 Kirpalani and Madsen Theoretical Economics 18 (2023) a high signal and no investment by firm −i. Finally,  Vi(t)will denote firm i’s expected time-tcontinuation value after obtaining a signal, given no investment by firm −i. Since obtaining a low signal leads to no investment, it follows that  Vi(t)=h(μi(t))Vi +(t). Pre-signal/investment By standard arguments, Viis the unique bounded, absolutely continuous function satisfying the HJB equation rV i(t)=λ Vi(t)−c−Vi(t)++1t<T −iνi(t)λ−i(t)h(π0)V−Vi(t)+˙ Vi(t), where νi(t)is firm i’s time-tbelief that firm −ihas not yet obtained a signal given that it has not yet invested. Using Lemma B.1, the second term on the right-hand side may be rewritten in terms of μi(t),firmi’s posterior belief that the state is good: rV i(t)=λ Vi(t)−c−Vi(t)+−˙μi(t) π+−μi(t)V−Vi(t)+˙ Vi(t). Note that the sign of  Vi(t)−c−Vi(t)determines firm i’s optimal prospecting rule: When it is positive, the firm optimally prospects at rate λ; when it is negative, the firm optimally prospects at rate 0; when it is zero, any prospecting rate is optimal. At various points in our analysis, it will be useful to express the HJB equation as Fi(Vi,t)=0, where Fi(w,t)≡rw(t)−λ Vi(t)−c−w(t)++˙μλ(t) π+−μλ(t)V−w(t)−˙ w(t) is a functional that may be applied to arbitrary test functions wto compute the remainder of the HJB equation evaluated at w. Post-investment The continuation value Vsolves a simple single-agent problem analogous to the autarky case of Section 2.2, but with a choice between prospecting and immediate investment rather than between prospecting and free-riding given that π+>1/R.ThetermVmay be characterized explicitly as V=maxπ+R−1, λ λ+rh(π+)(π++R−1)−c. Recall from Lemma 2that r∗is defined as the minimal discount rate at which firm i invests immediately after seeing firm −iinvest. Thus, r∗corresponds to the smallest r such that the first argument of the max operator dominates. Theoretical Economics 18 (2023) Strategic investment allocation 1161 Post-signal Following observation of a high signal, firm i’s continuation payoff  Vi(t)is bounded below by the payoff of investing immediately if the signal is high and never investing otherwise. Thus,  Vi(t)≥h(μi(t))(μi +(t)R−1). Some algebra yields the useful associated identity h(μ)(μ+R−1)−c=K(μ−πA), where K≡qH(R−1)+(1−qL)>0. Thus,  Vi(t)−c≥K(μi(t)−πA), and the inequality holds with equality if firm ioptimally invests immediately upon obtaining a high signal at time t. Lemma C.1. The post-investment continuation value satisfies V≤K(π+−πA). Proof.Ifr≤r∗,thenV=λ λ+rK(π+−πA), in which case V<K (π+−πA).Otherwise, V=π+R−1, and so by Assumption 3,V≤K(π+−πA). Appendix D: Proofs D.1 Proof of Lemma 1 The requirement that π0>π Afor every c≤cis equivalent to the condition h(π0)(π+R− c)>c. By the law of total probability, π+R−1=h(π+)(π++R−1)+l(π+)(π+−R−1), so that c=−l(π+)(π+−R−1)and the desired condition may be stated as φ(R)>0, where φ(R)≡h(π0)(π+R−1)+l(π+)(π+−R−1). Note that φ(R)is strictly increasing in R,andφ(1/π+)=l(π+)(π+−/π+−1)<0 while φ(1/π+− )=h(π0)(π+/π+− −1)>0. Further, for any R<1/π+−, l(π+)(π+−R−1)>l (π0)(π+−R−1)>l (π0)(π−R−1), in which case φ(1/π0)>π 0R−1. So if 1/π0<1/π+−,wehaveφ(1/π0)>0. Thus, φ(1/max{π0,π+−})>0. It follows that there exists a unique R0, bounded between 1/π+ and 1/max{π0,π+−},atwhichφcrosses zero, as desired. D.2 Proof of Proposition 1 Fix a firm iand suppose firm −ifollows its equilibrium strategy. We first show that firm i’s equilibrium investment policy is a best response. By Lemma D.10,firmi’s optimal policy must be a threshold rule, so it remains only to argue that T∗ i=∞is the optimal threshold. Consider any time t>T Aand history in which firm ihas obtained a high signal. Because μi(t)=πAand h(πA)(πA+R−1)>0, it follows that μi +(t)>1/R.So investing immediately when in possession of a high signal at any time, which yields a 1162 Kirpalani and Madsen Theoretical Economics 18 (2023) payoff of μi +(t)R−1>0, dominates waiting until firm −iinvests, which yields a payoff of 0 (because firm −inever invests). As this argument holds for arbitrary large t>TA,it must be that T∗ i=∞is optimal. It remains to verify that firm i’s optimal prospecting policy prior to obtaining a signal is a threshold policy with Ti=TA. Subsequent to the cutoff time TA,thefirmisin autarky with beliefs πA,soλi(t)=0 is trivially an optimal strategy from this point onward. So consider times prior to TA.LetV†(t)≡K(μλ(t)−πA),whereKis as defined in Appendix C. Inserting V†into the function Fidefined in Appendix Cand using the fact that  Vi(t)=V†(t)+cfor all tgiven that μi=μλand T∗ i=∞,wehave FiV†,t=rV †(t)+˙μλ(t) π+−μλ(t)V−K(π+−πA). Note that for t<T A,V†(t)>0and ˙μλ(t)<0. Meanwhile Lemma C.1 in Appendix C establishes the bound V≤K(π+−πA).SoFi(V†,t)>0 for times t<TA. Now note that V†(TA)=0 by definition of TA, while also Vi(TA)=0 given that firm iis in autarky with beliefs πAsubsequent to TA. Therefore, V†(TA)=Vi(TA). This boundary condition, combined with the fact that Fi(V†,t)>F i(Vi,t)=0forall t<T A, implies by a standard result regarding supersolutions of ordinary differential equations that V†(t)>Vi(t)for all t∈[0, TA].Thenas Vi(t)≥V†(t)+c, prospecting at the maximum rate prior to TAis an optimal strategy. D.3 Proof of Proposition 2 We first characterize the follower’s best response to the leader. This characterization is built around a pair of belief thresholds that pin down the times at which the follower stops prospecting and investing. Let I(μ)≡μ+−π+− π++ −π+− λ λ+r(π++R−1)−(μ+R−1). As will be shown later, Irepresents the difference in payoffs between waiting and investing immediately following receipt of a high signal when current beliefs are μ. Lemma D.1. The difference Iis a strictly decreasing function of μ,andI(π−)>0.Also, I(π0)=λ λ+rh(π+)(π++R−1)−(π+R−1). In particular, I(π0)>0whenever r≤r∗. Proof. Differentiating Iyields  I(μ)=1 π++ −π+− λ λ+r(π++R−1)−Rdμ+ dμ . Theoretical Economics 18 (2023) Strategic investment allocation 1163 By assumption, π+− <1/R < π++,so (μ)<−r λ+rRdμ+ dμ <0. Further, I(π−)=− (π+−R−1)>0. Finally, to simplify I(π0), use the law of total probability to write π+=h(π+)π++ +(1−h(π+))π+− or, equivalently, π+−π+− = h(π+)(π++ −π+− ). This identity may be used to write I(π0)in the desired form. In light of the previous lemma, define the investment belief threshold μ∗∈(π−,π0] as μ∗≡π0,I(π0)≥0, −1 I(0),I(π0)<0. Define the associated investment time threshold T∗ F≡(μλ)−1(μ∗). This threshold is uniquely defined given that μλis a strictly decreasing function satisfying μλ(0)=π0 and μλ(∞)=π−. Next, define P(μ)≡μ−π− π+−π− λ λ+rV−ˇ V(μ)−c, where ˇ V(μ)≡h(μ)maxμ+R−1, μ+−π+− π++ −π+− λ λ+r(π++R−1). We will see later that Prepresents the difference in payoffs between prospecting or not when current beliefs are μ,and ˇ V(μ)represents the average continuation value after obtaining a signal at beliefs μ. We note two important properties of ˇ V. First, the argument of the max operator that dominates depends on the size of μrelative to μ∗, with the first argument dominating when μ>μ ∗, while otherwise the second argument dominates. (When (π0)≤0, the two branches are equal when μ=μ∗. Otherwise, the second argument dominates when μ=μ∗.) Second, ˇ V(μ)can be rewritten using Lemma D.2 as ˇ V(μ)≡maxh(μ)(μ+R−1),μ−π− π+−π− λ λ+rh(π+)(π++R−1), a form that will be convenient for various proofs. Lemma D.2. For every μ∈[π−,π+], h(π+)μ−π− π+−π− =h(μ)μ+−π+− π++ −π+− . Proof. Note that both the left- and right-hand sides of the identity in the lemma statement are affine functions of μ. (The left-hand side is immediate, while the numerator of the right-hand side may be rewritten qHμ−π+−h(μ), which is affine in μgiven that 1164 Kirpalani and Madsen Theoretical Economics 18 (2023) h(μ)is.) It is, therefore, enough to show that they coincide at two distinct values of μ. Note that when μ=π−, both sides vanish, while when μ=π+,bothsidesreduceto h(π+), as desired. Lemma D.3. ThepayoffgapPis a strictly decreasing function and P(π−)>0. Proof.Let ˆ (μ)≡μ−π− π+−π− λ λ+rV−h(π+)(π++R−1)+c. Differentiate ˆ to obtain ˆ (μ)=1 π+−π− λ λ+rV−h(π+)(π++R−1). By Lemma C.1,V≤K(π+−πA), i.e., V−h(π+)(π++R−1)≤−c,andso ˆ (μ)<0for all μ. Clearly P(μ)=ˆ (μ)and therefore  P(μ)<0forμ≤μ∗. Meanwhile P(μ)≤ˆ (μ) for μ>μ ∗. Since Pis continuous at μ∗and an affine function of μon [μ∗,π0],toensure P≤ˆ ,itmustbethat P(μ)= P(μ∗ +)≤ˆ (μ∗)<0forμ∈(μ∗,π0].Hence,Pis a strictly decreasing function. Finally, note that P(π−)=c>0. In light of the previous lemma, define the prospecting belief threshold μ∈(π−,π0] by μ≡π0,P(π0)≥0, −1 P(0),P(π0)<0. Define the associated prospecting time threshold TF≡(μλ)−1(μ). We now show that the follower’s strategy is a best response to the leader’s strategy. We further show that the best response is unique, which will be important for proving Proposition 3. Lemma D.4. Suppose firm −ichooses the threshold strategy T∗ −i=T−i=∞.Thenfirmi’s unique best response is the threshold strategy characterized by T∗ i=T∗ Fand Ti=TF. Proof. Given firm −i’s strategy, firm i’s posterior beliefs satisfy μi(t)=μλ(t)for all time. Consider first firm i’s optimal investment policy. Lemma D.10 establishes that an optimal policy must be a threshold rule, and so at each point in time, either Vi +(t)= W†(t)≡μλ +(t)R−1orelseVi +(t)=W‡(t),whereW‡(t)is the value of investing immediately after the leader invests. The follower’s investment cutoff time is determined by the first time at which W†(t)falls below W‡(t). The value W‡(t)may be calculated explicitly as W‡(t)=νλ +(t)λ λ+rh(π+)(π++R−1), Theoretical Economics 18 (2023) Strategic investment allocation 1171 matter what information arrives, and so any delay is suboptimal. Thus, again firm i’s strategy must involve immediate investing prior to t+,0 i=∞. This lemma ensures that firms’ investment policies must take the form of threshold rules, except in the case that t0 i<t 00 i=∞. However, multiplicity of best replies in that case impacts outcomes only off the equilibrium path, as on-path the firm either obtained a high signal prior to t0 iand invested immediately or else obtained no signal prior to t0 i, after which a signal is valueless and the firm does not optimally acquire one. Therefore, any choice of a non-threshold investment policy in this case has no impact on equilibrium outcomes. Our proof will proceed by restricting attention to equilibria in threshold investment strategies, with each firm’s investment threshold denoted by T∗ i. This analysis will characterize all possible equilibrium paths and, in particular, will establish that either t00 i<∞or else t0 i=∞in any equilibrium, proving that all equilibria involve threshold investment policies. Now, assume that both players use pure prospecting strategies. We will maintain this assumption until the end of the proof, when we verify that no equilibria with mixed prospecting strategies can exist. We next establish an important technical result about the dynamics of the value of effort prior to time tA i. This result will be critical to establishing that firms follow a threshold prospecting rule in any equilibrium. For each firm i, define fi(t)≡ Vi(t)−K(μi(t)−πA).Notethatfi(t)≥Vi(t)− Vi(t)+c, with equality for all t<T∗ i. Lemma D.11. Fix any firm i. Then for almost every t∈[0, min{T∗ i,tA i}],eitherfi(t)<0or f i(t)>0. Proof.Fixafirmi. Suppose first that T∗ −i≤t<t A i.Thenattimet,firmiis in autarky with beliefs μi(t)>π A, meaning its continuation value is Vi(t)=λ λ+rK(μi(t)−πA)< K(μi(t)−πA).Thus,fi(t)<0 for all such times. So it is sufficient to establish the result for t<min{tA i,T∗ i,T∗ −i}. Note that whenever t<T ∗ i,wehavefi(t)=Vi(t)− Vi(t)+c. Then for almost every t<min{T∗ i,T∗ −i}such that fi(t)≥0, Vi(t)must satisfy the HJB equation rV i(t)=λ−i(t)μi(t)−π− π+−π−V−Vi(t)+˙ Vi(t). This may be rewritten in terms of fand fas f i(t)=rKμi(t)−πA−λ−i(t)μi(t)−π− π+−π−V−K(π+−πA)−fi(t)+rfi(t). The first term on the right-hand side of this expression is strictly positive for every t<t A i. Further, by Lemma C.1,V≤K(π+−πA). Finally, the coefficient on fi(t)on the righthand side is always nonnegative. Thus, whenever fi(t)≥0, we must have f i(t)>0. 1172 Kirpalani and Madsen Theoretical Economics 18 (2023) We proceed by splitting the analysis into two cases: Either T∗ i<∞for some firm i,or else T∗ 1=T∗ 2=∞. We will show that in the first case, the only permissible equilibrium behavior is the leader–follower strategy profile, while in the second case, the only permissible behavior is the symmetric equilibrium profile. Consider first the T∗ i<∞case. The following lemma establishes that the remaining firm −imust employ the leader strategy in any equilibrium. Lemma D.12. Suppose that T∗ i<∞for some firm i.Thenfirm−imust follow the threshold strategy T−i=T∗ −i=∞. To establish this result, we first prove an auxiliary lemma that restricts the permissible scope of equilibrium behavior and beliefs in response to a firm using a threshold investment rule with T∗ i<∞. Lemma D.13. Suppose that T∗ i<∞for some firm i.ThenT∗ −i=∞and μ−i(T∗ i)>π A. Proof. Suppose by way of contradiction that μ−i(T∗ i)<π A. Then beginning at time T∗ i,firm−iis in autarky with beliefs below the autarky threshold, implying that it does not invest on the equilibrium path after time T∗ i. Further, on the interval (tA −i,T∗ i],we have V−i(t)≥0>K (μi(t)−πA). Then at all such times, it cannot be optimal for firm i to both prospect and invest immediately upon acquiring a signal. Therefore, firm −iis in autarky beginning at time tA −i. However, since tA −i<T∗ iby continuity of μ−i, the fact that it is optimal for firm ito invest immediately at times in [tA −i,T∗ i)but wait after T∗ iimplies that μi +(tA −i)=μi +(T∗ i)= 1/R. Therefore, μi(tA −i)<π A,soλi(t)=0forallt≥tA −i. But then, on the equilibrium path, firm idoes not invest first after tA −i, implying firm −iis in autarky with constant beliefs μ−i(t)=μ−i(tA −i)=πAfor all times t>t A −i. This contradicts μ−i(T∗ i)<π A,soit must be that μ−i(T∗ i)≥πAand, in particular, T∗ i≥tA i. Subsequent to time T∗ i,firm−iis in autarky with fixed beliefs no lower than the autarky threshold. Therefore, μ−i +(t)>1/R for all t≥T∗ i, in which case immediate investing is strictly superior to waiting forever for every t≥T∗ i.Thus,firm−imust choose T∗ −i=∞. Now suppose by way of contradiction that μ−i(T∗ i)=πA, in which case tA −i≤T∗ i. As firm −i’s beliefs do not change over the interval [tA −i,T∗ i], it must be in autarky with constant beliefs πAfrom time tA −ionward, implying V−i(t)=0. Note that V−i(tA −i)=0andμ−i(tA −i)=πAimply f−i(tA −i)=0. But by Lemma D.11, for almost every t∈[0, tA −i],eitherf−i(t)<0orf −i(t)>0. These conditions imply that if f−i(t)=0forsomet<t A −i,thenf−i(t)>0forallt∈(t,tA −i].Hence,f−i(t)<0forall t<t A −i, implying that for all such times, the value of waiting is less than the value of prospecting and investing immediately upon obtaining a high signal. Since this investment strategy is a lower bound on the value of prospecting, it must be that λ−i(t)=λ a.e. on [0, tA −i]. This prospecting policy, combined with T∗ −i=∞, implies that μi(t)≤μ−i(t)for t∈[0, tA −i]and, therefore, tA i≤tA −i.IftA i<t A −i, then for every t∈(tA i,tA −i),firm−i’s Theoretical Economics 18 (2023) Strategic investment allocation 1173 prospecting and investment policies imply that μi(t)<π A, meaning it cannot be optimal for firm ito both prospect and invest immediately upon obtaining a signal at any such time. Thus, firm idoes not invest on the equilibrium path on this time interval, implying μ−iis constant on the interval, contradicting the definition of tA −i.SotA 1=tA 2=tA for some tA, which can only hold if T∗ i≥tAand λi(t)=λfor almost every t∈[0, tA]. If Vi(tA)>0, then given continuity of Viand μi, for sufficiently large t<t Ait would be the case that Vi(t)>K (μi(t)−πA). But then it cannot be optimal for firm ito both prospect and invest immediately at such times, a contradiction. So Vi(tA)=0. But as T∗ −i=∞, this can be true only if λ−i(t)=0fora.e.t>t A. But then subsequent to time TA,firmiis in autarky with beliefs μi(t)=πA, contradicting the optimality T∗ 1<∞.So μ−i(T∗ i)>π A, as desired. Proof of Lemma D.12. Lemma D.13 established that T∗ −i=∞and μ−i(T∗ i)>π A.The latter inequality implies that for t>T ∗ i,firm−iis in autarky with beliefs above the autarky threshold, meaning −i’s unique optimal prospecting policy subsequent to T∗ iis λ−i(t)=λ. It remains only to pin down firm −i’s optimal prospecting behavior prior to T∗ i. Define V† −i(t)≡K(μi(t)−πA). Since T∗ −i=∞,itmustbethat V−i(t)−c=V† −i(t)for all times. Then inserting V† −iinto the functional F−idefined in Appendix Cyields F−iV† −i,t=rV † −i(t)+˙μ−i(t) π+−μ−i(t)V−K(π+−πA). Note that V≤K(π+−πA)by Lemma C.1, so the second term on the right-hand side is nonnegative. Meanwhile for t≤T∗ i,μ−i(t)>π Aand, therefore, V† −i(t)>0. Thus, F−i(V† −i,t)>0 for all times t≤T∗ i. Now note that as firm −iis in autarky at time T∗ i, its value function at this point is V−i(T∗ i)=λ λ+rV† −i(T∗ i)<V† −i(T∗ i). This boundary condition, combined with the fact that F−i(V† −i,t)>0 while F−i(V−i,t)=0forallt∈[0, T∗ i], implies by a standard result regarding supersolutions of ordinary differential equations that V† −i(t)>V−i(t)for all t≤T∗ i.ThenasV† −i(t)= V−i(t)−c,firm−i’s unique optimal prospecting strategy prior to T∗ iis λ−i(t)=λ. Lemma D.12 establishes that in any equilibrium in threshold investment strategies in which some T∗ i<∞, the other firm must follow the leader’s strategy. Meanwhile Lemma D.4 establishes that the follower’s strategy is a unique best reply to the leader’s strategy. So there exists a unique equilibrium in threshold investment strategies with some T∗ i<∞, namely, the leader–follower equilibrium. The following lemma treats the remaining case in which T∗ 1=T∗ 2=∞. It establishes that the symmetric equilibrium strategies are the only ones consistent with equilibrium in this case. Lemma D.14. Suppose T∗ 1=T∗ 2=∞. Then both firms follow threshold prospecting policies with T1=T2=TA. 1174 Kirpalani and Madsen Theoretical Economics 18 (2023) Proof. Note that when T∗ 1=T∗ 2=∞,wehavefi(t)=Vi(t)− Vi(t)+cfor every iand t, and so a firm’s optimal prospecting rate depends only on the sign of fi(t).Further, fi(t)≥0whenevert≥tA i, and the inequality is strict if either μi(t)<π Aor Vi(t)>0. Also, by Lemma D.11, for each firm iand almost every t<t A i,eitherfi(t)<0orf i(t)>0. Suppose first that tA i<t A −ifor some firm i. Define tAA i≡inf{t:μi(t)<π A}.IftAA iis finite, then for each time t∈(tA i,tAA i]firm iexpects firm −ito invest at some point in the future with positive probability, meaning Vi(t)>0, and for each time t>t AA i,wehave μi(t)<π A. Thus, for all times t>t A i,wemusthavefi(t)>0andλi(t)=0. In this case, firm −iis in autarky with beliefs strictly above its autarky threshold beginning at time tA i, meaning λ−i(t)=λgoing forward. But then eventually firm i’s posterior beliefs must drop below μ∗, at which point it cannot be optimal for firm ito invest immediately after obtaining a signal, a contradiction of T∗ i=∞.SoitmustbethattAA i=∞, i.e., λ−i(t)=0 for all t>t A i. In that case Vi(tA i)=0and,thus,fi(tA i)=0. Now, if fi(t)≥0 on some positivemeasure subset of [0, tA i],thenforsomet<t A i,wemusthavefi(t)≥0andfi(t)>0, meaning that fi(t)>0fort>t sufficiently small. But for fito decline back to zero by time tA i, there must be a positive-measure set of times at which fiis both positive and has a negative derivative, a contradiction. So it must be that fi(t)<0a.e.on[0, tA i], i.e., λi(t)=λfor all such times. But then firm iprospects at the maximum rate at all times prior to tA i, meaning that μ−i(t)≤μi(t)for such times, a contradiction of tA i<t A −i.We conclude that tA 1=tA 2.LettAbe this common time. Suppose first that tA=∞. Then each firm imust prospect at less than full intensity on a positive-measure set of times, meaning there exists a time t iat which fi(t i)≥0and f i(t)>0. Thus, fi(t)>0fort>t  isufficiently small, and by reasoning similar to the previous paragraph, fi(t)>0forallt>t  i.Thus,λi(t)=0fort>t  i, meaning firm −iis in autarky with beliefs strictly above the autarky threshold. It therefore sets λ−i(t)=λfor t>t  i, contradicting tA i=tA=∞.SoitmustbethattA<∞. Next, suppose that fi(tA)>0forsomei. Then also fi(t)>0fortsufficiently close to tA, meaning λi(t)=0 for such times. But then μ−i(t)is constant on this interval, contradicting tA −i=tA.Sofi(tA)=0 for each firm i. By now-familiar arguments, it must, therefore, be that fi(t)<0 for almost all t<t A, i.e., λi(t)=λfor each iand a.e. t<t A. Therefore, tA=TA.Further,fi(tA)=0impliesVi(tA)=0, so λ−i(t)=0forallt>t A. Thus, each firm must use the threshold prospecting strategy Ti=TA. We complete the proof by ruling out mixed prospecting rules in equilibrium. This is accomplished by the following lemma, which establishes that any equilibrium involving randomization over prospecting implies existence of a pure-strategy equilibrium involving interior prospecting. As no pure-strategy equilibria exhibit such behavior, no mixed-strategy equilibria exist. Lemma D.15. Fix any equilibrium in threshold investment strategies. Then there exists a payoff-equivalent equilibrium in pure strategies, exhibiting interior prospecting whenever some firm randomized over prospecting rates in the original equilibrium. Theoretical Economics 18 (2023) Strategic investment allocation 1175 Proof. Fix an equilibrium involved randomized prospecting and fix a firm i. After time T∗ i,firmi’s prospecting rule does not affect firm −i’s payoffs or incentives; thus, λimay be replaced with any pure strategy maximizing i’s payoffs subsequent to time T∗ iwithout disturbing the equilibrium. So consider times t<T∗ i. Let i(t)≡Eexp−t 0 λi(s)ds be the ex ante probability that firm ihas obtained no signal by time t. Defineanew pure-strategy prospecting rule  λiby letting  λi(t)=−d dt logi(t)for all times (with the prospecting rule arbitrary at any point of non-differentiability of i). By construction, λiand  λiinduce the same distribution of investment times by firm iconditional on θ, and, thus, the same posterior beliefs for firm −iconditional on observing no investment. Therefore, firm −i’s incentives are unchanged by replacing λiwith λi. It remains to check that  λiis feasible and optimal for firm i.Notethat  λi(t)=1 i(t) Eλi(t)exp−t 0 λi(s)ds. The second factor on the right-hand side is bounded above by λi(t)and below by zero, hence,  λi(t)∈[0, λ], ensuring feasibility. As for optimality, suppose first that at time t, the action λi(t)is strictly optimal for firm i. Then it must be non-random, in which case the previous expression for λi(t)collapses to λi(t)=λi(t).Soatanytimesforwhichrandomization is not optimal for firm i, the modified prospecting rule specifies the same prospecting intensity as the original rule. Additionally, at all other times, any prospecting intensity is optimal; thus, in particular, the intensity specified by λiis optimal. So λi is an optimal prospecting rule. This argument shows that firm i’s randomized prospecting rule may be replaced by a non-random one that is also optimal for firm i, without disturbing firm −i’s payoffs or incentives. This procedure may be performed for both firms, yielding a pure-strategy equilibrium. Finally, for any time tat which λi(t)is not deterministic, it must be that Pr(λi(t)> 0)>0andPr(λi(t)<λ)>0, in which case the previous expression for λiimplies  λi(t)∈ (0, λ). So randomization in the original equilibrium implies an interior prospecting rate in the new equilibrium. D.6 Proof of Proposition 4 To prove the small-rresult, we show that whenever r≤r∗, welfare under the leader– follower equilibrium exceeds welfare in the symmetric equilibrium. Let VA(t)≡ λ λ+rK(μλ(t)−πA)be the autarky payoff under beliefs μλ(t). We first show that in the symmetric equilibrium, Vi(t)=VA(t)for each iand all t≤TA. First note that for all such times, μi(t)=μλ(t). Then trivially Vi(TA)=VA(TA),asattimeTA, each firm is in autarky with beliefs μλ(TA). So evaluate the functional Fidefined in Appendix C 1176 Kirpalani and Madsen Theoretical Economics 18 (2023) at VAfor any time t≤TA. Using the identity V=λ λ+rK(π+−πA), which holds whenever r≤r∗, as well as the identity  Vi(t)−c=K(μλ(t)−πA), which holds given that T∗ i=∞,yieldsFi(VA,t)=0. Then a standard verification argument establishes that VA(0)=VS. Now consider the leader–follower equilibrium. Recall that when r≤r∗, Lemma D.1 implies that T∗ F=0. Hence, the leader is in autarky for all times and VL=VA(0).So consider the follower’s strategy. Suppose firm iis the follower. Note that μi(t)=μλ(t)for all time and that  Vi(t)−c>K (μλ(t)−πA)for all time given that investing immediately is strictly dominated by waiting at all times. Hence, Fi(VA,t)<0 for all time. Then as VAis a bounded function, a standard verification argument establishes that VAis bounded strictly above by the payoff of the threshold strategy Ti=T∗ i=∞. Since VFis an upper bound on the payoff of any strategy followed by firm i,itmustbethatVF> VA(0).Thus,VL+VF>2VA(0)=2VS,asclaimed. We now prove the large-rresult. Going forward, we will assume that r>r ∗. We first establish that VF>VS>VL.WriteVF(t),VS(t),andVL(t)for the time-tcontinuation value of each firm in each equilibrium given no signal and no investment by the other firm. Let  TF≡min{TF,T∗ F}bethetimeatwhichthefollowerbecomespassiveinthe leader–follower equilibrium. The leader’s beliefs equal μλ( TF)at time  TFand, further, the leader is in autarky going forward. It follows that VL( TF)=VA( TF). Meanwhile, a firm in the symmetric equilibrium possesses posterior beliefs μλ( TF) at time  TFgiven that  TF<TA(as established in Proposition 2). Further, the autarky strategy is feasible but not optimal for that firm in the continuation after time  TF.This is because when r>r ∗, investing following observation of investment by the other firm improves on the autarky strategy of ignoring the other firm’s actions and continuing to prospect. Additionally, since  TF<TA, each firm invests with positive probability subsequent to time  TFin the symmetric equilibrium. It must, therefore, be that VS( TF)>VA( TF)>VL( TF). Next observe that the follower could achieve the symmetric equilibrium continuation value at time  TFby following the strategy of prospecting until time TA, investing immediately if it has obtained a signal or observed investment, and then halting all prospecting and investment subsequent to time TA, regardless of what it sees the other firm do. However, this strategy cannot be optimal, since the leader invests with positive probability after time TA, and the follower’s payoff would be improved by investing in such histories whenever it has not yet obtained a signal. It must, therefore, be that VF( TF)>VS( TF). To complete the argument, we show that VF( TF)>VS( TF)>VL( TF)implies that VF>VS>VL. Note that no firm in either equilibrium delays investment prior to time  TF, and posterior beliefs for all firms equal μλ(t)for all times prior to  TF. It follows that Fi(·,t)(as defined in Appendix C) is the same for a leader, follower, or firm in the symmetric equilibrium prior to  TF. Then since VF( TF)>VS( TF)>VL( TF),astandard comparison result implies that VF>VS>VL. To complete the proof, we perform a limiting payoff comparison as r→∞.Forthe remainder of the proof, we will make the dependence of variables on rexplicit. Note in Theoretical Economics 18 (2023) Strategic investment allocation 1177 particular that μ(r)and μ∗(r)are both functions of r, while πAis independent of r.We begin with two auxiliary lemmas. Lemma D.16. For sufficiently large r, the inequality max{μ∗(r),πA}<μ(r)<π 0holds. Further, limr→∞ μ(r)=πA. Proof.Notethatasr→∞,I(μ,r)converges uniformly to −(μ+R−1)for all μ∈ [π−,π0]and, thus, μ∗(r)approaches μ,whereμsolves μ+R−1=0. Since πA+R−1>0 and π+R−1>0, it must, therefore, be that μ∗(r)<min{π0,πA}for large r.Inparticular, μ∗(r)<π 0implies that Pμ∗(r),r=μ∗(r)−π− π+−π− λ λ+r(π+R−1)−Kμ∗(r)−πA. As the first term approaches zero for large rwhile μ∗(r)<π Afor large r,wemusthave P(μ∗(r),r)>0, i.e., μ(r)>μ ∗(r). Lemma D.6 then further implies that μ(r)>π Afor such r.Further,forlargerand μ>μ ∗(r),P(μ,r)converges uniformly to −K(μ−πA) and, thus, μ(r)converges to πA. Since πA<π 0, we therefore have μ(r)<π 0for rsufficiently large. Lemma D.17. The limit limr→∞ r(TA−TF(r)) =π+R−1 h(π+)(π++R−1)−cholds. Proof. Recall that TA=(μλ)−1(πA)while TF(r)=(μλ)−1(μ(r)).Sotofirstorder, TA−TF(r)=− 1 ˙μλTAμ(r)−πA+Oμ(r)−πA2. For large r,μ(r)∈(μ∗,π0)and so μ(r)solves μ−π− π+−π− λ λ+r(π+R−1)=K(μ−πA). The solution to this equation may be written to first order in r−1as μ(r)=πA+K−1λ(πA−π−) π+−π− (π+R−1)r−1+Or−2. Thus, TA−TF(r)=−K−1λ(πA−π−) ˙μλTA(π+−π−) (π+R−1)r−1+Or−2. Using Lemma B.2 to eliminate ˙μλ(TA)yields TA−TF(r)=π+R−1 K(π+−πA)r−1+Or−2=π+R−1 h(π+)(π++R−1)−cr−1+Or−2. Multiplying through by rand taking r→∞yields the desired identity. 1178 Kirpalani and Madsen Theoretical Economics 18 (2023) In light of Lemma D.16, going forward we will assume that ris sufficiently large that TF(r)<T∗ F(r),TA. Fix a strategy profile in which both firms play the leader’s strategy. Let νλ(t)be the associated time-tprobability that a firm’s opponent has obtained no signal, supposing it has not invested yet. Further let πP(t)≡λK(μλ(t)−πA)and πO(t)≡νλ(t)h(π0)(π+R−1)be each firm’s time-tflow profits from prospecting and observing investment, respectively, conditional on having obtained no signal and having observed no investment. (Note that when r>r ∗, each firm optimally invests immediately following observation of investment.) Finally, let δ(t)≡1−(1−e−λt )h(π0)be a firm’s probability of reaching time twithout having observed investment. Each firm’s profits in each equilibrium may be written using this notation. Symmetric equilibrium profits are VS(r)=TA 0 e−(r+λ)tδ(t)πP(t)+πO(t)dt, where the upper limit of integration accounts for the termination of flow profits at time TAsupposing no firm has acquired a signal or invested by that time. Meanwhile, the leader’s profits are VL(r)=TF(r) 0 e−(r+λ)tδ(t)πP(t)+πO(t)dt +e−(r+λ)TF(r)δTF(r)πPTF(r) λ+r, where the final term accounts for the transition to autarky supposing no firm has acquired a signal or invested by time TF(r).(RecallthatTF(r)<T∗ F(r),soTF(r)is the time of transition to autarky.) Finally, the follower’s profits are VL(r)=TF(r) 0 e−(r+λ)tδ(t)πP(t)+πO(t)dt +∞ TF(r) e−rte−λTF(r)δ(t)πO(t)dt, where the final term accounts for the termination of prospecting at time TF(r). Define V (r)≡rerT R(r)(2VS(r)−VL(r)−VF(r)). This expression may be written explicitly as V (r)=2TA TF(r) re−r(t−TF(r))e−λtδ(t)πP(t)+πO(t)dt −∞ TF(r) re−r(t−TF(r))e−λT F(r)δ(t)πO(t)dt −r λ+re−λTF(r)δTF(r)πPTF(r). We now take the limit r→∞. Recall that limr→∞ TF(r)=TAand πP(TA)=0. Thus, the final term vanishes in the limit. To evaluate the integrals, make the substitution t=r(t−TF(r)).AsπP,πO,andδare bounded functions, the resulting integrands are uniformly bounded for all tand r, and the bounded convergence theorem may be used to evaluate each integral in the limit. The first converges to Theoretical Economics 18 (2023) Strategic investment allocation 1179 2(1−exp(−limr→∞ r(TA−TF(r))))e−λT Aδ(TA)πO(TA), while the second converges to −e−λTAδ(TA)πO(TA). Combining these calculations and invoking Lemma D.17 yields lim r→∞ V (r)=1−2exp−π+R−1 h(π+)(π++R−1)−ce−λT AδTAπOTA. The sign of 2VS(r)−VL(r)−VF(r)for large rmust be the same as the sign of limr→∞ V (r). Hence, it is strictly positive whenever c>c≡h(π+)(π++R−1)−(π+R− 1)/log2. Note that cis independent of rand c < c given that log2<1, as claimed in the proposition statement. References Aghamolla, Cyrus and Tadashi Hashimoto (2020), “Information arrival, delay, and clustering in financial markets with dynamic freeriding.” Journal of Financial Economics. (Forthcoming). [1143] Akcigit, Ufuk and Qingmin Liu (2016), “The role of information in innovation and competition.” Journal of the European Economic Association, 14, 828–870. [1146] Ali, S. Nageeb (2018), “Herding with costly information.” Journal of Economic Theory, 175, 713–729. [1144] Bolton, Patrick and Christopher Harris (1999), “Strategic experimentation.” Econometrica, 67, 349–374. [1143] Bonatti, Alessandro and Johannes Hörner (2011), “Collaborating.” American Economic Review, 101, 632–663. [1143] Bonatti, Alessandro and Johannes Hörner (2017), “Career concerns with exponential learning.” Theoretical Economics, 12, 425–475. [1143] Campbell, Arthur, Florian Ederer, and Johannes Spinnewijn (2014), “Delay and deadlines: Freeriding and information revelation in partnerships.” American Economic Journal: Microeconomics, 6, 163–204. [1144] Chamley, Christophe and Douglas Gale (1994), “Information revelation and strategic delay in a model of investment.” Econometrica, 62, 1065–1085. [1143,1149,1152] Chari, V. V. and Patrick J. Kehoe (2004), “Financial crises as herds: Overturning the critiques.” Journal of Economic Theory, 119, 128–150. [1143] Décamps, Jean-Paul and Thomas Mariotti (2004), “Investment timing and learning externalities.” Journal of Economic Theory, 118, 80–102. [1144] Dong, Miaomiao (2021), “Strategic experimentation with asymmetric information.” Working paper. [1143] Fajgelbaum, Pablo D., Edouard Schaal, and Mathieu Taschereau-Dumouchel (2017), “Uncertainty traps.” The Quarterly Journal of Economics, 132, 1641–1692. [1144] 1180 Kirpalani and Madsen Theoretical Economics 18 (2023) Frick, Mira and Yuhta Ishii (2020), “Innovation adoption by forward-looking social learners.” Working paper. [1144] Graham, Paul (2013), “Investor herd dynamics.” http://paulgraham.com/herd.html (visited on 08/16/2019). [1141] Gul, Faruk and Russell Lundholm (1995), “Endogenous timing and the clustering of agents’ decisions.” Journal of Political Economy, 103, 1039–1066. [1143,1144,1156] Guo, Yingni and Anne-Katrin Roesler (2018), “Private learning and exit decisions in collaboration.” Working paper. [1143] Heidhues, Paul, Sven Rady, and Philipp Strack (2015), “Strategic experimentation with private payoffs.” Journal of Economic Theory, 159, 531–551. [1143] Hendricks, Kenneth and Dan Kovenock (1989), “Asymmetric information, information externalities, and efficiency: The case of oil exploration.” RAND Journal of Economics, 20, 164–182. [1141] Hendricks, Kenneth and Robert H. Porter (1996), “The timing and incidence of exploratory drilling on offshore wildcat tracts.” American Economic Review, 86, 388–407. [1141] Hörner, Johannes and Andrzej Skrzypacz (2017), “Learning, experimentation and information design.” In Advances in Economics and Econometrics: Volume 1: Eleventh World Congress (Bo Honoré, Ariel Pakes, Monika Piazzesi, and Larry Samuelson, eds.), 63–98, Cambridge University Press. [1143] Keller, Godfrey, Sven Rady, and Martin Cripps (2005), “Strategic experimentation with exponential bandits.” Econometrica, 73, 39–68. [1143,1145,1146] Keller, Godfrey and Sven Rady (2010), “Strategic experimentation with Poisson bandits.” Theoretical Economics, 5, 275–311. [1143] Keller, Godfrey and Sven Rady (2015), “Breakdowns.” Theoretical Economics, 10, 175– 202. [1143] Klein, Nicolas and Peter Wagner (2022), “Strategic investment and learning with private information.” Journal of Economic Theory, 204, 105523. [1144] Murto, Pauli and Juuso Välimäki (2011), “Learning and information aggregation in an exit game.” The Review of Economic Studies, 78, 1426–1461. [1143] Murto, Pauli and Juuso Välimäki (2013), “Delay and information aggregation in stopping games with private information.” Journal of Economic Theory, 148, 2404–2435. [1143] Rosenberg, Dinah, Eilon Solan, and Nicolas Vieille (2007), “Social learning in one-arm bandit problems.” Econometrica,75, 1591–1611. [1143] Co-editor Simon Board handled this manuscript. Manuscript received 22 March, 2021; final version accepted 9 May, 2022; available online 25 May, 2022.