Elsevier

Journal of Economic Theory

Volume 157, May 2015, Pages 973-1000
Journal of Economic Theory

Optimal entry timing

https://doi.org/10.1016/j.jet.2015.03.002Get rights and content

Abstract

A player of privately known strength chooses when to enter a market, and an incumbent chooses whether to compete or concede. Information about the potential entrant's type is revealed publicly according to an exogenous news process and the timing of entry. I analyze stationary equilibria using the public belief as a state variable. No equilibria in pure strategies exist, and smooth-pasting conditions need not hold. Under both D1 and a novel refinement, the informed player has nondecreasing value functions and her strategy has the following structure: for high states, both types enter with certainty; for a possibly empty interval of intermediate states, no type enters; and for low states, the high type enters while the low type mixes. I obtain closed form solutions and analyze comparative statics for such equilibria. The welfare effects of the presence of news, relative to no news, depend on the starting belief; however, for a fixed equilibrium, a marginal increase in news quality always helps the informed player regardless of her type and always hurts total welfare.

Introduction

In a wide variety of settings, an agent must choose the optimal time to take an aggressive action against an opponent. The aggressor naturally has private information about her strength, which may be high or low, and her opponent must choose how to respond to the action given his beliefs about the aggressor. If he is confident that the aggressor is weak, he may decide to fight back, knowing that he is likely to win. If instead he believes the aggressor is strong, he may prefer to concede, cutting his losses. Public information about the potential aggressor arrives through a continuous stream of news with noise; that is, both good news and bad news arrive in infinitesimal increments, but the likelihood of good news is correlated with the aggressor's strength. The aggressor's public reputation is influenced both exogenously by this news process and endogenously by the information inferred from the timing of her action. As the optimal response under uncertainty depends on the aggressor's reputation, the aggressor should optimize the timing of her action anticipating the path her reputation might take in the future.

Such games of aggression or “attack” occur in many specific settings. While for expositional purposes I will focus on an economic setting of market entry, it is worth highlighting some other applications of the model.

  • A military power or rebel organization decides when to invade a territory in the hope of gaining control or having some demands met. Prior to this decision, exogenous information arrives through central intelligence reports and social media. Its opponent may simply meet these demands, or it can engage in risky physical conflict.

  • A politician running for office decides when to launch a harsh negative ad campaign, and its opponent then either quits the race or responds with heightened campaign spending. Or, an activist group may choose when to publicly rally to influence a decision making authority, and the authority responds by changing its stance or not. Information arrives in either case through journalism, social media and polls.

  • An injured party may privately threaten a defendant with a lawsuit, and the defendant can try to negotiate out of court or accept a costly legal battle. Information arrives in the form of evidence, legal research, and traditional news outlets insofar as they influence a potential jury pool.

In each of these settings, the aggressor has private information about her strength and can exploit a strong reputation by intimidating an opponent. In this paper, I model such scenarios as a dynamic game of private information in continuous time with an exogenous news process.

Consider a technology firm, firm one, deciding when to enter a market by introducing a highly innovative product. Specifically, this could be a firm that develops an innovative smart phone or a start-up that creates a new way for consumers to rent movies and video games. Its competitor, firm two, is a traditional firm in the same industry. Naturally, firm one has an informational advantage over firm two about the specific features of the innovation and relevant consumer preferences. However, this information is gradually revealed publicly in two ways: an exogenous news process and the fact that firm one has not yet decided to enter. The exogenous news process represents the aggregate information revealed through channels such as financial statements, customer reviews, expert speculation and leaks. Since firm one's entry strategy depends on its private information, firm two also revises its beliefs based on the timing of entry.

Prior to the market entry, firm two receives flow profits from its continuing business while firm one receives nothing. If firm one introduces the product, firm two decides whether to compete with firm one and final payoffs are realized. Conceding and exiting the market is firm two's safe option, and firm one then receives a large payoff in the form of monopoly profits. If firm two chooses to compete, both firms must pay fighting costs in the form of unavoidable sunk costs. The outcome of competition depends on the true type of firm one. Firm one wins and receives a high payoff after competition if and only if it has a high quality product. For simplicity I assume that a single winner emerges with a natural monopoly. Fig. 1 shows the interpretation of the instantaneous period beginning at time t. In the final section of the paper, I consider variations of the model in which only firm two can enter or either firm can enter.

Stationary subgame perfect Bayesian equilibrium is a natural solution concept for this model, using the public belief about firm one's type as the state variable. As in other games of private information and learning, delay arises in equilibrium. The high (i.e., strong) type of firm one anticipates good news to arrive exogenously and improve its reputation, and thus has an incentive to wait. The low type wants to mimic the high type so it can enter and induce firm two to concede.

I first define a class of “interval equilibria” in which strategies have an appealing structure. For high beliefs, both types of firm one enter with certainty. For low beliefs, the high type enters with certainty while the low type mixes to make firm two indifferent to fighting back. For an interval of intermediate beliefs, both types wait for news to arrive, and the belief is driven solely by the exogenous news process. I present a novel equilibrium refinement that belief updating must be weakly belief order preserving (WBOP). The essence of this refinement is that, for any fixed action of firm one, the posterior belief upon observing that action should be nondecreasing in the prior belief. In the main result of the paper, I show that in a particular sense, the class of equilibria surviving this refinement is precisely the class of interval equilibria with nondecreasing value functions. The class of equilibria surviving D1 is found to be a nonempty, proper subclass of the class of WBOP equilibria.

One feature of the model is that a high type of firm one receives positive payoff from entry, even when firm two fights back with certainty. This feature limits the size of no-entry regions that can occur in equilibrium, because in the interior of such a region, a high type may prefer to deviate by entering rather than wait for beliefs to exit this region. It turns out that higher fighting costs, higher news quality and greater patience allow for larger delay regions by increasing the incentive of the high type to wait. For a fixed equilibrium, the value function of each firm one type is weakly decreasing in both the discount rate and the fighting cost. Since higher news quality hastens exit out of the delay region without affecting the distribution over exit boundaries, both firm one value functions are increasing in news quality, while the opposite is true for firm two and total welfare. The total welfare effect of the presence of news depends on the starting belief; news can introduce the possibility of fighting on the equilibrium path for some beliefs, but it can introduce the possibility of full concession for other beliefs.

The interpretation of the main findings depends on the externalities imposed by the game in the specific application. First, one can compare social welfare across equilibria for a fixed set of model inputs. In real market entry applications, the introduction of a new product to the market likely exerts a positive net externality on consumers. If this externality is sufficiently large, then equilibria with longer delay are worse for social welfare. In a military application, however, invasion and conflict presumably exert negative net externalities, so delay is socially desirable. For applications in politics, activism, and legal disputes, net externalities likely fall somewhere between these extremes and depend on further details. Second, one can compare social welfare by fixing an equilibrium and varying news quality; then quality of news determines the expected duration of delay, so in this sense higher news quality may help or hurt social welfare depending on the application.

This paper belongs to the literature on dynamic games and stopping problems. With thematic differences, the information structure of the model is closely related to Daley and Green [7] (“DG”). In their model, a seller has private information about the quality of an object and faces a competitive market of buyers who make continuous offers on the basis of exogenous news and the fact that sale has not yet occurred. A market for lemons arises even if there is common knowledge of gains from trade, since at certain reputation levels a high-value seller may prefer to wait, expecting upward drift from news; in DG, this results in inefficient “no-trade” regions. Similarly, equilibria arise in the current model which contain “no-entry” regions. However, the current model differs from DG in several respects. First, in DG flow payoffs to player 1 depend on her type but payoffs at termination do not; in this model the reverse is true. Second, a feature of the DG model is that two forces encourage the high type seller to delay trade: she not only anticipates good news but also earns higher utility from retaining the object. In the current model, this monotonicity is absent: a high type expects good news but earns a positive payoff even when player 2 fights back. Finally, in my model, player 1 chooses when to end the game, but player 2 makes the final move; in DG, the seller is responsible for both the timing of the end and the final move.

Aside from DG, a variety of papers have looked at inefficiency from delay. In Taylor [24], a home owner may want to set a high price to weaken the negative inference made by a prospective buyer about its quality upon observing no sale. In Bonatti and Hörner [5], team members procrastinate on a project that has uncertain potential payoff. In Ortner [22], a monopolist with stochastic time-varying costs may find it optimal to delay sale in equilibrium. When asymmetric information is added to the baseline model in Ambrus et al. [1], bidders in an eBay-like auction may delay high bids in order to manipulate their rivals' beliefs.

There is a large existing literature on dynamic games of one-sided incomplete information that is revealed through a player's actions. In the 1960s, Aumann and Maschler developed the first of such models, where player one observes the (persistent) state that determines the true payoff matrix, and player two only learns by observing player one's actions; see Aumann and Maschler [2]. Hörner et al. [16] generalize this work by allowing the state to follow a Markov process. Dynkin [9] introduced a general class of games in which players choose when to stop a stochastic process that determines their termination payoffs. Touzi and Vieille [25] have expanded this literature to include mixed strategies in continuous time.

A large number of recent economics papers have employed continuous time techniques in timing and/or reputation games. Faingold and Sannikov [10] use a general framework to study the reputation of a large player with imperfectly observed actions facing a continuum of small players. Gryglewicz [11] models a similar market entry interaction to that of the current paper, where instead of news, a stochastic process drives the value of the market. In Heinsalu [14], a news process arises endogenously from noisy signalling, but in that model there are no actions other than signalling. Gul and Pesendorfer [12] model a game between opposing political parties who choose how long to incur costs to spread information. Murto and Välimäki [21] characterize exit waves in a stopping game with informational externalities. In Moscarini and Squintani [20], privately informed firms in an R&D race decide when to abandon a project, and a survivor's curse arises in equilibrium.

The current paper also contributes to the literature on market entry. In the model of Loury [18], firms make investment decisions under rivalry and uncertainty; market failure arises in the form of excessive R&D expenditure and an excess of firms entering the market. In Dixit [8], firms facing volatile prices enter the market at higher thresholds than they exit, due to sunk costs. In Bergemann and Välimäki [4], an entrant adjusts the supply of its innovation as it learns about its quality. In Jovanovic [17], entrants have private information but they choose entry locations. On the empirical side of the literature, there is much work relating firm and industry characteristics to entry timing. Schoenecker and Cooper [23] use data from the minicomputer industry in the 1990s to find a positive correlation between early entry and higher R&D intensity, possession of a direct sales force, larger firm size and stronger involvement in a threatened market; the entrant's type in the current model can be thought to aggregate factors such as these. Mitchell [19] looks at the decision to enter an emerging field from the perspective of an incumbent, which is related to the variations of the current model. Other well-studied phenomena in this literature include preemption, deterrence, first-mover advantage or disadvantage, brand extensions and patent protection. For an extensive review of the market entry literature, see Helfat and Lieberman [15] and Hauser et al. [13].

The rest of the paper is organized as follows. Section 2 presents the model. Section 3 defines and characterizes equilibria and discusses several refinements. Section 4 provides comparative statics. Section 5 concludes and briefly introduces several variations, which are treated in detail in a separate, supplementary appendix. All proofs not provided in the main text are contained in Appendix A, Appendix B of this paper or in the supplement.

Section snippets

Model

I model the entry game as a dynamic game between two players played in continuous time with infinite horizon. Player 1 (“she”) is of privately known type, θΘ:={H,L}, and may be interpreted as a firm of high or low strength; player 2 (“he”) is an incumbent that places prior probability p0 on θ=H, which is common knowledge. Except for in the variations of the model discussed in Section 5, only player 1 can enter, and while the game continues, players 1 and 2 earn flow payoffs of 0 and 1,

Equilibria

The most general form of equilibrium I will consider is stationary subgame perfect Bayesian equilibrium, which I define in this section. Given a fighting strategy f, types H and L at time s face the optimal stopping problemsmaxτsEH[er(τs)(1kfτ)|Fs]maxτsEL[er(τs)(1(1+k)fτ)|Fs]. For all s0, define Aθ,s to be the support of θ's continuation strategy from time s; that is, the set of stopping times τ such that for all ϵ>0 and for all ωΩ,Aτ(θ,ω)+ϵθ,s(ω)Aτ(θ,ω)ϵθ,s(ω)>0. A strategy AH,s

Comparative statics

In this section I provide comparative statics, both within and across equilibria.20 Proposition 3 analyzes the delay, defined as the time until the belief process exits the no-entry region. This proposition makes use of the observation that an increase in the signal-to-noise ratio reduces the delay pathwise. The underlying intuition is that the only significance of noise is that it takes time to filter, slowing down the

Conclusions

In this paper I have analyzed the effect of news on an entry timing game of private information. News introduces delay as the high type has an incentive to wait for good news and the low type mimics. Under a novel equilibrium refinement, WBOP, equilibria share an appealing three-region structure and nondecreasing value functions. The length of the delay region is bounded above by the fact that a high type would prefer to enter early if the wait is too long otherwise. The D1 refinement produces

References (25)

  • Dirk Bergemann et al.

    Entry and vertical differentiation

    J. Econ. Theory

    (2002)
  • Giuseppe Moscarini et al.

    Competitive experimentation with private information: the survivor's curse

    J. Econ. Theory

    (2010)
  • Attila Ambrus et al.

    Gradual bidding in ebay-like auctions

    (2013)
  • Robert J. Aumann et al.

    Repeated Games with Incomplete Information

    (1995)
  • Jeffrey S. Banks et al.

    Equilibrium selection in signaling games

    Econometrica

    (1987)
  • A. Bonatti et al.

    Collaborating

    Amer. Econ. Rev.

    (2011)
  • In-Koo Cho et al.

    Signaling games and stable equilibria

    Quart. J. Econ.

    (1987)
  • B. Daley et al.

    Waiting for news in the market for lemons

    Econometrica

    (2012)
  • Avinash Dixit

    Entry and exit decisions under uncertainty

    J. Polit. Economy

    (1989)
  • E.B. Dynkin

    Game variant of a problem on optimal stopping

    Sov. Math. Dokl.

    (1969)
  • E. Faingold et al.

    Reputation in continuous-time games

    Econometrica

    (2011)
  • Sebastian Gryglewicz

    Signaling in a stochastic environment and dynamic limit pricing

    (2009)
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    The author is especially indebted to Attila Ambrus and Brendan Daley for thoughtful advice and conversations throughout the research process, and to two anonymous referees for numerous suggestions leading to significant improvements of the paper. The author also thanks Dirk Bergemann, Vincent Conitzer, Ehud Kalai, Rachel Kranton, R. Vijay Krishna, David McAdams, Wolfgang Pesendorfer, Curtis Taylor, Xiao Yu Wang for helpful comments.

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