On the interplay of informational spillovers and payoff externalities.

Informational spillovers induce agents to outwait each other's actions in order to make more-informed decisions. If waiting is costly, we expect the best-informed agent, who has the least to learn from other agents' decisions, to take the first action. I study the interplay between informational spillovers and a direct payoff externality. I show that when the payoff externality is positive or relatively weak, the above intuition is validated. On the other hand, if the externality is negative and strong, the best-informed agent has the most to gain from outwaiting the other.

1. Introduction

* A fundamental insight in information economics is that in a world where information is seldom perfect, one agent's actions often carry valuable information for others. Such informational spillovers are interesting both from a welfare perspective (like any externality, it gives rise to inefficiencies) and from a behavioral perspective, since agents take into account the informational value of their own actions as well as those of others.

In this article I extend the study on informational spillovers to situations where agents' payoffs are also directly linked through their actions. This payoff externality could represent numerous real-life phenomena, such as pollution, movements in asset prices, or network effects. Indeed, one can think of few circumstances where informational spillovers are present but payoff externalities are not. My focus is on how the externality, depending on its sign and size, affects the timing of actions.

I model a situation where two firms are about to enter a new product market. A firm must both decide where to position itself in product space (the firm's "niche"), as well as when to enter the market. Demand varies depending on which niche a firm chooses, but the exact demand structure is unknown. Both firms have some private information that they use--possibly together with the other firm's entry decision--to infer where demand is high. The payoff externality reflects how firms interact in the market: a negative externality means that products are strategic substitutes (which drives firms further apart in product space); a positive externality that products are strategic complements (which drives firms toward the same niche). Delay is costly, so the presence of informational spillovers leads to a waiting game between the firms. Importantly, the only asymmetry between firms ex ante is the precision of their private signals. My main result is that if products are close substitutes, the worst-informed firm enters first in equilibrium.

In the studies on herd behavior by Banerjee (1992) and Bikhchandani, Hirschleifer, and Welch (1992), agents make a binary decision in an exogenously given order. The outcome of the two alternatives is uncertain, and each agent receives an imperfect signal of which alternative is the better. Agents' interests are fully aligned, so the only impact of one agent's choice on another's is the informational value his or her action provides. Zhang (1997) employs the same setting but endogenizes the decision order. Moreover, each agent's accuracy is now private knowledge, so that any agent with less-than-perfect information could potentially benefit from observing another agent's action. By assuming delay to be costly, Zhang shows that the best- informed agent takes the first action in the unique symmetric perfect Bayesian equilibrium (PBE). The intuition is, of course, that poorly informed agents (in expectation) have more to gain from observing others' decisions. In fact, this result was conjectured by Bikhchandani, Hirschleifer, and Welch (1992).

Other articles that study waiting games with asymmetrically informed agents include Fudenberg and Tirole (1986), Hendricks and Kovenock (1989), Bolton and Farrell (1990), and Gul and Lundholm (1995). However, none of these articles studies the interplay of informational spillovers and direct externalities. (1) Fudenberg and Tirole (1986) study endogenous exit in a duopoly game. Firms differ with respect to their opportunity cost of exiting the market, and this cost is private knowledge. By assuming that there is an ex ante positive probability that duopoly is profitable (unlike in the classic war of attrition), the authors derive a unique equilibrium where high-cost firms exit before low-cost ones. In Gul and Lundholm, two agents receive a signal of the value of a project. Their task is to correctly estimate the size of the project, which always equals the sum of the two signals. The second agent will thus be able to forecast the project with certainty, hence a strong informational spillover is present. Due to discounting, a high signal--indicating high future profits--implies that delay is more costly. As a result, in the symmetric PBE, types with higher signals make their estimates first.

Less-related articles include Shaked and Sutton (1982) and Judd (1985). Judd models a market with two product niches, where an incumbent has the opportunity to occupy both niches (to "crowd" the market) in order to preempt entry. Judd shows that if exit costs are low and the products are substitutes--though not too close substitutes--the incumbent refrains from crowding and allows entry. Shaked and Sutton study entry and quality choice in a vertical product market. They show that if entry is costly and the market can sustain only two products, exactly two firms enter in equilibrium and they choose distinct product qualities.

This article is organized as follows. Section 2 sets up the model. Section 3 contains the results. I focus on a symmetric equilibrium where the firms' waiting strategy is strictly monotone and differentiable. These strategies are invertible, which makes it easy to characterize the information that transpires from the waiting game. Section 4 concludes and discusses some extensions. All proofs are found in the Appendix.

2. The model

* Two firms i [member of] {A, B} will enter a horizontal product market. Each firm must make two decisions: it must choose a product design [[theta].sub.i] [member of] R, and when to enter the market [t.sub.i] [member of] [0, [infinity]) = T. Entry decisions are irreversible, and the first firm's decision is observed by the other firm. Consider the following payoff function:

[[pi].sub.i]([[THETA].sub.i], [[THETA].sub.j], [t.sub.i]) = -[([[THETA].sub.i]-[rho]).sup.2]-[[alpha] [([theta].sub.i] -[[theta].sub.j]).sup.2]-[delta] [t.sub.i], i[not equal to]j.

Profits depend on entry decisions in two ways. First, firm i's profit decreases quadratically with the distance between [[theta].sub.1] and [rho], where [rho] is an unknown parameter. Firms receive an unbiased signal of [rho], such that [[rho].sub.i] = [rho] + [[epsilon].sub.i], where [[epsilon].sub.i] is normally distributed with mean zero and variance [v.sub.i]. Hence, each firm has two pieces of private information, a signal of the state and the precision of that signal. In words, [rho] represents the (a priori) most profitable market niche. If [[theta].sub.i] is far from [rho], the firm has chosen an unattractive product design, an event that is more likely the higher is [v.sub.i]. I assume that variances are drawn from a nonatomic distribution [PSI](v) with density function [psi](v) > 0 for all v [member of] [0, [bar]v 0] = V, where [bar]v is finite. All draws are conditionally independent.

Second, both firms' profits vary quadratically with the distance between [[theta].sub.A] and [[theta].sub.B], which captures the market interaction between firms. Parameter [alpha] characterizes how firms interact. If [alpha] is positive (negative), products are strategic complements (substitutes), since firms benefit from decreasing (increasing) the distance between [[theta].sub.A] and [[theta].sub.B]. If [alpha] = 0, there is no direct externality and we have a case of pure informational spillovers. For example, think of a market where product-specific marketing also increases generic demand. If advertising costs are significant, firms should launch similar products so as to maximally exploit the spillovers from each other's marketing. If advertising costs are relatively small, or the spillovers from the other firm's marketing are small, similar designs will only intensify competition, so firms should differentiate their products.

Finally, firm i's payoff decreases linearly in [t.sub.i], the time the firm enters the market. (2) This could, for example, reflect the fact that corporate resources are tied up as long as the decision is delayed, resources that could have been put to use elsewhere. Parameter [delta] > 0 measures the degree of urgency: the higher is [delta], the more costly it is to delay the entry decision.

* The waiting game. Since signals are unbiased, the signal realization has no effect on the incentive to learn the other firm's information and should, therefore, have no effect on the timing decision. This allows me to consider the entry decision (the choice of [theta]) and the timing decision (the choice of t) separately. Consider first the entry decision. There are two possibilities: firm i enters either as the leader or as the follower. Below, I impose sufficient conditions to ensure that the leader's and the follower's optimal decisions, as a function of their available information, are unique. In turn, this allows me to characterize expected payoffs in terms of variances [v.sub.A] and [V.sub.B] only, which is done in the next section. For now, denote the leader's and follower's expected payoff (excluding delay) by [L.sub.i]([v.sub.i], [v.sub.j]) and [F.sub.i] ([v.sub.i], [v.sub.j]), respectively.

Consider now a firm's waiting strategy. Since delay is costly, once one firm has entered, the other will follow as soon as possible. For simplicity I assume that there is no involuntary delay, so that both firms' delay is determined by the leader's choice. (3) Hence, it is sufficient to consider strategies that are conditioned on the fact that the other firm has not entered. Because [PSI] is atomless, we may as usual restrict attention to "stopping time" strategies [s.sub.i] : V [right arrow] T, i = A, B. That is, a (pure) waiting strategy is a mapping from a firm's variance to a nonnegative number: the time the firm will enter given that the other firm has not entered up to that moment. I restrict attention to equilibria where strategies are strictly monotone and differentiable. This simplifies the analysis in at least two important ways. First, since variances are drawn from a nonatomic distribution, the event that both firms enter at the same time is a set of probability measure zero, so that event can be ignored. Second, each strategy has an inverse function [[s.sup.-1.sub.i] (.) that maps each point in time to a unique variance. This means that when the first firm enters, the other firm can infer its variance.