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.

Suppose that firm i draws variance [v.sub.i] and that the firms use strategies [s.sub.i](v) and [s.sub.j](v). Firm i's expected payoff can then be written (i [not equal to] j)

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let [[mu].sub.i] be firm i's posterior over firm j's variance. A strategy profile s = {[s.sub.i], [s.sub.j]} and a belief system [mu] = {[[mu].sub.i], [[mu].sub.j]} constitute a PBE if[pi] and and [pi] are maximized given [mu] and the other firm's strategy, and if [mu] is consistent with s in terms of Bayesian updating. (4) In the following, I shall focus on the symmetric PBE, which is natural given the ex ante symmetry between firms. That is, I impose the condition

[s.sub.A](v) = [s.sub.B](v), [for all]v [member of] V.

3. Results

* Entry decisions. I now characterize the leader's and follower's expected payoff as a function of their variances. To guarantee interior equilibria, I must put a lower bound on [alpha], i.e., I have to assume that products are not too close substitutes. If they were, there would be an equilibrium where firms ignored their private information and resorted to "maximum differentiation," which in fact would result in infinite profits. Alternatively, the bound below ensures that the leader actually uses its private information. (5)

Assumption 1. [alpha] > ([square root of 5] - 3)/2 [approximately equal to] -.38.

Lemma 1. Suppose firm A is the leader. Let m denote B's conditional expectation of [rho] (on observing [[theta].sub.A]). Under Assumption 1, in a PBE firm A sets

(2) [[theta].sub.A] = [[rho].sub.A],

and firm B sets

(3) [[theta].sub.B] = m + [alpha][[theta].sub.A] / 1 + [alpha]

The expressions when B is the leader are analogous. The expression in (3) captures the follower's tradeoff between the informational value the leader's entry provides and the externality it imposes. If [alpha] is positive (the marketing spillover dominates the effects of competition), the follower will choose a design closer to the leader's as compared with its expectation of [rho]. If [alpha] is negative, the follower will choose a design less similar to the leader's. The fact that the leader chooses its design according to its signal is important: the follower thereby has de facto access to both signals. Since the follower also infers the leader's variance, its expectation of [rho] is simply a linear combination of [[rho].sub.A] and [[rho].sub.B]. Expected payoffs therefore take simple expressions.

Lemma 2. Excluding delay costs, firm A's expected payoff from being the leader and the follower is, respectively,

(4) [L.sub.A] = -[v.sub.A] - [alpha][v.sup.2.sub.A] / [(1+[alpha]).sup.2]([v.sub.A] + [v.sub.B])

and

(5) [F.sub.A] = -[v.sub.B]([b.sub.B][alpha]+(1+[alpha])[v.sub.A]) / (1+[alpha])([v.sub.A]+[v.sub.B])

The corresponding payoffs for firm B are analogous. To reiterate, under Assumption 1, the leader's expected payoff decreases with its variance ([differential] [L.sub.A]/[differential] [v.sub.A] < 0). However, how the leader's payoff varies with the follower's variance depends on the payoff externality. Intuitively, the less informed the following firm is, the more inclined it will be to imitate the leader's choice rather than trust its own information. Poorly informed followers therefore (on average) choose positions that are closer to the leader. Hence, if products are substitutes, the leader benefits from having a well-informed follower (?[L.sub.A]/?[v.sub.B] < 0). The latter effect is key to my main result, namely that if products are substitutes, the well- informed firm may have the stronger incentive to wait.

* Timing. When considering their waiting strategy, firms strike a tradeoff between expected delay costs and the possibility of being the follower instead of the leader. (This possibility is not necessarily of positive value, but in the symmetric equilibrium it is never negative.) Consider firm A. Conditional on B's strategy and its own variance [v.sub.A], A chooses an optimal entry time, t. Equivalently, A can choose the variance v that, given B's strategy, corresponds to t. For example, suppose firm B uses an increasing strategy, [s.sub.B](v). Using the payoff expressions (4) and (5) in the maximization problem (1) gives the v that must maximize the sum of

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

(7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The integral in (6) is firm A's expected payoff from being the leader, which happens if A picks a v that is smaller than [v.sub.B]. Likewise, the integral in (7) is the expected payoff from being the follower, which happens when v is larger than [v.sub.B]. The case of decreasing strategies is completely analogous. I am now ready to present my main result. The proposition uses the following definition:

Definition 1. [alpha]* = ([square root of 13] - 5)/4 [approximately equal to]-.35.

Proposition 1.

(i) If [alpha] > [alpha]*, the best-informed firm enters first. The equilibrium-stopping function is

s(v) = (2[alpha] + 1)/2[delta] [(1 + [alpha]).sup.2] [[bar]v ln[bar]v - v - [bar]v ln([bar]v - v)].

(ii) If [alpha] < [alpha]*, the worst-informed firm enters first. The equilibrium-stopping function is

s(v) = (s[alpha] + 1)/2[delta] [(1 + [alpha]).sup.2] ([bar]v - v).

(iii) If [alpha] = [alpha]*, both equilibria are possible.

When products are complements or weak substitutes, the time a firm is prepared to wait is increasing in its variance--much like in Zhang (1997). Here, Farrell and Saloner's (1986) "penguin effect" dominates: firms wait because they hope to get more information about the unknown demand parameter [rho]. Moreover, the function is exponential, so poorly informed firms are prepared to wait a disproportionately longer time than better- informed ones. The stopping function for case (i) is illustrated in Figure 1.

[FIGURE 1 OMITTED]

As [alpha] is reduced, i.e., as products become closer and closer substitutes, a well-informed firm's incentive to outwait a poorly informed firm becomes relatively stronger. The intuition is as follows. When products are substitutes, the follower always chooses a niche too close to the leader's--from the leader's point of view. Moreover, the worse informed the follower is relative to the leader, the more the follower relies on the leader's choice. Hence, a poorly informed follower imposes a larger externality on the leader than a well-informed one, and more so the lower is [alpha].

When [alpha] passes below [alpha]*, a well-informed firm's incentive to wait becomes stronger than that of a poorly informed firm. For this parameter region, the negative externality a poorly informed firm imposes as a follower, through its "penguin-like" behavior, is larger than the informational spillover a well-informed firm would generate as a leader. Alternatively, when competition is sufficiently harmful (or marketing spillovers are sufficiently small), a well-informed firm gains more from enjoying "monopoly" in a good market niche than what a poorly informed firm loses from choosing a bad niche. (6) The stopping function for case (ii) is illustrated in Figure 2.

[FIGURE 2 OMITTED]

* Delay. I conclude with some remarks on delay. Inspection of the stopping functions in Proposition 1 reveals that delay is proportional to the factor

(8) (2[alpha] + 1)/2[delta] [(1 + [alpha]).sup.2].

Note that expected delay decreases geometrically with [delta] and, thus, that delay costs are independent of the degree of urgency. Differentiating (8) with respect to [alpha] gives

- [alpha]/[delta][(1 + [alpha]).sup.3].

Ceteris paribus, the longest delay occurs when [alpha] = 0. Whenever an externality is present, the following firm to some extent chooses a niche based on the leader's choice, rather than according to market information. Hence, the stronger the externality--whether positive or negative--the less important (relatively) becomes product design per se. This reduces the incentive to observe the other firm's decision, and decreases delay. Note from (8) that delay appears to go to zero as [alpha] goes to -.5. However, Proposition 1 presumes that the leader enters in accordance with its signal, which is no longer optimal if [alpha] < -.38 (the limit in Assumption 1). (See Figure 3.)

[FIGURE 3 OMITTED]

Note that with a positive payoff externality, delay decreases relatively slowly as the externality grows stronger. Hence, firms suffer substantial delay costs despite there being large gains from sharing information. This is not very realistic: at some point these gains will induce firms to overcome any potential coordination costs. In particular, if firms can engage in cheap talk, they can reach the first-best solution whenever [alpha] [greater than or equal to] 0 by revealing their private information and entering without delay (at the same location).

Finally, though not modelled here, the presence of more than two firms should strengthen the effect of the payoff externality. Suppose that the leader in my model instead was followed by n firms of the same type. The higher is n, the more beneficial it becomes to generate an informational spillover if products are complements, and the more costly it becomes if products are substitutes. This effect is similar to that of magnifying [alpha], which reduces delay. As a consequence, adding more firms should also increase the threshold [alpha]*, requiring less- intense competition for the least-informed firm to enter first.

4. Conclusion

* Informational spillovers induce agents to outwait each other in order to make more-informed decisions themselves. If delay is costly, the presence of spillovers leads to a classic war of attrition between agents. Zhang (1997) showed that if agents have different informational precisions, the best-informed agent takes the first action in a symmetric equilibrium. In this article I combine informational spillovers with a direct payoff externality. Still, the only difference between agents ex ante is the quality of their private information.

The addition of the direct externality has two effects on the waiting game. First, it reduces delay per se. The stronger the externality--whether positive or negative--the smaller the (relative) importance of being well informed. This attenuates the second-mover advantage and decreases delay. Interestingly, the externality may have a more qualitative effect. When the externality is negative and very strong, it turns out that poorly informed agents take action before well-informed ones. The intuition is that poorly informed agents mimic the behavior of others to a larger extent. Hence, as a follower they impose a larger negative externality on the leader than do well-informed agents. If the externality is sufficiently strong, this effect outweighs informational concerns, which makes well-informed agents wait the longest.

I have illustrated this mechanism as an entry game between two firms. In this context, the direct externality has a straightforward interpretation as a measure of the strategic complementarity/substitutability between products. However, the model should apply to any situation where informational spillovers and payoff externalities co- exist. For example, the agents could be investors in the stock market. A trading decision has a direct effect on the price of the asset in question, but it also reveals something about the investor's private information or expectations. Will a purchase trigger other investors to buy or sell the stock? Gamblers in betting markets with moving odds face a similar situation. As a political application, consider candidates choosing which policy platform to adopt on a complex issue. Not only does a candidate want to endorse policies that appeal to a large share of the electorate, he may also be anxious to represent a policy that stands out from those of other politicians. Hence, the order in which politicians take stands may depend on how well informed they are as well as how badly they need publicity. It may be important to recognize that in some circumstances, the politicians who choose policies first are those with the least knowledge, and that the sooner a politician decides, the less he knows.

Appendix

* Proofs of Lemmas 1 and 2 and Proposition 1 follow. For ease of exposition, I let [v.sub.A] = a and [v.sub.B] = b in the entire Appendix. Let the cumulative distributions G([rho]) and H([[rho].sub.j]) denote firm i's posterior of [rho] and [[rho].sub.j] (i [not equal to] j), respectively.

Proof of Lemma 1. Given [[theta].sub.A], firm B solves the following problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Let m = E[[rho] | [[rho].sub.B], [[theta].sub.A]] denote B's expectation of [rho]. The first-order condition reads

(A1) -2[[theta].sub.B](1 + [alpha]) + 2m + 2[alpha][[theta].sub.A] = 0.

As long as, [alpha] > -1, the left-hand side of (A1) is everywhere decreasing in [[theta].sub.B] so that the first-order condition gives a global maximum. Rearrange (A1) to get

[[theta].sub.B]= m+[alpha][[theta].sub.A] / 1+[alpha],

which proves the second part of the lemma. Anticipating this, the leader (firm A) solves

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In a PBE, firm B must have the correct expectation of [[rho].sub.A]. Suppose, therefore, without loss of generality, that B's expectation of [rho] is a linear combination of the two signals, i.e., m = [lambda][[rho].sub.A] + (1 - [lambda])[[rho].sub.B] for some [lambda] [member of] [0,1] Firm A's expectation of [rho] is simply [[rho].sub.A]. Firm A's first-order condition then reads

(A2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Both estimators are unbiased, so E[[[rho].sub.B]] = [[rho].sub.A]. becomes

(A3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the expression in square brackets is positive, the derivative is everywhere decreasing in [[theta].sub.A] and (A3) gives a global maximum. This occurs as long as [alpha] > ([square root of 5] - 3)/2, i.e., as long as Assumption 1 is satisfied. The solution is, naturally, to set [[theta].sub.A] = [[rho].sub.A]. Q.E.D.

Proof of Lemma 2. Consider first the case when A is the follower. By Lemma 1, [[theta].sub.B] = [[rho].sub.B], so upon observing B's entry decision and its own signal [[rho].sub.A], firm A'S posterior distribution over [rho] is normal with expected value

m = b[[rho].sub.A] + a[[rho].sub.B] / a + b

and variance

w = ab / a + b.

Conditional on observing [[theta].sub.B], A's expected payoff is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substitute for E[[rho]] and E[[[rho].sup.2]] and the equilibrium expressions for [[theta].sub.A] and [[theta].sub.B] and extend the expression by (1 + [alpha]) to get

- (w+[m.sup.2])(1+[alpha]) + 2[alpha]m[[rho].sub.B] + [m.sup.2] + [alpha][[rho].sup.2.sub.B] / (1+[alpha])

Substituting for m and w and extending by [(a + b).sup.2] gives

b -ab - [alpha]ab - [a.sup.2] - [alpha][a.sup.2] - [alpha]b[[[rho].sup.2.sub.A] + 2[alpha]b[[rho].sub.A][[rho].sub.B] - [alpha]b[[rho].sup.2.sub.B] / [(a+b).sup.2](a + [alpha])

We want the "unconditional" expectation of this (i.e., before A observes [[theta].sub.B]). Since the two estimators are unbiased and conditionally independent, we have that, conditional on [[rho].sub.A], E[[[rho].sub.B]] = [[rho].sub.A] and E[[[rho].sup.2.sub.B]] = [[rho].sup.2.sub.A] + a + b, [for all][[rho].sub.A]. Hence, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which proves the second part of the lemma. Now suppose A is the leader. In equilibrium, its expected payoff is (ignoring delay costs)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Extend the integral by (1 + [alpha])[(a + b).sup.2] and rearrange to get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, substitute for E[[[rho].sup.2.sub.B]] to get

= -a - [alpha][a.sup.2] / [(1 + [alpha]).sup.2](a + b).

Proof of Proposition 1. Suppose first that [alpha] > [alpha]*. Suppose that firm B uses an increasing strategy s(v) so that firm A's posterior over B's variance at time t, given that no firm has entered, ranges over [[s.sup.-1] (t), [bar]v]. Firm A chooses v to maximize

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order condition with respect to v reads

(A4) 3[alpha][a.sup.2] + [a.sup.2] + [a.sup.2][[alpha].sup.2] - [v.sup.2][alpha] - [v.sup.2][[alpha].sup.2] / [(1 + [alpha]).sup.2](a+v) - [delta]s'(v)[[bar]v - v] = 0

Firm B naturally solves the analogous problem. Following Gul and Lundholm's (1995) style of proof, if there is a symmetric PBE, (A4) is satisfied for all v = a. In other words, in equilibrium both firms must find it optimal to use the strategy that they postulate the other firm uses. Let us confirm that (A4) indeed yields the maximum. The simplest way of doing this is to differentiate (A4) with respect to a instead of v. If the resulting second-order condition is positive at v = a, we know that (A4) gives a maximum. Differentiating (A4) with respect to a gives

3[alpha][a.sup.2] + 6a[alpha]v + [a.sup.2] + 2av + [a.sup.2][[alpha].sup.2] + 2a[[alpha].sup.2]v + [v.sup.2][alpha] + [v.sup.2] [[alpha].sup.2] / [(1 + [alpha]).sup.2][(a + v).sup.2].

Setting v = a gives

(A5) 10[alpha] + 3 + 4[[alpha].sup.2] / 4[(1 + [alpha])].sup.2].

As long as [alpha] > [alpha]*, (A5) is positive. Further, in the increasing equilibrium we have the boundary condition that s(0) = 0. Otherwise a firm with variance zero would suffer a positive delay cost yet enter first almost surely. Setting v = a in (A4) gives

a(2[alpha] + 1) / 2 [(1 +[alpha]).sup.2][[bar]v - a] = [delta]s' (a).

Integrate and use the boundary condition to get case (i) of the proposition. Suppose instead that [alpha] < [alpha]* and that firm B uses a decreasing strategy, so that firm A's posterior over B's variance at time t ranges over [0, [s.sup.-1](t)]. Firm A chooses v to maximize

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The first-order condition reads

(A6) -([a.sup.2] + 3[alpha][a.sup.2] + [a.sup.2][[alpha].sup.2] - [v.sup.2][alpha] - [[alpha].sup.2][v.sup.2] / [(1 + [alpha]).sup.2](a + v)) - [delta]s'(v)v = 0.

In analogy with case (i), (A6) gives a maximum as long as [alpha] < [alpha]*. Setting v = a in (A6) gives

- (1 + 2[alpha]) - / 2[(1 + [alpha]).sup.2] = [delta]s'(a).

In the decreasing equilibrium we have the boundary condition s([bar]v) = 0. Using this proves case (ii) of the proposition. Finally, if [alpha] = [alpha]*, the second derivative is exactly zero, so that both equilibria are possible. Q.E.D.

(1) Rob (1991) studies sequential entry in a market with demand uncertainty. In his model, as in mine, both informational spillovers and direct payoff externalities are present. In his model, however, firms are ex ante (before entry) identical.

(2) The linear cost simplifies matters but is not necessary. As long as delay costs are separable, my results hold for all strictly increasing and differentiable functions f([t.sub.i]). However, each function will give rise to a different stopping function (see Proposition 1).

(3) This assumption simplifies matters, since I do not have to be concerned with the "monopoly profits" the first firm would make before the other firm enters. However, the introduction of a period of monopoly profits would not change the results as long as these profits are not too large compared to overall profits.

(4) Formally, let maps [S.sup.V.sub.i] : [V.sub.i] [right arrow] [S.sub.i] be a firm's set of pure strategies in the "expanded game" (Fudenberg and Tirole, 1996), i.e., before [v.sub.i] is realized. The strategy profile s = {[s.sub.i], [s.sub.j]} is a PBE if, for each firm, [s.sub.i](.) [member of] arg max [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(5) See Neven (1985) for a location game where the outcome is maximum differentiation.

(6) Note that the decision to delay reveals something about the firm's informational quality ([v.sub.i]) only, not its information about market properties ([[rho].sub.i]). Hence, unlike in Mailath (1993), the (potential) second-mover advantage does not disappear because of the endogenous timing.

References

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Lars Frisel, Sveriges Riksbank, Stockholm; lars.frisell@riksbank.se

I thank James Dana, Joseph Harrington (the editor), Fredrik Heyman, Johan Stennek, Jonas Vlachos, Karl Warneryd, and Jorgen E. Weibull for help comments. Financial support from the Jan Wallanders and Tom Hedelius Foundation is grately acknowledge.