Entry and Preemption When the Competitor Waits and Sees∗ Stefano Comino Departament d’Economia i d’Història Econòmica Universitat Autònoma de Barcelona. July 18, 2000

Abstract In the paper, I characterize the first mover’s strategic behavior in a model in which two firms can enter two markets of unknown but correlated profitability. To collect information, the first mover can enter markets sequentially. However, this choice reveals the markets’ conditions to the competitor. I show that when the prior probabilities about markets’ profitability are favorable, then the first mover strategically chooses the sequential entry pattern. Indeed, this choice induces the competitor to wait and see and thus to postpone entry. That is, the competitor observes the first mover’s complete entry pattern, infers information about profitability and then decides about entry with a superior information. When the prior probabilities are less favorable, the first mover strategically enters both markets at the same time, reveals no information and tries to preempt the competitor. Keywords: Entry, Sequential Entry, Preemption, Wait and See, First Mover. J.E.L. classification: D83, L13.

∗ The address for correspondence is: Stefano Comino, Departament d’Economia i d’Història Econòmica - Universitat Autònoma de Barcelona - 08193 Bellaterra, Spain. E-mail: [email protected]. Telephone: 34 93 5811813. Fax: 34 93 5812012.

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1

Introduction

Entry in a market is a decision that typically involves uncertainty. Variables in question might be the production costs, the demand conditions, the suitability of the supplied products to the consumers’ tastes or the level of competition that a firm is going to experience. Uncertainty arises both when a firm is entering a yet unserved market or when it is planning to launch a new product. One strategy that a firm can use is to enter in a sequential way. It can start with a low scale and then increase it if market conditions have been favorable. Cases where firms seem to have chosen this kind of entry pattern abound. Matsushita, a Japanese consumer electronics producer, established its first venture in China in 1987 and then, in the period 1992-96, it expanded its presence in the Chinese territory with 30 new companies.1 Similarly, Fiat, an Italian car producer, launched its ”third world car”, the Palio, in Brazil as first and then, after succeeding, in similar markets such as Argentina and Venezuela.2 Further, firms use ”low-risk-countries” such as Austria, Canada and Taiwan as beachheads in order to learn business opportunities or to build capabilities for then entering the actual target markets, Germany, the US and China respectively.3 However, a sequential entry pattern might reveal information to other firms. Observing the increased size of the investment, a company can infer that market conditions are favorable and, therefore, it can follow the first entrant. That is, a sequential pattern might induce other firms to follow the leader. This combination among sequential entry and follow-the-leader behavior might be recognized in the entry pattern chosen by Japanese firms investing in the US market during the 80s. According to Chang (1995), Japanese firms typically adopted a sequential entry pattern. Furthermore, both Hennart and Park (1994) and Head et al. (1995) find that the presence of Japanese firms in the US territory increased the likelihood that other Japanese firms entered. That is, these authors find that a follow-the-leader behavior was employed.4 In my paper, I introduce these features in an entry model à la Dixit (1980). A first mover chooses its entry pattern into two markets of unknown but correlated size. It can enter with a full scale (enter the two markets at the same time) or with a smaller one (enter one market), learn the correlated size and then increase the scale (enter the second market) if profitable. However, the sequential pattern can reveal information to a competitor. Observing that the first mover enlarges its scale, the second entrant infers that the market conditions are favorable and then it enters markets. 1 The

Economist 20.09.97. Economist 13.09.97. 3 Financial times 30.01.98. 4 In particular, Hennart and Park (1994) find that this follow-the-leader behavior held among large groups. That is, if a large Japanese group entered the US market it was more likely that another large group from the same country followed. According to Head et al. (1995) initial investments by Japanese firms spurred subsequent Japanese investors in the same industry or industrial group to select the same states. 2 The

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The explicit consideration that the first mover behavior might reveal information to the competitor is the novel aspect of my analysis. Indeed, the literature about entry usually extends the two-period two-firms Dixit model to allow for uncertainty but not for learning among firms.5 In Somma (1999) and Brander and Spencer (1992), uncertainty resolves as time goes by. Between the first and the second period both firms learn the market conditions. In McGahan (1993), uncertainty is resolved just after firms have invested. The author compares the equilibrium under two regimes. In the first, both firms learn the market conditions as soon as the first mover invests. In the second, it is only the investing firm that learns the state of demand. In my paper, in some sense, I endogenize these two regimes. The first mover can decide whether to choose an entry pattern that reveals information to the competitor (sequential entry) or another that does not (enter the two markets at the same time). Inferring information by observing the behavior of informed agents is an issue that has been extensively addressed within contexts different from entry or investment decisions.6 Typically, these models consider situations in which the link among agents is purely informational. That is, each agent can learn valuable information by observing the behavior of the other players, but its pay-off does not depend on their actual choices. Each player would prefer the others to move as first to learn their private information, but once she decides to move (and then reveal her own information) she will not act in a strategic way since her pay-off does not depend on the choice of the others. However, when uncertainty is related to variables that are specific to the firm’s business, it is only the observation of the behavior of companies belonging to the same industry which is relevant for information acquisition. In this case, the assumption that the firm’s pay-off is unaffected by the choice of the other companies is no longer acceptable. Another strand of literature which is related to my paper is the so called market-experimentation.7 In these papers, competing firms, in order to improve their information about an unknown variable, distort their choices from the ones that maximize short-run profits. The main difference with my model relies on the nature of information. While I consider it as being privately acquired, in the market experimentation literature information is public so that firms are always symmetrically informed. As a benchmark, I first analyze (in section 3) the case in which the profits of each firm do not depend on the other firm’s choices; that is, firms are not rival. Still, the behavior of the first mover might reveal information. In this context, the first mover does not behave strategically and both firms face a trade-off between higher potential profitability (enter the two markets at the same time and as soon as possible) and additional information (enter markets sequentially or observe the first mover’s behavior before acting). 5 Maggi (1996) considers a framework different from the Dixit model in that there are no exogenous first and second mover. However, also in this case there cannot be any learning among firms. Indeed, uncertainty resolves as time goes by. 6 See, among others, Chamley and Gale (1994), Gul and Lundholm (1995). 7 See Harrington (1995) and Mirman et al. (1994).

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In section 4, I analyze what changes when firms are rival. In particular, by comparing with the benchmark, I focus on the strategic behavior of the first mover. This firm knows that entering markets sequentially it reveals information to the competitor, while entering them at the same time it does not. When the prior probabilities are such that firms believe that markets are likely to be profitable, then the first mover has no chance to preempt the competitor. In this case, the best it can do is to enter markets sequentially, reveal information trying to delay the competitor’s entry. Indeed, when information is going to be revealed, the second mover might choose to wait and see. That is, it observes the complete entry pattern of the first mover, infers the market conditions and takes any decision with a better knowledge about profitability. On the contrary, when the priors are such that firms believe that markets are not likely to be profitable, the first entrant tends to strategically choose to enter the two markets at the same time, reveal no information and preempt the competitor. In the subsequent section 5, I briefly consider the case of complementarities. That is, the case in which profits are higher when both firms are simultaneously present in the same market. This time, the first mover’s aim is to encourage the second firm to enter as soon as possible. I find that the first mover might strategically choose to enter both markets at the same time rather than enter them sequentially. Indeed, this last strategy might induce the second firm to wait and see and thus to postpone entry. In section 6, I consider an extension of my model. Entry models à la Dixit assume the presence of an exogenous first and second mover. The question I address in this section is when such a structure arises as an equilibrium outcome. That is, if both firms were free to move from the beginning, when would it happen that one firm enters as first and, some time later, the other follows? I find that the results crucially depend on the level of competition among firms and on the prior probabilities about markets’ profitability. The last section of the paper is devoted to the conclusions. The proofs of the various propositions, corollaries and lemmas can be found in the appendix.

2

The Model

Two firms, denoted by A and B, can enter two markets of perfectly positively correlated size. The size, s, is unknown before entry and it is perfectly revealed thereafter to the entering firm. Before entry takes place, firms believe that s = 1 with probability p and s = 0 with probability (1 − p). The per-period profits in a market are sπ if only one firm is present and s (π − r) when both firms are present. I assume that once entered a market a firm stays there forever. sπ and s(π−r) The present value of the profits from the period of entry on is 1−δ 1−δ respectively; where δ ∈ (0, 1) is the discount factor. To save notation, I call π r Π ≡ 1−δ . The parameters r and R measure the level of rivalry and R ≡ 1−δ among firms. To enter each market a firm pays a fixed cost F. The cost is paid at the entry 4

time and it is sunk. Clearly, if before entry a firm knows that s = 0, then it does not enter any market: F has to be paid and zero profits will be earned. On the contrary, I assume that if a firm knows that s = 1, then independently of the presence or absence of the other firm in the same market, entry is profitable. In other words, I assume that when s = 1 then the flow of profits that a firm earns in a market from the period of entry on is always sufficient to cover F . This is formalized by the following condition: Π − R ≥ F.

(1)

For future reference, I define R1 ≡ Π − F so that condition (1) can be re-written as R ≤ R1 . I assume that (1) holds throughout the whole paper. In the model, I just look at the entry decisions. The timing of the game is as follows. I assume that firm A moves in periods t = 1, 2 while firm B moves in periods t = 2, 3. Firm A can locate in the two markets in the same period and ¢ the only knowledge of the priors, p and (1 − p). I denote this choice ¡ with A , with i = 1, 2. Alternatively, A can enter markets in sequence and take EEt=i advantage of the correlation of the sizes. In this case, at t = 1, it locates in one A )), it observes s and, the next period, it conditions entry in the market ((Et=1 second upon this new information. The last option for firm A is to enter no A )). market at all ((Nt=1,2 Formally, the same choices are available to firm B in periods t = 2, 3. However, B has also the chance of taking advantage of the information that A’s behavior might reveal. When, at t = 1, firm A has entered only one market, then, at t = 2, it decides whether to locate in the second having perfect knowlB ), B observes A’s choice edge of s. Choosing to enter no market at t = 2, (Nt=2 and it infers information about s. In this case, I say that firm B chooses to wait and see. The advantage of this strategy is that firm B will decide about entry with a better knowledge of markets’ profitability. The disadvantage is that entry in any site is postponed till t = 3. Before starting the analysis, I characterize firm B’s behavior in two simple scenarios. First, from condition (1), whenever B has learned8 that s = 1 (s = 0), then it locates in all the markets (no market) which it has not entered yet. Second, consider the case where A enters the two markets at the same time. A’s behavior reveals no information and B expects 2 (p (Π − R) − F ) when enB tering the ¡ two ¢ markets at t = 2 (that is choosing (EEt=2 )). Entering one B market ( Et=2 ) B knows that, at t = 3, it will enter the other iff s = 1 and then the pay-off associated with this choice ¡ Bis p ¢(Π − R)−F +δp ((Π − R) − F ) . ), firm B expects a pay-off of 0. Choosing not to locate in any market ( Nt=2,3 One can check that B chooses to enter both markets at t = 2 when priors are 8 Either because it has entered one market or because it has inferred that s = 1 by observing A’s behavior.

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high: p ≥ p1 (R) . It chooses to enter markets in sequence (enter one market at t = £2 and then, at¢ t = 3, enter the second iff s = 1) for intermediate values: p ∈ p2 (R) , p1 (R) . It does not enter any market for low values of the priors: p < p2 (R) . The bounds p1 (R) and p2 (R) as well as all the other bounds for p that I will use later are defined in the appendix. For future reference, I will say that when p < p2 (R) , then by entering the two markets at the same time firm A preempts B’s entry.

3

The Benchmark: r = 0

As first, I consider the case in which firms are not rival. That is, r and R are equal to 0. Still, given that s is common for the two firms, B might have the chance of inferring the market conditions by observing A’s entry pattern. In this setting, firm A does not play strategically. Indeed, its pay-off does not depend on B’s choices and, given the timing of the game, A cannot take advantage of any information that B’s behavior might reveal. When, at t = 1, it has entered one market, then, at t = 2, firm A always separates. Indeed, by condition (1), ¢firm A locates in the second market if ¡ A (s = 1) ), while it does not if s = 0 (denote this E s = 1 (denote this choice t=2 ¡ A ¢ (s = 0) ). Therefore, in the benchmark, when entering one market choice Nt=2 at t = 1, firm A is truly choosing a sequential entry pattern in that, whenever market conditions are it will enter the other market too. This implies ¢ ¡ favorable, A , firm A always reveal the true state of the markets. that after choosing Et=1 Choosing their entry pattern firms face a trade-off between higher potential profitability and additional information. Entering the two markets as soon as possible, a firm earns profits from the start but it locates in the two markets with the only knowledge of priors. With the sequential pattern or by waiting and seeing, a firm postpones the period from which it starts earning profits but it decides upon entry in the second or both markets with perfect knowledge of s. Next proposition characterizes the optimal choices of the two firms. Proposition 1 When R = 0 , then: (i) A enters both markets at t = 1 and B enters both markets at t = 2, if p ≥ p1 (R = 0); markets sequentially and B chooses to wait and see, if p ∈ ¢ £ 2 (ii) A enters p (R = 0), p1 (R = 0) ; (iii) Neither A nor B enter any market, if p < p2 (R = 0).

Consistently with what I have said above, when the markets are likely to be profitable (p ≥ p1 (R = 0)), firms enter them as soon ¢ £ as they can and just knowing the priors. For intermediate values of p (p ∈ p2 (R = 0), p1 (R = 0) ) firms prefer to collect information. A chooses a sequential pattern while B takes advantage of the information that A’s behavior reveals. That is, it chooses to 6

wait and see. When the priors are so low that negative profits are expected (p < p2 (R = 0)), no firm will enter any market.

4

The Rivalry Framework: r > 0

Contrarily to the benchmark, when there is competition, firm A benefits from B not entering or postponing entry in the markets. Consequently, when choosing its entry pattern, A might behave in a strategic way. I start the analysis by characterizing the equilibrium of the signaling game that takes place at t = 2 once A has entered one market.

4.1

The Signaling Game played at t = 2

Even in the case of rivalry, if A has observed that the size is s = 0, then it is dominant for it not to enter the remaining market. However, when the size s = 1 has been observed, things might change. Condition (1), assures that entry in the second market is per se profitable, but this choice has a second effect too: firm A reveals that s = 1. Therefore, whenever B chooses to wait and see, firm A the¢ competitor if enters the second market. In other words, whenever ¡ attracts A Et=1 (s = 1) is decisive for B’s entry, then this is a costly choice for firm A: it involves an increase in competition in the two markets. Whether the first or the second effect dominates depends on the level of 2 2 rivalry. As I check in the ¡appendix, there ¢ is a cut-off value, R . If R ≤ R , then A the benefits of choosing Et=2 (s = 1) are greater than the costs due to the increase in competition. On the contrary, when R > R2 , the competition effect is dominant. The cut-off value R2 as well as all the other bounds for R that I will use later are defined in the appendix. I will say that competition is not harsh when R ≤ R2 , otherwise I will say that competition is harsh. Next two lemmas characterize the equilibrium of the signaling game for these two cases. Given that it is dominant not to enter the second market when s = 0, I only characterize A’s choices when s = 1 has been observed. Lemma 2 When R ≤ R2 , then the Bayesian equilibrium of the signaling game played at t = 2 once A has entered the first market is separating. In equilibrium, if s = 1 A enters the second market, while B enters both markets at t = 2 if p ≥ p1 (R) and it chooses to wait and see if p < p1 (R) . When waiting, firm B updates its beliefs in this way: if A has entered the second market, then B believes that s = 1 with probability 1; if A has not entered the second market, then B believes that s = 0 with probability 1. When competition is not harsh, in the signaling game, firm A behaves exactly in the same way as in the benchmark. Since the benefits of entry are higher than 7

the costs due to the increase in competition, firm A always locates in the second market when s = 1 has been observed. Therefore, when entering one market at t = 1, firm A is truly choosing a sequential entry pattern ¡ and it is¢ going to reveal the true state of s at t = 2. When priors are high p ≥ p1 ¡(R) , then ¢B enters the two markets as soon as it can. When priors are lower p < p1 (R) , then B prefers to wait and see and take advantage of the information that A’s behavior reveals. The equilibrium in the case of harsh competition is characterized in the lemma here below.9 Lemma 3 When R > R2 , then the Bayesian equilibrium of the signaling game played at t = 2 once A has entered the first market: (i) is separating, if p ≥ p1 (R). In equilibrium, if s = 1 A enters the second market, while B enters both markets at t = 2; ¢ £ (ii) is semi-separating, if p ∈ p3 (R) , p1 (R) . In equilibrium, if s = 1 A enters the second market with probability y ∗ and does not with probability (1 − y∗ ). Firm B enters the second market with probability x∗ and does not enter any market with probability (1 − x∗ ) ; (iii) is pooling, if p < p3 (R) . In equilibrium, A does not enter the second market and B does not enter any market.

When, at t = 2, it does not enter any market, firm B updates its beliefs in this way: if A has entered the second market, then B believes that s = 1 with probability 1; if A has not entered the second market, then B believes that ∗ )p s = 1 with probability 0 if p ≥ p1 (R), with probability (1−y(1−y ∗ )p+(1−p) when £ 3 ¢ p ∈ p (R) , p1 (R) and with probability p when p < p3 (R) . ³ ´ 1 p−p3 (R) F +2δR−Π ∗ Where y ∗ = p (R) 1 3 p p (R)−p (R) , x = R(2δ−1) .

When competition is harsh, the costs due to the increase in competition are high and then the preemption of the rival is particularly valuable. Therefore, firm A is willing not to enter the second market even when s = 1, if this preempts B. This happens when priors are low: p < p3 (R) . On the contrary, when priors are high (p ≥ p1 (R)) firm B enters both markets as soon as it can, the decision of A has no influence on B’s actions, so A locates in the second market if s = 1. For intermediate values of priors, A tries to £preempt the competitor by ¢ ”worsening” its beliefs about s. Indeed, when p ∈ p3 (R) , p1 (R) , firms play a mixed strategy equilibrium in which A randomizes among entering or not entering the second market, while B randomizes among entering one market or 9 Recalling that when it has observed s = 0 firm A does not enter the second market, I define the equilibrium of signaling game, separating, semi-separating or pooling depending on whether A enters, randomizes or does not enter the second market once it has observed s = 1.

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not.10 The probabilities according to which A and B randomize satisfy a further condition. If the result of the randomization is that no firm enters any market at t = 2, then the beliefs of firm B are so worsened that, the next period, it will choose not to enter any market. That is, B is preempted. Lemma 3 makes clear that, in the harsh competition case, entering one market at t = 1 does not always imply that A chooses a sequential pattern. Even when s = 1, firm A might refrain from entering¡ the second market in an ¢ A A does not always attempt of preempting B. Therefore, after choosing Et=1 take advantage of the information it has obtained and then it does not always reveal the true state of the markets. Simple differentiation leads to the following result. Lemma 4 The probability, (1 − x∗ ) , with which B chooses not to enter any market at t = 2 in the semi-separating equilibrium of the signaling game, is a decreasing function of the level of competition, r. In the semi-separating equilibrium, firm A is indifferent among entering and not entering the second market. The latter choice is valuable since with probability (1 − x∗ ) it preempts B. Clearly, the stronger the effect of competition (the higher r and R), the greater the advantage of preempting B and then the greater the advantage of choosing not to enter the second market. Therefore, to keep firm A indifferent among entering or not in the second market a decrease in (1 − x∗ ) has to be associated with an increase in r.

4.2

The Results

In this subsection, I first characterize, for some values of R, the entry pattern chosen in equilibrium by A and B. Then, I focus on the strategic behavior of the first mover. In particular, I say that firm A behaves strategically whenever its entry pattern differs from the one it chooses in the benchmark for the same values of the priors. Entering the two markets at the same time, firm A does not reveal any information about s. This choice is preemptive if priors are sufficiently low: p < p2 (R) . The strategic rational for entering only one market at t = 1, depends instead on the equilibrium of the signaling game and then on whether competition is or is not harsh. As I will check, in the latter case, firm A might strategically choose to enter one market in order to delay the entry of the competitor. On the contrary, when competition is harsh, this strategy might be chosen to preempt firm B via the worsening of its beliefs. 10 Note that the mixed strategy equilibrium ”approaches” the pure strategy ones. That is, when p = p3 (R) then y∗ is equal to 0 so that A chooses not to enter the second market with probability 1. When p = p1 (R) , then y∗ is equal to 1 and so A chooses to enter the second market with probability 1.

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4.2.1

Competition is not harsh: R ≤ R2

To completely characterize the entry pattern chosen in equilibrium by the two firms when R ≤ R2 I need to distinguish among several cases. The results do not significantly change from another. Therefore, here I present just the © one toª¤ £ case in which R ∈ R3 , min R2 , R4 .11 ª¤ © £ Proposition 5 When R ∈ R3 , min R2 , R4 , then:

(i) A enters both markets at t = 1 and B enters both markets at t = 2, if p ≥ p1 (R) ; (ii) £ A enters both ¢ markets at t = 1 and B enters markets sequentially, if p ∈ p4 (R) , p1 (R) ;

¢ markets sequentially and B chooses to wait and see, if p ∈ £ 2 (iii) A4 enters p (R) , p (R) ;

(iv) markets at t = 1and B does not enter any market, if £ A enters both ¢ p ∈ p5 (R) , p2 (R) ;

¢ markets sequentially and B chooses to wait and see, if p ∈ £ 6 (v) A 5enters p (R) , p (R) ;

£ 2 (vi) A does6 not¢enter any market and B enters markets sequentially, if p ∈ p (R = 0) , p (R) ; (vii) neither A nor B enter any market, if p < p2 (R = 0) .

As it was the case in the benchmark, firms face a trade-off between higher potential profitability and additional information. When priors are high, firms enter markets as soon as they can. As the probability of having a good state of the markets decreases, then firms try to collect additional information before entering. It is worth taking a closer look at A’s strategic behavior. That is, how the first mover’s behavior changes with respect to the benchmark in order to obtain a different response by firm B. Firm A’s Strategic Behavior Let’s first focus on From previous analyses we know that ¢ ¡ B’s responses. when priors are high p ≥ p1 (R) , firm B enters both markets as soon as it can and independently of A’s choices. In this case no strategic behavior can occur. 11 To characterize the equilibrium, I should consider four cases: R ≤ R3 , R ∈ ª¤ £ ª¤ ¤ © © £ R3 , min R2 , R4 , R ∈ R4 , min R2 , R6 or R ∈ R6 , R2 . The relation among R2 , R4 ¡ ¢1 and R6 is not established a priori. E.g. when δ < 13 2 , then R3 ≤ R4 ≤ R6 so that the third of the four cases is not defined. The results and proofs for the cases I omit in this version of the paper are available upon request.

£

10

¢ £ For intermediate values of the priors (p ∈ p2 (R) , p1 (R) ), firm B enters markets sequentially when A enters the two markets at t = 1, while it chooses to wait and see when A enters only one market at t = 1. Indeed, in this case, since R ≤ R2 , firm A separates at t = 2 so that by choosing to wait and see and thus by postponing any entry decision till period t = 3, firm B learns the true value of s. Therefore, a sequential entry pattern induces firm B to delay entry. This implies that firm A has a strategic to choose entry ¢ ¡ this ¢ ¡ A incentive A is preferred to EEt=1 for pattern and in the appendix I check that Et=1 any p < p4 (R) . A comparison with the benchmark leads to the following result. ª¤ © ¢ £ £ Corollary 6 When R ∈ R3 , min R2 , R4 , then for any p ∈ p1 (R = 0) , p4 (R) firm A enters markets sequentially for strategic reasons. The interval where the strategic behavior takes place enlarges as the level of competition increases; that is, as r increases. According to corollary 6, for intermediate values of the priors, under rivalry, firm A enters markets in £a sequential way more often than in the benchmark. ¢ In particular, when p ∈ p1 (R = 0) , p4 (R) it does so under rivalry while it locates in both markets at t = 1 in the benchmark. As said, the reason for this behavior relies on the possibility of delaying the entry of the competitor. Clearly, the higher r, the more beneficial is to delay B’s entry and, consistently, the larger the set of priors for which the strategic behavior arises. This result can be easily checked by observing that p4 (R) is increasing in r. An implication of the corollary is that: ¢ £ Remark 7 For any p ∈ p1 (R = 0) , p4 (R) , A perfectly reveals the true s under rivalry, while it does not reveal any information in the benchmark. In other words, under the previous conditions, rivalry among firms increases the cases where there is revelation of information. When priors are low (p < p2 (R)), firm B is preempted whenever A enters the two markets at once. This time, strategic reasons clearly induce¡ firm A¢to play A this strategy. In the appendix I check that A prefers to choose EEt=1 when5 ever p ≥ p (R) . Firm A’s strategic behavior when priors are low is summarized in the corollary here below: ª¤ © ¢ £ £ Corollary 8 When R ∈ R3 , min R2 , R4 , then for any p ∈ p5 (R) , p2 (R) firm A enters both markets at t = 1 for strategic reasons. The interval where the strategic behavior takes place enlarges as the level of competition increases; that is, as r increases. ¢ £ When p ∈ p5 (R) , p2 (R) , under rivalry, firm A enters both markets at t = 1 while, in the benchmark, it enters them sequentially. As r increases, the interval where the strategic behavior arises enlarges because of two reasons. 11

On the one hand, as competition becomes stronger it is more advantageous to preempt B; that is, p5 (R) decreases in r. On the other hand, the stronger the effects of competition, the larger the set of priors for which B is preempted; that is, p2 (R) increases in r. 4.2.2

Competition is harsh: R > R2

Even when R > R2 , the complete characterization of the equilibrium requires distinguishing among several cases.12 Here, I present just one of them. The results for the other cases are similar. ª ¢ ¡ © Proposition 9 When R ∈ max R2 , R4 , R5 , then:

(i) A enters both markets at t = 1 and B enters both markets at t = 2, if p ≥ p1 (R) ; (ii) ¢ markets at t = 1 and B enters markets sequentially, if £ A enters both p ∈ p8 (R) , p1 (R) ; (iii) A enters one market at t = 1 and, at£t = 2, the semi-separating equi¢ librium of the signaling game is played, if p ∈ p2 (R) , p8 (R) ;

(iv) ¢ both markets at t = 1 and B does not enter any market, if £ A enters p ∈ p7 , p2 (R) ;

£ 2 (v) A does7 ¢not enter any market and B enters markets sequentially, if p ∈ p (R = 0) , p ; (vi) neither A nor B enter any market, if p < p2 (R = 0) .

When competition is harsh, firm A chooses to enter one market at t = 1 only for limited range of priors. In other words, when R > R2 , A tends either to locate in both markets at the same time or to enter no market ¡ Aat ¢all. Recalling firm A does the analysis done in lemma 3, we know that after choosing Et=1 not always enter the second market even once s = 1 has been observed. That is, in an attempt of preempting the competitor, firm A does not always take ¡ A ¢ choice makes available. For this advantage of the information that the Et=1 reason, locating in one market at t = 1 tends to be non-optimal. Firm A’s Strategic Behavior Similarly to the previous case, for high values of the priors (p ≥ p1 (R)), firm A is not able to affect B’s£ choices so that ¢ no strategic behavior occurs. For intermediate values of p (p ∈ p2 (R) , p1 (R) ), the only chance to preempt the competitor is by entering one market at t = 1. In this case, at t = 2, firms 12 Three

max

©

¢ ¡ © ª ¢ ¡ cases should be considered: R ∈ R2 , R4 , R ∈ max R2 , R4 , R5 and R > ª .

R2 , R5

12

play the semi-separating equilibrium of the signaling game13 and if the result of the randomizations is such that no firm enters any market, then B’s posterior beliefs are so worsened that neither at t = 3 firm B enters any market at t = 3. Therefore, an attempt of preempting the competitor might induce A ¡to choose ¢ ¡ A ¢ A . E In I check that given B’s responses, A prefers Et=1 to ¡ t=1A ¢ the appendix EEt=1 for any p < p8 (R) . A comparison with proposition 1 leads to the following result: ª ¢ ¢ ¡ © £ Corollary 10 When R ∈ max R2 , R4 , R5 , then for any p ∈ p2 (R) , p8 (R) firm A enters only one market at t = 1 for strategic reasons. The interval where the strategic behavior takes place shrinks as the level of competition increases; that is, as r increases. Corollary 10 states that, for intermediate values of the priors, firm A might choose to enter only one market at t = 1 for reasons. In particular, ¡ strategic ¢ ¡ ¢ ¢ £ A A under rivalry and EEt=1 for any p ∈ p2 (R) , p8 (R) , firm A chooses Et=1 in the benchmark. However, the set of priors where this strategic behavior takes place shrinks as r increases. This last result is due to two effects. On the one hand, as pointed out in lemma 4, as r increases the probability that B enters no market at t = 2, (1 − x∗ ) , and then the probability of preempting B decreases. That is, the stronger the competition the less likely that entering one market at t = 1 leads to the preemption of the competitor and, therefore, the less advantageous for A to play this strategy. This, is reflected in the fact that p8 (R) is decreasing in r. On the other hand, an increase in the level of competition enlarges the set of priors for which entering both markets at t = 1 2 ¡preempts ¢ B. That is, an increase in r increases p (R) . As I argue below, when A EEt=1 is preemptive, then entering one market at t = 1 is a dominated choice.

For the case of low priors (p < p2 (R)) I obtain a result similar to corollary 8. Firm A, in order to preempt the competitor, chooses to locate in both markets at t = 1 more often under rivalry than in the benchmark. ª ¢ ¢ £ ¡ © Corollary 11 When R ∈ max R2 , R4 , R5 , then for any p ∈ p7 , p1 (R = 0) firm A chooses to enter both markets at t = 1 for strategic reasons. The interval where the strategic behavior takes place does not depend on the level of competition; that is, it does not depend on r.

Differently from corollary 8, this time, the interval where the strategic behavior arises does not depend on r. The reason for this is that the choice of entering one market at t = 1 is dominated. When priors are low, firm A can preempt the competitor ¡ A ¢ either by entering the two markets at the same time . However, this last strategy requires A to sacrifice the or by choosing Et=1 profits available in the second market in the case s = 1. Therefore, entering the two markets at the same time is a superior way of preempting the competitor so ¢ £ the appendix I check that p3 (R) ≤ p2 (R) and then for any p ∈ p2 (R) , p1 (R) , the equilibrium of the signaling game is the semi-separating. 13 In

13

that A chooses either this strategy or it enters no markets at all.14 Given that the benefits of these two choices do not depend on r, then neither the set where A behaves strategically does.

5

Complementarities: r < 0

To complete the analysis of the first mover’s strategic behavior, it is worth considering the case in which r < 0, that is, the case in which there are complementarities among firms. This time, the profits are higher when both A and B are located in the same market. I do not characterize the equilibrium of the game. Indeed, this would require particularly cumbersome calculations. The reason is that, contrarily to the previous frameworks, under complementarities, choosing to enter ¡both markets ¢ A at ¡ t A= ¢2 is not a dominated strategy for firm A. Given that EEt=1 and EEt=2 induce the same response by firm B, then , both in the benchmark and rivalry cases, the former choice is preferred to the second or, if not, then they both assure a negative pay-off. One can check that this does not occur in the case of complementarities. When r < 0, firm A gets low profits in the early periods, that is, when it is the only active firm in the markets. Therefore, entering both markets at t = 2 is preferred to the same choice at t = 1 and to not entering any market whenever the expected monopoly profits (pΠ) are not sufficient to cover F , while the expected duopoly profits (p (Π − R)) are. Even if I do not characterize the whole equilibrium, it is worth discussing the strategic behavior of firm A. With complementarities, the aim of the first mover is opposite to the one it has with rivalry: it wants to induce B to enter as soon as possible. Choosing a sequential entry pattern,15 firm A might encourage B to wait and see and then to postpone entry. Therefore, strategic reasons might induce the first mover to enter both market at the same time rather than sequentially. That is, firm A might strategically choose not to reveal any information to induce B to enter as soon as possible. In the appendix, I provide an example in which firm A chooses to enter both markets at time t = 2 only because of strategic reasons. That is, if its behavior were not to affect B’s choices, then A would prefer to enter markets sequentially. However, since this choice induces B to postpone entry, then in equilibrium firm A chooses to enter both markets at t = 2 so that B starts entering at t = 2. In the example, the parameters are such the first mover might prefer to enter both markets at t = 2 only because of strategic reasons. That is, A would enter markets sequentially if its choice were not to affect that of B. Further, 14 In

¢ ¡ A ¢ £ the case analyzed in corollary 10, when p ∈ p2 (R) , p1 (R) , Et=1 is not dominated.

Indeed, entering one market and then playing the semi-separating equilibrium of the signaling game is the only way to preempt B with positive probability. 15 Once entered the first market, firm A, at t = 2, always separates.

14

the parameter R is sufficiently negative (i.e. the level of complementarities is sufficiently high) that inducing firm B to enter earlier is particularly valuable. The choice of entering both markets at the same time rather than sequentially is the only case in which, with complementarities, the strategic behavior of firm A might arise. Indeed, the other two ways of strategically inducing firm B to enter do not play any role in equilibrium. First, firm A could attract B by being present in both markets, thus assuring it s (π − r) per period. However, independently of the entry pattern, by time t = 2, that is, by the time in which B moves, firm A is located in both markets in the relevant case; that is, when s = 1. Second, when priors are so low that B expects a negative pay-off, firm A could induce it to enter by choosing a sequential pattern and then revealing that s = 1. However, with complementarities, it is the first mover’s non-negativity constraint which is binding. That is, if ignoring the true value of s, firm B expects a negative pay-off, then firm A, when deciding about entry, anticipates a negative pay-off too. Therefore, neither firm A enters.

6

Extensions: Endogenous First and Second Mover

In the analysis made so far, I have employed the standard assumption of the entry models à la Dixit: there are an exogenous first and second mover. One might question whether this structure arises as an equilibrium outcome. That is, if both firms can move from the beginning of the game, when will it happen that, in equilibrium, one firm enters as first and, some time later, the other follows? Allowing both firms to move at any t = 1, 2...∞, it is possible to address this question within the framework of my model. In this setting, the only rational to postpone entry and to be the endogenous second mover arises when the other firm chooses an entry pattern that reveals the true state of the markets. This means that a first-second mover structure arises only when one firm enters markets sequentially and the other chooses to ”wait and see”. Before starting the analysis, some remarks are in order. First, the identity of the first and of the second mover is not determined That is, it is only possible to find conditions that assure that one firm enters as first and the other as second. As a convention, I consider firm A as being the first entrant. Second, the expected pay-off of each firm depends on its own information and on the markets already entered by the two firms, but it is independent of the time index, t. Thus, I concentrate the analysis on Markov perfect equilibria in which the strategies of each firm are not conditional on t. Third, when firm A has entered the first market and firm B the second, then two equilibria might follow. In the first, ”non-collusive”, equilibrium, whenever s = 1 each firm enters the other market too. This equilibrium exists for any values of parameters. The second

15

possible equilibrium is a ”collusive” one. When s = 1, firm A (B) does not enter the second (first) market unless B (A) enters the first (second). One can check that, this equilibrium exists only for a limited range of parameters.16 In the analysis that follows, I assume that when each firm has entered one market, then they always play the ”non-collusive” equilibrium. Finally, in what follows, I do not solve for the equilibrium of the whole game but I just check when a structure with a first and a second mover arises.

6.1

The Benchmark

In the benchmark, the next result holds: Proposition 12 When A and B can play at any t = 1, 2...∞, £then when R = 0, ¢ a first-second mover structure arises in equilibrium for any p ∈ p2 (R = 0) , p11 (R = 0) .

6.2

6.2.1

The Rivalry Case

Competition is not harsh R ≤ R2

The analysis of section 4 is largely applicable to this case. In particular, the equilibrium of the signaling game is unchanged. When priors probabilities about markets profitability are at an intermediate level, then the following result holds:17 Proposition 13 When A and B can £ play at any t¢ = 1, 2...∞, then when R ≤ R2 for intermediate values of p, p ∈ p2 (R) , p1 (R) , the set of priors for which a first-second mover structure arises in equilibrium is larger in the case of rivalry than in the benchmark. Proposition 13 makes an interesting prediction. When competition is not harsh and prior probabilities about markets’ profitability are at an intermediate level, then it is more likely to observe the emergence of a market structure with a first and a second mover when there is competition. The intuition for this result is the following. The emergence of this structure requires two conditions: one firm enters markets sequentially and the other waits and sees. As proved in section 4, when priors are intermediate, the first entrant is more willing to choose a sequential entry pattern under rivalry; in this way, it postpones the competitor’s entry. Therefore, strategic reasons, favors the appearance of a structure with a first and a second mover. Further, the opportunity cost of being an endogenous second mover is represented by the benefits that this firm gives up by postponing entry. In the benchmark, a second mover gives up monopoly 16 In the model of section 2, the ”collusive” equilibrium does not arise. Indeed, B always enters the first market as soon as s = 1. 17 For high values of priors the two firms enter the both markets at t = 1.

16

profits, while, under rivalry, it only gives up duopoly profits. Therefore, it is ”less costly” to wait and see under rivalry. For low values of the priors (p < p2 (R)), the effects due to the strategic behavior of the first mover and to the lower cost of waiting that characterizes the rivalry framework, work in opposite direction. From section 4, we know that the first mover is less willing to enter markets sequentially because in this way it gives up the opportunity of preempting the competitor. This fact is detrimental for the emergence of a structure with a first and a second mover. However, it is still true that under rivalry it is ”less costly” to wait and see. Proposition 14 When A and B can play at any t = 1, 2...∞, then when R ≤ R2 for low values of p, p < p2 (R) , the set of priors for which a first-second mover structure arises in equilibrium is £smaller ¤in the case of© rivalryªthan in the benchmark, provided that either R ∈ R7 , R2 or R ≤ min R2 , R8 . In all the other cases the two sets are not comparable. According to proposition 14, when p is low, the set of priors for which a first and a second mover arise in equilibrium is larger in the benchmark than under rivalry provided that the lower cost of waiting that characterizes the latter framework plays no role in the comparison. Indeed, one can check in the appendix that conditions R ≥ R7 or R ≤ R8 that appear in proposition 14 have the effect of neutralizing the lower cost of waiting. In all the cases that do not satisfy one of these two conditions the sets where a first-second mover structure arises are not comparable.

6.2.2

Competition is harsh: R > R2

One can check that when competition is harsh, then a first-second mover structure never emerges as an equilibrium outcome. The formal proof of this result would require long calculations so that I prefer just to give an intuition of it. The equilibrium of the signaling game played when A has entered the first market while B none has the same features as the one characterized in lemma 3. In particular, for high values of the priors, the equilibrium is separating with firm B that enters the two markets at t = 2. However, in this case, B is better-off entering the two markets at time t = 1 rather than waiting one period. Therefore, for high values of the priors both firms enter markets enter markets at t = 1. For low values of the priors, the equilibrium of the signaling game is pooling and firm B is preempted. Therefore, neither in this case a first-second mover structure arises. For intermediate values of p, the signaling game has a semi-separating equilibrium with firm B that randomizes among entering or not entering one market. However, if B is willing to enter one market when playing the signaling game, then it is better-off entering that market the previous period. This implies that the semi-separating equilibrium of the signaling game is never played in 17

an equilibrium of the whole game. Therefore, neither for intermediate values of the priors a first-second mover structure arises.

7

Conclusions

In the paper, I have extended an entry model à la Dixit to allow for uncertainty and learning among firms. In particular, I have considered a first and a second mover that have the opportunity of locating into two markets of unknown but correlated profitability. The first mover can enter the two markets at the same time and just knowing the prior probabilities about profitability, or it can enter them sequentially; that is, enter one market, observe the correlated profitability and, the next period, condition entry in the second upon this new information. The second mover has an additional possibility. When the first mover enters market sequentially, it can choose to wait and see. It observes the complete entry pattern of the first mover, infers information about the markets’ conditions and decides about entry with a superior knowledge regarding profitability. In this context, I have characterized the first mover strategic behavior in the cases of rivalry and complementarities. In the case of rival firms, I have shown that when prior probabilities are such that firms believe that markets are likely to be profitable, then the first mover cannot preempt the competitor and the best it can do is to enter markets sequentially. This might induce the second mover to wait and see, thus postponing entry in the markets. On the contrary, when priors are such that markets are not likely to be profitable, then the first mover tends to strategically enter the two markets at the same time, reveal no information and attempt to preempt the competitor. In the case of complementarities, that is, in the case profits are higher when both firms are active in the same market, the aim of the first mover is to encourage the other firm to enter as soon as possible. In this case, the first mover might strategically choose to enter both markets at the same time because otherwise it could induce the second firm to wait and see. Finally, in an extension of the model, I have questioned whether a structure with a first and a second mover can arise as an equilibrium outcome. That is, if both firms had the opportunity to play from the beginning, when would it be that a first-second mover structure arises? In this context, I have shown that the results crucially depend on the level of competition among firms and on the prior probabilities about profitability.

18

APPENDIX List of definitions for R : R1 ≡ Π − F ; ) R2 ≡ (Π−F 2δ ; ) ; R3 ≡ 2δ(Π−F 1+δ+2δ 2 R4 ≡ R5 ≡ R6 ≡

R7 ≡ R8 ≡

2δ(Π−F ) 1+δ ; δ(1−3δ)(Π−F ) ; (1−2δ−δ2 ) 2δ(Π−F ) ; 1+δ2 δ(Π−F ) 1−δ ; 2δ2 (Π−F ) . 1+δ

List of definitions for p : p1 (R) ≡ p2 (R) ≡ p3 (R) = p4 (R) ≡ p5 (R) ≡ p6 (R) ≡ p7 ≡ FΠ ; p8 (R) ≡

F (Π−R)(1−δ)+δF ; F (Π−R)(1+δ)−δF ; F Π+δ(Π−R)−δF ; F (Π−δR)(1−δ)+δF ; F ; Π(1−δ)+2δ 2 R+δF F ; Π(1+δ)−2δ 2 R−δF

F (2δ−1) ; δR(1−δ)−Π(1−δ)2 +δ2 F F (1−2δ) 9 p (R) = (Π+δR)(1−δ)−δF ; 2F ; p10 (R) = 2Π−R(1+δ) F p11 (R) = (Π+2δ(Π−R))(1−δ)+2δ 2 F −δF

.

Proof of Proposition 1 A ), pΠ − F + δp (Π − F ) Firm A expects 2 (pΠ − F ) when choosing (EEt=1 A A when choosing (Et=1 ), and 0 when choosing (Nt=1,2 ). The same pay-offs are ¡ ¢ ¡ B ¢ ¡ B ¢ B , Et=2 and Nt=2,3 expected by firm B when choosing EEt=2 respectively. ¡ B ¢ A ), B learns the true Choosing Nt=2 (that is, wait and see) when A plays (Et=1 s and, then, it enters both markets at t = 3 iff s = 1. The expected pay-off associated with this strategy is 2δp (Π − F ). Given these pay-offs, and since, by condition (1), p2 (R = 0) ≤ p1 (R = 0), then proposition 1 follows immediately. Q.E.D. Proof of Lemma 2 B ) In equilibrium, firm B expects 2 (p (Π − R) − F ) when choosing (EEt=2 B and p (Π − R) − F + δp ((Π − R) − F ) when choosing (Et=2 ) and entering the 19

B ) (i.e. wait and see), second market at t = 3 iff s = 1 is observed. Playing (Nt=2 B learns the true s and, at t = 3, enters the two markets iff s = 1 is revealed. The expected pay-off of this last strategy is 2δp ((Π − R) − F ) B - When p ≥ p1 (R), B chooses ¡ A (EEt=2 ¢ ). Condition (1) assures that firm A’s best response is to choose Et=2 (s = 1) . ¡ B ¢ ¡ A ¢ (s = 1) . Playing Et=2 - When p < p1 (R), in equilibrium, firm B chooses Nt=2 2π−F A ¢ +2δ (Π − R) . Deviating from its proposed strategy and playing ¡ obtains A (s = 1) , A obtains Π. Indeed, in this last case, firm B assigns probability Nt=2 B 2 1 to the event s¡ = 0 and, at t¢ = 3,¡ it chooses (N ¢ t=3 ). Condition R ≤ R assures A A that A prefers Et=2 (s = 1) to Nt=2 (s = 1) . Q.E.D.

Proof of lemma 318 already proved in lemma 1. - Part (i), p ≥ p1 (R): ¢ £ - Part (ii), p ∈ p3 (R) , p1 (R) : call pe (y) B’s posterior beliefs when A randomizes according to y and (1 − y) . As first, I check that y ∗ , as defined in lemma 3, is such that in the case neither ¡ AAnor B enter ¢ at t = 2 (that ¢ any¡market B (s = 1) and Nt=2 is the result of the randomizations is Nt=2 ), then B does pe (y ∗ ) Π < F ; not enter any market at t = 3 neither. This happens provided that ¢ £ 3 ∗ 7 7 that is, pe (y ) < p . For p ∈ p (R) , p this follows immediately. For p ∈ ¢ £ 7 1 p−p7 e. The p , p (R) , according to the Bayesian rule, it requires y∗ > p(1−p 7) ≡ y derivative with respect to p of (y∗ − ye) is equal to

p1 (R)p3 (R) (p)2 (p1 (R)−p3 (R))

2

−

p7 (p)2 (1−p7 )

(2δ−1)(Π−R−F ) which is positive iff (δ(Π−R−FF)−(Π−R))Π(Π+δ(Π−R−F )) > 0. Condition (1) assures that the last expression is positive when δ < 12 and it is negative otherwise. Therefore the derivative with respect to p of (y ∗ − ye) is either always positive or always negative. Further, (y ∗ − ye) is positive both £when p = ¢p7 and when p = p1 (R) . Therefore, y ∗ is greater than ye for any p ∈ p7 , p1 (R) . When, in the randomizations, B enters one market, then it enters the second at t = 3 iff s = 1. When, in the randomizations, A enters the second market, then B learns that s = 1 and, at t = 3, it enters all the markets it has not entered yet. x∗ and (1 − x∗ ) make firm A indifferent between playing ¡ AThe probabilities ¢ Et=2 (s = 1) , pay-off x∗ (π + (π − r) − F + 2δ (Π − R))+(1 − x∗ ) (2π − F + 2δ (Π − R)) , A (s = 1)), pay-off x∗ (π + δ (Π − R))+(1 − x∗ ) (Π) . Conditions (1) and and (Nt=2 2 ∗ ∗ R > R assure that x∗ ∈ (0,¡1) . The ¢ probabilities y and (1 − y ) make firm B B indifferent between playing Et=2 , pay-off y ∗ (p (Π − R) − F + δp (Π − R − F ))+(1 − y∗ ) (pΠ − F + δp £ (Π − R − F ))¢ , B ), pay-off y ∗ (2δp ((Π − R) − F )) . Condition p ∈ p3 (R) , p1 (R) , and (Nt=2 assures that y ∗ ∈ (0, 1) . ¡ A ¢ (s = 1) is chosen no information is Part (iii), p < p3 (R): when Nt=2 revealed. Therefore, in equilibrium, B does not update the beliefs and, for 18 Note

that condition (1) assures that p3 (R) ≤ p7 ≤ p1 (R) .

20

p < p3 (R) , any pattern of entry assures it a negative pay-off. Thus, in the pooling equilibrium B does not enter any market and A obtains Π. Out of A Et=2 equilibrium, A chooses ¡ ¢ (s = 1), B believes that s = 1 with probabilB ity 1 and chooses EEt=3 . In this case, A obtains 2π − F + 2δ (Π − R) . 2 When ¡ A R > R¢ , then A prefers to play according to the pooling equilibrium: Nt=2 (s = 1) .Q.E.D. Proof of Proposition£ 5 ª¤ © In proposition 5, R ∈ R3 , min R2 , R4 and by (1) it £has to ¤be R ≤ R1 . These conditions are simultaneously verified for any R ∈ R3 , R4 when δ ≤ √ √ £ 3 2¤ 1 1 when δ > 18 + 18 17. 8 + 8 17 and for any R ∈ R , R The following inequalities hold: - p4 (R) ≤ p1 (R) and p2 (R = 0) ≤ p6 (R) , since R ≥ 0; - p6 (R) ≤ p5 (R) since R ≤ R2 ; - p5 (R) ≤ p2 (R) , since R ≥ R3 ; - p2 (R) ≤ p4 (R) , since R ≤ R4 . Therefore, overall: p2 (R = 0) ≤ p6 (R) ≤ p5 (R) ≤ p2 (R) ≤ p4 (R) ≤¡p1 (R) .¢ B - If p ≥ p1 (R) , as checked in the¡ text B¢ chooses EEt=2 independently of A EE 2 (p (π + δ (Π − R)) − F ), A’s¡ choice. Given this, firm prefers , pay-off t=1 ¢ A to Et=1 , pay-off p (π + δ (Π − R))−F +δp (Π − R − F ), for any p ≥ p1 (R = 0) . Since 1 R ≥ 0, then p1 (R =¡0) ≤ p1¢(R) and so for any ¡ pB ≥¢ p (R) the equilibrium is A such that A£chooses EEt=1 ¡ B ¢ ¢ and B chooses EEt=2 ; , as checked ¡in the¢ text B chooses Et=2 - If¡ p ∈ p¢2 (R) ¡, p1 (R) A ¡ when ¢ ¢ A B A A . Given this, A prefers EEt=1 plays EEt=1 and Nt=2 when A plays Et=1 , pay-off ¡ A ¢ ¢ ¢ ¡ ¡ p 2π + δπ + δ (π − r) + 2δ 2 (Π − R) −2F , to Et=1 , pay-off p π + δπ + δ 2 (Π − R) − F + δp (π +£ δ (Π − R) − F¢), whenever ¡p ≥ p4¢(R) . Therefore, ¡ B ¢ in equilibrium, 4 1 A Et=2 and when p ∈ (R) , p (R) EE p ∈ p when and B A chooses ¡ t=1 ¢ ¢ ¡ A ¢ £ 2 B ; p (R) , p4 (R) A chooses Et=1 and B Nt=2 ¡ B ¢ - If p < p2 (R) , as checked in the text, B chooses Nt=2,3 (that is, it is ¡ ¢ ¡ B ¢ ¡ A ¢ A EEt=1 and it choose Nt=2 when A plays Et=1 . preempted) when ¡ AA plays ¢ When A plays Nt=1,2 , B faces the same scenario as in the benchmark and then ¡ B ¢ ¢ ¡ B ¢ £ it chooses Et=1 if p ∈ p2 (R = 0) , p2 (R) and Nt=2,3 if p < p2 (R = 0) . ¢ ¡ ¢ £ A , pay-off 2 (pΠ − F ), if p ∈ p5 (R) , p2 (R) , Given this, firm A chooses EEt=1 ¡ A ¢ ¢ ¡ + ¢δ 2 (Π − R) − F + δp (π + δ (Π − R) − F ), if p ∈ £ E6t=1 , pay-off ¢ p π ¡+ δπ 5 A p (R) , p (R) and Nt=1,2 , pay-off 0, if p < p6 (R) .19 Therefore, in equilib¢ ¡ B ¢ ¢ ¡ £ A , when p ∈ rium, when p ∈ p5 (R) , p2 (R) A chooses EEt=1 and B Nt=2,3 £ 6 ¢ ¡ ¢ £ ¢ ¡ ¢ A B p (R) , p5 (R) A chooses Et=1 , when p ∈ p2 (R = 0) , p6 (R) and B Nt=2 ¡ A ¢ when B’s that p ≥ p6 (R) assures that A gets a positive pay-off by playing Et=1 ¡ B ¢ response is Nt=2 . 19 Note

21

¡ A ¢ ¡ B ¢ A chooses Nt=1,2 and B chooses Et=2 and when p < p2 (R = 0) A chooses ¡ B ¢ ¡ A ¢ Nt=1,2 and B chooses Nt=2,3 . Q.E.D Proof of Corollaries 6 and 8 One can check that - p1 (R = 0) ≤ p4 (R), since R ≥ 0. - p2 (R) ≤ p1 (R = 0) , since R ≤ R4 . Therefore, combining these relations with the ones used in the proof of proposition 5:

p2 (R = 0) ≤ p6 (R) ≤ p5 (R) ≤ p2 (R) ≤ p1 (R = 0) ≤ p4 (R) ≤ p1 (R) . Propositions 5£ and 1 assure that:¢ ¡ A ¢ 1 4 (R = 0) , p (R) Et=1 under rivalry and p ∈ p for any firm A chooses: ¡ ¢ A EEt=1 in the benchmark;

¢ ¢ ¡ £ 5 2 A ¡ A- for ¢ any p ∈ p (R) , p (R) firm A chooses: EEt=1 under rivalry and Et=1 in the benchmark.Q.E.D.

Proof of Proposition ¡9 ª ¢ © In proposition 9, R ∈ max R2 , R4 , R5 and by (1) it has ¢ R ≤ ¡ to be R1 . These conditions are simultaneously verified for any R ∈ R2 , R5 when h¡ ¢ 1 √ ´ √ ¢ ¡ δ ∈ 13 2 , 18 + 18 17 and for any R ∈ R4 , R5 when δ ≥ 18 + 18 17. For ¡ ¢1 δ < 13 2 the above conditions cannot be simultaneously verified. The following inequalities hold: ¡ ¢1 - p8 (R) ≤ p1 (R) , since R ≥ R2 and δ > 13 2 ; ¡ ¢1 - p2 (R) ≤ p8 (R) , since R ≤ R5 and δ > 13 2 ; - p7 ≤ p2 (R) , since R ≥ R2 ; - p3 (R) ≤ p7 , since R ≤ R1 ; - p2 (R = 0) ≤ p3 (R) , since R ≥ 0. Overall:

p2 (R = 0) ≤ p3 (R) ≤ p7 ≤ p2 (R) ≤ p8 (R) ≤ p1 (R) . - If p ≥ p£1 (R) , the proof ¡ B ¢ ¢ is the same as in proposition 5. 1 , p (R) , as checked in the text, B chooses Et=2 - If p¡ ∈ p2 (R) when ¢ A A plays EEt=1 and the semi-separating equilibrium of the signaling game is ¡ ¡ A ¢ ¢ A . Then, A prefers EEt=1 played when A chooses Et=1 , pay-off ¡ A ¢ ¢ ¡ p 2π + δπ + δ (π − r) + 2δ 2 (Π − R) − 2F , to Et=1 , pay-off 22

p (π + δ (x∗ (π + (π − r) − F + 2δ (Π − R)) + (1 − x∗ ) (2π£− F + 2δ (Π −¢R))))− F , when p¢8 (R) . Therefore, in equilibrium, when p ∈ ¢ p8 (R) , p1 (R) ¡ ¢ ¡ p≥ ¡ A A¢ £ A B and B Et=2 and when p ∈ p2 (R) , p8 (R) A chooses Et=1 chooses EEt=1 and the semi-separating equilibrium of the signaling game follows. ¡ B ¢ ¢ £ - If p ∈ p3 (R) , p2 (R) , as checked in the text, B chooses Nt=2,3 when ¡ ¢ A A plays EEt=1 and the equilibrium ¡ semi-separating ¢ ¡ A ¢of the signaling game is A played when A chooses Et=1 . When A plays Nt=1,2 then B faces the same ¡ B ¢ E scenario ¡as in the benchmark and then it chooses ¢ ¢firm A ¡ this, £ 7 2 t=2 ¢. Given A A chooses EEt=1 , pay-off 2 (pΠ − F ), for p ∈ p , p (R) and Nt=1,2 , pay¢ £ ¢ £ p ∈ p7 , p2¡ (R) A¢ in equilibrium, when off 0, for¡ p ∈ p¢3 (R) , p7¡ . Therefore, ¢ ¢ £ A B A and B Nt=2,3 and when p ∈ p3 (R) , p7 A chooses Nt=1,2 chooses EEt=1 ¡ B ¢ and B Et=2 . ¡ B ¢ - If p < p3 (R), as checked in the text, B chooses Nt=2,3 when A plays either ¡ A ¢ ¢ ¡ ¢ ¡ A A . However, given that p < p7 , firm A plays Nt=1,2 EEt=1 or Et=1 since ¡ ¢ ¡ ¢ A A both EEt=1 and Et=1 yield a¡ non-positive pay-off. In this case, firm B is ¢ ¢ ¡ B the¢ £ 2 B 3 only entrant and then it chooses Et=2 if p ∈ p (R = 0) , p (R) and Nt=2,3 £ ¢ in equilibrium, when p ∈ p2 (R = 0) , p3 (R) firm if p < p2 (R¡= 0) . Therefore, ¢ ¡ ¢ A B and B Et=2 and when p < p2 (R = 0) firm A chooses A chooses Nt=1,2 ¡ B ¢ ¡ A ¢ Nt=1,2 and B Nt=2,3 . Q.E.D Proof of Corollaries 10, 11 One can check that: - p7 ≤ p1 (R = 0) , since R ≤ R1 ; - p1 (R = 0) ≤ p2 (R) , since R ≥ R4 . Therefore, combining these with the inequalities used in proposition 9: p2 (R = 0) ≤ p3 (R) ≤ p7 ≤ p1 (R = 0) ≤ p2 (R) ≤ p8 (R) ≤ p1 (R) . Propositions 9 £and 1 assure that: ¡ A ¢ ¢ 2 8 ¡ - Afor¢ any p ∈ p (R) , p (R) , firm A chooses: Et=1 under rivalry and EEt=1 in the benchmark; ¡ ¢ ¢ £ 7 1 A ¡ A- for ¢ any p ∈ p , p (R = 0) , firm A chooses: EEt=1 under rivalry and Et=1 in the benchmark.Q.E.D. Example for the case of Complementarities

¡ ¢1 δF , δ ≥ 13 2 and that the level of complementarities Assuming that Π ≥ 1−δ is sufficiently high is, R is ´ sufficiently negative), then: h¡ ¢(that 1 √ £ ¢ 1 2 1 1 * when δ ∈ 3 , 8 + 8 17 , in equilibrium, for any p ∈ p9 (R) , p1 (R) , ¢ ¡ A only because of strategic reasons; A chooses EEt=2 ¢ £ 10 ¢ £1 1√ 1 + 17, 1 , p ∈ p (R) , p (R) δ ∈ in equilibrium, for any ,A * when 8 8 ¢ ¡ A chooses EEt=2 only because of strategic reasons. 23

Proof −Π(1−δ) . Condition R ≤ R9 assures that the denominator of Define R9 ≡ δFδ(1−δ) p9 (R) is negative. One can check that: −Π) p1 (R) ≤ p7 , when R ≤ δ(F1−δ ; o n −Π) 9 1 9 ; , R p (R) ≤ p (R) , when R ≤ min 2δ(F 1−δ √ 1 1 10 9 9 p (R) ≤ p (R) , when R ≤ R and δ ≥ 8 + 8 17; n o ³¡ ¢ 1 √ ´ −Π) 1 2 1 1 9 , R , + δ p9 (R) ≤ p10 (R) , when R ≤ min 2δ(F and 2 3 8 8 17 ; 1+δ−4δ

p10 (R) ≤ p1 (R) , when R ≤ 0 and δ ≥ 13 ; ¡ ¢1 p2 (R) ≤ p9 (R) , when R ≤ 0 and δ ≥ 13 2 ; p2 (R) ≤ p10 (R) , when R ≤ 0.

√ ¢ £ Therefore when R is sufficiently negative and δ ∈ 18 + 18 17, 1 : p2 (R) ≤ p10 (R) ≤ p9 (R) ≤ p1 (R) ≤ p7h; √ ´ ¡ ¢1 when R is sufficiently negative and δ ∈ 13 2 , 18 + 18 17 : p2 (R) ≤ p9 (R) ≤ p10 (R) ≤ p1 (R) ≤ p7

¡ B ¢ ¢ £ E¡t=2 when p ∈ ¢p2 (R) , p1 (R) , then B chooses, that A plays either ¡ ¡ Given ¢ ¡ ¢ ¢ A A A B EEt=1 . When A chooses Et=1 or EEt=2 B plays Nt=2 since A separates ¢ ¡ 2 (Π − R) − 2F p 2π + δπ + δ (π − r) + 2δ at t = 2. Therefore, firm A expects ¢ ¡ A when choosing EEt=1 , δ (p (π + (π − r) + 2δ (Π − R)) − 2F ) when choosing ¡ ¢ ¢ ¡ A EEt=2 + − +¢ δ 2 (Π − R) − F + and p π + ¡ δp (π ¢ δ (Π ¡ − R) ¢ F) ¡ δπ A A A 7 EE . EE E Firm A prefers to when choosing t=1 t=2 t=1 for any p < p ¡ ¢ ¡ ¢ A A 10 and EEt=2 to Et=1 for any p ≥ p9 (R) .20 ¡ Further, ¢ for any p ≥ p (R) A A expects a non-negative pay-off when playing EEt=2 . √ ¢ £ ¢ £ Therefore, when δ ∈ 18 + 18 17, 1 , then for every p ∈ p9 (R) , p1 (R) firm ´ h √ ¡ ¢ ¡ ¢1 A A chooses EEt=2 in equilibrium. When δ ∈ 13 2 , 18 + 18 17 , then for any £ ¢ ¢ ¡ A p ∈ p10 (R) , p1 (R) firm A chooses EEt2 in equilibrium. ¢ ¡ A is only due to strategic reasons.¡ Indeed,¢ if A’s behavThe choice of EEt=2 ¡ A ¢ A ior were not to affect that of B, then firm A would prefer EEt=2 to Et=1 iff δF p (δF − Π (1 − δ))+F (1 − 2δ) ≥ 0; this last inequality does not hold if Π ≥ 1−δ ¡ 1 ¢ 12 and δ ≥ 3 . Q.E.D Proof of Proposition 12 From proposition 1, when B waits and sees, A enters sequentially for any 20 Note

that this last condition requires R ≤ R9 .

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£ ¢ p ∈ p2 (R = 0) , p1 (R = 0) . In turn, when A enters sequentially, firm B prefers to wait and see21 , pay—off 2δ 2 p (Π − F ), rather than entering sequentially for any p < p11 (R = 0) . Condition (1) assures that: p2 (R = 0) ≤ p11 (R = 0) ≤ p1 (R = 0) . Q.E.D. Proof of Proposition 13 Under rivalry, when A enters markets sequentially, then B prefers to wait and see, pay-off 2δ 2 p (Π − R − F ), rather then entering sequentially, pay-off p (π + δ (Π − R)) −£ F + δp (Π − R¢ − F ), for any p < p11 (R) . When B waits and sees, then for p ∈ p2 (R) , p1 (R) , A face the same situation as in proposition 5 and it prefers to enter markets sequentially for any p < p4 (R) . When R ≥ 0, p11 (R) ≤ p4 (R) ≤ p1 (R) and £ then © a first-second ª mover ¢structure arises in equilibrium for any p ∈ IR = min p2£(R) , p11 (R) ,¢p11 (R) .22 In the benchmark, focusing£ on p©∈ p2 (R) , p1 (R) , ªthen a first and ¢ a second 2 11 11 mover arise for any p ∈ IB = min p (R) , p (R = 0) , p (R = 0) . Since p11 (R = 0) ≤ p11 (R) for any R ≥ 0, then IB ⊆ IR . Q.E.D. Proof of Proposition 14 Under rivalry, as in the previous proposition, when A enters markets sequentially, B chooses to wait and see for any p < p11 (R) . When B waits and sees, then for p < p2 (R) , A face the same ¢ in proposition 5 and £ situation as enters markets sequentially for any p ∈ p6 (R) , p5 (R) . One can check that p6 (R) ≤ p2 (R) , by R ≥ 0, p6 (R) ≤ p5 (R) by R ≤ R2 and p6 (R) ≤ p11 (R) by R ≥ 0 and R ≤ R1 . arises in Therefore, when p < p£2 (R) , then a© first-second mover structure ª¢ equilibrium for any p ∈ IR p6 (R) , min p2 (R) , p5 (R) , p11 (R) . In the benchmark, focusing on p < p2 (R) and since p2 (R = 0) ≤ p2 (R) for any R ≥ 0. then £ a first-second©mover structure arises ª¢ in equilibrium for any p ∈ IB = p2 (R = 0) , min p2 (R) , p11 (R = 0) . Since© p2 (R = 0) ≤ p6 (R) for that ª any R ≥©0, then IR ⊆ IB provided ª min p2 (R) , p5 (R) , p11 (R) ≤ min p2 (R) , p11 (R = 0) . If the last inequality does not hold then IR and IB are not comparable. When R ≥ 0 then p11 (R = 0) ≤ p11 (R) and then the inequality holds iff either p5 (R) ≤ p11 (R = 0) or p2 (R) ≤ p11 (R = 0) . Condition p5 (R) ≤ p11 (R = 0) holds iff R ≥ R7 . Therefore, the existence 2 of this case requires R ≥ R7 , R ≤£ R1 and ¤ R ≤ 1R . These conditions are 7 2 simultaneously verified for any R ∈ R , R iff δ ≥ 2 . Condition p2 (R) ≤ p11 (R = 0) holds iff R ≤ R8 . Therefore, the existence of this case requires R ≤ R8 , R ≤ R1 and R ≤ R2 . These conditions are simultaneously verified for any R ≤ R8 when δ ≤ 0.76 and for any R ≤ R2 for δ > 0.76. 21 Note that in this case a wait and see strategy requires not to enter any market for two periods: t = 1 and t = 2. 22 Note that when min {x, y} = y, then [min {x, y} , y) represents the empty set.

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Q.E.D.

Acknowledgments I would like to thank my advisor David Pérez-Castrillo, Inés Macho-Stadler, Pierre Regibeau and Konrad Stahl for very helpful comments. I acknowledge the TMR program - contract number ERBFMBICT 960669 - and the Consiglio Nazionale delle Ricerche - grant number 203.10.38 - for financial support.

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