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Quantity-setting oligopoly with endogenous sequencing
International Journal of Industrial Organization 17 (1999) 289–296
Quantity-setting oligopoly with endogenous sequencing Toshihiro Matsumura* Department of Social Engineering, Tokyo Institute of Technology, 2 -12 -1 O-okayama, Meguro-ku, Tokyo 152 -8552, Japan Received 27 November 1996; received in revised form 1 November 1997; accepted 21 March 1998
Abstract This paper investigates endogenous sequencing in a quantity-setting oligopoly. I formulate an n-firm, m-period model where each firm chooses both how much to produce and when to produce it. I find that at least n 2 1 firms simultaneously produce in the first period in every pure strategy equilibrium. This result shows that the generalized Stackelberg-type outcome never appears in equilibrium except in a duopoly. 1999 Elsevier Science B.V. All rights reserved. Keywords: Cournot; Stackelberg; Endogenous sequencing JEL classification: C72; L13
1. Introduction Cournot and Stackelberg models have occupied important positions in oligopoly theory. The Cournot model involves simultaneous moves, while the Stackelberg model involves sequential moves. If we allow for more than two firms, we can analyze more varied situations: a pure simultaneous-move model in which all firms choose their actions at the same time (Cournot-type), a pure sequential-move model in which all firms choose their actions at different times (generalized
* Tel. / fax: 1 81-3-5734-3318; e-mail: [email protected] 0167-7187 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 98 )00021-6
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Stackelberg-type),1 and mixed-type models in which there is a sequence of periods in each of which some players choose their actions simultaneously. Each of these models produces a different equilibrium outcome, that is, the equilibrium outcome crucially depends on the sequence of each firm’s choice. The aim of this paper is to endogenize the sequence. Other economists have investigated similar problems. Robson (1990b), Albæk (1992), and Mailath (1993) discussed duopoly games with endogenous timing and emphasized that Stackelberg-type outcomes appear in endogenous timing games but a Cournot-type outcome does not. These papers suggested that the Stackelberg model is more plausible than the Cournot model if firms can choose when to take their actions.2 Here let us take a close look at the model of Hamilton and Slutsky (1990) (hereinafter referred to as H-S), which they call the ‘extended game with action commitment’. This is a standard model with endogenous sequencing. The game runs as follows. In period 1, each duopolist chooses whether to take its action this period or to wait until the next period. Any firm choosing period 1 takes its action without knowing whether its rival chooses period 1 or 2. At the beginning of period 2, each firm knows its rival’s action. A firm must act in period 2 if it elected to wait in period 1. The payoff for each firm depends on its and its rival’s actions in the basic market game, but not directly on the period in which either firm produces. In the H-S duopoly model there are four possible outcomes: 1. 2. 3. 4.
firm 1 acts in period 1 and firm 2 acts in period 2 (Stackelberg type); firm 2 acts in period 1 and firm 1 acts in period 2 (Stackelberg type); both firms act in period 1 (Cournot type); and both firms act in period 2 (Cournot type).
They found that under moderate conditions three cases ((1)–(3)) are supported as subgame perfect equilibria (Theorem VII). They also found that the outcome of case (3) is weak in the sense that it is supported by weakly dominated strategies, while the outcomes of cases (1) and (2) are not (Theorem VIII). In this paper I elaborate on the H-S duopoly framework to allow for oligopoly with more than two periods. I formulate an n-firm, m-period, quantity-setting oligopoly model in which each firm can choose when to take its action as well as what action to take. Each firm’s payoff depends only on its own output level and that of other firms. I find that there are two types of equilibria: either all firms
1
For the generalized Stackelberg model, see Robson (1990a) and Anderson and Engers (1992). This statement may be misleading. We should note that the above articles restrict the firms to pure strategies. If the firms are allowed to choose mixed strategies, the probability of ex post Cournot outcome is significant (see Pal, 1996). 2
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simultaneously produce, or n 2 1 firms simultaneously produce first and thereafter one firm produces, and that no other equilibrium exists. In other words, most firms (all or all but one) produce simultaneously in every equilibrium. This result shows the weakness of the sequential-move model (the generalized Stackelberg model). The remainder of this paper is organized as follows: Section 2 formulates a model; Section 3 discusses the equilibria of the model; and Section 4 concludes the paper.
2. The model In this section, I formulate an (m 1 1)-stage, n-firm oligopoly game. Both m and n are larger than one. The set of firms is denoted by N where N ; h1,2, . . . ,nj. In the first stage (period 0), each firm i [ N chooses the timing e i [ h1,2, . . . ,mj where e i 5 t implies that firm i chooses to produce in period t. The set of firms choosing to produce in period t is denoted by N t . At this time, each firm j ± i does not observe e i . Each firm i [ N 1 independently chooses the output quantity x i [ [0,`) in period 1. At the beginning of period t( > 2), each firm observes actions taken in period t 2 1; as a result, each firm knows which firms belong to N 1 , N 2 , . . . ,N t 21 . In period t, each firm j [ N t independently chooses the output quantity x j [ [0,`). At the end of period m, the market opens and each firm sells its own output. Define X2hi j ; o j[N •hi j x j , where N•hij ; h1,2, . . . ,i 2 1,i 1 1, . . . ,nj. Firm i’s payoff Ui : R 21 → R is given by Ui (x i , X2hi j ). Definition 1. (leader, follower, intermediate, and the last intermediate). I call firms producing in period 1 leaders, firms that produce in period t9 . 1 with no firm producing in period t . t9 followers, and other firms intermediates. I call firms the last intermediates if they are intermediates and no intermediate produces after them. Before investigating the equilibrium outcomes of the game with endogenous sequencing, I wish to discuss two-stage games with exogenous sequencing as a benchmark.3 From here on, I shall restrict our attention to pure strategy equilibria. Consider the following two-stage games with exogenous timing. The set of players is S L < S F , where S L < S F # N and S L > S F 5 [. S L is a set of leaders (namely firms producing in the first stage) and S F is a set of followers (namely firms producing in the second stage). Each firm h [ ⁄ S L < S F has already produced before the game and the output of it is common knowledge.
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Similar two-stage games with exogenous timing are discussed in the context of multiple leaders. See Sherali (1984) and Daughety (1990).
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In the first stage, each firm i [ S L simultaneously chooses its output. In the second stage, after observing the outputs of firms that belong to S L , each firm j [ S F simultaneously chooses its output. We now make the following three assumptions.4 Assumption 1. There exists a pure strategy equilibrium and the equilibrium is unique in the two-stage game with exogenous sequencing. We call the equilibrium outcome where each firm i [ S L < S F simultaneously chooses its output the Cournot outcome. Assumption 2. Suppose that the number of followers is one. Then the follower strictly prefers the Cournot outcome to the follower’s outcome. Assumption 3. Suppose that the number of leaders is one. Then the leader strictly prefers the leader’s outcome to the Cournot outcome.
3. Equilibrium outcomes In this section I discuss the equilibrium outcomes of the model with endogenous sequencing. I use perfect Bayesian equilibrium 5 as the solution concept.6 I restrict my attention to pure strategy equilibria. Proposition 1. Suppose that Assumptions 1 – 3 are satisfied. Then the number of followers is at most one in every equilibrium. Proof. I shall prove Proposition 1 by contradiction. Suppose that there is more than one follower in an equilibrium. Without loss of generality, assume that followers produce in period t9 . 1. Suppose that one of the followers unilaterally deviates from the equilibrium strategy and produces in period t9 2 1. From the definition of followers, all players other than followers produce before observing this deviation, so the deviation never affects the actions of leaders and intermediates. From Assumption 3, it is clear that the deviation strictly increases the deviator’s payoff, a contradiction. Q.E.D.
4
The following assumptions are not restrictive. Assumptions 2 and 3 are satisfied under moderate conditions in the context of quantity-setting oligopoly. Matsumura (1996) shows sufficient conditions for Assumptions 2 and 3. 5 For the concept of perfect Bayesian equilibrium see, e.g. Gibbons (1992) chap. 3. 6 I use perfect Bayesian equilibrium rather than subgame perfection because the proper subgame which begins with the beginning of period 1 cannot be defined. See Myerson (1991, pp. 183–185) for an example where even among complete information games unnatural equilibrium outcomes survive if we use subgame perfection.
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Proposition 2. Suppose that Assumptions 1 – 3 are satisfied. Then no intermediate exists in equilibrium. Proof. By the definition of intermediate, an intermediate never exists if m (the number of periods) is two. Thus, let us restrict our attention to cases in which m is larger than two. I shall prove Proposition 2 by contradiction. Suppose that there is at least one intermediate in an equilibrium. From the definitions of intermediate, it is certain that at least one follower exists. From Proposition 1, it is known that exactly one follower exists. Without loss of generality, I assume that the last intermediates produce in period t9 where m . t9 . 1. Suppose that the follower unilaterally deviates from the equilibrium strategy and produces in period t9 2 1. From the definition of the last intermediates and followers, all players other than followers and the last intermediates produce before observing this deviation, so this deviation affects the actions of last intermediates only. From Assumptions 2 and 3, it can be seen that the deviation strictly increases the deviator’s payoff, a contradiction. Q.E.D. Proposition 2 states that each firm is either a leader or a follower in an equilibrium; thus, pure sequential-move outcomes (generalized Stackelberg outcomes) where no firms simultaneously produce in the same period never appear in an equilibrium except in duopoly. Proposition 3 states that the number of leaders is n or n 2 1 in every equilibrium. Proposition 3. Suppose that Assumptions 1 – 3 are satisfied. Then: ( i) there exists an equilibrium where all firms become leaders; ( ii) there exist equilibria where all but one firm become leaders, and ( iii) no other pure strategy equilibrium exists. Proof. (i) Suppose that each firm i [ N produces the Cournot output in period 1. Obviously, the above strategies construct an equilibrium. (ii) I show that there is an equilibrium in which firms 2,3, . . . n produce in period 1 and firm 1 produces in period t . 1. The deviation of firm 1 never affects the actions of other firms, so firm 1 has no incentive for deviating from the equilibrium strategy. Suppose that firm i ± 1 deviates from the equilibrium strategy. Note that the deviation never affects the actions of leaders. Suppose that firm i chooses period t9 (1 , t9 , t). Then the deviation gives it the same payoff as before the deviation. Suppose that firm i chooses period t > t9. Then firm 1 and i are faced with Cournot-type competition, or the Stackelberg duopoly in which firm i is a follower. From Assumptions 2 and 3 (considering the case of S F 5 h1j and S L 5 hij), it can be seen that the deviation decreases firm i’s payoff. Therefore, no deviation by firm i strictly improves its payoff. (iii) This is derived from Propositions 1 and 2. Q.E.D.
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Some readers may think that a refinement of the equilibria discussed by H-S (1990) (elimination of weakly dominated strategies) can be applied to this model. If the number of periods is two, we can eliminate the Cournot-type equilibrium by this refinement. Suppose that n 5 m 5 2. Let Ci denote the output of firm i in the Cournot duopoly model, and Li (resp. Fi ) denote the output of firm i in the Stackelberg duopoly model when firm i is a leader (a follower). As is shown by H-S (1990), for firm 1, producing C1 in period 1 is weakly dominated by not producing in period 1. Suppose that firm 2 also produces in period 1. For firm 1, not producing in period 1 is never worse than producing C1 in period 1, because firm 1 can optimally respond to x 2 when it does not produce in period 1. Suppose that firm 2 does not produce in period 2. If firm 1 produces C1 in period 1, firm 2 produces C2 in period 2. If firm 1 does not produce in period 1, firm 1 and firm 2 are faced with Cournot competition in period 2, and each firm i also chooses Ci . Thus, the above two strategies are indifferent in regard to firm 1. In short, for firm 1, not producing in period 1 is never worse than producing C1 in period 1, regardless of the choice of firm 2. If the number of periods is more than two, we cannot eliminate the Cournot-type equilibrium by this refinement. Suppose that n 5 2 and m 5 3. Suppose that firm 2 produces in period 1. Then, as well as the cases in which m 5 2, for firm 1, not producing in period 1 is never worse than producing in period 1. Suppose that firm 2 does not produce in period 1. When firm 1 produces C1 in period 1, firm 2 produces C2 in period 2 or in period 3. When firm 1 does not produce in period 1, x 2 depends on e 2 and the expectation about e 1 by firm 2. If e 2 5 2 and firm 2 believes that e 1 5 3, firm 2 produces L2 in period 2. Since by assumption U1 (C1 , C2 ) . U1 (F1 , L2 ) > U(x 1 , L2 ) ; x 1 [ R 1 , not producing in period 1 is strictly worse than producing C1 in period 1 when firm 2 chooses the strategy above.7 I now want to emphasize that the existence of equilibria with one follower depends on the informational structure of the game. If firm i [ N t chooses its output after observing N s (s 5 1, . . . ,m), the result is completely changed. The game where N s is observable before each firm chooses its output is closely related to ‘the extended game with observable delay’ discussed by H-S (1990). They showed that in duopoly cases the equilibrium outcomes of this game are different from those of ‘the extended game with action commitment’, and we can apply similar principles to oligopoly models. Suppose that in an equilibrium, firm 1 becomes a follower and all other firms become leaders. Then if firm 1 deviates from the strategy above and chooses period 1, N 1 also changes, and each leader changes its output, resulting in an improvement of payoff of firm 1. Therefore, in this case, an outcome with one follower is never found in an equilibrium.
7
Some readers may think that firm 1 can prevent firm 2 from producing L2 in period 2 by choosing e 1 5 2. However, this is impossible, because in my model, firm 2 cannot observe e 1 . So firm 2 produces x 2 without knowing whether e 1 5 2 or 3. Therefore, firm 1’s choosing e 1 5 2 does not prevent firm 2 from producing L2 in period 2.
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In this paper I assume that firm i [ N t chooses it output before observing N s (s . t). Under this assumption, a follower cannot improve its payoff by deviating from the equilibrium strategy. If a possible follower producing in period t deviates from the equilibrium strategy and produces in period t9 ( , t), the above deviation is never observed by each firm i [ N s (s < t9); thus, the deviation never affects the output of other firms. Note that by the definition of a follower, all other firms produce until period t. Therefore, the above deviation never improves the payoff of a follower. This is why a firm may accept the role of a follower in an equilibrium. Finally, I must emphasize that my results are crucially dependent on the assumption that the payoff for each firm depends on its own output level and that of other firms, but not directly on the period in which either firm produces. As Anderson and Engers (1994) showed, in a model with discounting, the generalized Stackelberg-type outcome (sequential-move outcome) can appear in equilibrium.8
4. Concluding remarks In many economic situations, it is more reasonable to assume that firms choose not only what actions to take, but also when to take them. Some economists have investigated this problem and emphasized that Stackelberg-type outcomes appear but a Cournot-type outcome does not. They suggested that the Stackelberg model is more plausible than the Cournot model if firms can choose when to take their actions. In this paper I find that most firms (all or all but one) take their actions in the first period and the number of followers is at most one.9 In other words, the equilibrium outcome is either Cournot-type or joint leadership, and a hierarchical Stackelberg outcome never appears in an equilibrium. This paper shows the importance of joint leadership.
Acknowledgements I am grateful to Osamu Kanda, Murdoch MacPhee, Masuyuki Nishijima, Makoto Okamura, Masahiro Okuno-Fujiwara, Yoshiyasu Ono and participants of the workshops at Chuba University, Toyo University and the University of Tokyo for their helpful discussions. I am also indebted to two anonymous referees and an
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For another important model with discounting, see Robson (1990b). Independently, Nishijima (1995) formulated a different oligopoly model with endogenous sequencing and derives a result similar to my Proposition 3. In addition to my assumptions he requires another condition, which is very restrictive in the sense that it is not satisfied in a broad class of standard quantity-setting oligopoly models. For example, it is not always satisfied even in linear-demand, linear-cost cases. 9
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editor for their valuable and constructive suggestions. Needless to say, I am responsible for any remaining errors. The financial support of the Seimei Foundation and of the Grant-in-Aid for Encouragement of Young Scientists from the Japanese Ministry of Education, Science and Culture are greatly acknowledged.
References Albæk, S., 1992. Endogenous timing in a game with incomplete information. Center for Economic Research Discussion paper, Tilburg University, No. 9211. Anderson, S.P., Engers, M., 1992. Stackelberg versus Cournot oligopoly equilibrium. International Journal of Industrial Organization 10, 127–135. Anderson, S.P., Engers, M., 1994. Strategic investment and timing of entry. International Economic Review 35, 833–853. Daughety, A.D., 1990. Beneficial concentration. American Economic Review 81, 1231–1237. Gibbons, R., 1992. Game Theory for Applied Economists. Princeton University Press. Hamilton, J.H., Slutsky, S.M., 1990. Endogenous timing in duopoly games: Stackelberg or Cournot equilibria. Games and Economic Behavior 2, 29–46. Mailath, G.J., 1993. Endogenous sequencing of firm decisions. Journal of Economic Theory 59, 169–182. Matsumura, T., 1996. Essays in Oligopoly with Endogenous Sequencing. Ch. 3. in Doctoral Dissertation, University of Tokyo. Myerson, R.B., 1991. Game Theory: Analysis of Conflict. Harvard University Press. Nishijima, M., 1995. N-person Endogenous Leader-Follower Relations, mimeo. Yokohama City University. Pal, D., 1996. Endogenous Stackelberg equilibria with identical firms. Games and Economic Behavior 12, 81–94. Robson, A.J., 1990a. Stackelberg and Marshall. American Economic Review 81, 69–82. Robson, A.J., 1990b. Duopoly with endogenous strategic timing: Stackelberg regained. International Economic Review 31, 263–274. Sherali, D.H., 1984. A multiple leader Stackelberg model and analysis. Operations Research 32, 390–404.
Quantity-setting oligopoly with endogenous sequencing Toshihiro Matsumura* Department of Social Engineering, Tokyo Institute of Technology, 2 -12 -1 O-okayama, Meguro-ku, Tokyo 152 -8552, Japan Received 27 November 1996; received in revised form 1 November 1997; accepted 21 March 1998
Abstract This paper investigates endogenous sequencing in a quantity-setting oligopoly. I formulate an n-firm, m-period model where each firm chooses both how much to produce and when to produce it. I find that at least n 2 1 firms simultaneously produce in the first period in every pure strategy equilibrium. This result shows that the generalized Stackelberg-type outcome never appears in equilibrium except in a duopoly. 1999 Elsevier Science B.V. All rights reserved. Keywords: Cournot; Stackelberg; Endogenous sequencing JEL classification: C72; L13
1. Introduction Cournot and Stackelberg models have occupied important positions in oligopoly theory. The Cournot model involves simultaneous moves, while the Stackelberg model involves sequential moves. If we allow for more than two firms, we can analyze more varied situations: a pure simultaneous-move model in which all firms choose their actions at the same time (Cournot-type), a pure sequential-move model in which all firms choose their actions at different times (generalized
* Tel. / fax: 1 81-3-5734-3318; e-mail: [email protected] 0167-7187 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 98 )00021-6
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Stackelberg-type),1 and mixed-type models in which there is a sequence of periods in each of which some players choose their actions simultaneously. Each of these models produces a different equilibrium outcome, that is, the equilibrium outcome crucially depends on the sequence of each firm’s choice. The aim of this paper is to endogenize the sequence. Other economists have investigated similar problems. Robson (1990b), Albæk (1992), and Mailath (1993) discussed duopoly games with endogenous timing and emphasized that Stackelberg-type outcomes appear in endogenous timing games but a Cournot-type outcome does not. These papers suggested that the Stackelberg model is more plausible than the Cournot model if firms can choose when to take their actions.2 Here let us take a close look at the model of Hamilton and Slutsky (1990) (hereinafter referred to as H-S), which they call the ‘extended game with action commitment’. This is a standard model with endogenous sequencing. The game runs as follows. In period 1, each duopolist chooses whether to take its action this period or to wait until the next period. Any firm choosing period 1 takes its action without knowing whether its rival chooses period 1 or 2. At the beginning of period 2, each firm knows its rival’s action. A firm must act in period 2 if it elected to wait in period 1. The payoff for each firm depends on its and its rival’s actions in the basic market game, but not directly on the period in which either firm produces. In the H-S duopoly model there are four possible outcomes: 1. 2. 3. 4.
firm 1 acts in period 1 and firm 2 acts in period 2 (Stackelberg type); firm 2 acts in period 1 and firm 1 acts in period 2 (Stackelberg type); both firms act in period 1 (Cournot type); and both firms act in period 2 (Cournot type).
They found that under moderate conditions three cases ((1)–(3)) are supported as subgame perfect equilibria (Theorem VII). They also found that the outcome of case (3) is weak in the sense that it is supported by weakly dominated strategies, while the outcomes of cases (1) and (2) are not (Theorem VIII). In this paper I elaborate on the H-S duopoly framework to allow for oligopoly with more than two periods. I formulate an n-firm, m-period, quantity-setting oligopoly model in which each firm can choose when to take its action as well as what action to take. Each firm’s payoff depends only on its own output level and that of other firms. I find that there are two types of equilibria: either all firms
1
For the generalized Stackelberg model, see Robson (1990a) and Anderson and Engers (1992). This statement may be misleading. We should note that the above articles restrict the firms to pure strategies. If the firms are allowed to choose mixed strategies, the probability of ex post Cournot outcome is significant (see Pal, 1996). 2
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simultaneously produce, or n 2 1 firms simultaneously produce first and thereafter one firm produces, and that no other equilibrium exists. In other words, most firms (all or all but one) produce simultaneously in every equilibrium. This result shows the weakness of the sequential-move model (the generalized Stackelberg model). The remainder of this paper is organized as follows: Section 2 formulates a model; Section 3 discusses the equilibria of the model; and Section 4 concludes the paper.
2. The model In this section, I formulate an (m 1 1)-stage, n-firm oligopoly game. Both m and n are larger than one. The set of firms is denoted by N where N ; h1,2, . . . ,nj. In the first stage (period 0), each firm i [ N chooses the timing e i [ h1,2, . . . ,mj where e i 5 t implies that firm i chooses to produce in period t. The set of firms choosing to produce in period t is denoted by N t . At this time, each firm j ± i does not observe e i . Each firm i [ N 1 independently chooses the output quantity x i [ [0,`) in period 1. At the beginning of period t( > 2), each firm observes actions taken in period t 2 1; as a result, each firm knows which firms belong to N 1 , N 2 , . . . ,N t 21 . In period t, each firm j [ N t independently chooses the output quantity x j [ [0,`). At the end of period m, the market opens and each firm sells its own output. Define X2hi j ; o j[N •hi j x j , where N•hij ; h1,2, . . . ,i 2 1,i 1 1, . . . ,nj. Firm i’s payoff Ui : R 21 → R is given by Ui (x i , X2hi j ). Definition 1. (leader, follower, intermediate, and the last intermediate). I call firms producing in period 1 leaders, firms that produce in period t9 . 1 with no firm producing in period t . t9 followers, and other firms intermediates. I call firms the last intermediates if they are intermediates and no intermediate produces after them. Before investigating the equilibrium outcomes of the game with endogenous sequencing, I wish to discuss two-stage games with exogenous sequencing as a benchmark.3 From here on, I shall restrict our attention to pure strategy equilibria. Consider the following two-stage games with exogenous timing. The set of players is S L < S F , where S L < S F # N and S L > S F 5 [. S L is a set of leaders (namely firms producing in the first stage) and S F is a set of followers (namely firms producing in the second stage). Each firm h [ ⁄ S L < S F has already produced before the game and the output of it is common knowledge.
3
Similar two-stage games with exogenous timing are discussed in the context of multiple leaders. See Sherali (1984) and Daughety (1990).
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In the first stage, each firm i [ S L simultaneously chooses its output. In the second stage, after observing the outputs of firms that belong to S L , each firm j [ S F simultaneously chooses its output. We now make the following three assumptions.4 Assumption 1. There exists a pure strategy equilibrium and the equilibrium is unique in the two-stage game with exogenous sequencing. We call the equilibrium outcome where each firm i [ S L < S F simultaneously chooses its output the Cournot outcome. Assumption 2. Suppose that the number of followers is one. Then the follower strictly prefers the Cournot outcome to the follower’s outcome. Assumption 3. Suppose that the number of leaders is one. Then the leader strictly prefers the leader’s outcome to the Cournot outcome.
3. Equilibrium outcomes In this section I discuss the equilibrium outcomes of the model with endogenous sequencing. I use perfect Bayesian equilibrium 5 as the solution concept.6 I restrict my attention to pure strategy equilibria. Proposition 1. Suppose that Assumptions 1 – 3 are satisfied. Then the number of followers is at most one in every equilibrium. Proof. I shall prove Proposition 1 by contradiction. Suppose that there is more than one follower in an equilibrium. Without loss of generality, assume that followers produce in period t9 . 1. Suppose that one of the followers unilaterally deviates from the equilibrium strategy and produces in period t9 2 1. From the definition of followers, all players other than followers produce before observing this deviation, so the deviation never affects the actions of leaders and intermediates. From Assumption 3, it is clear that the deviation strictly increases the deviator’s payoff, a contradiction. Q.E.D.
4
The following assumptions are not restrictive. Assumptions 2 and 3 are satisfied under moderate conditions in the context of quantity-setting oligopoly. Matsumura (1996) shows sufficient conditions for Assumptions 2 and 3. 5 For the concept of perfect Bayesian equilibrium see, e.g. Gibbons (1992) chap. 3. 6 I use perfect Bayesian equilibrium rather than subgame perfection because the proper subgame which begins with the beginning of period 1 cannot be defined. See Myerson (1991, pp. 183–185) for an example where even among complete information games unnatural equilibrium outcomes survive if we use subgame perfection.
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Proposition 2. Suppose that Assumptions 1 – 3 are satisfied. Then no intermediate exists in equilibrium. Proof. By the definition of intermediate, an intermediate never exists if m (the number of periods) is two. Thus, let us restrict our attention to cases in which m is larger than two. I shall prove Proposition 2 by contradiction. Suppose that there is at least one intermediate in an equilibrium. From the definitions of intermediate, it is certain that at least one follower exists. From Proposition 1, it is known that exactly one follower exists. Without loss of generality, I assume that the last intermediates produce in period t9 where m . t9 . 1. Suppose that the follower unilaterally deviates from the equilibrium strategy and produces in period t9 2 1. From the definition of the last intermediates and followers, all players other than followers and the last intermediates produce before observing this deviation, so this deviation affects the actions of last intermediates only. From Assumptions 2 and 3, it can be seen that the deviation strictly increases the deviator’s payoff, a contradiction. Q.E.D. Proposition 2 states that each firm is either a leader or a follower in an equilibrium; thus, pure sequential-move outcomes (generalized Stackelberg outcomes) where no firms simultaneously produce in the same period never appear in an equilibrium except in duopoly. Proposition 3 states that the number of leaders is n or n 2 1 in every equilibrium. Proposition 3. Suppose that Assumptions 1 – 3 are satisfied. Then: ( i) there exists an equilibrium where all firms become leaders; ( ii) there exist equilibria where all but one firm become leaders, and ( iii) no other pure strategy equilibrium exists. Proof. (i) Suppose that each firm i [ N produces the Cournot output in period 1. Obviously, the above strategies construct an equilibrium. (ii) I show that there is an equilibrium in which firms 2,3, . . . n produce in period 1 and firm 1 produces in period t . 1. The deviation of firm 1 never affects the actions of other firms, so firm 1 has no incentive for deviating from the equilibrium strategy. Suppose that firm i ± 1 deviates from the equilibrium strategy. Note that the deviation never affects the actions of leaders. Suppose that firm i chooses period t9 (1 , t9 , t). Then the deviation gives it the same payoff as before the deviation. Suppose that firm i chooses period t > t9. Then firm 1 and i are faced with Cournot-type competition, or the Stackelberg duopoly in which firm i is a follower. From Assumptions 2 and 3 (considering the case of S F 5 h1j and S L 5 hij), it can be seen that the deviation decreases firm i’s payoff. Therefore, no deviation by firm i strictly improves its payoff. (iii) This is derived from Propositions 1 and 2. Q.E.D.
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Some readers may think that a refinement of the equilibria discussed by H-S (1990) (elimination of weakly dominated strategies) can be applied to this model. If the number of periods is two, we can eliminate the Cournot-type equilibrium by this refinement. Suppose that n 5 m 5 2. Let Ci denote the output of firm i in the Cournot duopoly model, and Li (resp. Fi ) denote the output of firm i in the Stackelberg duopoly model when firm i is a leader (a follower). As is shown by H-S (1990), for firm 1, producing C1 in period 1 is weakly dominated by not producing in period 1. Suppose that firm 2 also produces in period 1. For firm 1, not producing in period 1 is never worse than producing C1 in period 1, because firm 1 can optimally respond to x 2 when it does not produce in period 1. Suppose that firm 2 does not produce in period 2. If firm 1 produces C1 in period 1, firm 2 produces C2 in period 2. If firm 1 does not produce in period 1, firm 1 and firm 2 are faced with Cournot competition in period 2, and each firm i also chooses Ci . Thus, the above two strategies are indifferent in regard to firm 1. In short, for firm 1, not producing in period 1 is never worse than producing C1 in period 1, regardless of the choice of firm 2. If the number of periods is more than two, we cannot eliminate the Cournot-type equilibrium by this refinement. Suppose that n 5 2 and m 5 3. Suppose that firm 2 produces in period 1. Then, as well as the cases in which m 5 2, for firm 1, not producing in period 1 is never worse than producing in period 1. Suppose that firm 2 does not produce in period 1. When firm 1 produces C1 in period 1, firm 2 produces C2 in period 2 or in period 3. When firm 1 does not produce in period 1, x 2 depends on e 2 and the expectation about e 1 by firm 2. If e 2 5 2 and firm 2 believes that e 1 5 3, firm 2 produces L2 in period 2. Since by assumption U1 (C1 , C2 ) . U1 (F1 , L2 ) > U(x 1 , L2 ) ; x 1 [ R 1 , not producing in period 1 is strictly worse than producing C1 in period 1 when firm 2 chooses the strategy above.7 I now want to emphasize that the existence of equilibria with one follower depends on the informational structure of the game. If firm i [ N t chooses its output after observing N s (s 5 1, . . . ,m), the result is completely changed. The game where N s is observable before each firm chooses its output is closely related to ‘the extended game with observable delay’ discussed by H-S (1990). They showed that in duopoly cases the equilibrium outcomes of this game are different from those of ‘the extended game with action commitment’, and we can apply similar principles to oligopoly models. Suppose that in an equilibrium, firm 1 becomes a follower and all other firms become leaders. Then if firm 1 deviates from the strategy above and chooses period 1, N 1 also changes, and each leader changes its output, resulting in an improvement of payoff of firm 1. Therefore, in this case, an outcome with one follower is never found in an equilibrium.
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Some readers may think that firm 1 can prevent firm 2 from producing L2 in period 2 by choosing e 1 5 2. However, this is impossible, because in my model, firm 2 cannot observe e 1 . So firm 2 produces x 2 without knowing whether e 1 5 2 or 3. Therefore, firm 1’s choosing e 1 5 2 does not prevent firm 2 from producing L2 in period 2.
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In this paper I assume that firm i [ N t chooses it output before observing N s (s . t). Under this assumption, a follower cannot improve its payoff by deviating from the equilibrium strategy. If a possible follower producing in period t deviates from the equilibrium strategy and produces in period t9 ( , t), the above deviation is never observed by each firm i [ N s (s < t9); thus, the deviation never affects the output of other firms. Note that by the definition of a follower, all other firms produce until period t. Therefore, the above deviation never improves the payoff of a follower. This is why a firm may accept the role of a follower in an equilibrium. Finally, I must emphasize that my results are crucially dependent on the assumption that the payoff for each firm depends on its own output level and that of other firms, but not directly on the period in which either firm produces. As Anderson and Engers (1994) showed, in a model with discounting, the generalized Stackelberg-type outcome (sequential-move outcome) can appear in equilibrium.8
4. Concluding remarks In many economic situations, it is more reasonable to assume that firms choose not only what actions to take, but also when to take them. Some economists have investigated this problem and emphasized that Stackelberg-type outcomes appear but a Cournot-type outcome does not. They suggested that the Stackelberg model is more plausible than the Cournot model if firms can choose when to take their actions. In this paper I find that most firms (all or all but one) take their actions in the first period and the number of followers is at most one.9 In other words, the equilibrium outcome is either Cournot-type or joint leadership, and a hierarchical Stackelberg outcome never appears in an equilibrium. This paper shows the importance of joint leadership.
Acknowledgements I am grateful to Osamu Kanda, Murdoch MacPhee, Masuyuki Nishijima, Makoto Okamura, Masahiro Okuno-Fujiwara, Yoshiyasu Ono and participants of the workshops at Chuba University, Toyo University and the University of Tokyo for their helpful discussions. I am also indebted to two anonymous referees and an
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For another important model with discounting, see Robson (1990b). Independently, Nishijima (1995) formulated a different oligopoly model with endogenous sequencing and derives a result similar to my Proposition 3. In addition to my assumptions he requires another condition, which is very restrictive in the sense that it is not satisfied in a broad class of standard quantity-setting oligopoly models. For example, it is not always satisfied even in linear-demand, linear-cost cases. 9
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editor for their valuable and constructive suggestions. Needless to say, I am responsible for any remaining errors. The financial support of the Seimei Foundation and of the Grant-in-Aid for Encouragement of Young Scientists from the Japanese Ministry of Education, Science and Culture are greatly acknowledged.
References Albæk, S., 1992. Endogenous timing in a game with incomplete information. Center for Economic Research Discussion paper, Tilburg University, No. 9211. Anderson, S.P., Engers, M., 1992. Stackelberg versus Cournot oligopoly equilibrium. International Journal of Industrial Organization 10, 127–135. Anderson, S.P., Engers, M., 1994. Strategic investment and timing of entry. International Economic Review 35, 833–853. Daughety, A.D., 1990. Beneficial concentration. American Economic Review 81, 1231–1237. Gibbons, R., 1992. Game Theory for Applied Economists. Princeton University Press. Hamilton, J.H., Slutsky, S.M., 1990. Endogenous timing in duopoly games: Stackelberg or Cournot equilibria. Games and Economic Behavior 2, 29–46. Mailath, G.J., 1993. Endogenous sequencing of firm decisions. Journal of Economic Theory 59, 169–182. Matsumura, T., 1996. Essays in Oligopoly with Endogenous Sequencing. Ch. 3. in Doctoral Dissertation, University of Tokyo. Myerson, R.B., 1991. Game Theory: Analysis of Conflict. Harvard University Press. Nishijima, M., 1995. N-person Endogenous Leader-Follower Relations, mimeo. Yokohama City University. Pal, D., 1996. Endogenous Stackelberg equilibria with identical firms. Games and Economic Behavior 12, 81–94. Robson, A.J., 1990a. Stackelberg and Marshall. American Economic Review 81, 69–82. Robson, A.J., 1990b. Duopoly with endogenous strategic timing: Stackelberg regained. International Economic Review 31, 263–274. Sherali, D.H., 1984. A multiple leader Stackelberg model and analysis. Operations Research 32, 390–404.