Coalitions and Networks in Industrial Organization

Coalitions and Networks in Industrial Organization Francis Bloch May 28, 2001 1 Introduction Recent years have witnessed a surge of interest in the formation of coalitions and networks in Industrial Organization. Most of this interest stems from the emergence of new forms of cooperation and competition between rms. The development of strategic alliances, the acceleration in the creation of joint ventures and joint production and research facilities have given rise to a new strategic environment in which rms cooperate in some domains and compete in others. At the same time, new approaches have been developed in game theory to analyze the endogenous formation of coalitions and networks, providing simple tools which can be applied to study the formation of alliances and groups of rms. The objective of this paper is to provide a selective survey on recent approaches to coalition and network formation in Industrial Organization, and to o er a uni ed framework in which the di erent approaches can be compared. While the range of cooperative agreements signed by rms in reality is quite large, this paper focuses on two extreme forms of cooperation collusive agreements and cost-reducing alliances. In those two simple models, we analyze the formation of groups and networks of rms under various institutional arrangements. Collusive agreements cover cartels choosing quotas (as in many commodity markets), price- xing agreements, bidding rings as well as market sharing or exclusive territories agreements. The striking feature of collusive agreements, already pointed out by Stigler (1950), is that the formation of a cartel is a public good, inducing positive externalities to rms outside the cartel. Hence, typically, rms have an incentive to free-ride on the cartels formed by other rms, and collusive agreements are highly unstable. However, we will see that this conclusion can only be reached under some institutional procedures of group GREQAM and Ecole Superieure de Mecanique de Marseille. Address: 2 rue de la Charite, 13002 Marseille, FRANCE. Email: bloch@ehess.cnrs-mrs.fr 1

and network formation, but that other procedures may lead to the formation of stable coalitions. Cost-reducing alliances have recently been the focus of attention of many industrial economists and policy-makers. These alliances may pursue di erent objectives, such as the development of new products and processes, the de nition of common standards or the joint use of facilities. As opposed to collusive agreements, cost-reducing alliances typically induce negative externalities on other rms in the industry. Hence, the analysis of the formation of these alliances usually leads to very di erent conclusions than the study of the stability of cartels. By considering only the two pure cases of collusive groups and cost-reducing alliances, we only cover two extreme points on the spectrum of cooperative agreements between rms. It is widely believed that most cooperative agreements embody both a collusive and a cost-reducing aspect. For example, typical collusive agreements, such as mergers, are often justi ed by the existence of synergies and e ciency gains between the merging rms. On the other hand, cost-reducing agreements such as the formation of research joint ventures often provide a way for rms to exchange information and collude on the market. Many of the results obtained in the two polar cases that we study should thus be considered with caution before they can be applied to real cooperative agreements. As we only consider two very simple models of cooperation between rms, we are able to contrast the outcomes of various institutional rules of coalition and network formation. This approach enables us to answer the following questions, which are rarely addressed in the literature on cooperation between rms. Is cooperation more prevalent or more wide-spread when rms sign multilateral or bilateral agreements? Is the level of cooperation increased when cooperative agreements are open to all rms or restricted to the original signatories? How does the group formed depend on the sequentiality of the procedure of coalition formation? Should coalitions be formed on an industrywide basis or should competing coalitions be allowed to form? The rest of the paper is organized as follows. Section 2 presents the two basic models of collusive agreements and cost-reducing alliances, both in the case of multilateral agreements (coalitions) and bilateral agreements (networks). Section 3 studies the formation of coalitions under two di erent institutional procedures: a model of open membership and a sequential model of coalition formation. Section 4 studies the formation of networks. Section 5 contains bibliographical references and concluding comments. 2 Collusion and Alliances We consider a basic oligopoly model with n identical rms indexed by i = 1; 2; ::; n. In order to keep the analysis tractable, we assume that demand is 2

linear, and let denote the common degree of di erentiation of the products. We then have p i = q i X where p i denotes the price of good i and q j thequantitysoldofgoodj. Alternatively, we can write the demand system as j6=i q j q i = p i +» X j6=i p j (1+(n 2) ) (1 )(1+(n 1) ). where = 1+(n 1) ; = (1 )(1+(n 1) ) and» = We assume 0 1 and let c i denote the constant marginal cost of rm i. We consider two di erent models of interaction: a Cournot game where rms set quantities and a Bertrand game where they choose prices. In the Cournot game, each rm chooses its quantity q i to maximize its pro t ¼ i = (p i c i )q i = ( q i X q j c i )q i Standard arguments show that the unique Nash equilibrium of the Cournot game is given by P q i = 2+ (n 1) c i (2 + n) (2 )(2 + (n 1)) + j6=i c j (2 )(2 + (n 1)) j6=i In the Bertrand game, rm i chooses its price p i to maximize ¼ i = (p i c i )q i = (p i c i )( p i +» X j6=i p j ) resulting in equilibrium prices P p i = (2»(n 1)) + c i (2 n») (2»)(2»(n 1))» j6=i c j (2»)(2»(n 1)) : 3

2.1 Collusion We interpret collusion as the coordination of rms strategies, in order to reduce competition and increase pro ts on the market. Collusion can take many forms, ranging from explicit price and quota rules in cartels such as OPEC, to implicit agreements to share markets or abide by price standards. We need to distinguish between collusive agreements in multilateral settings, where rms form coalitions and bilateral settings, where rms form links. In multilateral settings, a collusive group is a cartel where rms agree on market prices and on production quotas. To model this situation, we consider a coalition (or cartel) structure, which is a partition B on the set of rms. Each element of B is a cartel K of rms which agree on a vector of quantities (in the Cournot setting) or a vector of prices (in the Bertrand setting) that will be chosen on the market. To keep the analysis simple, we suppose that in the collusion case, all rms have identical costs and we normalize c =0. 1 As rms are identical ex ante, we also assume that inside each cartel, the division of pro ts is equitable. To x notations, for any coalition structure B = fk 1 ;K 2 ;:;K j ::K b g, we let b denote the number of coalitions in B and k j the number of rms in coalition K j. As rms are ex ante identical, the numbers b and k j are su cient parameters to describe the cartel structure. Each cartel chooses the quantities (or prices) in order to maximize the total cartel pro t, taking as given the choice of other cartels. 2 Formally, in the Cournot setting, the pro t of cartel K j is written as j = X X X ( q i q j q k )q i i2k j j6=i;j2k j k2k k ;k6=j When products are homogeneous ( =1), we can explicitly compute the equilibrium quantity of a rm i in cartel K j as q i = (b +1)k j resulting in equilibrium pro ts ¼ i = 2 (b +1) 2 k j : When products are di erentiated ( <1), the computation of equilibrium quantities in the genral case is intractable. Instead, we can compute equilibrium quantities and pro ts in the single cartel case, where one cartel of size k is formed on the market, and all other rms are independent. Let q K denote the equilibrium quantity of a cartel member and q I the equilibrium quantity of an 1 We will discuss in the Conclusion di erent attempts to study cartel formation with heterogeneous rms. 2 The equilibrium concept is thus a hybrid equilibrium, including both cooperative (inside the cartel) and noncooperative (across cartels) features. 4

independent rm. We have q K = q I = (2 ) 4+2 (k + n 3) 2(k 2 +2n kn 2) ; (2 + (k 2)) 4+2 (k + n 3) 2(k 2 +2n kn 2) yielding equilibrium pro ts ¼ C K = ¼ C I = 2 (2 )(2 + (2k 3) + 2(1 k)) (4 + 2 (k + n 3) 2(k 2 +2n kn 2)) 2 ; 2 (2 + (k 2)) 2 (4 + 2 (k + n 3) 2(k 2 +2n kn 2)) 2 : The study of cartel formation in a Bertrand model with di erentiated commodities can also only be done in the case of a single cartel of size k facing (n k) independent rms. The cartel selects prices p i ;i2 K in order to maximize K = X X p i ( p i +» p k ) i2k j6=i;j2k p j +» X k=2k whereas individual rms choose prices to maximize individual pro ts. Tedious but straightforward computations show that the equilibrium prices chosen by rms inside the cartel and outside the cartel are given by p K = (2 +») 4 2 2»(k + n 3) +» 2 (2 k 2 2n + kn) ; p I = yielding equilibrium pro ts (2»(k 2)) 4 2 2»(k + n 3) +» 2 (2 k 2 2n + kn) ¼ B K = 2 (2 +»)(2 2»(2k 3) +» 2 (1 k)) (4 2 2»(k + n 3) +» 2 (2 k 2 2n + kn)) 2 ; ¼ B I = 2 ((2»(k 2)) 2 (4 2 2»(k + n 3) +» 2 (2 k 2 2n + kn)) 2 The interpretation of collusion in a bilateral setting is more complex. Clearly, rms cannot sign bilateral agreements on production quotas or market prices independently of the overall collusion structure. It doesn t make sense for a rm to commit to di erent production quotas with di erent rms! We thus consider a di erent form of collusion in bilateral settings, namely the signature of bilateral market sharing agreements. Suppose that rms are initially present on di erent geographical markets. (We assume that there is exactly one rm 5

on each market.) By signing market sharing agreements, rms commit not to enter each other s market. Hence, for any graph g on the set n of rms, we can compute n i (g), the number of rms present on market i. (Alternatively, if d i (g) denotes the degree of vertex i in the graph, n i (g) =n d i (g)+1.) Supposing that all rms are identical, we let ¼(n i (g)) denote the pro t made by each rm on market i. Forany rmi, the total pro t she makes is the sum of pro ts on each market where it is present, i.e. i (g) =¼(n i (g)) + X ¼(n j (g)) j;ij =2g Assuming again that the marginal cost of production is zero, we obtain, in the Cournot setting i (g) C = andinthebertrandsetting i (g) B = 2 (2 + (n i (g) 1)) 2 + X j;ij =2g 2 (2»(n i (g) 1)) 2 + X 2.2 Cost-Reducing Alliances j;ij =2g 2 (2 + (n j (g) 1)) 2 2 (2»(n j (g) 1)) 2 : Firms form alliances for a variety of reasons: to lobby for pieces of legislation, agree on common standards, develop new products and processes, launch joint marketing campaigns, produce common inputs, use jointly some facilities... In spite of the variety of objectives assigned to alliances, there is a common goal to all alliances formed: they should induce a reduction in the production cost of all their members. Hence, to model the role of alliances on a market, we suppose that rms can bene t from synergies which lead to a reduction in the constant marginal cost of production. In the multilateral setting, we let A denote a structure of alliances (i.e. a partitiononthesetof rms)witha = fa 1 ;A 2 ; :::; A r g. All rms are assumed to be identical ex ante, so that the identity of rms does not matter, and the sizes of the di erent alliances, a 1 ;a 2 ; :::; a r characterize the association structure A: We suppose, keeping the analysis as simple as possible, that the constant marginal cost of production of a rm is linearly decreasing in the size of the alliance it belongs to. Formally, if rm i belongs to association A j,itsmarginal cost of production is given by c i = ¹a j Hence, each alliance structure determines a vector of marginal costs for the di erent rms. These di erences in marginal costs then lead to di erences in 6

equilibrium quantities, prices and pro ts. 3 In the Cournot setting, the equilibrium pro t of a rm i belonging to association A j is given by: ¼ C i =[ (2 + (n 1) ) + ¹a P j (2 ) ¹ a 2 k (2 )(2 + (n 1) ) ]2 andinthebertrandsetting ¼ B + i = [ (2 (n 1)») ¹a P j 2 +»»¹ a 2 k (2 (n 1)»)(2 +») + ¹a j] 2 In the multilateral setting, we suppose that rms sign bilateral agreements to reduce their marginal cost of production. Hence, the constant marginal cost of production is linearly decreasing in the number of agreements signed by each rm. For any graph g, weletd i (g) denote the degree of vertex i, namelythe number of agreements signed by rm i. Wethenhave c i = ¹d i (g); and the equilibrium pro ts are given by C i =[ (2 + (n 1) ) + ¹d P i(g) (2 ) ¹ dj (g) (2 )(2 + (n 1) ) ]2 ; B + i = [ (2 (n 1)») ¹d P i(g) 2 +»»¹ dj (g) (2 (n 1)»)(2 +») + ¹d i(g)] 2 : 3 Models of Coalition Formation In this Section, we characterize the formation of cartels and alliances in the multilateral setting. As the above expressions for equilibrium pro ts show, the pro t of a rm depends not only on the coalition it belongs to, but on the entire coalition structure formed on the market. In the presence of externalities across coalitions, we cannot apply the classical models of cooperative games, and we need to develop new tools to characterize the endogenous coalition structure. The main di culty stems from the fact that, when a group of rms considers a deviation from a coalition structure, it must anticipate the reaction of other rms to the deviation. Di erent models of reactions have been proposed in the literature, but they all rely on ad hoc speci cations of the behavior of external players. In this paper, we restrict our attention to two simple situations: (i) the formation of a single coalition and (ii) the sequential formation of multiple coalitions. 3 We assume thorughout that di erences in marginal costs are not high enough to drive some rms out of the market. This assumption amounts to placing an upper bound on the parameter ¹: 7

3.1 Formation of a single coalition When a single coalition is formed, the reaction of external players is irrelevant, as they are all assumed to remain independent. The single coalition case corresponds to the formation of an industrywide cartel or of a trade association, and the only decision that rms make is whether they want to join the coalition. We implicitly assume that membership in the coalition is open: rms cannot exclude other rms from the coalition. We consider a simultaneous game, where all rms announce whether they want to join the coalition. The strategy set of the rm is thus the pair f0; 1g and the coalition C is formed of all the agents who have announced 1. The payo s of the agents correspond to the pro ts they make, given the formation of the coalition C. We recall that a Nash equilibrium is a strategy pro le such that no rm has an incentive to deviate unilaterally given the strategy choice of the other rms. Proposition 1 Consider the formation of a cartel on a Cournot market with homogeneous goods. If n 3, in the unique Nash equilibrium of the simultaneous game of cartel formation, all rms remain independent. If n =2, in the unique Nash equilibrium of the simultaneous game of cartel formation, the two rms form a cartel. Proof. Suppose that a cartel of size k is formed. Each cartel member receives a pro t ¼ = 2 k(n k+2). If a rm leaves the cartel, it receives a pro t 2 ¼ 0 = 2 (n k+3). It is easy to see that, as long as n 3, for any 1 <k 2 n; k(n k +2) 2 > (n k +3) 2,so¼ 0 >¼. This establishes that the unique equilibrium of the game is for all rms to remain independent. If n =2,the two rms have an incentive to form a cartel. Proposition 1 illustrates the well-known free-riding problem associated with the formation of a cartel. Since external rms bene t from the increase in price due to the formation of the cartel, all rms prefer to stay outside the cartel and free ride on the public good provided by cartel members. In a linear Cournot market, this incentive is so strong that no cartel can be formed. When products are di erentiated, the same free-riding incentives appear, but computations are too intricate to obtain a simple characterization of the equilibrium of the game of cartel formation. Instead, we graph below the pro t of cartel members and of outsiders in a speci c numerical example, setting = 1=2, =1and n =10: The dashed line represents the pro t of cartel members for values of k ranging from 1 to 10, whereas the full line represents the pro ts of outsiders. Outsiders pro ts are always increasing in the cartel size, re ecting the positive externality that the formation of a cartel induces on outsiders. Insiders pro ts, on the other hand, are U-shaped, as cartel members su er the cost of reducing quantities to increase prices, and this cost is proportionately 8

pi 0.07 0.06 0.05 0.04 0.03 2 4 6 8 10 k Figure 1: higher for lower cartel sizes. As in the case of homogeneous goods, it appears that a rm always has an incentive to defect from the cartel, as the pro t of an outsider when a cartel of size k 1 is formed is always larger than the pro t of an insider for a cartel of size k. Figure 2 graphs the pro ts of outsiders and insiders for the same numerical values in the case of Bertrand competition. Again, outsider pro ts are represented by a full line and insiders pro ts by a dashed line, for k ranging from 0 to 10. Not surprisingly, outsiders pro ts are always increasing in the size of the cartel. For insiders, the picture is somewhat di erent from the case of Cournot competition. Collusion among insiders leads them to raise their prices, inducing a higher demand for external rms and reducing their own demand. The cost of raising prices appears to be proportionately increasing with the size of the cartel. Hence, the pro t of insiders is rst increasing in the size of the cartel, reaches a peak and then becomes decreasing. Note that the free-riding incentives remain extremely strong in the case of Bertrand competition. No cartel is stable, as a rms always obtains a higher pro t as an outsider with a cartel of size k 1 than as an insider in a cartel of size k. Proposition 2 Consider the formation of a cost-reducing alliance. In the unique equilibrium of the simultaneous game of alliance formation, all the rms form a single alliance. 9

pi 0.018 0.017 0.016 0.015 0.014 0.013 0.012 2 4 6 8 10 k Figure 2: Proof. Consider an association structure with an association A and independent rms. We verify that any independent rm has an incentive to join the association. In the Cournot case, computations show that by joining the a¹(2+ (n 3)) association, the rm increases its equilibrium quantity by (2 )(2+(n 1) ). As equilibrium pro ts are increasing in equilibrium quantities, the independent rm has an incentive to join the association. In the Bertrand case, when it joins the association, the independent rm experiences an increase in its mark-up, p c, of ¹a(2 2 (n 1)» 2 (n 1)» (2 (n 1)»)(2 +») : As equilibrium pro ts are increasing in the equilibrium mark-up, this deviation is pro table. Proposition 2 highlights the fact that rms always have an incentive to join a cost-reducing alliance. There are both direct and strategic bene ts from joining the alliance. The reduction in cost directly a ects the rms pro ts but also confers a strategic advantage over the rms who do not belong to the alliance. However, the result of Proposition 2 depends on the assumption that membership is open: otherwise,aswewillseebelow,some rmshaveanincentiveto exclude others from the alliance. 3.2 Sequential Formation of Coalitions We now consider the more di cult problem of endogenous formation of multiple coalitions. When multiple coalitions can be formed, and each rm s pro t depends on the entire coalition structure, rms must anticipate the reaction of other rms when choosing whether to enter a coalition. In order to take into 10

account this forward-looking behavior, we analyze here the equilibria of a sequential game of coalition formation, where each rm takes into account the e ect of its decision on the behavior of rms choosing actions subsequently in the game. More precisely, we consider the following extension of games of coalitional bargaining. At each period t, one rm is chosen to make a proposal (a coalition to which it belongs), and all the prospective members of the coalition respond in turn to the proposal. If the proposal is accepted by all, the coalition is formed and another rm is designated to make a proposal at t +1 ;ifoneof the rms rejects the proposal, she makes a countero er at period t +1. The identity of the di erent proposers is given by an exogenous rule of order. There is no discounting in the game but all players receive a zero payo in case of an in nite play. As the games i a sequential game of complete information and in nite horizon, we use as a solution concept stationary perfect equilibria. When rms are ex ante identical, it can be shown that the coalition structures generated by stationary perfect equilibria can also be obtained by analyzing the following simple nite game. The rst rm announces an integer k 1, corresponding to the size of the coalition she wants to see formed, rm k 1 +1 announces an integer k 2, etc;, until the total number n of rms is exhausted. An equilibrium of the nite game determines a sequence of integers adding up to n, which completely characterizes the coalition structure as all rms are ex ante identical. We state below (without proof) two Propositions describing the coalition structures formed at the equilibrium of the sequential game in a Cournot oligopoly with homogeneous products. The characterization of endogenous association structures can be obtained for models with di erentiated products, but there is no simple way to characterize endogenous cartel structures when products are di erentiated. Proposition 3 Consider a Cournot oligopoly with homogeneous products. In theuniqueequilibriumofthesequentialgameofcartelformation,the rst(n k) rms remain independent and the last k rms form a cartel. The integer k is the rst integer following 2n+3 p 4n+5 2 : Proof. See Bloch (1996). Proposition 3 can easily be interpreted. From Figure 1, recall that the pro t of a cartel member is a U shaped function. Hence, there exists a minimal cartel size for which rms prefer to form a cartel than remain independent. This minimal pro table cartel size is exactly equal to k in a linear Cournot model with homogeneous products. Hence, Proposition 3 shows that in a sequential game of cartel formation, the rst rms choose to remain independent and freeride on the cartel formed by subsequent rms. When the number of remaining market participants is exactly equal to the minimal pro table cartel size, rms will choose to form a cartel, as it becomes impossible for a rm to stay out and free ride on the cartel formed by the others. 11

It is interesting to contrast the outcome of the sequential game of coalition formation with that of the simultaneous game. In the simultaneous game, free riding incentives prevent the formation of a cartel, as any rm has an incentive to defect. In the sequential game, rms can commit to stay out of the cartel, and hence a cartel is formed in equilibrium. Notice furthermore that the minimal pro table cartel size k is usually very high (representing around 80% of the rms in the industry). This implies that there will always be at most one cartel formed in equilibrium, and the issue of the formation of multiple competing cartels does not arise. Proposition 4 Consider a Cournot oligopoly with homogeneous products. In theuniqueequilibriumofthesequentialgameofallianceformation,the rsta 1 rms form an alliance, and the last (n a 1 ) rms form a competing alliance. The integer a 1 is the integer closest to 3n+1 4 : Proof. See Bloch (1995). To understand the result of Proposition 4, recall that rms may have an incentive to exclude other rms from the alliance. This incentive will be stronger, the larger the number of rms already in the alliance, as a newcomer would bene t from a huge cost reduction, but only contribute a small cost reduction to the standing alliance members. Hence, the pro t of an alliance member is a concave function of the alliance size, and in the linear Cournot case, it reaches a peak at n+1 n+1 2. This implies that the rst alliance formed will contain at least 2 members. In fact, the rst alliance will be even larger because its members anticipate that all the subsequent players will also form an alliance, and they will enlarge the rst alliance in order to reduce the size of the second. On balance, this leads the rst rms to form an alliance grouping around three quarters of the members of the industry and the second one quarter. Comparing the outcomes of the sequential and simultaneous games of coalition formation, two striking di erences appear between the models. First, in the sequential game, rms can limit membership and exclude other rms from the alliance. Second, in the sequential game, rms anticipate that a competing association may be formed by the other players, and choose the size of the rst association accordingly. 4 Stable Networks In this Section, we analyze the formation of collusive networks and networks of alliances. While coalition structures can simply be represented by partitions on the set of rms, networks are more complicated objects, which can be characterized by a set of pairwise links, i.e. a symmetric and re exive binary relation on the set of rms. The complexity of networks makes it harder to characterize the 12

formation of a network as the outcome of a noncooperative of link formation. We follow here a di erent approach. A network g is called pairwise stable if no pair of rms has an incentive to form a new link, and no rms has an incentive to unilaterally destroy an existing link. Formally, network g is stable if (i) i (g) > i (g ij) and j (g) > j (g ij) (ii) If i (g + ij) > i (g) then j (g + ij) j (g ij) Pairwise stability is clearly a weak notion of stability (for example, it doesn t allow rms to form more than one link nor to destroy di erent links at once.) However, pairwise stable networks remain an important benchmark, especially when as in the case of collusive networks and networks of alliances, few networks pass the test of pairwise stability. Proposition 5 Consider the formation of market sharing agreements in a Cournot oligopoly with homogenous goods. If n 3;there are exactly two pairwise stable networks: the empty network and the complete network. If n =2,thecomplete network is the only stable network. Proof. To show that the empty network is stable, compute the bene t that two rms would have from forming a link i (g + ij) i (g) = 2 n 2 2 2 (n +1) 2 = 2 ( n 2 +2n +1) n 2 (n +1) 2 For n 3; i (g + ij) i (g) < 0 and the empty network is stable. Now suppose that two rms have an incentive to form a link, i.e. i (g) i (g ij) > 0 j (g) j (g ij) > 0 This system of inequalities is equivalent to 2 (n i (g)+1) 2 > 2 (n j (g)+1) 2 > 2 (n i (g)+2) 2 + 2 (n j (g)+1) 2 2 (n j (g)+2) 2 + 2 (n i (g)+1) 2 which can only be satis ed if n i (g) =n j (g) =1; showing that if a nonempty network is formed, it must be the complete network, and that the complete network is indeed stable. Proposition 5 shows that, if collusive agreements are negotiated bilaterally, they may actually lead to full collusion, with every rm a monopoly on its own market. The free-riding incentives remain strong: no smaller collusive network can be sustained because rms have an incentive to defect from any market 13

sharing agreement which involves less rms than the entire industry. When products are di erentiated, competition may be less strong and the incentive to sign market sharing agreements is reduced. Proposition 6 Consider the formation of market sharing agreements in a Cournot oligopoly with di erentiated products, =1; 1 >0:For >2( p 2 1), there exist two stable networks: the empty network and the complete network. For 2( p 2 1), the only stable network is the empty network. Proof. See Belle amme and Bloch (2001). In the case of Bertrand competition with di erentiated products, di erent network architectures may arise, as the following Proposition demonstrates. The threshold values for can only be obtained by numerical computations. Proposition 7 Consider the formation of market sharing agreements in a Bertrand oligopoly with di erentiated products, = 1; 1 > 0:For > 0:715, there exist three stable networks: the empty network, the complete network and a network with one isolated rm and a complete component of n 1 rms. For 0:611 < 0:715, there exist two stable networks: the empty network and the complete network. For 0:611, the only stable network is the empty network. Proof. See Belle amme and Bloch (2001). Proposition 7 re ects the fact that price competition results in lower equilibrium pro ts than quantity competition, and increases the incentive to collude. When products are highly substitutable, a network with one isolated free-riding rm and a collusive network of n 1 rms appears to be stable, along with the complete and empty networks. As products become more and more di erentiated, market competition is less erce, and market sharing agreements become less stable. The formation of networks of alliances results in a very di erent outcome. Proposition 8 Consider the formation of bilateral alliances in a Cournot oligopoly with homogeneous products. The only stable network is the complete network. Proof. Consider the e ect of the formation of a link to player j on the equilibrium quantity of rm i: (n 1)¹ q i = > 0: n +1 As equilibrium pro ts are increasing in equilibrium quantities, any link is pro table, and the only stable network is the complete network. 14

Proposition 8 shows that a rms always has an incentive to form a bilateral alliance. In fact, the formation of a link results in a simultaneous cost reduction for only two rms on the market, and this will always be pro table for the two rms. This results stands in sharp contrast to the multilateral case, where large alliances have no incentive to include new members, as the inclusion of a new member implies that the newcomer forms simultaneous links to all the rms in the alliance. It is not di cult to see that the e ect of the formation of a bilateral alliance is qualitatively similar in Cournot and Bertrand models with di erentiated commodities. Proposition 9 Consider the formation of bilateral alliances in a Cournot or Bertrand model with di erentiated commodities. The only stable network is the complete network. Proof. See Goyal and Joshi (2000). 5 Bibliographical Notes The study of collusion and cartel formation has a long history in Industrial Organization. Stigler (1950) was probably the rst to point out that free-riding incentives may prevent the formation of a cartel. Studies of cartel formation in Cournot oligopoly include Salant, Switzer and Reynolds (1983), who were the rst to de ne the minimal pro table cartel size, and d Aspremont et al. (1983) who were the rst to apply a formal model of coalition formation to the study of cartels. In particular, the simultaneous game of formation of a single coalition used in this paper is due to d Aspremont et al. (1983). Deneckere and Davidson (1985) study the case of a Bertrand oligopoly with di erentiated commodities. Donsimoni (1985) studies cartel formation with heterogenous cost functions, Donsimoni, Economides and Polemarchakis (1986) analyze the stability of cartels and Perry and Porter (1985) consider a model with conjectural variations. More recently, Bloch (1996) analyzes the sequential game of coalition formation presented in this paper. Thoron (1998) studies the endogenous formation of coalition-proof stable cartels. The analysis of cost-reducing alliances can be traced back to the literature on research joint ventures initiated by Katz (1986) and d Aspremont and Jacquemin (1988) in the duopoly case. Bloch (1995) considers an extension of the model to an oligopoly and analyzes the outcome of the sequential game of coalition formation presented here. In the same spirit, Yi (1998) and Yi and Shin (2000) propose a general model of cost reducing alliances and research joint ventures. Morasch (2000) studies the formation of alliances which commit to transfer schemes among their members. Models of endogenous coalition formation have been proposed in the game theoretic literature ever since von Neumann and Morgenstern (1944) s pioneering book. In games characterized by externalities across coalitions, Aumann and 15

Drèze (1974) and Shenoy (1979) were the rst to describe and analyze cooperative solution concepts. In an in uential paper, Hart and Kurz (1983) propose di erent models of stability of coalition structures which rely on di erent assumptions on the behavior of external players. Yi (1997) proposes a complete study of games with externalities across coalitions based on di erent models of coalition formation. Bloch (1996) and Ray and Vohra (1999) were the rst to analyze the extensive form game of coalitional bargaining with a xed rule of order described in this paper. Montero (2000) proposes a related extensive form, where the choice of proposers at every period is random. The study of stable networks in economics started with Jackson and Wolinksy (1996). The de nition of pairwise stability is due to them. Qin (1996), Dutta et al. (1998) and Bala and Goyal (2000) study noncooperative games of link formation where every player announces the set of links she wants to form. The analysis of these games is usually very complex, and di cult to apply to models in Industrial Organization. The formation of bilateral alliances is studied in Goyal and Joshi (2000) and Goyal and Moraga (2001). Belle amme and Bloch (2001) introduce the model of market sharing agreements and study collusive networks. Most of the analyses of cartels and network formation in Industrial Organization are done under the assumption that rms are ex ante symmetric. While this symmetry assumption is clearly unrealistic, models of asymmetric rms are often intractable. The existing studies either consider a very small number of rms or assume a speci c form of asymmetry. For example, Belle amme (1998) considers the formation of two alliances among asymmetric rms, and Belle- amme (2000) fully analyzes cost reducing alliances among three asymmetric rms. Fauli-Oller (2000) considers the formation of cartels among asymmetric rms, which can either have a high or a low production cost. 6 References References [1] D Aspremont, C., A. Jacquemin, J.J. Gabszewicz and J. Weymark, «On the Stability of Collusive Price Leadership,» Canadian Journal of Economics 16 (1983), 17-25. [2] D Aspremont, C. and A. Jacquemin, «Cooperative and Noncooperative R&D in a Duopoly with Spillovers,» American Economic Review 78 (1988), 1133-1137 [3] Aumann, R. and J. Drèze, «Cooperative Games with Coalition Structures,» International Journal of Game Theory 3 (1974), 217-237. [4] Bala, V. and S. Goyal «A Noncooperative Model of Network Formation,» Econometrica 68 (2000), 1181-1229. 16

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[22] Qin, C.Z. «Endogenous Formation of Cooperation Structures,» Journal of Economic Theory 69 (1996), 218-226. [23] Ray, D. and R. Vohra, «A Theory of Endogenous Coalition Structures,» Games and Economic Behavior 26(1999), 286-336. [24] Salant, S., S. Swizer and J. Reynolds (1983), «Losses from Horizontal Mergers : The E ects of an Exogenous Change in Industry Structure on Cournot Nash Equilibrium», Quarterly Journal of Economics 98 (1983), 185-199. [25] Shenoy, P. «On Coalition Formation: A Game Theoretic Approach,» International Journal of Game Theory 8 (1979), 133-164. [26] Stigler, J. G. «Monopoly and Oligopoly by Merger,» American Economic Review 40 (1950), 23-34. [27] Thoron, S. «Formation of a Coalition Proof Stable Cartel», Canadian Journal of Economics 31 (1998), 63-76. [28] Van Neumann, J. and O. Morgenstern, Theory of Games and Economic Behavior, Princeton, Princeton University Press, 1944. [29] Yi, S.S. «Endogenous Formation of Joint Ventures with E ciency Gains,» Rand Journal of Economics 29(1998), 610-631. [30] Yi, S.S. «Stable Coalition Structures with Externalities,» Games and Economic Behavior 20 (1997), 201-237. [31] Yi, S.S. and H. Shin «Endogenous Formation of Research Coalitions with Spillovers,» International Journal of Industrial Organization 18 (2000), 229-256. 18