Coalitions and networks in industrial organization

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Coalitions and Networks in IndustrialOrganization¤

Francis Blochy

August 30, 2001

Abstract

This paper provides a selective survey of recent approaches to coali-tion and network formation in Industrial Organization, and o¤ers a uni…edframework in which the di¤erent approaches can be compared. We focuson two extreme forms of cooperation – collusive agreements and cost-reducing alliances. We show that bilateral negotiations yield higher levelsof cooperation than multilateral agreements, that the formation of a carteldepends on the sequentiality of the procedure of coalition formation, andthat the size of alliances depends on the membership rule. JEL Classi-…cation Number: D43, L13. Keywords: strategic alliances, cartels,networks, coalitions.

1 IntroductionRecent years have witnessed a surge of interest in the formation of coalitionsand networks in Industrial Organization. Most of this interest stems from theemergence of new forms of cooperation and competition between …rms. The de-velopment of strategic alliances, the acceleration in the creation of joint venturesand joint production and research facilities have given rise to a new strategicenvironment in which …rms cooperate in some domains and compete in oth-ers. At the same time, new approaches have been developed in game theory toanalyze the endogenous formation of coalitions and networks, providing simpletools 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 of recent ap-proaches to coalition and network formation in Industrial Organization, and too¤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

¤I am grateful to the guest editor, Rabah Amir and to an anonymous referee for helpfulcomments on an earlier draft.

yGREQAM and Ecole Superieure de Mecanique de Marseille. Address: 2 rue de la Charite,13002 Marseille, FRANCE. Email: [email protected]

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large, this paper focuses on two extreme forms of cooperation – collusive agree-ments and cost-reducing alliances. In those two simple models, we analyze theformation of groups and networks of …rms under various institutional arrange-ments.

Collusive agreements cover cartels choosing quotas (as in many commoditymarkets), price-…xing agreements, bidding rings as well as market sharing orexclusive territories agreements. The striking feature of collusive agreements,already pointed out by Stigler (1950), is that the formation of a cartel is apublic good, inducing positive externalities on …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 thatthis conclusion can only be reached under some institutional procedures of groupand network formation, but that other procedures may lead to the formation ofstable coalitions.

Cost-reducing alliances have recently been the focus of attention of manyindustrial economists and policy-makers. These alliances may pursue di¤erentobjectives, such as the development of new products and processes, the de…nitionof common standards or the joint use of facilities. As opposed to collusiveagreements, cost-reducing alliances typically induce negative externalities onother …rms in the industry. Hence, the analysis of the formation of these alliancesusually leads to very di¤erent conclusions than the study of the stability ofcartels.

By considering only the two pure cases of collusive groups and cost-reducingalliances, we only cover two extreme points on the spectrum of cooperativeagreements between …rms. It is widely believed that most cooperative agree-ments embody both a collusive and a cost-reducing aspect. For example, typi-cal collusive agreements, such as mergers, are often justi…ed by the existence ofsynergies and e¢ciency gains between the merging …rms. On the other hand,cost-reducing agreements such as the formation of research joint ventures of-ten provide a way for …rms to exchange information and collude in the market.Many of the results obtained in the two polar cases that we study should thusbe considered with caution before they can be applied to real cooperative agree-ments.

We should also note that there are a number of cooperative agreementsthat are not covered in our analysis. We only consider horizontal agreements,and do not discuss vertical arrangements or contracts between upstream anddownstream …rms. Hence, licensing agreements are not discussed in the paper,as they involve a vertical relation between licensor and licensee. (On the otherhand, cross-licensing agreements are covered by our analysis, as they involveeither a reduction in cost, or an attempt to share markets between horizontalcompetitors.)

As we only consider two very simple models of cooperation between …rms, weare able to contrast the outcomes of various institutional rules of coalition andnetwork formation. This approach enables us to answer the following questions,which are rarely addressed in the literature on cooperation between …rms. Iscooperation more prevalent or more wide-spread when …rms sign multilateral

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or bilateral agreements? Is the level of cooperation increased when cooperativeagreements are open to all …rms or restricted to the original signatories? Howdoes the group formed depend on the sequentiality of the procedure of coalitionformation? Should coalitions be formed on an industry-wide basis or shouldcompeting coalitions be allowed to form?

The rest of the paper is organized as follows. Section 2 analyzes the formationof multilateral agreements under two di¤erent institutional procedures: a modelof open membership and a sequential model of coalition formation. We presentand contrast the results obtained for collusive agreements and cost-reducingalliances. Section 3 studies the formation of bilateral agreements, both in thecollusive and cost-reduction cases. Section 4 contains bibliographical referencesand concluding comments.

2 Formation of Multilateral AgreementsIn this Section, we study the formation of coalitions of …rms which sign a mul-tilateral agreement to collude or form a strategic alliance. We let N denotethe set of …rms active on the market. A coalition is a nonempty subset S ofN . A coalition structure B is a partition of the set of …rms into nonempty dis-joint coalitions. A typical aspect of the formation of coalitions on oligopolisticmarkets is that each …rm’s pro…t is a function of the whole coalition structureformed on the market. For example, the pro…t of an independent …rm dependson the sizes of the cartels formed on the market ; similarly, the pro…t of analliance member depends on the sizes of the competing alliances. As there arewidespread externalities among coalitions, we cannot use the tools developed incooperative game theory to analyze the stability of a coalition structure, butneed to develop new tools to determine endogenously the coalitions formed inequilibrium. We present here two di¤erent models of coalition formation, re-cently proposed in the literature, to analyze the formation of coalitions in thepresence of externalities.Game 1: Formation of a Single Coalition

The …rst model we propose is a very simple model, which will be used to in-troduce the problems of multilateral externalities, and will serve as a benchmarkfor comparison with the next model of coalition formation. In game 1, …rmsform a single coalition, which can be interpreted as an industry-wide cartel ora trade association. The only decision that …rms make is whether they wantto participate in the coalition. We implicitly assume that membership in thecoalition is open: …rms cannot exclude other …rms from the coalition.

We consider a simultaneous game, where all …rms announce whether theywant to join the coalition. The strategy set of the …rm is thus the pair f0; 1gand the coalition C is formed of all the agents who have announced 1. Thepayo¤s of the agents correspond to the pro…ts they make, given the formationof the coalition C. We recall that a Nash equilibrium is a strategy pro…le such

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that no …rm has an incentive to deviate unilaterally given the strategy choiceof the other …rms.Game 2: Sequential Formation of Coalitions

In the second game, we consider the more di¢cult problem of endogenousformation of multiple coalitions. When multiple coalitions can be formed, andeach …rm’s pro…t depends on the entire coalition structure, …rms must anticipatethe reaction of other …rms when choosing whether to enter a coalition. In orderto take into account this forward-looking behavior, we analyze here the equilibriaof a sequential game of coalition formation, where each …rm takes into accountthe e¤ect of its decision on the behavior of …rms choosing actions subsequentlyin the game.

More precisely, we consider the following extension of games of coalitionalbargaining. At each period t, one …rm is chosen to make a proposal (a coalitionto which it belongs), and all the prospective members of the coalition respond inturn to the proposal. If the proposal is accepted by all, the coalition is formedand another …rm is designated to make a proposal at t + 1 ; if some of the…rms reject the proposal, the coalition is not formed, and the …rst …rm to rejectthe o¤er makes a countero¤er at period t + 1. The identity of the di¤erentproposers and the order of response are given by an exogenous rule of order.There is no discounting in the game but all players receive a zero payo¤ in caseof an in…nite play. As the game is a sequential game of complete informationand 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 struc-tures generated by stationary perfect equilibria can also be obtained by ana-lyzing the following simple …nite game. The …rst …rm announces an integer k1,corresponding to the size of the coalition she wants to see formed, …rm k1 + 1announces an integer k2, etc.;, until the total number n of …rms is exhausted.An equilibrium of the …nite game determines a sequence of integers adding upto n, which completely characterizes the coalition structure as all …rms are exante identical.

Games 1 and 2 describe two speci…c procedures of coalition formation, andour analysis clearly depends on the details of the procedures. Other modelsof endogenous coalition formation have been proposed, but they often rely onad hoc assumptions on the outcome functions of the game (for example, in asimultaneous model of coalition formation, what happens if some but not allmembers of a coalition announce that they want to form a coalition? Willa subcoalition be formed with those players who agree to get together, or nocoalition be formed?). Notwithstanding the wide range of models of coalitionformation that can be constructed, we believe that games 1 and 2 cover twosimple, natural procedures of coalition formation. They enable us to discuss thedi¤erences between games of open (game 1) and exclusive (game 2) member-ships, and between simultaneous (game 1) and sequential (game 2) proceduresof coalition formation.

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2.1 Collusion

We analyze the formation of coalitions and networks in a standard oligopolymodel with di¤erentiated products. In order to keep the analysis tractable, weassume that demand is linear, and let ¯ denote the common degree of di¤eren-tiation of the products. We then have

pi = ® ¡ qi ¡ ¯Xj 6=i

qj

where pi denotes the price of good i and qj the quantity sold of good j. Alter-natively, we can write the demand system as

qi = ° ¡ ´pi + »Xj 6=i

pj

where ° = ®1+(n¡1)¯ ; ´ = (1+(n¡2)¯)

(1¡¯)(1+(n¡1)¯) and » = ¯(1¡¯)(1+(n¡1)¯) .

We assume 0 · ¯ · 1 and let ci denote the constant marginal cost of …rmi. We consider two di¤erent models of interaction: a Cournot game where…rms set quantities and a Bertrand game where they choose prices. Standardcomputations show that equilibrium quantities in the Cournot game are givenby

q¤i =

®

2 + ¯(n ¡ 1)¡ ci(2 + ¯n)

(2 ¡ ¯)(2 + ¯(n ¡ 1))+

¯P

j 6=i cj

(2 ¡ ¯)(2 + ¯(n ¡ 1))

In the Bertrand game, the equilibrium prices are obtained as

p¤i =

°

(2´ ¡ »(n ¡ 1))+

´ci(2´ ¡ n»)

(2´ ¡ »)(2´ ¡ »(n ¡ 1))¡ ´»

Pj 6=i cj

(2´ ¡ »)(2´ ¡ »(n ¡ 1)):

We interpret collusion as the coordination of …rms’ strategies, in order toreduce competition and increase pro…ts on the market. Collusion can take manyforms, ranging from explicit price and quota rules in cartels such as OPEC, toimplicit agreements to share markets or abide by price standards. To keep theanalysis simple, we suppose that in the collusion case, all …rms have identicalcosts and we normalize c = 0. 1 In the multilateral setting, we focus on theformation of cartels on the market, and use the notation K for a cartel. Inside acartel, …rms choose quantities or prices in order to maximize the joint pro…t ofcartel members. As …rms are identical ex ante, we also assume that inside eachcartel, the division of pro…ts is equitable. To …x notations, for any coalition

1 We will discuss in the Conclusion di¤erent attempts to study cartel formation with het-erogeneous …rms.

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structure B = fK1; K2; :; Kj ::Kbg, we let b denote the number of cartels in Band kj the number of …rms in cartel Kj . As …rms are ex ante identical, thenumbers b and kj are su¢cient parameters to describe the cartel structure.

Each cartel chooses the quantities (or prices) in order to maximize the totalcartel pro…t, taking as given the choice of other cartels.2 Formally, the pro…t ofcartel Kj is written as

¦j =X

i2Kj

(® ¡ qi ¡ ¯X

j 6=i;j2Kj

qj ¡ ¯X

k2Kk;k 6=j

qk)qi

When products are homogeneous (¯ = 1), we can explicitly compute theequilibrium quantity of a …rm i in cartel Kj as q¤

i = ®(b+1)kj

, resulting in equi-librium pro…ts

¼¤i =

®2

(b + 1)2kj:

The expression for pro…ts in the homogeneous Cournot case embodies most ofthe crucial features of collusion on oligopolistic markets. Notice …rst that pro…tsare higher when the cartel structure (as measured by the number of cartels) ismore concentrated. This is the positive externality induced by collusion: insidea cartel, …rms reduce their output, raising the price and resulting in a higherpro…t for all the external …rms. Second, it is clear that …rms belonging to largercartels have a lower pro…t. This is the free-riding phenomenon associated tocartel formation: insiders typically incur the cost of output reduction whereasoutsiders bene…t from the raise in price without paying any cost.

When products are di¤erentiated (¯ < 1), the computation of equilibriumquantities in the general case is intractable. Instead, we can compute equilib-rium quantities and pro…ts in the single cartel case, where one cartel of size k isformed on the market, and all other …rms are independent. Let q¤

K denote theequilibrium quantity of a cartel member and q¤

I the equilibrium quantity of anindependent …rm. We have

q¤K =

®(2 ¡ ¯)

4 + 2¯(k + n ¡ 3) ¡ ¯2(k2 + 2n ¡ kn ¡ 2);

q¤I =

®(2 + ¯(k ¡ 2))

4 + 2¯(k + n ¡ 3) ¡ ¯2(k2 + 2n ¡ kn ¡ 2)

yielding equilibrium pro…ts

¼CK =

®2(2 ¡ ¯)(2 + ¯(2k ¡ 3) + ¯2(1 ¡ k))

(4 + 2¯(k + n ¡ 3) ¡ ¯2(k2 + 2n ¡ kn ¡ 2))2;

¼CI =

®2(2 + ¯(k ¡ 2))2

(4 + 2¯(k + n ¡ 3) ¡ ¯2(k2 + 2n ¡ kn ¡ 2))2:

2 The equilibrium concept is thus a ”hybrid” equilibrium, including both cooperative (insidethe cartel) and noncooperative (across cartels) features.

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The study of cartel formation in a Bertrand model with di¤erentiated com-modities can also only be done in the case of a single cartel of size k facing(n ¡ k) independent …rms. Equilibrium prices chosen by …rms inside the carteland outside the cartel are given by

p¤K =

°(2´ + »)

4´2 ¡ 2´»(k + n ¡ 3) + »2(2 ¡ k2 ¡ 2n + kn);

p¤I =

°(2´ ¡ »(k ¡ 2))

4´2 ¡ 2´»(k + n ¡ 3) + »2(2 ¡ k2 ¡ 2n + kn)

yielding equilibrium pro…ts

¼BK =

°2(2´ + »)(2´2 ¡ ´»(2k ¡ 3) + »2(1 ¡ k))

(4´2 ¡ 2´»(k + n ¡ 3) + »2(2 ¡ k2 ¡ 2n + kn))2;

¼BI =

°2´((2´ ¡ »(k ¡ 2))2

(4´2 ¡ 2´»(k + n ¡ 3) + »2(2 ¡ k2 ¡ 2n + kn))2:

We now use the expressions for equilibrium pro…ts to compute the coalitionstructures formed on the market. We start by analyzing game 1, where a singlecartel is formed on the market.

Proposition 1 Consider the formation of a single cartel on a Cournot marketwith homogeneous goods. If n ¸ 3, in the unique Nash equilibrium of the simul-taneous game of cartel formation, all …rms remain independent. If n = 2, inthe unique Nash equilibrium of the simultaneous game of cartel formation, thetwo …rms form a cartel.

Proof. Suppose that a cartel of size k is formed. Each cartel memberreceives a pro…t ¼ = ®2

k(n¡k+2)2 . If a …rm leaves the cartel, it receives a pro…t

¼0 = ®2

(n¡k+3)2 . It is easy to see that, as long as n ¸ 3, for any 1 < k ·n; k(n ¡ k + 2)2 > (n ¡ k + 3)2, so ¼0 > ¼. This establishes that the uniqueequilibrium of the game is for all …rms to remain independent. If n = 2, thetwo …rms have an incentive to form a cartel.

Proposition 1 illustrates the well-known free-riding problem associated withthe formation of a cartel. Since external …rms bene…t from the increase in pricedue to the formation of the cartel, all …rms prefer to stay outside the cartel andfree ride on the public good provided by cartel members. In a linear Cournotmarket, this incentive is so strong that no cartel can be formed.

When products are di¤erentiated, computations become intractable, andwe can only obtain numerical estimates on the sizes of the cartels. Figures 1and 2 graph the equilibrium pro…ts of insiders and outsiders in a Cournot andBertrand game, setting ¯ = 1=2, ® = 1 and n = 10: The dashed line represents

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2 4 6 8 10k

0.03

0.04

0.05

0.06

0.07

pi

Figure 1:

the pro…t of cartel members for values of k ranging from 1 to 10, whereas thesolid line represents the pro…ts of outsiders.

From the …gures, it appears that both the positive externalities and free-riding aspects of cartel formation are present in the two models. In both cases,pro…ts of outsiders are increasing in the size of the cartel, and always higher thanthe pro…ts of insiders. Pro…ts of cartel members are U-shaped in the Cournotcase, re‡ecting the fact that cartel members su¤er the cost of reducing quantitiesto increase prices, and this cost is proportionately higher for lower cartel sizes.On the other hand, in the Bertrand case, pro…ts of insiders and outsiders arealways increasing in the size of the cartel.

We next turn to the sequential model of coalition formation. As compu-tations for models with di¤erentiated products are intractable, we restrict ourattention to a Cournot oligopoly with homogeneous products.

Proposition 2 (Bloch, 1996)Consider a Cournot oligopoly with homogeneousproducts. In the unique equilibrium of the sequential game of cartel formation,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+52 :

Proposition 2 can easily be interpreted. From Figure 1, recall that the pro…tof a cartel member is a U shaped function. Hence, there exists a minimal cartel

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2 4 6 8 10k

0.03

0.04

0.05

0.06

0.07

pi

Figure 2:

size for which …rms prefer to form a cartel than remain independent. Thisminimal pro…table cartel size is exactly equal to k in a linear Cournot modelwith homogeneous products. Hence, Proposition 2 shows that in a sequentialgame of cartel formation, the …rst …rms choose to remain independent and free-ride on the cartel formed by subsequent …rms. When the number of remainingmarket participants is exactly equal to the minimal pro…table cartel size, …rmswill choose to form a cartel, as it becomes impossible for a …rm to stay out andfree ride on the cartel formed by the others.

It is interesting to contrast the outcome of the sequential game of coalitionformation with that of the simultaneous game. In the simultaneous game, freeriding incentives prevent the formation of a cartel, as any …rm has an incentiveto defect. In the sequential game, …rms can commit to stay out of the cartel, andhence a cartel is formed in equilibrium. Notice furthermore that the minimalpro…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 cartelformed in equilibrium, and the issue of the formation of multiple competingcartels does not arise.

2.2 Strategic Alliances

Firms form alliances for a variety of reasons: to lobby for pieces of legisla-tion, agree on common standards, develop new products and processes, launchjoint marketing campaigns, produce common inputs, use jointly some facilities...While there is a large variety of types of strategic alliances, their objective al-ways seems to be to reduce the cost of production, or increase market demandfor alliance members. Hence, to model the role of alliances on a market, we

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suppose that …rms can bene…t from synergies which lead to a reduction in theconstant marginal cost of production (or equivalently to an increase in the scaleparameter of demand).

In the multilateral setting, we let A denote a structure of alliances withA = fA1; A2; :::; Arg. All …rms are assumed to be identical ex ante, so thatthe identity of …rms does not matter, and the sizes of the di¤erent alliances,a1; a2; :::; ar characterize the association structure A: We suppose, keeping theanalysis as simple as possible, that the constant marginal cost of production ofa …rm is linearly decreasing in the size of the alliance it belongs to. Formally, if…rm i belongs to association Aj , its marginal cost of production is given by

ci = ¸ ¡ ¹aj

Hence, each alliance structure determines a vector of marginal costs for thedi¤erent …rms. These di¤erences in marginal costs then lead to di¤erences inequilibrium quantities, prices and pro…ts.3 In the Cournot setting, the equilib-rium pro…t of a …rm i belonging to association Aj is given by:

¼Ci = [

® ¡ ¸

(2 + (n ¡ 1)¯)+

¹aj

(2 ¡ ¯)¡ ¯¹

Pa2

k

(2 ¡ ¯)(2 + (n ¡ 1)¯)]2

and in the Bertrand setting

¼Bi = ´[

´¸ + °

(2´ ¡ (n ¡ 1)»)¡ ´¹aj

2´ + »¡ ´»¹

Pa2

k

(2´ ¡ (n ¡ 1)»)(2´ + »)¡ ¸ + ¹aj ]2:

Two important features of alliance formation can be deduced from the pro…texpressions given above. First, the formation of an alliance induces negative ex-ternalities on outsiders. Members of an alliance enjoy a lower marginal cost, andare thus more aggressive on the market, reducing the pro…ts of their competi-tors. Second, members of a larger alliance have higher pro…ts, as they bene…tfrom a cost advantage with respect to their competitors. Using game 1, we …rstshow that these incentives imply that all …rms belong to a single alliance whenmembership to the alliance is open.

Proposition 3 Consider the formation of a cost-reducing alliance. In theunique equilibrium of the simultaneous game of alliance formation, all the …rmsform a single alliance.

Proof. Consider an association structure with an association A and inde-pendent …rms. We verify that any independent …rm has an incentive to jointhe association. In the Cournot case, computations show that by joining theassociation, the …rm increases its equilibrium quantity by a¹(2+¯(n¡3))

(2¡¯)(2+(n¡1)¯) . As

3 We assume throughout that di¤erences in marginal costs are not high enough to drivesome …rms out of the market. This assumption amounts to placing an upper bound on theparameter ¹:

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equilibrium pro…ts are increasing in equilibrium quantities, the independent …rmhas an incentive to join the association. In the Bertrand case, when it joins theassociation, the independent …rm experiences an increase in its mark-up, p¡c, of¹a(2´2¡(n¡1)»2¡(n¡1)´»

(2´¡(n¡1)»)(2´+») : As equilibrium pro…ts are increasing in the equilibriummark-up, this deviation is pro…table.

Proposition 3 highlights the fact that …rms always have an incentive to join acost-reducing alliance. There are both direct and strategic bene…ts from joiningthe alliance. The reduction in cost directly a¤ects the …rms’ pro…ts but alsoconfers a strategic advantage over the …rms who do not belong to the alliance.We now turn to the case of sequential alliance formation, where membership toan alliance is exclusive.4

Proposition 4 (Bloch, 1995) Consider a Cournot oligopoly with homogeneousproducts. In the unique equilibrium of the sequential game of alliance formation,the …rst a1 …rms form an alliance, and the last (n ¡ a1) …rms form a competingalliance. The integer a1 is the integer closest to 3n+1

4 :

To understand the result of Proposition 4, note that …rms may have an in-centive 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 wouldbene…t from a huge cost reduction, but only contribute a small cost reductionto the standing alliance members. Hence, the pro…t of an alliance member is aconcave function of the alliance size, and in the linear Cournot case, it reaches apeak at n+1

2 . This implies that the …rst alliance formed will contain at least n+12

members. In fact, the …rst alliance will be even larger because its members an-ticipate that all the subsequent players will also form an alliance, and they willenlarge 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 ofthe members of the industry and the second one quarter.

Comparing the outcomes of the sequential and simultaneous games of coali-tion formation, two striking di¤erences appear between the models. First, inthe sequential game, …rms can limit membership and exclude other …rms fromthe alliance. Second, in the sequential game, …rms anticipate that a competingassociation may be formed by the other players, and choose the size of the …rstassociation accordingly.

3 Formation of Bilateral Agreements

We now consider the formation of bilateral agreements, resulting in the for-mation of a collusive network or a network of collaboration between …rms. A

4 We only consider the case of a Cournot model with homogeneous products. The study ofoligopolies with di¤erentiated commodities reaches similar conclusions.

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network is represented by a graph g on the set N of …rms. A graph is a setof pairwise links, denoted ij between …rms i and j. Formally, a graph can berepresented by a symmetric and re‡exive binary relation on the set N . Thecomplexity of networks makes it harder to characterize the formation of a net-work as the outcome of a noncooperative of link formation. We follow here adi¤erent approach. A network g is called pairwise stable if no pair of …rms hasan incentive to form a new link, and no …rms has an incentive to unilaterallydestroy an existing link. Formally, following obvious notations, network g isstable 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’tallow …rms to form more than one link nor to destroy di¤erent links at once.)However, pairwise stable networks remain an important benchmark, especiallywhen as in the case of collusive networks and networks of alliances, few networkspass the test of pairwise stability.

3.1 Collusive Networks

The interpretation of collusion in a bilateral setting is not immediate. Clearly,…rms cannot sign bilateral agreements on production quotas or market pricesindependently 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 thusconsider a di¤erent form of collusion in bilateral settings, namely the signatureof bilateral market sharing agreements. Suppose that …rms are initially presenton di¤erent geographical markets. (We assume that there is exactly one …rmon each market.) By signing market sharing agreements, …rms commit not toenter each other’s market. Hence, if ij 2 g, …rm i withdraws from market j and…rm j withdraws from market i. For any graph g on the set N of …rms, we cancompute ni(g), the number of …rms present on market i. (Alternatively, if di(g)denotes the degree of vertex i in the graph, ni(g) = n ¡ di(g) + 1.) Supposingthat all …rms are identical, we let ¼(ni(g)) denote the pro…t made by each …rmon market i. For any …rm i, the total pro…t she makes is the sum of pro…ts oneach market where it is present, i.e.

¦i(g) = ¼(ni(g)) +X

j;ij =2g

¼(nj(g))

Assuming again that the marginal cost of production is zero, we obtain, inthe Cournot setting

¦i(g)C =®2

(2 + ¯(ni(g) ¡ 1))2+

Xj;ij =2g

®2

(2 + ¯(nj(g) ¡ 1))2

12

and in the Bertrand setting

¦i(g)B =´°2

(2´ ¡ »(ni(g) ¡ 1))2+

Xj;ij =2g

´°2

(2´ ¡ »(nj(g) ¡ 1))2:

When deciding whether to form bilateral links, …rms must thus trade o¤ twoe¤ects. On the one hand, the formation of a link enables a …rm to increase thepro…t it makes on its own market ; on the other hand, it forces it to forego thepro…t it makes on the market of the other …rm. The balance between those twoe¤ects leads to the following characterization of stable collusive networks.

Proposition 5 Consider the formation of market sharing agreements in a Cournotoligopoly with homogenous goods. If n ¸ 3;there are exactly two pairwise stablenetworks: the empty network and the complete network. If n = 2, the completenetwork is the only stable network.

Proof. To show that the empty network is stable, compute the bene…t thattwo …rms would have from forming a link

¦i(g + ij) ¡ ¦i(g) =®2

n2¡ 2

®2

(n + 1)2=

®2(¡n2 + 2n + 1)

n2(n + 1)2

For n ¸ 3; ¦i(g + ij) ¡ ¦i(g) < 0 and the empty network is stable. Nowsuppose 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

(ni(g) + 1)2>

®2

(ni(g) + 2)2+

®2

(nj(g) + 1)2

®2

(nj(g) + 1)2>

®2

(nj(g) + 2)2+

®2

(ni(g) + 1)2

which can only be satis…ed if ni(g) = nj(g) = 1; showing that if a nonemptynetwork is formed, it must be the complete network, and that the completenetwork 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 ownmarket. The complete network is the only nonempty stable network, as any …rmhas an incentive to defect from a network of smaller size. Hence, the free-ridingincentives remain strong, but do not prevent the formation of a fully collusivenetwork. When products are di¤erentiated, competition may be less strong andthe incentive to sign market sharing agreements is reduced.

13

Proposition 6 (Belle‡amme and Bloch, 2001) Consider the formation of mar-ket sharing agreements in a Cournot oligopoly with di¤erentiated products, ® =1; 1 ¸ ¯ > 0:For ¯ > 0:828, there exist two stable networks: the empty networkand the complete network. For ¯ · 0:828, the only stable network is the emptynetwork.

In the case of Bertrand competition with di¤erentiated products, di¤erentnetwork architectures may arise, as the following Proposition demonstrates. Thethreshold values for ¯ can only be obtained by numerical computations.

Proposition 7 (Belle‡amme and Bloch, 2001) Consider the formation of mar-ket sharing agreements in a Bertrand oligopoly with di¤erentiated products,® = 1; 1 ¸ ¯ > 0:For ¯ > 0:715, there exist three stable networks: the emptynetwork , the complete network and a network with one isolated …rm and a com-plete component of n ¡ 1 …rms. For 0:611 < ¯ · 0:715, there exist two stablenetworks: the empty network and the complete network. For ¯ · 0:611 , theonly stable network is the empty network.

Proposition 7 re‡ects the fact that price competition results in lower equilib-rium 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 thecomplete and empty networks. As products become more and more di¤erenti-ated, market competition is less …erce, and market sharing agreements becomeless stable.

3.2 Networks of collaboration

We now analyze networks of collaboration, where …rms sign bilateral agreementsto reduce their marginal cost of production (or, alternatively, increase theirscale parameter of demand). We suppose that the constant marginal cost ofproduction is linearly decreasing in the number of agreements signed by each…rm. For any graph g, we let di(g) denote the degree of vertex i, namely thenumber of agreements signed by …rm i. We then have

ci = ¸ ¡ ¹di(g);

and the equilibrium pro…ts are given by

¦Ci = [

® ¡ ¸

(2 + (n ¡ 1)¯)+

¹di(g)

(2 ¡ ¯)¡ ¯¹

Pdj(g)

(2 ¡ ¯)(2 + (n ¡ 1)¯)]2;

14

¦Bi = ´[

´¸ + °

(2´ ¡ (n ¡ 1)»)¡ ´¹di(g)

2´ + »¡ ´»¹

Pdj(g)

(2´ ¡ (n ¡ 1)»)(2´ + »)¡ ¸ + ¹di(g)]2:

In the case of networks of collaboration, the formation of a link betweentwo …rms induces a simultaneous decrease in the marginal cost of production ofthe two …rms. The following Proposition shows that the formation of a link isalways pro…table for the two …rms, so that a stable network is always complete.

Proposition 8 (Goyal and Joshi, 2000) Consider the formation of bilateralalliances. The only stable network is the complete network.

Proof. In a Cournot oligopoly, consider the e¤ect of the formation of a linkto player j on the equilibrium quantity of …rm i:

¢qi =(2 + (n ¡ 2)¯)¹

(2 ¡ ¯)(2 + (n ¡ 1)¯> 0:

As equilibrium pro…ts are increasing in equilibrium quantities, any link ispro…table, and the only stable network is the complete network. Similarly, in aBertrand oligopoly, consider the e¤ect of the formation of a link to player j onthe equilibrium mark-up of …rm i;

¢Mi =¹[(´ + »)(2´ ¡ (n ¡ 1)») ¡ ´»]

(2´ ¡ (n ¡ 1)»)(2´ + »)> 0:

As equilibrium pro…ts are increasing in equilibrium markups, any link ispro…table, and the only stable network is the complete network.

Proposition 8 shows that a …rms always has an incentive to form a bilateralalliance. This results stands in sharp contrast to the multilateral case, wherelarge alliances have no incentive to include new members, as the inclusion ofa 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 formationof a bilateral alliance is qualitatively similar in Cournot and Bertrand modelswith di¤erentiated commodities.

Proposition 9 (Goyal and Joshi, 2000) Consider the formation of bilateralalliances in a Cournot or Bertrand model with di¤erentiated commodities. Theonly stable network is the complete network.

15

4 Conclusion and Bibliographical Notes

In this paper, we have analyzed the formation of coalitions and networks of…rms in two simple models of cooperation: collusive groups and cost reducingalliances. We have characterized the level of cooperation under three alternativemodels: an open membership model of cartel formation, a sequential model ofcoalition formation, and a model of stable networks. Our results show that thelevel of cooperation will often be higher when negotiations are bilateral. Stablecollusive networks typically embrace all the …rms in the industry, whereas car-tels either fail to form (in the open membership game) or only contain a fractionof the …rms (in the sequential formation game). Similarly, networks of collabo-ration encompass all the …rms in the industry, whereas strategic alliances onlygroup a fraction of the …rms when exclusion is possible. Our analysis also shedslight on the di¤erence between sequential and simultaneous games of coalitionformation, and open and exclusive membership games. In the case of cartels,sequential procedures of coalition formation enable some …rms to commit to stayout of the cartel, and results in the formation of a collusive group ; instead, insimultaneous procedures, all cartels are unstable as every …rm has an incentiveto defect and free-ride on the decisions of other cartel members. In the caseof strategic alliances, the crucial feature of the coalition formation procedure isthe openness or exclusivity of membership. In models with open membership, alarge, e¢cient, alliance is formed whereas in models with exclusive membershipa fragmented, ine¢cient structure with multiple competing alliances emerges.

While this paper has no pretense to provide a complete survey of the litera-ture on cartels and alliances, we end by o¤ering some bibliographical notes on themost relevant research on coalition and network formation in oligopolies. Thestudy of collusion and cartel formation has a long history in Industrial Organiza-tion. Stigler (1950) was probably the …rst to point out that free-riding incentivesmay prevent the formation of a cartel. Studies of cartel formation in Cournotoligopoly include Salant, Switzer and Reynolds (1983), who were the …rst tode…ne the minimal pro…table cartel size, and Selten (1973) and d’Aspremont etal. (1983) who were the …rst to apply a formal model of coalition formation tothe study of cartels. In particular, the simultaneous game of formation of a sin-gle coalition used in this paper is due to d’Aspremont et al. (1983). Deneckereand Davidson (1985) study the case of a Bertrand oligopoly with di¤erentiatedcommodities. Donsimoni (1985) studies cartel formation with heterogenous costfunctions, Donsimoni, Economides and Polemarchakis (1986) analyze the sta-bility of cartels and Perry and Porter (1985) consider a model with conjecturalvariations. More recently, Bloch (1996) analyzes the sequential game of coali-tion formation presented in this paper. Thoron (1998) studies the endogenousformation of coalition-proof stable cartels. Nocke (1999) provides an interestinganalysis of cartel formation when …rms face capacity constraints.

The analysis of cost-reducing alliances can be traced back to the litera-ture on research joint ventures initiated by Katz (1986) and d’Aspremont andJacquemin (1988) in the duopoly case. Kamien, Muller and Zang (1992) pro-

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pose a di¤erent model of research joint ventures and apply it to general marketconditions. Kamien and Zang (1993) explicitly consider the formation of com-peting research joint ventures in oligopolies. Bloch (1995) analyzes the outcomeof 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 reducingalliances and research joint ventures. Morasch (2000) studies the formation ofalliances which commit to transfer schemes among their members. Amir (2000)tries to reconcile the approaches of d’Aspremont and Jacquemin and KamienMuller and Zang and provides a critique of the two models.

Models of endogenous coalition formation have been proposed in the gametheoretic literature ever since von Neumann and Morgenstern (1944)’s pioneer-ing book. In games characterized by externalities across coalitions, Aumann andDrèze (1974) and Shenoy (1979) were the …rst to describe and analyze cooper-ative solution concepts. In an in‡uential paper, Hart and Kurz (1983) proposedi¤erent models of stability of coalition structures which rely on di¤erent as-sumptions on the behavior of external players. Yi (1997) proposes a completestudy of games with externalities across coalitions based on di¤erent modelsof coalition formation. Bloch (1996) and Ray and Vohra (1999) were the …rstto analyze the extensive form game of coalitional bargaining with a …xed ruleof order described in this paper. Montero (2000) proposes a related extensiveform, where the choice of proposers at every period is random.

The study of stable networks in economics started with Jackson and Wolin-sky (1996). The de…nition of pairwise stability is due to them. Qin (1996),Dutta et al. (1998) and Bala and Goyal (2000) study noncooperative gamesof link formation where every player announces the set of links she wants toform. The analysis of these games is usually very complex, and di¢cult to ap-ply to models in Industrial Organization. The formation of bilateral alliances isstudied in Goyal and Joshi (2000) and Goyal and Moraga (2001). Belle‡ammeand Bloch (2001) introduce the model of market sharing agreements and studycollusive networks.

Most of the analyses of cartels and network formation in Industrial Organi-zation are done under the assumption that …rms are ex ante symmetric. Whilethis symmetry assumption is clearly unrealistic, models of asymmetric …rms areoften 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.

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Coalitions and networks in industrial organization

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