Coalitions and networks in oligopolies Francis Bloch∗ June 26, 2016

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Introduction

This chapter discusses the formation of coalitions and networks in oligopolies. It weaves together a literature in game theory on cooperation and a literature in Industrial Organization on the formation of groups of oligopolistic firms. The literature on coalitions in oligopolies started with Stigler ’s discussion of cartel instability in the 1950’s. It was particularly active in the late eighties and the nineties, spurred by new regulations on cooperative research, new merger policies and the emergence of ”co-opetition”, to use a phrase coined by Brandenburger and Nalebuff [10], as a new form of interaction where oligopolistic firms cooperate on some dimensions and compete on others. At the same time, a number of empirical studies on collaborative projects established the prevalence of new forms of cooperation across oligopolistic firms. From its inception, the literature on cooperation in oligopolies has made use of solution concepts derived from game theory. The initial solution concepts of cartel stability were borrowed from the literature on cooperative games, focusing on notions of internal and external stability dating back from von Neumann and Morgenstern [67]. Gradually, these cooperative solution concepts gave way to equilibrium outcomes of non-cooperative games – either simultaneous or sequential – providing strategic foundations for cooperation. More recently, the emphasis has been placed on bilateral rather than multilateral cooperation, leading to the development of new models of network formation, which replace models of coalition formation. Our discussion of cooperation in oligopolies starts with a brief presentation of the game-theoretic models used to predict the formation of coalitions and networks. We then consider two different forms of cooperation. We start by analyzing ∗ Universit´e Paris 1 and Paris School of Economics, 106-112 Boulevard de l’Hopital, 75647 Paris CEDEX 13, France, [email protected].

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collusion, discussing the formation of cartels and horizontal mergers in oligopolies. The last part of the chapter is devoted to the analysis of strategic alliances, which encompass both research joint ventures and information exchange platforms. The literature on cooperation in oligopolies covers a large fraction of theoretical research in industrial organization, and it is of course impossible to cover it all in one chapter. We have chosen to be very selective, restricting attention to theoretical models which aim at predicting whether cooperation will arise, and what are the sizes of groups and the architecture of networks which are likely to emerge among oligopolistic firms. By lack of space, we cannot cover the empirical literature on collusion and collaboration, the theoretical literature on supergames and tacit collusion, nor the more recent theoretical literature on the interaction between antitrust policies and cooperation decisions.

2

Models of coalition and network formation

In this Section, we review models of coalition and network formation developed in game theory, which have been applied to Industrial Organization. The models fall into three categories: simultaneous models of coalition formation which are played in one shot and where all players announce the coalitions they want to form, dynamic games of coalition formation, where coalitions are formed sequentially through extensive form games and models of network formation where pairs of players form bilateral links. We start with some simple notations. Let N = {1, 2., , , n} be a set of players with typical element i and 2N \ ∅ the set of all nonempty coalitions of players with typical element C. A partition π is a collection of coalitions which are disjoint, nonempty and cover the entire set N . We denote by Π the set of all possible partitions. We also consider the set G of all undirected networks over n players. An undirected network cam be identified with a symmetric n × n matrix of 0, 1 with gij = 1 if and only if there is a link between i and j. n(n−1) This formulation shows that there are 2 2 networks on the set of players. We let ij ∈ g denote the fact that players i and j are directly linked in network g.

2.1

Open membership games

In the open membership game, members cannot prevent other players from joining a coalition. In the simplest model considered by D’Aspremont et al. [22], all players select a strategy in S = {0, 1}. All players who announce 1 belong to the coalition, and all players who announce 0 remain independent. Hence C = {i|si = 1} and the partition formed is π = {C, {j}j ∈C / }. Following D’Aspremont et al. [22], we say that a coalition is internally stable if no player wants to leave the coalition 2

and externally stable if no player wants to join the coalition. A coalition which is internally and externally stable is called stable. Alternatively, we can define a stable coalition as the Nash equilibrium outcome of the open membership game. The cartel formation game of D’Aspremont et al. [22] only allows for one coalition to form. A natural extension of this game is the address game discussed by Yi [92]. Players choose addresses in the state S = {0, a1 , .., an }. All players who choose the same address aj form the coalition Cj . All players who choose 0 remain independent. The partition formed is thus π = {C1 , ..CJ , {k}k∈C / j ∀j }.

2.2

Exclusive membership games

In exclusive membership games, players announce the coalition they want to form and can thus prevent the entry of other members into the coalition. The earliest game of coalition formation was proposed by von Neumann and Morgenstern [67], (pp. 243-244). Each player i announces a coalition Ci to which she wants to belong. The outcome function assigns to any vector of announcements C1 , .., Cn , a coalition structure π as follows: C 6= {i} ∈ π if and only if, for all agents i ∈ C, Ci = S. A singleton i belongs to the coalition structure π if and only if (i) either Ci = {i} or Ci = C and there exist j ∈ C such that Cj 6= C. In this procedure, a coalition is formed if and only if all its members unanimously agree to form the coalition. This procedure was rediscovered by Hart and Kurz [45], who labeled it ‘model γ’. They contrast it with another procedure, labeled ‘model δ’, where unanimity is not required for a coalition to form. In the δ procedure, the outcome function assigns to any vector of announcements C1 , ..., Cn , a coalition structure π where: C ∈ π if and only if Ci = Cj ⊇ C for all i, j ∈ C. In other words, coalitions are formed by any subset C of players who coordinate and announce the same coalition Ci . In this procedure, the announcement serves to coordinate the actions of the players, and indicates what is the largest coalition that players are willing to form.

2.3

Bidding game

The bidding game of coalition formation was proposed by Kamien and Zang [51]. Every agent i submits a vector of bids, bij over all agents j in N . A bid bij for i 6= j is interpreted as the amount of money that agent i is willing to put to acquire the resources of agent j. The bid bii is interpreted as the asking price at which agent i is willing to sell her resources. Given a matrix B = [bji ] of nonnegative bids, one can assign the resources of every agent i either to another agent j or to agent i herself, if she remains a singleton. Formally, let

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S(i) = {j ∈ N, j 6= i, bji ≥ bki ∀k 6= j} denote the set of players other than i such that (i) the bid they offer is no smaller than the bid of any other player and (ii) the bid they offer is higher than the asking price. If S(i) is a singleton, the assignment of the resources of player i to the unique player in S(j) (and hence the formation of a coalition S containing {i, j}) is immediate. If S(i) is not a singleton, one needs to define an exogenous tiebreaking rule to assign the resources of player i to some member of S(i). As a result of this bidding procedure, resources of some players are bought by other players, resulting both in the formation of a coalition structure π and in transfers across players given by tji = bji and tij = −bji if player j acquires the resources of player i. Perez Castrillo [73] independently proposed a procedure of coalition formation which bears close a resemblance to Kamien and Zang [51]’s bidding game. The main difference is that Perez Castrillo introduces competitive outside players (the ”coalition developers”) who simultaneously bid for the resources of the players.

2.4

Sequential formation of coalitions

Sequential games of coalition formation are based on Rubinstein [82]’s model of alternative offers bargaining. As in Rubinstein [82]’s model, the representative model has an infinite horizon, players discount future payoffs, and at each period in time, one of the players (the proposer) makes an offer to other players (the respondents) who must approve or reject the proposal. Different variants of this scenario have been proposed. Chatterjee, Dutta, Ray and Sengupta [14] propose a rejector-proposer version. Players are ordered according to an exogenous protocol. At the initial stage, player 1 chooses a coalition C to which she belongs and a vector of payoffs for all members of C, xC satisfying P i∈C xi = v(C), where v(C) describes the coalitional surplus of the coalition C. Players in C then respond sequentially to the offer. If all accept the offer, the coalition C is formed, and the payoff vector xC is implemented. The first player in N \C is chosen as proposer with no lapse of time. If one of the players in C rejects the offer, one period elapses and the rejector becomes the proposer at the following period. Okada [71] analyzes a coalitional bargaining game where the proposer is selected at random after every rejection. In the context of coalition formation, payoffs depend on the entire coalition structure, and underlying gains from cooperation depend on the coalitions formed by other players. In this context, Bloch [6] proposes a coalitional bargaining game capturing this forward-looking behavior when the division of the surplus across coalition members is fixed. At any stage player i announces a coalition Ci that she 4

wants to form. If all players in Ci agree the coalition is formed and the next player is chosen to make a proposal. If one of the members of Ci rejects the proposal, she becomes the proposer next period. Consider symmetric games where payoffs only depend on the size distribution of coalitions. In that case, the equilibrium coalition structures of the infinite horizon bargaining game can be computed by using the following finite procedure. Let players be ordered exogenously. The first player announces an integer k1 , corresponding to the size of the coalition she wants to form. Player k1 + 1 then announces the size k2 of the second P coalition formed. The game ends when all players have formed coalitions, i.e. kt = n. While Bloch [6] assumes that the division rule of the surplus is fixed, Ray and Vohra [79] consider a model of coalitional bargaining with externalities, where the division of coalitional surplus is endogenous, and payoffs are represented by an underlying game in partition function form. Ray and Vohra [79] first establish the existence of stationary equilibria in mixed strategies, where the only source of mixing is the probabilistic choice of a coalition by each proposer. Their main theorem establishes an equivalence between equilibrium outcomes of the game and the result of a recursive algorithm. This algorithm, in four steps, characterizes equilibrium coalition structures for symmetric games. It can easily be implemented on computers and has been successfully applied in Ray and Vohra [80] to study the provision of pure public goods.

2.5

Successive formation of coalitions

In successive games of coalition formation, players meet in pairs and decide whether to merge. If the players agree on a merger, one of the players acquires the resources of the other, and forms a single entity which continues to take part in the process. In successive games of coalition formation, coalitions are thus formed by successive acquisition of the resources of the other players. Gul [44] proposed the first game of successive formation of coalitions and showed that the equilibrium payoff converges to the Shapley value of the underlying cooperative game. The set of active players in the game varies over time, as the resource of players are acquired by other players. At any period, a pair of active players is selected and one of the players is chosen at random to make a take-it-or-leave-it offer to acquire the resources of the other player. If the offer is accepted, the set of active players is reduced by one (the player whose resources have been acquired) and the process continues. If the offer is rejected, the set of active players does not change, one period of time elapses, and a new pair is chosen.

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2.6

Formation of networks

We now consider the formation of links in networks. Myerson [66] proposed a game of undirected network formation which is very similar to models γ and δ. Agents simultaneously announce the set of agents with whom they want to form links. Hence, a pure strategy in the game is a subset Ci ⊆ N \ {i} for every agent i. The formation of a link requires consent by both parties. Link ij is formed if and only if i ∈ Cj and j ∈ Ci . Given the typical indeterminacy of Nash equilibrium in models of undirected networks, it is not surprising that other equilibrium notions have been considered in the literature. Because it takes agreement of both players i and j to form the link ij, it is natural to consider coalitions of size two since this is the minimal departure from a purely non-cooperative equilibrium concept. Jackson and Wolinsky [49] specify a very weak notion of stability for networks. A network g is pairwise stable if for all i, j ∈ N , (i) Yi (g) ≥ Yi (g − ij) (ii) Yi (g + ij) > Yi (g) implies that Yj (g + ij) < Yj (g). This concept of stability is very weak because it restricts deviations to change only one link at a time - either some agent can delete a link or a pair of agents can add the link between them. This notion of stability is not based on any specific procedure of network formation. A stronger concept of stability based on bilateral deviations uses Myerson’s network formation game: a pairwise Nash equilibrium is a Nash equilibrium if it is a Nash equilibrium of the Myerson game which is immune to the formation of a new link by a pair of players. In a Pairwise Nash equilibrium, players can delete any subset of links, and pairs of players can coordinate on the formation of a new link.

3 3.1

Cartels and mergers Cartel formation in Cournot oligopolies

In Cournot oligopolies, the formation of a cartel leads its members to reduce quantities in order to increase the selling price. This provides a public good to firms which do not belong to the cartel – the outsiders, who benefit from the price increase without paying the cost of a limitation in quantities. Hence firms may be reluctant to form or join cartels, resulting in a ”puzzle” of cartel formation which was first noted by George Stigler [86], 25-26) in his discussion of mergers: The major difficulty in forming a merger is that it is more profitable to be outside a merger than to be a participant. The outsider sells 6

at the same price but at a much larger output at which marginal cost equals price. Hence the promoter of a merger is likely to receive much encouragement from each firm – almost every encouragement in fact except participation. The ”Stigler effect” can easily be observed in a linear Cournot oligopoly. Let n firms on the market, with zero marginal cost, produce homogeneous products with a linear demand P = 1 − Q. The profit of each firm only depends on the number 1 of active firms on the market and is given by R = (n+1) 2 . Now suppose that a cartel of size k forms on the market, with the remaining n − k firms remaining independent. The total number of active firms in the market reduces to n − k + 1. As cartel members share equally the profit of the cartel, the profit of an insider is 1 1 o Ri (k) = k(n−k+2) 2 , whereas the profit of an outsider is R (k) = (n−k+2)2 . We immediately observe that the profit of an outsider is always greater than the profit of an insider: Ro (k) > Ri (k) for all k. Following D’Aspremont et al. [22], a cartel of size k is stable if Ri (k) ≥ Ro (k − 1) (internal stability) and Ro (k) ≥ Ri (k + 1) (external stability). These conditions amount to (n − k + 3)2 ≥ k(n − k + 2)2 , (k + 1)(n − k + 1)2 ≥ (n − k + 2)2 It is easy to check that the first inequality (internal stability) cannot be satisfied for k ≥ 2. Hence, the only stable partition is the partition of singletons, where no cartel is formed and all firms remain independent. This simple computation suggests that free riding in the formation of cartels is so strong as to prevent the formation of cartels or mergers on any market. A closer inspection of the profits of insiders and outsiders, due to Salant et al. [83] shows that Ro is increasing in k but Ri is non-monotonic in k and assumes a U-shape, first decreasing, then increasing in k. The minimal profitable cartel size is defined as the unique value of k for which Ri (k) = Ri (1), namely the solution to the equation (n + 1)2 = k(n − k + 2)2 , √

giving k ∗ = 2n+3−2n 4n+5 , or around 80% of the size of the market. The observation made by Salant et al. [83] is thus that mergers must involve a very large fraction of the firms in the industry to become profitable. The computation of the minimal profitable cartel size also has important implications for the study of exclusive membership and sequential models of coalition formation. In the γ game, when a firm leaves the cartel, the cartel dissolves. Hence any cartel of size k ≥ k ∗ is an equilibrium outcome, because no player wants to 7

deviate from the cartel and obtain Ro (1) instead of Ri (k). By contrast, in the δ model, when a firm leaves the cartel, other cartel members remain together, so that the deviating firm compares Ri (k) with Ro (k − 1) and always has an incentive to deviate: no cartel of size greater than one can be formed at equilibrium. In the model of sequential coalition formation, Bloch [6] and Ray and Vohra [79] show that the only subgame equilibrium outcome is for the first firms to remain outsiders, and the last k ∗ firms to agree to form a coalition. Hence, in equilibrium, the minimum profitable cartel size is formed. Macho Stadler et al. [61] use the successive coalition formation model where firms meet bilaterally and decide whether or not to merge. They show that the equilibrium outcome is either that all firms remain singletons or that they merge into a single coalition. Merger to monopoly arises for a specific region of the parameters which is described by a complex recursive formula. Mauleon and Vannetelbosch [63] explore the formation of cartels in the linear Cournot oligopoly using a farsighted solution concept: the largest consistent set of Chwe [15]. They observe that any coalition structure where the size of the cartel is above the minimal profitable cartel size belongs to the largest consistent set. However, other coalition structures can also be sustained, including some involving the formation of multiple cartels. Kamien and Zang [51] and [52] propose a different approach to the study of horizontal mergers. In their acquisition game, firms announce a bidding price for the assets of all other firms and an asking price for their own assets. They observe that no merger will arise at equilibrium. To understand this point, notice that if a firm forms a cartel of size k, it must compensate all k−1 firms for their participation in the cartel at a price π o (k − 1). now clearly, kπ i (k) < (k − 1)π o (k − 1), so that the cartel cannot profitably acquire k − 1 other firms at their asking price π o (k − 1). This line of reasoning is reminiscent of a classical argument on the difficulty of successful take-overs in the corporate finance literature, when an investor must acquire shares from different shareholders of the target firm. The fact that cartels are inherently unstable, and unlikely to emerge as equilibrium outcomes of a game of coalition formation is a puzzle, as cartels are mergers are indeed observed on many markets. The puzzle can be solved by enriching the model in order to give an advantage to cartels over independent firms. In D’Aspremont et al. [22], Donsimoni [27] and Donsimoni et al. [26], Thoron [88] , Schaffer [84] and Prokop [77], cartels are dominant firms fixing prices and independent firms form a competitive fringe, responding in quantity to the price of the dominant firm. In all these papers, cartels are formed by a simultaneous open membership game and a nontrivial stable cartel size exists. Diamantoudi [25] considers a more general farsighted solution concept, where firms anticipate the sequence of moves following deviations. Using an indirect dominance relation, she shows that von Neumann Morgenstern stable sets always exist in the cartel game, and singles 8

out the smallest stable cartel as the most appealing prediction in the game of cartel formation. In the same spirit, Kuipers and Olaizola [58] define a different dynamic process of cartel formation where firms move from one cartel structure to another considering myopic improvements, but moves that can be countered immediately are excluded. With this alternative model of transitions, Kuipers and Olaizola [58] shows that stable cartels have a size larger than the largest stable cartel – a conclusion which stands in sharp contrast to Diamantoudi [25]. Konishi and Lin [56] generalize the analysis of the Stackelberg game where the cartel chooses its quantity first to arbitrary demand and cost functions. They offer a conjecture on the size of the stable cartel and numerically compute it for small values of n. Recently, Zu et al. [95] have provided an exact formula for the size of the stable cartel in Konishi and Lin [56]’s model (which results in higher cartel sizes than originally conjectured). Perry and Porter [74] assume that costs are quadratic and that firms own capital units which can be recombined after a merger. Hence the merged entity can produce more efficiently than outsiders by distributing production in the plants of the constituent firms. In this model of convex costs, the cartel benefits from a cost advantage over the outsiders, and profitable mergers can form. Farrell and Shapiro [30] move away from the homogeneous linear Cournot oligopoly and assume a general demand and cost structure. They highlight the fact that cartels benefit from synergies among their members so that the marginal cost of a cartel is lower than the marginal cost of independent firms. Hence, in the presence of synergies, concentration and welfare may move in the same direction, casting a new light on the antitrust treatment of horizontal mergers. Any other strategic advantage given to cartel members over independent firms would help explain the formation of cartels on actual oligopolistic markets. Brown and Chiang [12] and Banal Estanol and Ottaviani [1] suppose that firms face idiosyncratic shocks in demand and costs. When firms are risk-averse, a merger allows firms to diversity risk, and the merged entity has a strategic advantage over independent firms, which leads it to increase production. Hence, mergers occur more frequently when firms are risk averse and the environment more variable. Brown and Chiang [12] study a three-firm environment and a sequential merger process, and observe that since mergers of two firms are more likely to be profitable, merger to monopoly is less likely to emerge with risk-averse firms. Banal Estanol and Ottaviani [1] consider a general environment and characterize the sharing contract among participants to the merger, showing that under Cournot competition, firms have an incentive to equalize the number of shares they possess in each other’s firm. Davidson and Ferrett [23] suppose that cartel members can share the benefit of R& D investments and show that this allows for profitable cartels. Nocke [68] studies the formation of cartels when firms face capacity constraints. 9

Because cartels have larger capacity, they enjoy a strategic advantage when firms are capacity-constrained. Large cartels are easier to sustain when demand is high and the capacities of individual firms are low. Espinosa and Macho Stadler [29] incorporate moral hazard in the Cournot model, assuming that production is realized by independent teams. The moral hazard problem becomes more stringent when teams are larger (free-riding incentives are higher) so that, at first glance, cartels are less likely to form. However, as intermediate cartels are unlikely to form, large cartels are easier to sustain – firms realize that by leaving the cartel, they will lead to an unstable intermediate cartel which unravels so that in the end all firms become independent. Horn and Persson [48] study cartel formation through a cooperative game theoretic solution concept. They assume that mergers are profitable and show that the most concentrated coalition structure will always emerge: firms form monopolies if monopolies are allowed, duopolies if duopolies are allowed but not monopolies, etc.. When firms have heterogeneous costs, the formation of cartels becomes harder, because the gains from cooperation cannot be divided equally among cartel members. Characterizing conditions under which cartels are profitable, and the equilibrium coalition structure becomes a difficult exercise, and results have only been obtained for small numbers of firms or small numbers of types. Donsimoni [27] studies cartels among heterogeneous firms in the dominant firm - competitive fringe model when firms have different quadratic costs. Barros [2] and Brown and Chiang [13] discuss the formation of cartels in the linear Cournot model among three firms with different costs. Fauli Oller [31] analyzes a four firm model where two firms have low cost and two firms have high cost. When costs are privately known, the cartel must in addition elicit information about cost from the cartel members. Cramton and Palfrey [18] provide a complete analysis of the mechanism design problem faced by a cartel when firms have to reveal their production costs.

3.2

Cartel formation in Bertrand and spatial oligopolies

The intuition underlying the instability of cartels in Cournot oligopolies is related to the fact that quantities are strategic substitutes: a reduction in quantity by cartel members leads outsiders to expand their own quantity, thereby depressing the price and reducing the profit of cartel members. In a Bertrand oligopoly, prices are strategic complements, and the increase in price resulting from collusion among cartel members leads outsiders in turn to increase their prices, resulting in an equilibrium with higher prices and profits for all cartel members. Deneckere and Davidson [24] were the first to make this observation in a model of symmetric product differentiation, when firms set prices rather than quantities. They note that both cartel members and outsiders benefit from the formation of a merger, even 10

though outsiders benefit more than insiders. They compute the profit functions Ri (k) and Ro (k) and show that they are both strictly increasing in k. However, it remains true that outsiders obtain higher profits than insiders (or more generally members of smaller cartels obtain higher profits than mergers of larger cartels), so that the formation of mergers is not guaranteed. Deneckere and Davidson [24] provide a numerical example to show that firms may be unwilling to merge even under Bertrand competition, but note that, when the degree of product differentiation becomes small and fierce Bertrand competition erodes the firms’ profits, merger to monopoly is obtained as the equilibrium outcome of an open membership game of cartel formation. The formation of cartels in spatial models has been studied both in the circular city and the line. Following early work by Levy and Reitzes [59], Brito [11] computes the effect of a merger between two consecutive firms. He shows that the profit of insiders always goes up, providing a positive incentive to merger as in the model of symmetric product differentiation of Deneckere and Davidson [24]. When two consecutive firms merge around the circle, the pricing game is no longer symmetric, and firms’ equilibrium prices depend on their proximity to the merged entity. Not surprisingly, firms closer to the merged entity are more affected by the merger, and hence raise their prices more and benefit from a larger increase in profits than firms at a higher distance. However, the effect of the merger ripples through the entire circle, and all firms effectively raise their prices and experience an increase in profit. Giraud-H´eraud et al. [37] use the same model of a circular city but assume that one of the firm sells products at all locations (the multi-product firm). They analyze the incentives of the multi-product firm to merge with some of its independent rivals. One difficulty that they highlight is that merging firms are no longer ex ante symmetric, and the profitability of the merger depends on the post-merger division of the gains from cooperation. Studies of mergers on the line have also led to significant insights. Braid [8] studies mergers between two adjacent stores on an infinite line. He shows that when prices are set simultaneously, collusion among stores only has an effect if the two stores are nearest neighbors, and that affects the prices of all other stores on the infinite line. If the merged entity acts as a Stackelberg leader, merger has an effect even when stores are not adjacent. Braid [9] builds on this model to study mergers between two stores located on a two-dimensional space and computes numerically the effect on equilibrium prices. On the Hotelling line, Rothschild, Heywood and Monaco [81] analyze a three-firm model, where two firms have the opportunity to merge. The innovation of their paper is that they consider how the possibility of merger affects the firms’ location decisions. They thus consider a three-stage model where firms initially choose locations, then two of the three firms decide whether to merge and finally firms compete in prices. In this model, 11

the two merging firms obtain a higher gain than the outsider. Heywood, Monaco and Rothschild [46] extend the analysis to n firms, and distinguish between corner cases where the merging firms are at the extremity of the Hotelling segment and interior cases. They show that outsiders are always harmed in corner cases but not in interior cases. In the context of vertical differentiation, the analysis of mergers has so far been restricted to oligopolies with three firms. Norman et al. [69] analyze a model where the two merging firms sell the goods of lowest qualities. They show that the merged entity will always choose to sell the good of lowest quality and argue that the post-merger equilibrium may lead to higher market prices. Gabszewicz et al. [34] analyze general mergers among three competing firms and show that the only stable mergers involve the firms producing the bottom two qualities or the firm producing the high quality and the firm producing the low quality. We conclude by noting that three papers have attempted to characterize equilibrium coalition structures in an abstract context encompassing both the Cournot and Bertrand games. Currarini and Marini [20] explore the difference between situations where the competitive game among firms has strategic complements (Bertrand) or substitutes (Cournot). They show that nontrivial coalition structures emerge in games with strategic complements and provide conditions under which nontrivial coalition structures also emerge in games with strategic substitutes. Yi [92] and Finus and Rundshagen [32] consider a general model with positive externalities which encompasses mergers and cartels. They obtain interesting results comparing the sizes of cartels formed under different processes of coalition formation.

3.3

Dynamic mergers

Dynamic models of mergers emphasize the changing environment under which firms interact, the interplay between entry, exit and merger decisions, and the role of repeated interactions on the enforcement of collusion. The seminal model proposed by Gowrisankaran [38], in the spirit of Ericsson and Pakes [28], analyzes a dynamic model where firms choose to enter, invest, merge and exit at every period. Each firm evaluates the outcome of its decision based on expected discounted profit calculations, and the equilibrium concept is a Markov perfect equilibrium outcome in a complex environment where the state captures all relevant information about the industry. Equilibrium is showed to exist, and can be computed using numerical techniques. Computations show that the possibility of mergers greatly affects the structure of the industry, reducing the number of active firms in equilibrium. Once mergers are introduced, production, prices and profits go up, but consumer surplus decreases. Gowrinsankaran and Holmes [39] analyze a dynamic model with a dom12

inant firm and a competitive fringe, where, as in Perry and Porter [74], merging firms can reallocate productive capital to reduce production costs. In the dynamic environment, capital is not given but results from firms’ investment decisions. The analysis shows that both perfect competition and monopoly are absorbing states. In some situations, fringe firms acquire the capital of the dominant firm, in others, the dominant firm successively acquires all the capital of the dominant firm, resulting in a monopoly. Pesendorfer [75] studies a simpler model of mergers and entry and provides an explicit characterization of Markov Perfect equilibria. He assumes that, at every period, a single firm has the opportunity to enter, and that firms make offers to merge as in Kamien and Zang [51]. Firms are identical and profits only depend on the number of active firms every period. In this simple setting, conditions are obtained under which no merger ever takes place, and under which mergers result in monopoly. Mergers may occur because firms anticipate that other mergers will follow – this is the preemptive role of mergers. Pesendorfer [75] establishes the existence of merger cycles, under which k mergers happen at some period, followed by k − 1 periods with no mergers. Preemptive mergers also occur in Fridolfsson and Stennek [33] who analyze a three-firm model, where following a shock, firms race to merge with another firm. In their model, firms merge not only to increase prices as in the classical framework but also in order to guarantee that they will not be left out of the wave of mergers. Mergers have also been analyzed in the context of repeated interaction between oligopolistic firms. When firms have different capacity constraints, mergers allow firms to recombine capacities, and change the environment under which collusive agreements can be enforced. Compte et al. [17], Vasconcelos [89] and Kuhn and Motta [57] analyze different models of collusion with mergers. They focus attention on situations where all firms in the industry (the merged firm and the independent firms) collude. Both Compte et al. [17] and Kuhn and Motta [57] observe that collusion is easier to sustain when firms have equal capacities, so that any merger of small firms which leads to an equalization of capacities may help collusion. By contrast, mergers involving large firms, may make collusion harder to sustain and hence be procompetitive. In addition, by reducing the number of active firms, a merger helps sustain collusion in a repeated interaction. Vasconcelos [89] generalizes the analysis by allowing merged firms to recombine capital as in Perry and Porter [74] and allowing for more general punishment schemes in the repeated game. In all the previous papers, collusion involves all the firms in the industry, By contrast, Bos and Harrington [7] allow for collusion to involve a subset of firms, and propose a model which combines endogenous cartel formation, enforcement through repeated interaction and asymmetric capacities. They show that a cartel is stable if the smallest firm finds it optimal to be in the cartel and the largest firm 13

finds it optimal to be outside the cartel. This characterization yields a formula to compute stable cartels. A merger of firms under partial collusion produce complex effects, as it simultaneously affects firms outside the cartel and the incentives to collude inside the cartel. After a merger, the set of stable cartels may change, and hence post-merger equilibrium prices and quantities may be difficult to analyze. Bos and Harrington [7] use numerical computations to evaluate the effects of mergers in their model.

3.4

Bidding rings

Bidding rings are groups of buyers who submit their bids cooperatively in auctions. Graham and Marshall [42] and Mailath and Zemsky [62] have analyzed bidding rings in second price private value auctions. Suppose that values are independently distributed according to a common distribution F with density f . In a second-price auction, the optimal bidding strategy is to bid one’s valuation and the expected profit is given by Z ∞Z z R= (z − y)(n − 1)F (y)n−2 f (y)f (z)dydz. 0

0

n−2

where (n−1)F (y) f (y) is the distribution of the highest bid among n−1 bidders. If a bidding ring of size k forms, the distribution of the highest bid among ring members is kF (y)k−1 f (y) and of the highest bid among independent bidders (n − k)F (y)n−k−1 f (y). Hence the expected profit of a ring member (insider) is Z Z 1 ∞ z i kF (y)k−1 f (y)(n − k)F (y)n−k−1 f (y)dydz, R (k) = k 0 0 whereas the expected profit of an independent bidder (outsider) remains Ro (k) = R. In the special case where the distribution of values is uniform on [0, 1], the 1 1 profits are Ri (k) = (n−k+1)(n+1) and Ro = n(n+1) . We immediately observe that Ro is independent of k, Ri increasing in k and Ri (k) > Ro for all k > 1. Hence the only stable bidding ring is the complete bidding ring including all the bidders. This is also the unique equilibrium outcome of the γ and δ games and of the sequential game of coalition formation. In sharp contrast to the oligopoly case, bidding rings in auctions are always profitable to all the bidders. Mailath and Zemsky [62] consider a general situation where values are drawn from different distributions and prove a stronger result. They show that the sum of utilities of bidders is increasing and convex in the size of the bidding ring. Hence, in cooperative game-theoretic terms, the coalitional function is convex, so that the core of the game is non-empty. Even when bidders are heterogeneous, there always

14

exists a distribution of the surplus of the bidding ring which will be accepted by all the bidders. The analysis of bidding rings in first-price auctions is much more complex, as it requires to compute the equilibrium payoff of a first-price auction with asymmetric bidders (the bidding ring and independent bidders), a notoriously complex task. Mac Afee and Mac Millan [64] compute equilibrium strategies when values are independent and identically distributed according to a binary distribution, v = 1 with probability p and v = 0 with probability 1 − p. In that case, the distribution of values of all bidders is a binomial distribution with parameter p, allowing for simple computations of the distributions of order statistics. The expected profit of a member of the bidding ring is Ri (k) = k1 (1 − p)n−k (1 − (1 − p)k ) whereas the expected profit of an independent bidder is Ro (k) = p(1−p)n−k . As opposed to the second-price auction, but in line with the oligopoly models, the profit of an insider is always smaller than the profit of an outsider, π i (k) < π o (k). Stable bidding rings exist when π i (k) > π o (k − 1), a condition which holds for the unique value k ∗ such that 1 − (1 − p)k+1 1 − (1 − p)k ≥ p(1 − p) ≥ . k k+1 Mac Afee and Mac Millan [64] show that k ∗ is always larger than 3, increasing in p and converges to infinity when p converges to 1.

3.5

Collusive networks

Collusive networks emerge when firms form reciprocal market sharing agreements whereby they refrain from entering each other’s market. This geographical division of markets has been analyzed by Belleflamme and Bloch [4]. Suppose that firm i is based on market i. By forming a link ij with firm j, firm i refrains from entering market i and firm j refrains from entering market j. For any graph g, ni (g) = n − di (g) is the number of active firms on i’s market, where di (g) denotes the degree of firm i in the collusive network g. Assuming that firms are symmetric, the profit that each firm makes on market i is given by π(ni (g)) and the total profit of firm i is X R = π(ni (g)) + π(nj (g)). j|ij∈g

Belleflamme and Bloch [4] characterize pairwise-stable collusive networks when profit functions are decreasing and log-convex in the number of active firms. They show that (i) every stable network must include complete components – when firms sign market-sharing agreements, they must sign them with all other firms in 15

their component – (ii) that components must be of different sizes and (iii) that every component must be of size greater than m∗ where m∗ is the solution to π(n−m+1) = 2. π(n−m+2) 1 ∗ In a linear Cournot oligopoly where π(n) = (n+1) 2 , it is easy to see that m = n so that there are only two candidates for collusive networks: the complete network and the empty network. In a second-price auction with uniform distributions, the 1 and we find that m∗ = n − 1 so expected profit of a bidder is π(n) = n(n+1) that there exists three possible stable collusive networks: the empty network, the complete network, and an asymmetric configuration where one independent bidder faces a bidding ring of n − 1 bidders.

4 4.1

Alliances, RJVs and trade associations Research joint ventures and alliances

Cost-reducing alliances have been extensively studied in the context of research joint ventures (RJVs). The seminal papers by Katz [54] and D’Aspremont and Jacquemin [21] considered the incentives of two firms to cooperate in cost-reducing research before competing on the market. This line of research was prompted by a change in the regulatory environment, with programs aimed at stimulating cooperative research among firms both in the US and in Europe in the mid eighties - the National Cooperative Research Act of 1984 and the National Cooperative Production Amendments of 1993 in the US and the block exemption to collusion in R & D of Regulation 418/85 in 1985 in the European Union. The main trade-off embodied in these models compares the direct benefit of the cost reduction experienced by a firm with the indirect cost of facing a competitor who also experiences a cost reduction and thus behaves more aggressively in a Cournot market. D’Aspremont and Jacquemin [21] discuss how this trade-off is affected by the presence of spillovers, when some part of the research output of one firm are leaked to the other firm. Because there are only two firms involved in the models, the formation of an RJV has no external effect on other firms in the industry. Suzumura [87] and Kamien et al. [50] consider an industry with an arbitrary number n of firms. Kamien et al. [50] distinguish between different types of alliances: RJVs where firms share their R & D results but do not coordinate their investments, R & D cartels, where firms coordinate their investments but do not share research outputs and RJV cartels where firms coordinate their investments and share research outputs. Both Suzumura (suzumura1992cooperative and Kamien et al. [50] restrict attention to situations where the formation of an RJV has no external effects by considering alliances covering all the firms in the industry. Kamien and Zang [53] analyze a 16

model with symmetric alliances. Poyago-Theotoky [76] considers partial alliances which only cover a fraction of the firms in the industry but supposes that only one alliance is formed. Hinloopen [47] contrasts cooperative research with R & D subsidies and concludes that research subsidies are a more effective policy tool than allowing firms to cooperate in research. Bloch [5] proposes a model to endogenously derive the structure of cost-reducing alliances in oligopolies. Consider a linear Cournot oligopoly with inverse demand P = 1 − Q. Let ak denote the size of the alliance Ak . We suppose that firms have complementary assets in R & D so that the marginal cost of production of a firm is linearly decreasing in the size of the association it belongs to. Formally, if firm i belongs to association k(i), its marginal cost of production is given by ci = λ − µak(i) . The equilibrium profit of a firm belonging to an association of size ai is given by P µ k a2k 2 1−λ + µai − ]. Ri = [ n+1 n+1 We observe significant differences between the formation of alliances and cartels. First, the formation of an alliance has a negative externality on the profit of outsiders. An increase in ak , k 6= i reduces the profit of firm i. Second,P in a fixed coalition structure, members of larger alliances have higher profits, as a2k is constant, but profit increasing in ai . These differences lead to very different predictions on equilibrium coalition structures. For example, in an open membership game, firms always have an incentive to join a larger alliance so that the only equilibrium outcome is for all firms to join in a single RJV. On the other hand, in exclusive membership games, the equilibrium alliance structure will not be the grand coalition. To understand this fact, notice that, when a single firm joins an alliance, benefits are asymmetric. The single firm benefits from a large cost reduction whereas alliance members only experience a small reduction in costs as the size of the alliance only increases by one unit. This implies that, when an alliance is very large, it will be reluctant to admit new members. A careful look at the profit function shows that Ri is increasing in ai until ai = n2 and decreasing afterwards. The symmetric association structure with two associations of size n cannot be an equilibrium either. In order to increase the cost difference with 2 members of the rival association, any association has an incentive to accept more than n2 members. Anticipating that the remaining players will form an association of size n − a, members of the first association optimally choose a coalition size of a∗ = 3n+1 . In the sequential game of coalition formation, the unique equilibrium 4 association structure thus results in the formation of two associations of unequal sizes, one with 3n+1 members and the other with n−1 members. 4 4 Bloch [5] discusses the extension of the model to Cournot and Bertrand com17

petition with differentiated products. As the level of product differentiation increases, competition on the market is less fierce and the dominant association becomes larger. Interestingly, the sizes of equilibrium associations are identical under Cournot and Bertrand competition. Yi [93] and Yi and Shin [94] generalize the model by studying arbitrary demand and cost functions. They identify conditions on demand and cost functions for which the grand coalition emerges in an open membership game. Greenlee [43] considers a general linear model with intraRJV and industrywide spillovers and characterizes the equilibrium outcomes of the open membership game and the sequential game of coalition formation. He finds that the grand coalition always forms in the open membership game, but that a more fragmented coalition structure with different alliances arises in the sequential game. However, the number of alliances is bounded above by 3 for all n. Numerical computations are used to illustrate the size of alliances as a function of the two spillover parameters. Belleflamme [3] extends the model to asymmetric firms and shows that when cost reductions are not symmetric, the grand coalition may fail to form in the open membership game. Morasch [65] considers a related model where heterogenous firms in a strategic alliance propose output-based transfer payments. Under this formulation of profit-sharing contracts, he computes numerically the equilibrium association structures for small values of n.

4.2

Networks of collaboration

As an alternative to multilateral alliances, Goyal and Joshi [40] propose a model of networks of bilateral collaboration among firms. They assume that the marginal cost of production is linearly decreasing in the number of bilateral alliances a firm has formed ( rather than the size of the alliance it belongs to). We then have ci = λ − µdi (g), where di (g) denotes again the degree of firm i in the graph g. Equilibrium profits are given by P µ j dj (g) 2 1−λ + µdi (g) − ]. Ri = [ n+1 n+1 As in the model of alliances, the formation of a link between two firms i and j hurts all other competitors, and for a fixed network structure, firms with higher degree obtain a higher profit. If the formula for the profit is very similar to the formula in strategic alliances, the analysis of the model of network formation is very different. When two firms sign a bilateral agreement, they enjoy a symmetric reduction in production cost. One computes the marginal effect of an additional agreement on firm i’s equilibrium quantity as ∆qi =

nµ > 0. n+1 18

As equilibrium profits are increasing in quantities, all bilateral agreements thus raise the firms’ profits so that the only pairwise stable network is the complete network. When firms face a significant fixed cost of link formation, the complete network ceases to be stable. Goyal and Joshi [40] show that stable networks have a dominant group architecture, with one complete component and singleton firms. Goyal and Moraga Gonzales [41] extend the analysis by supposing that firms endogenously choose their research effort. Research effort will be decreasing in the number of links a firm has formed, and hence the addition of a new bilateral agreement may result in lower R & D on the market. In a linear Cournot market, Goyal and Moraga Gonzales [41] show that research efforts are maximized when competitors. However, as the marginal benefit of every firm is linked to exactly n−1 2 an additional link remains positive, firms have an incentive to form the complete network. Hence, in a model with endogenous research efforts, firms engage in excessive collaborative activities.

4.3

Exchange of information and trade associations

Another important instance of collaboration among firms is the exchange of information. We distinguish between two types of information: common value information (about market demand) and private value information (about idiosyncratic costs). Information exchange has been studied in the context of trade associations – groups covering all firms in the industry. The first strand of papers by Novshek and Sonnenschein [70], Clarke [16], Vives [90] and Gal-Or [35] consider information sharing about an unknown parameter of demand. Novshek and Sonnenschein [70] and Vives [90] focus on a duopoly model. Novshek and Sonnenschein [70] solve for the partial pooling of information, when each firm chooses to pool some of the signals they receive. Vives [90] compares the incentive to share information under Cournot and Bertrand, and under substitutes and complements, showing that it is optimal not to pool information in games of strategic substitutes (Cournot with substitutes and Bertrand with complements) but optimal to share information in games of strategic complements (Cournot with complements and Bertrand with substitutes). The same result – that information sharing is never optimal under Cournot with substitutes – is obtained by Clarke [16] and Gal-Or [35]) in an oligopoly model with n firms, quadratic payoffs and normally distributed signals. Li [60] extends the model by allowing for more general signal distributions, and considers also information sharing about private cost parameters. He finds that firms never have an incentive to share information about common market demand but always have an incentive to share information about private costs in a Cournot oligopoly. Shapiro [85] also notes that information sharing about private costs arises as an equilibrium. Gal-Or [36] analyzes information sharing 19

about market demand and private costs under Cournot and Bertrand and shows that there is a stark distinction between Cournot and Bertrand and common value and private value, with no information sharing emerging as the equilibrium outcome for Cournot under common values and Bertrand under private values, and full information sharing for Bertrand under common values and Cournot under private values. Okuno et al. [72] offer a general argument to show that unraveling results in all firms revealing their private cost information. Raith [78] provides a useful guide to the literature and a generalization of all existing models, indicating exactly which conditions are required for information sharing. Most of the literature considers information sharing with all other firms in the industry. One exception is the paper by Kirby [55] which allows for information sharing among a subset of firms, and consider the formation of information pools. Building on Clarke [16]’s model, she shows that information pooling among a subset of firms may be an equilibrium behavior for some subset of parameters – in sharp contrast to the case where firms must exchange information with all other firms in the industry, where no information is ever shared. This result suggests that allowing firms to form smaller exclusive trade associations may lead to more information sharing, increasing the profit of firms and the expected consumer surplus. Vives [91] compares different disclosure rules in trade associations. He allows trade associations to use exclusionary disclosure rules – the aggregate signal on the market is only distributed to a fraction of the firms in the industry – and shows that exclusionary disclosure rules restore the firms’ incentives to share information, but does not necessarily lead to all firms joining the trade association. Currarini and Feri [19] analyze information sharing as bilateral agreements among firms, and characterize the stable networks of information sharing in Cournot oligopolies. They show that in the case of private values, pairwise stable networks are connected components with some isolated firms. In the case of correlated signals, they show that pairs of firms always have an incentive to exchange information so that the empty network is never pairwise stable and the complete network is always stable. Hence, as in the case of strategic alliances, there is a sharp contrast between coalition and network formation in information sharing, and firms will more easily share information about demand when agreements are bilateral than when they are multilateral.

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