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# Stable Coalition Structures with Open Membership and Asymmetric Firms

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## Abstract

I study games of coalition formation with open membership where firms form associations in order to decrease their costs before competing on the market. According to previous analyses, only the grand coalition forms at the Nash equilibrium of such games. I show that this result hinges on the assumption of symmetric firms. I therefore introduce asymmetric firms in a game where only two associations can form. I demonstrate that there exists a coalition-proof Nash equilibrium coalition structure in this game, and that when the equilibrium involves two associations, all the members of an association have a higher taste for this association than all nonmembers do. Journal of Economic Literature Classification Numbers: C70, C72, L13.

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... A configuration of component owners in and outside of the bargaining group has to exhibit stability in equilibrium, in the sense that no member wants to leave the group, and no outsider wishes to join (D'Aspremont et al. 1983). We consider open membership in which no component owner can refuse others the right to join the bargaining group (see Belleflamme 2000;Yi 1997, for a survey). ...
... 6 In determining the composition of the components in the bargaining group endogenously, our work is also related to that of coalition formation. The literature distinguishes between coalitions in collusive markets (Stigler 1950) in which coalition formation induces a positive externality on the firms outside of the cartel by restricting production and increasing prices; coalitions in cost-reducing markets on the other hand induce a negative externality on non-members since the competitive position of the cartel is strengthened (Bloch 2002;Belleflamme 2000). Often simplifying assumptions will have to be made when considering endogenous coalition formation. ...
... We also consider the case in which a single coalition can form. 7 Asymmetry in coalition formation is examined by Belleflamme (2000) who considers firms that can form cost-reducing cartels prior to competition. The asymmetry has two dimensions, one relating to the firm itself and one relating to the association to which the firm is attached, i.e. that the cost reduction may be relationship-specific. ...
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We consider a vertical supply chain in which a monopoly retailer produces a good by assembling a number of essential components each of which is owned by a monopoly. Rather than making the common assumption that the component price is set in the same way for each owner, we investigate the possibility that the retailer may profit by bargaining with some owners in a group, whilst others set their component price to maximize own profit. Furthermore, component owners can self-select into one of these groups, and the retailer can affect group formation by adjusting the order of negotiations. We present conditions under which the retailer can encourage the formation of a bargaining group, and thereby improve its own and industry profit.
... The myerson value for graph 1 is The Myerson value for graph 5 is (2 1 6 , 5 4 6 , 4 1 6 ) The Myerson value for graph 6 is (4 4 6 , 3 1 6 , 4 1 6 ) The Myerson value for the complete graph is (3,4,5) Conclusion: ...
... The myerson value for graph 1 is The Myerson value for graph 5 is (2 1 6 , 5 4 6 , 4 1 6 ) The Myerson value for graph 6 is (4 4 6 , 3 1 6 , 4 1 6 ) The Myerson value for the complete graph is (3,4,5) Conclusion: ...
... The Myerson value for the complete graph is(3,4,5) Claim 1: The endogenous game of link formation with the Myerson value as fixed valuation has 3 natural subgame perfect equilibria where either coalition 13, 12 or 23 form but never the two or three-link graph (for a general proof see claim 2 below). ...
Article
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The Coase theorem is evaluated when there is sequential strategic coalitional behavior and more than two agents are involved in the externality. Links or networks, needed for gains of cooperation, are proposed and payoffs are eval-uated according to "who is more connected". We use the Aumann-Myerson's bargaining solution (1988). Two firms pollute a third one. Without liabilities, once the first link is agreed, the two players choose not to form extra links be-cause they foresee a decrease in their bargaining power and hence lower payoffs; so only two-firm coalitions form even with strict superadditivity and costless links. If the latter is interpreted as zero transaction costs, then the latter re-sult is inconsistent with the Coase theorem as with liability the grand coalition forms (thus efficiency results). We give sufficient and necessary conditions for related results in general three-person games. It is pointed out that for policy purposes it is crucial to distinguish inefficiency induced by factors others than transaction costs. Finally we give a counterexample in which allowing for exter-nalities in coalition induces the grand coalition to form. As a solution concept, we use the extended Shapley value in Partition function form (Myerson 1977) and the link formation game of Aumann and Myerson.
... Sections 5 and 6 analyze two special cases of district independence and firm independence of stand-alone marginal costs. In Section 7 we examine the case with two 22 23 Before proceeding with the model we relate our results to the existing literature. ...
... Subsequently, Belleflamme [22] has shown the existence of a Nash equilibrium in the case of firm independence. Belleflamme [23] introduces a model with heterogeneous players. Assuming linearity of spillovers, he demonstrates the existence of an agglomeration equilibrium and indicates a possibility of dispersed equilibrium in the case of two districts. ...
... We do not impose any restrictions on the shape of the local social capital spillover functions µ α . In particular, they can be either concave or convex, and are not necessarily linear as in Belleflamme [23] and BPT. Although for the most results of the paper we will assume that the local spillover effects are positive and the functions µ α (·) are strictly increasing in the number of firms in every district α, we would like to point out that our main result on the existence of a Nash equilibrium, Proposition 2.1, as well as Lemma 3.2, hold for arbitrary spillover functions. ...
Article
This paper considers a model of district formation as a local socio-economic system incorporating the mix of local cooperation and competition, termed co-opetition. There are heterogeneous firms distinguished by their “stand-alone” district-dependent production and transportation cost. Every firm chooses its location when its production cost is affected by local socio-economic spillovers generated by other firms in the district. The firms take into account the reciprocal nature of local spillovers: while reducing their own costs, the firms also reduce the costs of their rivals. We show that the location game with a linear demand function yields an equilibrium for any number of firms and districts. We characterize both “agglomeration” equilibria, when all firms locate in one district, and “dispersed” equilibria, when firms locate in different districts. We demonstrate that a dispersed equilibrium can emerge only if firms' and districts' characteristics possess a sufficient degree of heterogeneity.
... Ebina and Shimizu, 2009;Watanabe and Matsubayashi, 2013;Zhao, 2013); (ii) works looking at the stability of mergers of heterogeneous firms in sequential oligopolies (e.g. Escrihuela-Villar and Faulí-Oller, 2008;Driessen et al., 2011); (iii) works looking at the formation of cost-reducing alliances by heterogeneous firms in simultaneous-move oligopolies (Belleflamme, 2000). ...
... (iii) Adopting Bloch's (1995) model of cost-reducing alliances in a Cournot oligopoly, Belleflamme (2000) explores the consequences of heterogeneous firms in a simultaneous open membership game of coalition formation. It is well known (e.g. ...
... This is because the benefits of cooperation increases linearly with the size of the coalition and, therefore, in an open membership game there is no reason to remain out of the industry-wide association of firms. Belleflamme (2000) shows that this result extends to the case with asymmetric associations but not to the case with symmetric associations and asymmetric firms. In particular, when firms obtain different benefits in joining an association, several associations might form or a pure strategy Nash equilibrium coalition structure might even fail to exists. ...
Article
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In this paper we review a number of coalitional solution concepts for the analysis of cartel and merger stability in oligopoly. We show that, although so far the industrial organization and the cooperative game-theoretic literature have proceeded somehow independently on this topic, the two approaches are highly interconnected. We …rst consider the basic problem of the stability of the whole industry association of …rms under oligopoly and, for this purpose, we introduce the concept of core in oligopoly games. We show that di¤erent assumptions on the behaviour as well as on the timing of the coalitions of …rms yield very di¤erent results on the set of allocations which are core-stable. We then consider the stability of associations of …rms organized in coalition structures di¤erent from the grand coalition. To this end, various coalition formation games recently introduced by the so called endogenous coalition formation literature are critically reviewed. Again, di¤erent assumptions concerning the timing and the behaviour of …rms are shown to yield a wide range of di¤erent results. We conclude by reviewing some recent extensions of the coalitional analysis to oligopolistic markets with heterogeneous …rms and incomplete information. JEL Classi…cation: C70, C71, D23, D43.
... , where r ≤ t is the period in which coalition S formed. 5 This payoff may be negative. ...
... Assuming that some time elapses in both cases would not alter the results in Sects. 3 and 4. 5 Thus utility is transferable, and proposers pay responders immediately after the proposal to form S is accepted. Assuming that proposers pay responders only after a complete coalition structure is formed would not change the results in Sects. ...
... The assumption of exogenous payoff division is common in applications (e.g. [5,48,67]). Kóczy [44] finds a connection between the equilibria of a modification of Bloch's [9] game and the recursive pessimistic core. ...
Article
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This paper studies an extensive form game of coalition formation with random proposers in a situation where coalitions impose externalities on other players. It is shown that an agreement will be reached without delay provided that any set of coalitions profit from merging. Even under this strong condition, the formation of the grand coalition is not guaranteed. Therefore, the resulting coalition structure will not necessarily be efficient. The results of this model are compared with the related work of Ray and Vohra (GEB, 1999), which assumes that players move in a predetermined order. The game with random proposers tends to give a large advantage to the proposer, whereas the game with a rule of order tends to favour the responders and may not capture the competition between players. The game with random proposers yields more efficient results for some specific classes of games. However, the results of the two games cannot be ranked in general in terms of efficiency.
... After having presented the rules of the game in Section 2, we now focus on the formal definition of the two-stage coalition formation game. 14 This construction may not always be realistic. For example, in the case where the physical game Γ is a Cournot oligopoly, one may imagine that the resulting cost function for a meta-player C in Γ C is not obtained as a sum ∑ i∈C c i (k C;i ) of the individual cost functions (see, for example, [25]). 15 Note that we do not assume that R is injective. ...
... For every C ∈ C and C ∈ C with #C = #C , the actions k C and k C are the same constant. 25 25 By parts 1 and 4, these are indeed constant. ...
Article
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Coalition formation is often analysed in an almost non-cooperative way, as a two-stage game that consists of a first stage comprising membership actions and a second stage with physical actions, such as the provision of a public good. We formalised this widely used approach for the case where actions are simultaneous in each stage. Herein, we give special attention to the case of a symmetric physical game. Various theoretical results, in particular, for cartel games, are provided. As they are crucial, recent results on the uniqueness of coalitional equilibria of Cournot-like physical games are reconsidered. Various concrete examples are included. Finally, we discuss research strategies to obtain results about equilibrium coalition structures with abstract physical games in terms of qualitative properties of their primitives.
... Three rules of exit are commonly used in alliances: (i) exit without breach via a deadlock implemented by the contractual board where only unanimous decisions are taken (a unanimity decision rule), 2 (ii) exit via breach of the agreement subject to damages (a unanimity decision rule with side payments), (iii) exit at the will of the larger party subject to forewarning (a simple majority decision rule). 3 We analyze how di¤erent rules for exiting an alliance will a¤ect the formation of strategic alliances. ...
... See Smith (2005). 3 Forewarning is usually required for an alliance agreement to be terminable at the will of the larger party. For instance, the Exclusive License and Collaboration Agreement between MedImmune Inc. and Critical Therapeutics Inc. (July 30, 2003) provided that MedImmune had the right to terminate on six months notice. ...
Article
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... If instead the principal colludes with the other identical agent 2, he will implement the symmetric allocation on the frontier F (P − a, a) = (y 1 , y 2 ). In our cooperative game, these simultaneous "offers" would imply v13 = v23 = a and v1 = v2 = P − a. 3 Note that if assumption in proposition 1 (m > 2(P −a)) and harrasing costs of production would be positive then we would have a strictly superaditive game. Intuitively, in the grand coalition the principal doesn't harrase; thus, harrasing costs incurred would be zero and more output would be produced. ...
... In the particular case analized, we have ε = 0. 3 If the two ranchers collude: The last partition, the grand coalition has one element, itself that is worth 4. w N {1,2,3},{{1,2,3}} = 4 = P Let Φ 1 (w N ) be the extended Shapley value for games with externalities in coalition formation for player 1 for the complete graph N. Following Myerson (1978) we have: ...
Article
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We add a Principal (abstract enforcer) to any 2 person TU game in strategic form because of transaction costs or extorsion. The game yields in reasonable cases an empty core when there are no externalities in coalition formation and when the principal can choose any allocation on the frontier of the individual, rational and feasible set. The principal does so by inducing "The Simultaneous and Double Extorsion Game". Only one of the identical players and the princi-pal collude, iff the identical agents combined "might" is more than a threshold of the principal's but less than 100%, i.e."divide and rule". The threshold de-pends on the limit to the degree of double extorsion. Strikingly, he would prefer to induce an empty core game with an extreme level of extortion instead of a nonempty core one. The result, ie. that only an identical agent and the prin-cipal collude, is robust to allowing for variable principal "might" or harrasing ability that might induce a cooperative game with externalities in coalition for-mation. As solution concept, we use an extension of Myerson's (1978) Shapley value generalization to partition function games and the Aumann-Myerson link (1988) formation game. * Ricardo Nieva is Visiting Faculty at Concordia University. Thanks to Leonid Hurwicz and Maria Montero for her sharp insights on the issue of externalities in coalition formation. This paper is the mathematical appendix to Nieva's (October 2002) results on enforcer games suggested by R. Myerson. We use principals instead of enforcers as we agree with a comment by K. Binmore.
... In this aspect, we follow Farrell and Scotchmer (1988), who define a partnership as a coalition that divides its output equally. 4 Bloch (1995) and Belleflamme (2000) consider the case where firms form associations in order to decrease their costs. This aspect is also taken into account in the existing theory of partnerships, where partners get together to exploit economies of scale (see, for example, Farrell and Scotchmer, 1988; and Sherstyuk, 1998). ...
Article
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We analyze the formation of partnerships as a sequential choice-of-sizes game with moral hazard within coalitions; once formed, partnerships compete a la Cournot in the marketplace. We show that when moral hazard within coalitions is very severe, no partnership will form. However, when moral hazard is not too severe the coalition structure will be either similar or more concentrated than without moral hazard. We also show that, while without moral hazard too many coalitions are formed in equilibrium as compared to the efficient outcome, moral hazard may be responsible for an inefficiency of opposite sign.
... 4 A recent literature on endogenous coalition formation deals with efficiency gains (e.g. Belleflamme [1], Bloch [2] and Yi [19]), but also these authors model efficiency gains as exogenous. Yi [20] lets firms decide on their investment in R&D, but the level of product market collusion is determined by a social planner. ...
Article
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We analyse the effects of investment decisions and firms' internal organisation on the efficiency and stability of horizontal mergers. In our framework economies of scale are endogenous and there might be internal conflict within merged firms. We show that often stable mergers do not lead to more e.ciency and may even lead to efficiency losses. These mergers lead to lower total welfare, suggesting that a regulator should be careful in assuming that possible efficiency gains of a merger will be effectively realised. Moreover, the paper offers a possible explanation for merger failures.
... The firms play a two-stage game, where in the first stage each firm chooses its location 5 Unlike in Belleflamme (2000) and BPT, we do not require the functions µ d to be linear. ...
Article
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This paper considers a model of district formation that incorporates a notion of regional industrial systems. Each firm chooses its location from the set of existing industrial districts. The heterogeneous firms are distinguished by its "stand alone" district-dependent production and transportation cost.
... This article is related to the literature on the noncooperative theory of endogenous formation of coalitions the literature which deals with situations in which players first  form coalitions and then, given the coalition structure determined, engage in noncooperative competition. Papers in this literature include Bloch (1995 Bloch ( , 1996), Yi (1996 Yi ( , 1997 Yi ( , 1998), Konishi et al. (1997), Belleflamme (2000 (1996) examines the equilibrium structure of customs unions in a situation in which countries can form customs unions freely. Morasch (2000) obtains the equilibrium structures of strategic alliances of firms and, using them, examines the loose competition policy toward strategic alliances as an alternative to the strategic trade policy. ...
Article
We consider a rent-seeking contest in which players can form strategic groups before expending their outlays. We examine the profitability of endogenous group formation and the effect of such group formation on rent dissipation. We show the following: When just one strategic group is formed in equilibrium, group formation is beneficial both to the group members and to the nonmembers, and rent dissipation is smaller than with usual individual rent seeking. However, when more than two strategic groups are formed in equilibrium, group formation is never profitable to any players, and rent dissipation is greater than with individual rent seeking. Copyright 2001 by Oxford University Press.
... In this section we analyze the bargaining failure inherent in patent pools and explore possible solutions. Coalition formation literature has shown that even with open membership, the grand coalition may not form in equilibrium when there is asymmetry among firms (Belleflamme (2000)). We do note that the premises of his analysis is quite specific (firms are Cournot competitors and coalitions reduce marginal costs), not applicable to extent of asymmetry in our analysis. ...
Article
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We examine patent pools in the context of a consortium standard. Although such pools of complementary technologies are approved by antitrust authorities, the actual implementation has proved to be problematic. We identify two possible obstacles: free riding and bargaining failure. We also examine the traditional RAND (reasonable and non-discriminatory) licensing condition. We suggest formation, licensing and rent distribution methods more conducive to a successful patent pool operation. 科学研究費補助金（特定領域研究） = Grant-in-Aid for Scientific Research on Priority Areas
... In this context, the present paper tries to contribute towards the understanding of the process of endogenous union formation. In our model, in the first stage workers participate in an open membership game of union formation (Yi and Shin, 1995; Yi, 1997; Belleflamme, 2000), i.e. each worker decides which union she wants to belong to, and no worker can be excluded from a union. After the number and composition of unions is so determined, the unions and the firm engage in a wage and employment determination game under the rules of the monopoly union model, that is, we consider a bilateral monopoly in the labor market in which unions make a ''take-it-or-leave-it'' offer concerning the wage and the firm decides employment from each union (Dunlop 1944; Farber, 1986; Oswald, 1985). ...
Article
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This paper analyzes the process of endogenous union formation in the context of a sequential bargaining model between a firm and several unions and tries to explain why workers may be represented by several unions of different sizes. We show that the equilibrium number of unions and their relative size depend on workers' attitudes toward the risk of unemployment and union configuration is independent of labor productivity.
... Specifically, for low values of the spillover parameter, competing RJVs 31 Endogenous formation of coalitions has been extensively studied in other contexts (not that of innovation). Among many others, see Bernheim et al. (1987b), Belleflamme (2000) and Kim and Shin (2002). 32 A research coalition C i is a subset of the N firms, and a coalition structure C = {C 1 , C 2 , . . . ...
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The paper examines the main factors that affect the incentive to cooperate in R&D, inquiring into the effects of cooperation on incentives to innovate in both a complete and an incomplete contract framework. It considers several forms of cooperative agreements and studies the circumstances that make one type of cooperation, more likely than others, to emerge.Theoretical considerations suggest that two of the main factors are uncertainty and spillovers. Further, the incentive to cooperate may be greater or less among symmetric than among asymmetric firms, depending on the source of the asymmetry.When firms cooperate, in most cases they prefer a research joint venture, but because of transaction costs, moral hazard and adverse selection problems other forms of cooperation in R&D may occur. Uncertainty and spillovers also affect the size and the nature of coalitions, and in some circumstances competing research joint ventures may be formed.Finally, the paper surveys the empirical evidence and discusses its consistency with the theoretical conclusions.
... For example, the model developed by Hart and Kurz (1983) evaluates various coalition structures and, based on this evaluation, determines the stable ones. Similar approaches can be found in articles by Ray and Vohra (1999) or Belleflamme (2000). These studies address the coalition formation from a perspective of a central planner and aim for an outcome advantageous to all players. ...
Thesis
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For logistics companies, operating in an economically efficient manner is becoming more and more challenging. Collaboration is one of the ways to improve efficiency, reduce costs and environmental impact, etc. However, its implementation is not always easy and many collaboration efforts fail to meet participants' expectations. Cooperative game theory provides tools to recognize when companies' incentives to cooperate exist as well as a framework to answer questions revolving around fairness, finding partners to cooperate with, etc. This thesis consists of four chapters addressing some of these questions within a scope of specific problems from transportation and logistics. The first chapter investigates optimal strategies of customers in a cooperative version of the traveling salesman problem with profits. The key contribution is finding that the set of optimal prizes customers need to offer to ensure being visited by a carrier coincides with the core of the underlying traveling salesman game whenever this core is nonempty. In the second chapter, several variants of the collaborative version of the location-routing problem are formulated and classified within a cooperative game-theoretic framework in terms of subadditivity, convexity, and the core. The theoretical results are supported by numerical experiments. The third chapter deals with the question of fairness in sports scheduling. A two-step approach is formulated in order to find a fair tournament schedule with respect to the distances traveled by the teams. First, by means of well-established cost allocation methods, an ideal situation is determined. Afterwards, the closest feasible schedule is found. The fourth chapter investigates coalition formation from the companies' perspective. Several approaches are formulated to determine which coalitions are optimal to pursue while taking into account the subsequent payoff or cost allocation. In addition to a novel approach to the coalition formation, the models also take into account possible uncertainty in the problem.
... Many other works have considered a participation model that has a similar structure to the participation game. The examples include participation in an international environmental treaty (Barrett, 1994; Siniscalco, 1993, 1998; Ecchina and Mariotti, 1998; Hoel and Schneider, 1997) and the formation of cartels (d'Asprempnt et al., 1983; Thoron, 1998; Belleflamme, 2000) ...
Article
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Doctor of Economics 博士（経済学） 乙第354号 Hitotsubashi University
... In this aspect, we follow Farrell and Scotchmer (1988), who define a partnership as a coalition that divides its output equally. 4 Bloch (1995) and Belleflamme (2000) consider the case where firms form associations in order to decrease their costs. This aspect is also taken into account in the existing theory of partnerships, where partners get together to exploit economies of scale (see, for example, Farrell and Scotchmer, 1988; and Sherstyuk, 1998). ...
Article
We analyze the formation of competing partnerships as a sequential game with moral hazard within coalitions. In a linear Cournot model, we show that when moral hazard is very severe, no partnerships will form. However, when moral hazard is not too severe, the coalition structure may be more concentrated than it is in the absence of moral hazard. Concerning industry profits, in the absence of moral hazard too many coalitions are formed in equilibrium as compared to the efficient outcome, but moral hazard may be responsible for an inefficiency of opposite sign.
... In this context, the present paper tries to contribute towards the understanding of the process of endogenous union formation. In our model, in the first stage workers participate in an open membership game of union formation (Yi and Shin, 1995;Yi, 1997;Belleflamme, 2000), i.e. each worker decides which union she wants to belong to, and no worker can be excluded from a union. After the number and composition of unions is so determined, the unions and the firm engage in a wage and employment determination game under the rules of the monopoly union model, that is, we consider a bilateral monopoly in the labor market in which unions make a ''take-it-or-leave-it'' offer concerning the wage and the firm decides employment from each union (Dunlop 1944;Farber, 1986;Oswald, 1985). ...
Article
Full-text available
This paper analyzes the process of endogenous union formation in the context of a sequential bargaining model between a firm and several unions and tries to explain why workers may be represented by several unions of different sizes. We show that the equilibrium number of unions and their relative size depend on workers' attitudes toward the risk of unemployment and union configuration is independent of labor productivity.
... However, regarding global climate change, the policy of 1 Yi (1997) presents number of properties of those different coalition-formation rules, in the presence of either positive or negative externalities. Belleflamme (2000) allows for asymmetric countries in an open membership game with negative externalities and McGinty (2007) for one with positive externalities. Some of these models have been applied to the context of climate change and global pollution (see for instance Carraro and Siniscalco, 1993, Barrett, 1994or Barrett, 2005. ...
Article
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We study strategic decentralization in the provision of a global public good. A federation, with the aim of maximizing the aggregate utility of its members, may find it advantageous to decentralize the decision-making, so that its members act au- tonomously to maximize their own utility. If the political struc- tures are chosen simultaneously, federations that are larger or more sensitive to the public good have more incentives to re- main centralized. This is not always true if the political struc- tures are chosen sequentially. A federation that always remains centralized in the simultaneous game may choose to commit to decentralization if it moves first.
... On the other hand, 2 We also note that Reddy and Zaccour (2016) propose another type of core that can be obtained with considerably less computation, and apply it to an economy with multilateral externalities examined by Chander and Tulkens (2006). Belleflamme (2000) explores a coalition-proof Nash equilibrium in a Cournot oligopoly wherein each player has an asymmetrically different preference for joining an association, which has differing impacts on the association's cost reduction. More recently, Zhao (2013) examined the core of a monopoly merger between asymmetric firms in terms of constant marginal costs in a three-firm Cournot market, finding that while the γ -core always exists, the δ-core is nonempty if the firms have lower average marginal costs and higher variance. ...
Article
This study analyzes the stability of horizontal mergers in a Cournot oligopoly market. Although many researchers have addressed this issue, most previous studies assume symmetric firms in terms of demand and cost structures due to analytical tractability. We attempt to find a stable merger in a general [Formula presented]-firm oligopoly in which we allow for asymmetric substitutability between firms. To ensure analytical tractability, we follow the related literature and employ a simple core allocation for a monopoly merger as a stability concept. We analyze several typical markets with asymmetric substitutability and show that although [Formula presented]-core is always nonempty in every setting that we examine, the [Formula presented]-core is very likely to be empty—it is always empty in a market with three or more symmetric firms. Nevertheless, we present an example of a market with a nonempty [Formula presented]-core, regardless of the number of firms in a market. The market has at most two symmetric firms in terms of substitutability (e.g., a linear city). Furthermore, substitutability is so low across the market that each firm competes with only two neighboring firms. Therefore, contrary to the conventional view suggested in previous studies, we show that a monopoly merger can be stable in a Cournot oligopoly market, even if there are many firms in the market.
... This paper is also related to the literature on the noncooperative theory of endogenous formation of coalitions: See, for example, Bloch (1995), Yi (1998), Belleflamme (2000), Morasch (2000), and Yi and Shin (2000). These papers examine the equilibrium structures of coalitions specifically, associations, R&D joint ventures, or strategic alliances of firms in   ...
Article
First, we study contests with unobservable sharing rules in which the number of groups, their sizes, and the number of nonmembers are exogenously given, and compare the outcomes of the case of unobservable sharing rules with those of the case of observable sharing rules. Next, we study contests with unobservable sharing rules in which the number of groups, their sizes, and the number of nonmembers are endogenously determined. We obtain the equilibrium numbers of groups, the equilibrium group sizes, and the equilibrium number of nonmembers, and then examine the effect of endogenous group formation on total effort level and the profitability of endogenous group formation.
... 14 Stable alliances differ when assuming different rules for joining. Yi (1997) finds that only the grand coalition is stable in an open membership game, but this result is not always robust when agents are not identical, Belleflamme (2000) and Yi and Shin (2000). 15 See Mauleon, Sempere-Monerris and Vannetelbosch (2015) and Caulier et al. (2013) for a model where firms can be both members of RJV and form bilateral links with other firms. ...
Chapter
This chapter aims to contribute to the better understanding of R&D by scholars and practitioners. It includes a first section where the concept of innovation is defined and its public good nature and cumulative dimension are analysed. Next, the incentives that firms have to undertake R&D to attain a competitive edge upon rivals are considered. This entails the consideration of both ex ante and ex post incentives to undertake R&D. Since innovation is costly and derives important external effects, cooperation in R&D activities is prominent in several industries where firms enter into research joint ventures, or form research networks. The effect of cooperation is that, under some circumstances, the industry performance is better as compared to full competition. The final section addresses the complementarities and conflicts among the different microeconomic policies (trade, industrial and competition policy) faced by governments when considering the support of R&D activities.
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We derive an almost non cooperative (ANC) analytical payoff function for all three-agent Aumann-Myerson (1988) games, and tractable ones exist for all three-agent A-M-like network games with any fixed valuation, in contrast to restricted results in the literature, if at all. Unlike link proposal game A-M and Myerson (1986), ANC has dynamic bilateral cooperation as we assume bilateral long cheap talk among three agents (differing from A-Hart (2003)), i.e., (1) pairs "smooth" Nash bargain during link discussions over credible expected payoffs induced by equilibria of a Nash demand-like game-where a link forms if the two agents match (2) double proposals, i.e., payoffs that sum up to their Myerson values in the prospective graph, and future bilateral coordination schemes. Thus, payoffs in the final graph of ANC yield a "variable Myerson value pair" which accounts for future possibilities of link formation. Instead, A-M has (a) "fixed" Myerson values (1977). In ANC, key (b) multiple equilibria in A-M-with conflicting requests of two agents to an indifferent third one-are solved, as coordination on requests by earlier linked pairs prevails if credible. This follows from (3) almost assuming that schemes are reminded chronologically "behind closed doors" as there is almost a natural first-mover advantage. (c) Inefficiency is possible in A-M, but not in ANC. We state some of the complete analytics for A-M. Also, in strictly superadditive games, only two-link graphs form. If only a link-coalition of two-forms then they achieve the grand coalition's worth
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We present a theory for predicting how business firms form alliances to develop and sponsor technical standards. Our basic assumptions are that the utility of a firm for joining a particular standard-setting alliance increases with the size of the alliance and decreases with the presence of rivals in the alliance, especially close rivals. The predicted alliance configurations are simply the Nash equilibria, i.e., those sets of alliances for which no single firm has an incentive to switch to another alliance. We illustrate our theory by estimating the choices of nine computer companies to join one of two alliances sponsoring competing Unix operating system standards in 1988.
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A sufficient condition for the existence of a pure-strategy coalition-proof Nash equilibrium in a strategic game is established.
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In an important class of “noncooperative” environments, it is natural to assume that players can freely discuss their strategies, but cannot make binding commitments. In such cases, any meaningful agreement between the players must be self-enforcing. Although the Nash best-response property is a necessary condition for self-enforceability, it is not sufficient—it is in general possible for coalitions arrange plausible, mutually beneficial deviations from Nash agreements. We provide a stronger definition of self-enforceability, and label the class of efficient self-enforcing agreements “coalition-proof.”
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We provide a selective overview of the literature on standardization. We first summarize the essential mechanisms underlying the economics of product compatibility and in so doing we review the various frameworks that have been used. Then we survey existing work about the consequences of compatibility on entry deterrence and technological progress. We finally investigate some implications of the literature for trade policy in the presence of network externalities and suggest some directions for future research.
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This paper examines the conditions which guarantee that the set of coalition-proof Nash equilibria coincides with the set of strong Nash equilibria in the normal form games without spillovers. We find that population monotonicity properties of the payoff functions, when the payoff of a player changes monotonically when the size of the group of players choosing the same strategy increases, are crucial to obtain the equivalence of these two solution concepts. We identify the classes of games, satisfying population monotonicity properties, which yield the equivalence of the set of coalition-proof Nash equilibria and the set of strong Nash equilibria. We also provide sufficient conditions for the equivalence result even when the population monotonicity assumptions are relaxed.
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The formulation of associations of firms in an oligopoly with linear demand is analyzed as a two-stage noncooperative game. In the first stage, firms form associations in order to decrease their costs, and in the second stage they compete on the market. Examples of associations include R&D joint ventures and groups of firms adopting common standards. In equilibrium, the associations formed exhibit two general features: they are asymmetric and inefficient.
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In this paper we show that a non-cooperative game with a finite set of players and common finite strategy sets possesses a strong Nash equilibrium in pure strategies whenever individuals' preferences satisfyindependence of irrelevant choices,anonymity, andpartial rivalry. Moreover, if any of these assumptions is violated, then even a pure strategy Nash equilibrium may fail to exist. Further- more, we demonstrate that even with a continuum of players, the same three assumptions yield the existence of a pure strategy strong Nash equilibrium and, in addition, the equivalence of the sets of Nash and strong Nash equilibria in pure strategies.Journal of Economic LiteratureClassification Numbers: C72, D62.
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This paper argues that the sign of external effects of coalition formation provides a useful organizing principle in examining economic coalitions. In many interesting economic games, coalition formation creates either negativee externalities or positive externalities for nonmembers. Examples of negative externalities are research coalitions and customs unions. Examples of positive externalities include output cartels and public goods coalitions. I characterize and compare stable coalition structures under the following three rules of coalition formation: the Open Membership game of Yi and Shin 1995 , the Coalition Unanimity game of Bloch 1996 , and the Equilibrium Binding Agreements of Ray and Vohra 1994.
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We analyze a two-stage non-cooperative game where the firms choose first to adopt (either simultaneously or sequentially) one of two network technologies, and then compete on the market. The two-stage procedure and the assumption that firms have heterogeneous tastes with respect to the technologies lead to a novel treatment of network externalities. In particular, as the network of some firm enlarges, the change in this firm's payoff is shown to depend both on the newcomer's identity and on the composition of the networks and, as a result, is not necessarily positive.
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This paper analyzes a sequential game of coalition formation when the division of the coalitional surplus is fixed and the payoffs are defined relative to the whole coalition structure. Gains from cooperation are represented by a valuation which maps coalition structures into payoff vectors. I show that any core stable coalition structure can be attained as a stationary perfect equilibrium of the game. If stationary perfect equilibria may fail to exist in general games, a simple condition is provided under which they exist in symmetric games. Furthermore, symmetric stationary perfect equilibria of symmetric games generate a coalition structure which is generically unique up to a permutation of the players. A general method for the characterization of equilibria in symmetric games is proposed and applied to the formation of cartels in oligopolies and coalitions in symmetric majority games.Journal of Economic LiteratureClassification Numbers : C78, C71.
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The authors derive a necessary and sufficient condition for the solution set of an optimization problem to be monotonic in the parameters of the problem. In addition, they develop practical methods for checking the condition and demonstrate its applications to the classical theories of the competitive firm, the monopolist, the Bertrand oligopolist, consumer and growth theory, game theory, and general equilibrium analysis. Copyright 1994 by The Econometric Society.
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Formation of coalitions in oligopolies is modelled as a non-cooperative game in which firms' strategies have a binary form (to cooperate or not). We demonstrate a one-to-one correspondence between stable cartels defined by d'Aspremont et al. (1983) and the Nash equilibria of this game. Using a Coalition-Proof Nash Equilibrium' (CPNE), we define the concept of a coalition-proof stable cartel and prove that there is a unique equilibrium. /// Le comportement cyclique des salaires et des profits en régime de concurrence monopolistique. Cet article propose de modéliser la formation des coalitions dans les oligopoles comme un jeu non coopératif dans lequel les stratégies des firmes ont une forme binaire (coopérer ou ne pas coopérer). Nous montrons qu'il existe une correspondance une à une entre les cartels stables définis par d'Aspremont et al. (1983) et les équilibres de Nash du jeu. En utilisant l'équilibre de Nash robuste aux déviations des coalitions (plus communément désigné par le sigle CPNE), nous définissons le concept de cartel stable robuste aux déviations des coalitions et démontrons l'existence d'un unique équilibre. Classification JEL: C71, C72, L22
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This paper identifies a domain of payoff functions inno spillovernoncooperative games withPositive externalitywhich admit a pure strategy Nash equilibrium. Since in general a Nash equilibrium may fail to exist, in order to guarantee the existence of an equilibrium, we impose two additional assumptions,AnonymityandOrder preservation. The proof of our main result is carried out by constructing, for a given gameG, a potential function Ψ over the set of strategy profiles in such a way that the maximum of Ψ yields a Nash equilibrium in pure strategies ofG.Journal of Economics LiteratureClassification Numbers: C72, D62, H73.
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The ability of the members of a coalition to communicate secretly determines whether the coalition can coordinate to deviate from a proposed strategy and thus affects which strategies are “coalition proof.” We show that the existence of a Pareto-best element in the set of strategies that survive iterated elimination of dominated strategies implies the existence of a coalition-proof correlated equilibrium for any specification of coalitional communication possibilities that always permits individual deviations. Such an element exists in games with strategic complementarities if either (1) there is a unique Nash equilibrium or (2) each player's payoff is nondecreasing in the others' strategies.Journal of Economic Literatureclassification number: C72.
Acceptable points in general cooperative n-person games Coalition formation in standard-setting alliances
• Ž Aumann
• R Axelrod
• R Mitchell
• W Thomas
• R Bennett
• D S Bruderer
Ž. Aumann, R. 1959 .   Acceptable points in general cooperative n-person games,'' Ann. Math. Stud. 40, 287?324. Axelrod, R., Mitchell, W., Thomas, R., Bennett, D. S., and Bruderer, E. 1995 .  ` Coalition formation in standard-setting alliances,'' Management Sci. 41, 1493?1508
Equivalence of strong and coalition-proof Nash equilibria in games without spillovers
• Konishi