Stability of Cartels in Multimarket Cournot Oligopolies

We investigate the stability of cooperation agreements, such as those agreed by cartels, among firms in a Cournot model of oligopolistic competition embedded

Other authors have demonstrated the sensitivity of the result on the precise order or sequencing of decisions and the presence of outside competitive pressure. Following an approach pioneered by d 'Aspremont, Jacquemin, Gabszewicz, and Weymark (1983) for example, Shaffer (1995), Konishi and Lin (1999) and Zu, Zhang, and Wang (2012) demonstrate the existence of a stable cartel where the cartel is a Stackelberg quantity leader and all non-member firms of the cartel are Cournot competitive with respect to the residual demand.
Instead, we focus on environments in which firms operate on multiple, strongly related marketsdenoted as multimarket oligopolies. In these multimarket oligopolies, we investigate the endogenous composition and location of cartel structures. The setting of multiple markets on which these firms operate can be interpreted as their presence at multiple geographical locations or as representing the separation between different media through which trade is conducted. The latter might refer to a traditional sales technology through stores versus virtual sales through online web stores. We make a contribution to this literature by demonstrating that stable agreements are sustained in a standard Cournot oligopoly if firms operate on multiple (strongly related) markets and that these agreements involve a strict subset of firms operating on those markets.
For this insight, we do not rely on the agreeing cartel coalition having a quantity-setting leadership position. We apply convex production costs, creating both strategic substitutes and diseconomies of scope across both markets-using the terminology of Bulow, Geanakoplos, and Klemperer (1985). 1 Our model is therefore related to that of Zhang and Zhang (1996). These authors provide conditions for the existence of Cournot-Nash equilibrium in multimarket environments. However, they do not consider the possibility of cooperative agreements among market participants. Such cartel formation is instead the focus of our work.
A key element in any analysis of cartel formation is what constitutes a sustainable cooperation agreement leading to a stable cartel. For this purpose, we choose to employ the notion of core stability as it best captures the range of deviation possibilities of the firms-unilaterally or as a group; and the concept of self-enforcing, binding agreements. A core stable configuration is one in which there are no incentives for any group of firms to deviate, either by forming an alternative cartel or individually. We build our notion on the premise that the deviating firms would expect the non-deviating firms to maintain the status-quo, i.e. by definition the non-deviating firms are assumed to be passive and, therefore, the market structures involving non-deviating players are not expected to change. Belleflamme and Bloch (2008) study conditions for sustainable cooperation between two firms in a symmetric two-market setting. Our work differs from theirs in that we allow for cartel formation among three firms in possibly asymmetric markets, i.e. cartels may be formed by fewer than all market participants and the two markets may differ in size. This allows us to provide further insights on the degree of cooperation and the location of the cartel based on the relative market size.

| Related literature
While multimarket oligopolies have been used to analyse corporate espionage (Billand, Bravard, Chakrabarti, & Sarangi, 2016) and taxation (Lapan and Hennessy, 2011), we believe we are the first to examine cartel stability in multi-market oligopolies.

CHAKRABARTI eT Al.
We note here that our underlying cartel formation game is akin to a partition function form game-a model that has been studied in cooperative game theory. In a partition function form game-as in our cartel formation setup-the value a coalition of players can generate in cooperation depends on the structure of cooperative agreements among the players outside of this coalition. Our multimarket cartel formation model, however, is more general than a partition function form game as players operate on multiple distinctive market settings: each firm may be part of multiple agreements-one on each market; the structures of cooperative agreements on each market may differ; and all market structures affect the payoffs of all cartels and all individual firms.
Concerning the fundamental notion of core stability as pursued here, we note that this notion originates in the theory of cooperative games in partition function form, seminally, by Chander and Tulkens (1997). More recently, Abe and Funaki (2017) examine the optimistic and pessimistic core of such games. In the pessimistic core, the deviating coalition assumes the worst reaction from the remaining non-deviating players. In the optimistic core, it assumes the best reaction from the non-deviating players. They also consider notions of the core where the non-deviating players are expected to dissolve their coalitions into singletons or form the largest possible coalition. This approach is not directly applicable to our setup, however, as in our model profits among cartel members are non-transferable.
As an alternative approach to the one we have taken-build on a standard cooperative game theoretic concept-one could consider a non-cooperative game theoretic approach to coalition formation. We mention a few papers that have taken this approach. A comprehensive review of this literature is available in Yi (2003). Bloch (1996) puts forward the following non-cooperative procedure. Based on an exogenous order, players propose coalitions which other coalition members can accept or reject. The first player to reject the offer makes a counter-offer in the next period and so forth. If all players agree, the coalition is formed and the coalition is not allowed to subsequently break-up or accept new members, and the remaining players continue the coalition formation process. Bloch (1997Bloch ( , 2002 shows that the equilibrium coalition structure corresponds to the equilibrium of a certain size announcement game and involves a no-delay equilibrium in which all proposals are accepted. In the homogeneous Cournot model, this procedure results in the formation of a single cartel. Yi and Shin (2000) and Bloch (2002) consider a different procedure where players simultaneously decide whether or not to join the coalition: Each player announces an address simultaneously, and players with the same address belong to the same coalition. Ray and Vohra (1997) analyse a coalition formation game in which coalitions can only break up into smaller sub-coalitions. They define stable coalition structures as follows: A degenerate coalition structure in which all coalitions are singletons is stable by definition. A non-degenerate coalition structure is stable if no coalition has a sub-coalition-called a coalition of leading perpetratorswhose members have incentives to initiate a break-up of the coalition structure. Players are far-sighted and foresee the final stable coalition structure that will form after further subsequent break-ups. The leading perpetrators will only initiate a break-up if they are better off in the final stable coalition structure. Ray and Vohra (1999) define an extensive form bargaining game similar to Bloch (1996) that yields such a stable coalition structure in the subgame perfect stationary equilibrium. One difference with Bloch (1996) is that while in the latter, coalitional worth is divided according to a fixed rule, in the former, how the worth is divided is part of the bargaining process. In the homogeneous Cournot model, once again a unique cartel emerges.
The advantage of the approach we take compared over the non-cooperative game theoretic analysis, is in its generality: Unlike non-cooperative concepts, the core is not tied to a specific procedure for the sharing of the coalitional value or pre-determined sequence of deviations.
In addition, our analysis is built on the premises on non-transferability of payoffs. In this respect, our work is related to the literature on hedonic coalition formation games. These game theoretic models were introduced by Bogomolnaia and Jackson (2002), further analysed by Banerjee, Konishi, and Sonmez (2001). In this setting, the players' preferences are ordinal and they are defined over coalition memberships. Various notions of stability have been defined in this setting. Individual stability refers to the situation where no player wants to leave her coalition for another one (including the empty coalition). Nash stability is a stronger version of individual stability where players can join any new coalition without the permission of existing members. More demanding than these is core stability, where multiple players can deviate to form a new coalition. While in hedonic games, a player's payoff depends only on the membership of her coalition, the approach can be extended to non-hedonic games where a player has a preference mapping over all possible partitions of the player set. Core stability can be extended to this setting which is what we are doing here.

| Structure of the paper
The next section presents the multimarket Cournot model and key concepts for its analysis. In Section 3, we present the main results on core stable structures and discuss the role of convexity. Section 4 contains a brief discussion of other assumptions and outlays paths for future work.

| THE MODEL
We explore a setting where three firms are selling an identical product in multiple, separate markets. The firms are labelled a, b and c and we denote the set of firms by N = {a, b, c} with i being a generic firm in the set N. We let the set of markets be given by  = {M 1 , M 2 , … , M m }, using the indicator k ∈ {1, …, m} to refer to market M k ∈ .
We assume that all firms in N are quantity-setters; that is in the absence of cooperation agreements these firms compete á la Cournot in quantities on all markets in . We denote by q ik , the quantities sold by firm i in market M k . Furthermore, we let p k stand for the market price emerging on market M k .
We assume that competitors' products are substitutes in all markets and there are diseconomies of scope across these markets. More specifically, markets are characterized by inverse linear demand functions specified for M k as where 0 < 1 < 2 < ⋯ < m are demand parameters capturing each market's size, respectively. Hence, all markets are ordered in terms of their size, where market M 1 is the smallest market and market M m is the largest market.
We assume that firms produce under an identical quadratic cost function given by We remark that from the postulated linearity of the market demand, it follows that the total revenues of a firm are simply the sum of the firm's revenues in each market. In contrast, there is no such market separability in the postulated quadratic cost function. This implies that firm i's total profit can be expressed as For this linear-quadratic formulation, the second-order conditions for a maximum are always satisfied if k 's are not too different. 2 Hence, one obtains a unique interior maximum through consideration of the first-order conditions.

| Cartel formation
We consider a general cooperation framework in which any subset of the postulated three firms may choose to form a cartel in any market. Thus, a cartel is any coalition of at least two firms in any one of the m markets in .
Note that in principle any given cartel operates in a single market, but we emphasize that the same group of firms may form a cartel in other markets as well. 3 Equally, there may be distinct groups that form cartels on different markets. Thus, whereas each firm can be a member of only one cartel in a given market, 4 the same firm may be a member of more than one cartel, each of which operates in a different market and has a distinct membership.
Such cartel formation is formalized as follows.

Definition 1 A market configuration is a listing of individual market structures
For every firm i ∈ N, we denote by i( k ) the status of firm i in the structure k in market M k given by the group A market structure imposes on each market a partitioning of all firms that describes the competitive structure in that particular market. For example, consider m = 2 and the following market structure: For instance, if k = 2, we require 1 > 1 5 2 , otherwise all firms will produce zero output in market M 1 . 3 We re-visit our modelling strategy of a cartel as being bound to a single market in the discussion section.

| Cartel objectives
The objective of a cartel operating on a specific market is in principle to maximize the joint profits of its members. Thus, members of the cartel commit to an agreed production level-output quotasdetermined for this particular market, given their output levels in all other markets. Here, we assume that the standard Cournot competitive hypothesis applies: All members of a cartel operating in a certain market decide their production levels jointly in order to maximize their total joint profits subject to the production decision of every firm outside the cartel (if any) on this particular market and the decisions made by all firms in the other markets.
Throughout, we assume that monetary transfers among cartel members are not possible. In particular, we focus our analysis on symmetric equilibria, making this assumption innocuous. Indeed, all firms in our model face the same demand in each market and use the same production technology; thus, all cartel members earn equal profits in the market where the cartel operates. Hence, there is no scope for transfers.
We formalize the behaviour of cartels and individual firms as follows.
Assumption 1 Let ∅ ≠ S ⊂ N = {a, b, c} be a group of firms in market M k . Denote by q k S = (q ik ) i∈S the vector of output levels of group members in market M k and by q k −S = (q jk ) j∉S the vector of output levels of all firms outside S in market M k .
Then the group S forms a cartel in market M k by maximizing the group's collective profit over the production decisions q k S , solving the optimization problem given by Consider a market configuration = ⟨ 1 , … , m ⟩. Then for every k ∈ {1, …, m} every S ∈ k is assumed to determine its collective output levels by solving the objective problem stated in (2). The immediate consequence of Assumption 1 is that every cartel or individual firm in a market configuration for market M k acts to maximize its collective profits over the output levels in that particular market given the output decisions of all non-cartel members in market M k and of all firms in all other markets.
Example 1 To illustrate the importance of the market configuration in deriving firms' optimal decisions in context of the decision objective (2) In market configuration Ω firm a acts competitively in both markets and, thus, firm a chooses its quantity vector (q a1 , q a2 ) to maximize its total profits as given by: Like firm a, firms b and c choose their respective outputs on market M 1 unilaterally. Thus, each firm i ∈ {b, c} chooses q i1 by solving the following maximization problem: (2) max In market M 2 , instead, firms b and c choose their respective outputs (q b2 and q c2 ) jointly. Thus, they consider their joint profit maximization problem: The optimization problems stated above form an exhaustive and proper description of all decision processes in this particular configuration.

| Core stable market configurations
To determine the equilibrium number, size, composition, and location of cartels, we adopt the notion of core stability based on the equilibrium notion of the core to the specific cartel formation model we study here. Before we present a formal definition of our core stability concept, we introduce some auxiliary notions that are used to clarify possible deviations by firms.
Consider k ∈ {1, …, m} and S⊂N. We say that the coalition S can transition from structure k to structure ̂ k on market M k if for all members i ∈ S : i( � k ) ⊆ S and for all non-members When coalition S can transition from k to ̂ k in market M k , we denote this by k → Ŝ k . A coalition S can transition from one structure to another in a certain given market if its members can abandon existing groups that they are member of and form alternative cartels with other coalition members.
A market configuration Ω is now "core stable" if there is no coalition S in any market that has the ability as well as the proper incentives to transition to an alternative structure. This is formalized as follows.
Definition 3 A market configuration = 1 , … , m is core stable if there does not exist a coalition (either a cartel or a singleton) S ⊆ N and an alternative market configuration ̂ = ⟨̂ 1 , … ,̂ m ⟩ such that 1. for every market M k with ̂ k ≠ k , coalition S can transition to ̂ k from k , i.e. k → Ŝ k , and 2. i (̂ ) ⩾ i ( ) for every i ∈ S and j ( � ) > j ( ) for at least one j ∈ S.
Our core stability notion presumes that a deviating coalition S can form any arbitrary partition among its own members in any market. If, by doing so, it can make one of its members strictly better off and the other members no worse off, the original market configuration is not core stable.

| IDENTIFICATION OF CORE STABLE CARTELS
To set up a benchmark for our multimarket analysis, we first present the result for core stable cartel configurations in a single market. This confirms that indeed a paradox emerges about the benefit of max q b2 , q c2 cartel formation under Cournot competition. Subsequently, we consider the case of two markets and show that a unique core stable configuration emerges in which cartels with the same membership form on each market.

| Core stability in one market
In the single market case with linear demand and quadratic cost functions, we show that there are no core stable market configurations. The merger paradox lies at the very foundation of this non-existence result, as firms find it profitable to transition away from a two-firm cartel. 5 The required analysis considers three fundamental market structures: The reaction function of each firm i simplifies to: where we adopt the notation Q −i = ∑ j≠i q j . The solution to the system of three equations gives us the equilibrium quantities q i = 5 . Thus, each firm's profit in this fully competitive environment equals i = 3 2 50 . 2 = {N}: Next, we consider the case when all three firms form a single monopolistic cartel and agree on their output quotas. The joint maximization problem is formally given by: The optimal quantity profile is given by the solution of the following system of first-order conditions: Thus, we arrive that at the optimum quantities: q a = q b = q c = 7 with respective profits i = 2 14 for all cartel members i ∈ N. 3 = { {a}, {b, c} }: Last, we consider a market configuration with a two-firm cartel. There are three possible cartels consisting of two firms, {a, b}, {a, c}, and {b, c}. Given the symmetry of firms, however, payoffs for cartel members and the outsider firm do not depend on the identity of the firms in the cartel. We therefore present here only the case when firms b and c operate in a cartel and firm a behaves competitively. 5 We acknowledge that a comprehensive analysis of the single market for a finite number of firms with linear demand and quadratic costs has been conducted by Amir and Stepanova (2009). Our set up of market demand and firm cost function are a special case of those considered by Amir and Stepanova (2009) with the value of their model parameters being b = 1, c = 0, and d = 1 2 . 6 We use the simplified notation = 1 in the case of a single market.
The profit maximization problem of the cartel members is given by: The two first-order conditions of the cartel optimization problem are: The solution to these two linear equations implies that at the optimum q b = q c = −q a 5 . The optimization problem of firm a which is outside of the cartel is given by (4) with the reaction function given by (5). Using these reaction functions, we arrive at the optimal quantity for the non-cooperating firm a being q a = 3 13 and for the cartel firms being q b = q c = 2 13 . Thus, the cartel members earn b = c = 10 2 169 , while firm a earns a = 27 2 338 . We now turn to the analysis of the stability properties of each market configuration: 1. Fully competitive conditions lead to payoffs that are lower than monopolistic cartel formation, i.e. i ( 1 ) = 3 2 50 < i ( 2 ) = 2 14 .
We conclude that in each of the three possible market configurations of a single market there is a deviating coalition: The grand coalition provides deviating possibility in the absence of cartel; any one firm will deviate from the grand coalition with the other two firms in a cartel; and any firm member of a two-firm cartel deviates to be a singleton. This implies that there indeed does not exist a core stable market configuration in a single market with three firms.

| Core stability in two markets
Consider the specific case where m = 2 with  = {M 1 , M 2 }. Furthermore, we recall that 1 < 2 . Even in this simplified setting, there emerge a relatively large number of non-trivial cartel formation scenarios regarding the degree of cooperation and its market location. Exploiting the homogeneity of firms, we note that agreements are distinguishable along two dimensions-the size of the cooperating group and the market on which cooperation occurs. In the case of two markets and three firms considered here, we arrive at 64 possible market configurations. Combining these possible market structures, we arrive at 10 exhaustive market configurations, denoted by n = ⟨ n 1 , n 2 ⟩ for n ∈ {1, …, 10}. These configurations are collected in Table 1.
Alternatively, we can dispense with the symbolism ϕ and show directly how these ten market configurations emerge as a result of different market structures in markets M 1 and M 2 . This is represented in the following

| Payoff schedules
An individual firm's decision on whether to enter into a cartel agreement with other firms, where to locate the agreed cartel, and whether to leave a cartel is based solely on that firm's profits. Before developing a complete analysis, we illustrate the computations involved by considering the particular market configuration discussed in Example 1. Example 1 considers a market configuration that is denoted as 6 = ⟨{ {a}, {b}, {c} },{ {a}, {b, c} }⟩ in the discussion above. In the series of profit maximization problems in Example 1 that describe how firms take their decisions in the context of market configuration 6 , result into a system of six first-order conditions that are relevant for this case. More specifically, the optimization problem for firm a, which acts non-cooperatively on both markets, results in the following reaction functions:  Similarly, the optimization problems that describe the decision making for firms b and c, respectively, on market M 1 yield the following reaction functions for market M 1 : Last, the joint maximization problem that the cartel bc resolves on market M 2 leads to the following first-order conditions: Solving the resulting system of six equations, we arrive at the following optimal quantity profiles: Clearly, firms b and c curtail their outputs at the optimum in the interest of their joint profits. This results into the following total firm profits: Similarly straightforward, though fairly tedious, computations result in the determination of all profits for cartel members and outsiders in each market configuration. These payoff schedules are collected in Table 2. 7 In the table, the index n ∈ {1, …, 10}, n refers to the market configuration listed in Table 1. Some configurations accommodate multiple Cournot equilibria. Here, in the interest of comparability across configurations and following the adopted convention in the literature, we focus on symmetric equilibria only. This leads to a unique payoff for each player in the game for each configuration. Due to the symmetry of the firms, we only make a distinction between the profits of cartel members and outsiders. For the sake of clarity we use a subscript C to denote the profits of a cartel member and a subscript O to denote the profits of an outsider. Hence, subscript C indicates a firm that is a member of at least ( 2 − 1 ) q b2 = q c2 = 3 98 (5 2 − 1 ) a ( 6 ) = 297 2 1 − 128 1 2 + 418 2 2 4802 b ( 6 ) = c ( 6 ) = 3(375 2 1 − 26 1 2 + 359 2 2 ) 19208 one cartel; subscript O identifies a firm that is not a member of cartel in any market. Furthermore, we use superscripts to denote the location of the cartel in the reported equilibrium: superscripts 1 or 2, refer to cartels on markets M 1 or M 2 , respectively (but not the other), while superscript 1, 2 indicates that firm is a member of cartels on both markets.

| Core Stable configurations
In contrast to the single market case, in the two-market setting a stable cartel agreement exists provided that markets are sufficiently similar in size. 8 Interestingly, that core stable agreement is unique and hinges upon the existence of cartels with the same membership in both markets. Intuitively, our result is achieved without the need to rely on punishment strategies, entry deterrence, or intertemporal considerations discussed in the literature. Below we present formally our main result.
Proposition 1 For 379 523 2 < 1 < 2 there exists a unique core stable market configuration given by Sketch of a Proof Assuming 379 523 2 < 1 < 2 we determine exactly the incentivized transitions that coalitions can establish between the different structures in the two markets. Using tedious computations, the transitions between configurations in 1 , … , 10 can be determined that are feasible-stated as condition (i)