Equilibrium in a Failed Cartel: Set-Valued Reactions and Sources of Market Asymmetry

1. Introduction

Many mathematical models arising in economics, game theory, operations research, engineering, and network sciences can be formulated as systems of coupled optimization problems. Rather than maximizing a single objective, several decision makers optimize individual objective functions whose values depend on the decisions of the remaining participants. Consequently, an equilibrium is characterized by the simultaneous compatibility of several interacting optimization problems. This viewpoint underlies a wide spectrum of mathematical theories, including competitive equilibrium, Nash games, generalized Nash equilibrium problems, variational inequalities, complementarity problems, and hierarchical optimization [1,2,3,4].

A central mathematical issue is that an optimization problem need not possess a unique solution. Given a continuous function on a compact feasible set, the Weierstrass theorem guarantees the existence of a maximizer, but it does not guarantee that this maximizer is unique. Consequently, the assignment that associates each parameter value with the set of all optimal solutions is generally a set-valued mapping, or correspondence. In particular, a parametric maximization problem of the form ${max}_{y\in K\left(x\right)}f(x,y)$ generates the argmax correspondence $A\left(x\right)={arg\; max}_{y\in K\left(x\right)}f(x,y)$.

Such correspondences arise naturally as reaction sets, best-response sets, optimal control sets, solution mappings of variational problems, and sets of admissible optimal decisions. Thus, the study of optimization problems with non-unique solutions leads naturally to set-valued analysis [4,5,6].

Parametric optimization also raises the question of how optimal solution sets change when the parameters of the problem vary. Berge’s Maximum Theorem provides a classical answer: under appropriate continuity and compactness conditions, the optimal value depends continuously on the parameters, while the corresponding argmax mapping is upper semicontinuous and compact-valued [5,6]. Upper semicontinuity expresses the stability of optimal solutions under perturbations: if a sequence of parameter values converges and a corresponding sequence of maximizers also converges, then its limit remains a maximizer of the limiting optimization problem. This property is essential when several parametric optimization problems are combined into a common equilibrium system.

Fixed point theory provides the natural mathematical framework for such systems. When every optimization problem has a unique solution, the associated best-response operator is single-valued, and the existence of an equilibrium may be studied through classical fixed point results such as Brouwer’s theorem. When optimal solutions are not unique, however, the best-response operator becomes set-valued. In that case, Brouwer’s theorem is replaced by fixed point principles for correspondences, most notably the Kakutani–Fan–Glicksberg theorem [7,8,9,10]. These results guarantee a fixed point for an upper semicontinuous correspondence with nonempty, compact, and convex values defined on a nonempty compact convex set. The fixed-point formulation therefore converts a system of mutually dependent optimization problems into an equilibrium existence problem.

The distinction between single-valued and set-valued responses is important both mathematically and conceptually. A single-valued best response assigns one optimal decision to each configuration of the other participants’ decisions. A set-valued best response retains all optimal decisions and therefore describes the optimization problem without imposing an artificial selection rule. Moreover, single-valuedness of all reaction mappings does not by itself imply uniqueness of equilibrium: a continuous single-valued best-response operator may have several fixed points. Additional conditions, such as contractivity, strict monotonicity, or diagonal strict concavity, are required to obtain uniqueness [3,4].

Cartels provide a particularly relevant economic setting in which these mathematical issues arise. Cartels occupy a central position in industrial organization because they illustrate the fundamental tension between cooperation and competition. By coordinating production decisions, firms may collectively restrict output and obtain profits exceeding those achieved under non-cooperative competition. Such agreements, however, are inherently unstable. Each participating firm faces an incentive to deviate from the agreed production plan, expanding its own output while continuing to benefit from the reduced aggregate production of the remaining cartel members. Understanding the formation, stability, and collapse of collusive agreements has therefore been a longstanding objective of oligopoly theory and competition economics [11,12,13,14,15,16,17].

The classical framework for quantity competition is the Cournot model, in which firms simultaneously choose production levels while treating the outputs of their competitors as fixed [18,19,20]. An alternative strategic organization is provided by the Stackelberg model, where firms move sequentially and later players optimally respond to the production decisions of earlier movers [19,20,21]. Numerous extensions of these models have incorporated heterogeneous technologies, asymmetric information, dynamic interaction, and mixed sequential–simultaneous competition [22,23,24]. Despite this extensive literature, comparatively little attention has been devoted to markets in which the original collusive structure only partially survives after one participant abandons the cartel.

Such partially coordinated markets arise naturally in practice. Cartels need not collapse instantaneously. One firm may abandon an agreement because of changing market conditions, regulatory intervention, financial incentives, or expectations of higher individual profit, while the remaining firms preserve some elements of their previous strategic relationship. The resulting market is neither a classical Cournot oligopoly nor a fully coordinated cartel. Rather, it represents an intermediate strategic organization combining features of cooperation, simultaneous competition, and sequential response. Understanding the equilibrium structure of such markets is important for explaining how production, market shares, and strategic advantages are redistributed following the partial breakdown of a cartel.

The present paper develops a mathematical framework for analysing this hybrid market structure, which we refer to as a failed cartel1. We consider a homogeneous-good market with three firms. Two firms act as Cournot leaders, while the third firm is distinguished as a follower whose production level belongs to the set of optimal responses to the quantities selected by the first two firms. The leaders’ and follower’s decisions are therefore represented by mutually dependent optimization problems. This construction connects Cournot-type simultaneous optimization with a follower-response mechanism, while allowing the strategic asymmetry generated by cartel defection to be studied in a unified setting.

A fundamental difficulty is that the optimal production decisions of the firms need not be unique. In standard presentations of Cournot and Stackelberg competition, firms are frequently represented by single-valued reaction functions obtained from first-order optimality conditions. Such a representation is justified only when every firm’s optimization problem has a unique maximizer. If strict concavity or differentiability is absent, several production levels may yield the same maximal profit. The follower’s reaction is then a set

$$R_3(x_1,x_2)=\underset{x_3\in X_3}{arg\; max}{\Pi }_3(x_1,x_2,x_3),$$

and the leaders’ best responses are likewise described by argmax correspondences. The equilibrium problem must consequently be formulated as a fixed-point problem for a product of set-valued mappings.

This issue is not merely a technical possibility. The existence and uniqueness of Cournot equilibria have been the subject of extensive research. A large body of literature has established conditions under which equilibrium exists or is unique by imposing generalized concavity assumptions on inverse demand, monotonicity properties of marginal revenue, supermodularity, or diagonal strict concavity [3,25,26,27,28,29]. These studies identify structural assumptions that guarantee quasi-concavity of firms’ profits, existence of maximizers, single-valued reactions, or uniqueness of equilibrium. They also demonstrate that these properties are logically distinct: the existence of an equilibrium does not require that every best response be unique, and the uniqueness of every individual maximizer does not automatically imply that the equilibrium itself is unique.

Our perspective is therefore to retain, rather than eliminate, the possible multiplicity of optimal responses. We represent each participant’s behavior by the full set of maximizers of its payoff function. This formulation preserves all optimal decisions and avoids imposing an arbitrary selection from an argmax set. From the mathematical point of view, the model becomes a system of interacting set-valued optimization correspondences. From the economic point of view, it allows a firm to possess several equally profitable production responses to the same market configuration.

The analysis is deliberately developed in two stages. First, we formulate an abstract existence theorem solely in terms of three continuous payoff functions

$${\Pi }_i:X_1\times X_2\times X_3\to \mathbb{R},i=1,2,3,$$

and their associated argmax correspondences. Under compactness, convexity, continuity, and quasi-concavity assumptions, the three optimization problems generate nonempty, compact, convex-valued, and upper semicontinuous response correspondences. Their product therefore possesses a fixed point by the Kakutani–Fan–Glicksberg theorem. This fixed point simultaneously solves the three individual optimization problems.

Second, the abstract theorem is applied to the profit functions

$${\Pi }_i(x_1,x_2,x_3)=x_{i}P(x_1+x_2+x_3)-c_i\left(x_i\right),i=1,2,3.$$

Generalized concavity of the inverse demand function, together with convexity and monotonicity of the cost functions, is used to verify the quasi-concavity conditions required by the abstract result. In this manner, the economic existence theorem is obtained as a direct consequence of a more general optimization and fixed-point principle. The separation between the abstract result and its Cournot application also makes the mathematical framework applicable to other systems of interdependent payoff functions.

The abstract formulation is not restricted to oligopoly theory. Similar structures arise whenever several agents solve coupled parametric optimization problems and the solution of each problem may be non-unique. Examples include network and congestion games, resource-allocation models, hierarchical decision systems, engineering design problems, generalized Nash equilibrium problems, and control models with set-valued optimal feedback. Although the present paper focuses on a failed-cartel application, its main existence argument concerns fixed points of product correspondences generated by interacting optimization problems.

The use of set-valued responses also provides a transparent separation between existence, single-valuedness, and uniqueness. General continuity, compactness, convexity, and quasi-concavity assumptions are sufficient for existence. If each argmax set is a singleton, the response correspondences reduce to continuous reaction functions, and the equilibrium problem becomes a fixed-point equation for a single-valued operator. Nevertheless, uniqueness of this fixed point requires further assumptions. In differentiable concave games, Rosen’s diagonal strict concavity condition provides one important route to uniqueness [3,4]. Thus, uniqueness appears as an additional structural property rather than as a prerequisite for defining the model.

Beyond the abstract existence problem, we investigate the economic consequences of a partial cartel breakdown. We identify two distinct sources of asymmetry in equilibrium production levels. The first is technological asymmetry, arising from differences in firms’ production costs. In this case, firms with lower marginal costs may produce larger equilibrium quantities independently of their strategic roles. The second is strategic asymmetry, which may persist even when all firms possess identical production technologies. In the failed-cartel market, the different strategic positions of the firms affect the structure of their optimization problems and may therefore alter the ordering of equilibrium quantities. Technological and strategic effects may reinforce one another, offset each other, or arise independently.

The contributions of the paper are both mathematical and economic. First, we establish an abstract equilibrium existence theorem for three interacting optimization problems whose solution mappings may be genuinely set-valued. Second, we derive a failed-cartel Cournot model as an application of this theorem by verifying the required quasi-concavity properties through generalized concavity of inverse demand and convexity of costs. Third, we examine the single-valued case separately and explain why uniquely determined reaction functions do not, without additional assumptions, guarantee a unique equilibrium. Fourth, we provide illustrative examples in which the reaction sets contain nondegenerate intervals and the model possesses more than one equilibrium with positive profits. Finally, we analyse the roles of technological and strategic asymmetry in determining equilibrium production levels.

The remainder of the paper is organized as follows. Section 2 introduces the mathematical preliminaries and the failed-cartel framework, including parametric optimization, generalized concavity, Berge’s Maximum Theorem, and the Kakutani–Fan–Glicksberg fixed point theorem. Section 3 formulates the abstract equilibrium theorem and derives the Cournot failed-cartel result as a consequence. It also discusses the reduction to single-valued reaction functions and conditions related to uniqueness. Section 4 presents illustrative examples, including cases with multivalued reaction correspondences, multiple equilibria, and strictly positive equilibrium profits. Section 5 studies the effects of technological and strategic asymmetry. Section 6 concludes the paper and outlines directions for future research.

2. Materials and Methods

The mathematical framework developed in this paper is based on the interaction of three coupled optimization problems. Each firm seeks to maximize its own payoff while taking into account the decisions of the remaining firms. Since the corresponding optimization problems may admit multiple optimal solutions, the natural mathematical objects are set-valued reaction correspondences rather than single-valued reaction functions. This viewpoint allows the equilibrium problem to be formulated as a fixed-point problem for a product correspondence.

We begin by introducing the market structure and the associated payoff functions. These functions determine the follower’s reaction correspondence and the leaders’ best-response correspondences through parametric maximization. The existence of an equilibrium will subsequently be derived from general results on parametric optimization, upper semicontinuous correspondences, and fixed-point theory.

Throughout the paper, we use the standard notation ${\mathbb{R}}_{+}=\{x\in \mathbb{R}:x\ge 0\}$ for the set of nonnegative real numbers and ${\mathbb{R}}_{++}=\{x\in \mathbb{R}:x>0\}$ for the set of strictly positive real numbers.

2.1. Market Structure and Profit Functions

Three firms compete in a homogeneous-good market, each producing quantities $x_1,x_2,x_3\ge 0$. The aggregate market output is $Q=x_1+x_2+x_3$ and the market price is $P\left(Q\right)$, where $P:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is inverse demand function. The profit, or payoff, of firm i is given by ${\Pi }_i(x_1,x_2,x_3)=x_{i}P(x_1+x_2+x_3)-c_i\left(x_i\right)$, for $i=1,2,3$, where each firm i has a cost function $c_i:{\mathbb{R}}_{+}\to \mathbb{R}$.

This general formulation includes the classical Cournot model, in which firms choose quantities simultaneously and non-cooperatively, and the Stackelberg model, in which firms move sequentially and later movers react to the decisions of earlier movers [12,18,19,20,21]. It also allows one to describe hybrid oligopoly structures, such as Cournot–Stackelberg competition and failed-cartel arrangements, where elements of simultaneous competition, sequential reaction, and partial coordination appear together [22,23,24].

In the present paper, the central object is a failed-cartel market structure. Such a structure arises when an initial cartel agreement is broken by one firm, while the remaining firms continue to coordinate their production decisions. This setting is motivated by the economic instability of collusive agreements: although cooperation may increase joint profits, individual firms often have incentives to deviate [11,13,14,15,16,17].

Throughout the paper, unless otherwise specified, the inverse demand function is assumed to be positive and decreasing, while the cost functions are assumed to be continuous and convex. When first-order equilibrium conditions are used, the corresponding differentiability of P and $c_i$, $i=1,2,3$, is additionally assumed.

2.2. Reaction Functions and Nash Equilibria

Let $X_i\subset {\mathbb{R}}_{+}$ for $i=1,2,3$, be the admissible strategy set of firm i, and put $X=X_1\times X_2\times X_3$.

In the Cournot framework, each firm chooses its own output while treating the output levels of the other two firms as fixed. Thus, the three firms solve ${max}_{y_1\in X_1}{\Pi }_1(y_1,x_2,x_3)$, ${max}_{y_2\in X_2}{\Pi }_2(x_1,y_2,x_3)$, and ${max}_{y_3\in X_3}{\Pi }_3(x_1,x_2,y_3)$, respectively.

This formulation follows the classical interpretation of quantity competition introduced by Cournot and subsequently developed in oligopoly theory [12,18,19,20].

The sets of optimal reactions of the firms are defined by

$$R_1(x_2,x_3)=arg\underset{y_1\in X_1}{max}{\Pi }_1(y_1,x_2,x_3),$$

$$R_2(x_1,x_3)=arg\underset{y_2\in X_2}{max}{\Pi }_2(x_1,y_2,x_3),$$

and

$$R_3(x_1,x_2)=arg\underset{y_3\in X_3}{max}{\Pi }_3(x_1,x_2,y_3).$$

In general, the sets $R_1(x_2,x_3)$, $R_2(x_1,x_3)$, and $R_3(x_1,x_2)$ need not be singletons. Therefore, we regard the reaction operators as set-valued mappings rather than as single-valued reaction functions. This distinction is essential for the approach developed below, since the existence analysis is based on the existence of maximizers and fixed points of set-valued mappings, rather than on first-order optimality conditions or contractive properties.

A vector $(x_1^{*},x_2^{*},x_3^{*})\in X$ is called a Cournot–Nash equilibrium if

$$x_1^{*}\in R_1(x_2^{*},x_3^{*}),x_2^{*}\in R_2(x_1^{*},x_3^{*}),x_3^{*}\in R_3(x_1^{*},x_2^{*}).$$

Equivalently, define the set-valued mapping $R:X\rightrightarrows X$ by

$$R(x_1,x_2,x_3)=R_1(x_2,x_3)\times R_2(x_1,x_3)\times R_3(x_1,x_2).$$

Then a Cournot–Nash equilibrium is a fixed point of R, i.e, $(x_1^{*},x_2^{*},x_3^{*})\in R(x_1^{*},x_2^{*},x_3^{*})$.

The use of reaction correspondences is natural when optimal reactions are not necessarily unique. Under compactness of the strategy sets and continuity of the profit functions, the existence and regularity of the sets of maximizers can be studied by means of the maximum theorem [5]. If, in addition, the reaction correspondences have nonempty, compact, and convex values and satisfy the appropriate upper semicontinuity assumptions, the existence of an equilibrium can be obtained from fixed point principles for set-valued mappings [7,8,9].

2.3. A Cartel Structured Market

A cartel is a coalition of firms that coordinates production decisions in order to increase the aggregate profit of its members. In contrast to the Cournot model, where each firm chooses its output independently while taking the quantities of its competitors as given, cartel members coordinate their production decisions and act collectively [11,13,15,16,17].

There are two closely related ways to describe the production decision of a cartel.

In the first formulation, the cartel directly chooses the production vector $(x_1,x_2,x_3)\in X$ so as to solve ${max}_{(y_1,y_2,y_3)\in X}\left\{{\Pi }_1(y_1,y_2,y_3)+{\Pi }_2(y_1,y_2,y_3)+{\Pi }_3(y_1,y_2,y_3)\right\}.$ Equivalently, the cartel maximizes the aggregate profit

$${\Pi }^C(x_1,x_2,x_3)=(x_1+x_2+x_3)P(x_1+x_2+x_3)-{\displaystyle \sum _{i=1}^3}{c}_i\left(x_i\right).$$

The set of cartel-optimal production vectors is therefore

$$C=arg\underset{(y_1,y_2,y_3)\in X}{max}{\Pi }^C(y_1,y_2,y_3).$$

A second formulation is obtained when the three firms have the same cost structure and the cartel determines its aggregate output as a single decision-making unit. Let $c\left(t\right)$ denote the cost associated with producing the total quantity t. The cartel then acts, with respect to market output, as a monopolist and solves ${max}_{t\in X_C}\left\{tP\left(t\right)-c\left(t\right)\right\},$ where $X_C\subset {\mathbb{R}}_{+}$ is the set of admissible aggregate production levels. We denote the set of monopoly-optimal quantities by $M=arg{max}_{t\in X_C}\left\{tP\left(t\right)-c\left(t\right)\right\}.$

After an aggregate quantity $t^{*}\in M$ has been selected, the cartel members divide the market according to an agreed allocation rule. If ${\alpha }_1,{\alpha }_2,{\alpha }_3\ge 0$, ${\alpha }_1+{\alpha }_2+{\alpha }_3=1$, denote the agreed market shares, then $x_1={\alpha }_1{t}^{*}$, $x_2={\alpha }_2{t}^{*}$, and $x_3={\alpha }_3{t}^{*}$. The shares may be determined by bargaining, production capacity, historical market shares, or any other agreement among the cartel members. In particular, an equal division of the market corresponds to ${\alpha }_1={\alpha }_2={\alpha }_3={\displaystyle \frac{1}{3}}$.

In both formulations, the cartel internalizes the effect of aggregate production on the common market price and coordinates the output decisions of its members. The difference lies in the level at which coordination is introduced. In the first formulation, the individual production quantities are chosen jointly. In the second, the cartel first determines the aggregate monopoly quantity and subsequently allocates this quantity among its members according to an agreed sharing rule. Both approaches represent fully coordinated market structures and provide natural cooperative benchmarks for the failed-cartel model developed below [15,19,20].

The existence of a cartel production decision does not require a characterization by first-order conditions. If the corresponding admissible strategy set is compact and the objective function is continuous, the existence of a maximizing production vector or aggregate quantity follows from the classical extreme value theorem.

The central question of the present paper is what happens when cartel coordination is only partially preserved: one firm leaves the agreement and optimizes its own profit, while the remaining two firms continue to coordinate their production decisions.

2.4. Market Structure Following Cartel Breakdown

Collusive agreements are inherently vulnerable to unilateral deviations. Although coordination may increase the aggregate profit of the participating firms, an individual cartel member may have an incentive to abandon the agreement and choose its production level independently. The stability of collusive arrangements and the incentives to deviate from them constitute classical problems in oligopoly theory [11,13,15,16,17].

The breakdown of a cartel may occur under either of the two cartel structures described in the previous subsection. If the cartel jointly determines the individual production levels of its members, one firm may leave the agreement and cease to follow the coordinated production decision. Alternatively, if the cartel first determines an aggregate monopoly quantity and subsequently divides the market according to agreed shares, one of its members may abandon the agreed allocation rule and independently choose its own output. In both cases, the original fully coordinated market structure is destroyed.

We consider a three-firm market and suppose, without loss of generality, that firm 3 leaves the cartel agreement. After firm 3 leaves the cartel, the collusive agreement is no longer enforced. Firms 1 and 2 simultaneously choose their production quantities according to the Cournot principle, while firm 3 reacts optimally to their decisions.

Although firms 1 and 2 originate from the former cartel, they no longer maximize their aggregate profit. Instead, each firm maximizes its own profit, taking the production level of the other leader and the follower’s response as given. Consequently, the strategic interaction between firms 1 and 2 is a Cournot game, whereas firm 3 acts as a follower whose production belongs to its reaction correspondence.

The terminology "failed cartel" refers to the origin of the market structure rather than to the optimization criterion adopted by the remaining firms. The cartel has already failed because one participant has abandoned the agreement. The remaining firms preserve their strategic advantage of moving before the follower, but they no longer cooperate by maximizing their joint profit. Instead, they compete non-cooperatively according to the classical Cournot principle.

For fixed production levels $x_1\in X_1$ and $x_2\in X_2$, the deviating firm chooses its output from the set $R_3(x_1,x_2)=arg{max}_{z\in X_3}\left\{zP(x_1+x_2+z)-c_3\left(z\right)\right\}$. Thus, $R_3(x_1,x_2)$ is the set of optimal production responses of firm 3 to the coordinated production levels of firms 1 and 2. The set of optimal responses need not contain a unique element. Consequently, $R_3$ is treated as a set-valued mapping.

The former cartel members, firms 1 and 2, no longer coordinate their production decisions. Instead, they act as Cournot competitors and choose their quantities simultaneously, each maximizing its own profit while taking the quantity of the other firm and the follower’s production level as given.

For a fixed follower quantity $z\in X_3$, firm 1 solves ${max}_{x_1\in X_1}{\Pi }_1(x_1,x_2,z)$, treating $x_2$ and z as fixed. Similarly, firm 2 solves ${max}_{x_2\in X_2}{\Pi }_2(x_1,x_2,z)$, treating $x_1$ and z as fixed.

Using the profit functions introduced above, these optimization problems take the form ${max}_{x_1\in X_1}\left\{x_{1}P(x_1+x_2+z)-c_1\left(x_1\right)\right\}$ and ${max}_{x_2\in X_2}\left\{x_{2}P(x_1+x_2+z)-c_2\left(x_2\right)\right\}$, respectively.

The follower, firm 3, chooses an optimal response to the quantities selected by firms 1 and 2. Its reaction set is $R_3(x_1,x_2)={arg\; max}_{z\in X_3}{\Pi }_3(x_1,x_2,z)$.

Thus, the market equilibrium is determined by the simultaneous compatibility of the two Cournot best-response problems and the follower’s optimization problem.

Hence, the failed-cartel structure combines two different forms of strategic behaviour. Firm 3 acts non-cooperatively and selects a best response to the production levels of firms 1 and 2, while the latter preserve their coordination and act as a coalition. The resulting market is therefore neither a classical Cournot oligopoly nor a fully cooperative cartel. It is an intermediate market structure generated by the partial breakdown of collusion.

An important feature of this model is that the coordinated firms do not necessarily face a uniquely determined reaction of the deviating firm. For a given pair $(x_1,x_2)$, several production levels may belong to $R_3(x_1,x_2)$, and these responses may lead to different aggregate profits for firms 1 and 2. Therefore, the optimization problem of the remaining coalition must explicitly take into account the set of possible optimal reactions of firm 3.

This observation leads naturally to an optimization problem involving a set-valued response correspondence. The mathematical approach developed in the next section is based on the existence of maximizers and on fixed point principles for set-valued mappings. In particular, no differentiability of the profit functions, first-order optimality conditions, or contractive assumptions are required in the main existence result. The relevant existence arguments rely instead on maximum theorems and fixed point principles for correspondences [5,7,8,9].

2.5. Generalized Concavity

Generalized concavity provides a useful framework for extending classical concavity arguments to functions that need not be concave in the ordinary sense. Various forms of generalized concavity, including quasi-concavity, log-concavity, and $\rho$-concavity, have been widely used in optimization and economic analysis [30,31,32].

In oligopoly theory, generalized concavity assumptions have been employed to establish existence and uniqueness properties of Cournot equilibria under substantially weaker conditions than ordinary concavity of the inverse demand function. In particular, log-concavity, biconcavity, concave price flexibility, and $\alpha$-concavity have been used to study the shape of revenue functions and the existence or uniqueness of equilibrium [25,26,27,28,29].

We recall the notion of $\alpha$-concavity, which will be used below to formulate sufficient conditions for the existence and regularity of the maximizers arising in the failed-cartel model.

Let $p:\Omega \to {\mathbb{R}}_{+}$ be a positive function defined on a convex set $\Omega \subseteq {\mathbb{R}}_{+}$.

Definition 1.([30,33]) A positive function p defined on a convex set $\Omega \subseteq {\mathbb{R}}_{+}$ is said to be α-concave, where $\alpha \in [-\infty ,+\infty ]$, if $p(\lambda x+(1-\lambda )y)\ge m_{\alpha }(p\left(x\right),p\left(y\right),\lambda )$ for all $x,y\in \Omega$ and all $\lambda \in [0,1]$, where

$$m_{\alpha }(a,b,\lambda )=\left\{\begin{array}{cc}{a}^{\lambda }{b}^{1-\lambda },\hfill & \alpha =0,\hfill \\ max\{a,b\},\hfill & \alpha =+\infty ,\hfill \\ min\{a,b\},\hfill & \alpha =-\infty ,\hfill \\ {\left(\lambda a^{\alpha }+(1-\lambda )b^{\alpha }\right)}^{1/\alpha },\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

For $\alpha =1$, this notion reduces to ordinary concavity, whereas $\alpha =0$ corresponds to log-concavity. Moreover, the generalized means $m_{\alpha }$ are monotone with respect to the parameter $\alpha$. Consequently, $\alpha$-concavity implies $\beta$-concavity whenever $\beta \le \alpha$ [29,30].

Of particular importance for the present analysis is the case of negative values of $\alpha$. For $n\ge 1$, a positive function p is $(-1)/n$-concave if

$$p(\lambda x+(1-\lambda )y)\ge {\left(\lambda p{\left(x\right)}^{-1/n}+(1-\lambda )p{\left(y\right)}^{-1/n}\right)}^{-n}.$$

Equivalently, p is $(-1)/n$-concave if and only if ${\displaystyle \frac{1}{p^{1/n}}}$ is convex.

This class contains, in particular, positive concave and positive log-concave functions and has proved useful in the analysis of revenue functions in Cournot models [29].

2.6. Fixed Point Methods

The classical Banach contraction principle guarantees the existence and uniqueness of a fixed point for a contraction on a complete metric space [34]. Numerous extensions of this principle have subsequently been developed, including fixed point results in partially ordered spaces and coupled, tripled, and more general n-tupled fixed point structures [35,36,37,38].

Fixed point methods have also proved useful in the analysis of market equilibria. Reaction functions may be interpreted as operators whose fixed points represent production profiles that are consistent with the strategic responses of all market participants. This viewpoint has been applied to nonlinear and non-differentiable response functions, as well as to oligopoly and Stackelberg-type market structures [39,40,41].

The classical fixed point problem concerns a single-valued mapping $T:X\to X$. A point $x^{*}\in X$ is called a fixed point of T if $T\left(x^{*}\right)=x^{*}$.

In equilibrium models, this equality expresses the consistency between a given production profile and the production profile generated by the corresponding reaction operator.

In many optimization and equilibrium problems, however, the optimal response to a given strategy profile need not be unique. The natural mathematical object is then a set-valued mapping, or correspondence, $F:X\rightrightarrows X$, where $F\left(x\right)$ is a nonempty subset of X. A point $x^{*}\in X$ is a fixed point of F if $x^{*}\in F\left(x^{*}\right)$. Thus, the fixed point condition for a correspondence requires the point to belong to the set of admissible responses generated by its own state.

Fixed point theory for set-valued mappings has a long history. Kakutani’s fixed point theorem extends Brouwer’s theorem to upper semicontinuous correspondences with nonempty compact convex values on compact convex subsets of finite-dimensional Euclidean spaces [7]. Fan and Glicksberg subsequently developed infinite-dimensional extensions for correspondences defined on compact convex subsets of locally convex topological vector spaces [8,9]. These results have become fundamental tools in mathematical economics and game theory, particularly in existence proofs for Nash equilibria and other equilibrium concepts [1,10,42].

For the purposes of the present paper, the importance of set-valued fixed point methods comes from the structure of best-response mappings. If an optimization problem admits several maximizers, the corresponding reaction cannot, in general, be represented by a single-valued function. Instead, it is naturally described by an argmax correspondence. For example, the reaction of the deviating firm in the failed-cartel model is given by

$$R_3(x_1,x_2)=arg\underset{z\in X_3}{max}\left\{zP(x_1+x_2+z)-c_3\left(z\right)\right\}.$$

Even when the objective function is continuous and a maximum exists, the set $R_3(x_1,x_2)$ may contain more than one element.

This distinction is essential for the approach developed below. We do not impose differentiability assumptions in order to derive first-order conditions, nor do we assume that reaction mappings are contractions. Instead, the existence of maximizers is used to construct set-valued response correspondences. Equilibrium is then formulated as a fixed point problem for these correspondences.

Under appropriate compactness, convexity, and continuity assumptions, maximum theorems guarantee suitable properties of the argmax correspondence, while set-valued fixed point principles provide the existence of a production profile consistent with the corresponding optimal responses [5,7,8,9].

Consequently, the fixed point approach adopted in this paper differs from contraction-based methods used in some earlier oligopoly models. The central role is played not by metric contractivity or differentiability, but by the existence of maximizers, the structural properties of the associated argmax correspondences, and fixed point principles for set-valued mappings.

2.7. Berge’s Maximum Theorem

We recall the form of Berge’s maximum theorem used below. This result is a standard tool for proving the existence and stability of optimal solutions in parametric optimization problems. In the present paper it is used to study argmax correspondences generated by the profit-maximization problems of the firms [5,6].

Definition 2 

([6,43]). Let X and Y be topological spaces and let $F:X\rightrightarrows Y$ be a set-valued mapping. The mapping F is called upper semicontinuous at a point $x_0\in X$ if for every open set $V\subset Y$ satisfying $F\left(x_0\right)\subset V$, there exists a neighbourhood U of $x_0$ such that $F\left(x\right)\subset V$ for all $x\in U$.

Definition 3 

([6,43]). Let X and Y be topological spaces and let $F:X\rightrightarrows Y$ be a set-valued mapping. The mapping F is called lower semicontinuous at a point $x_0\in X$ if for every open set $V\subset Y$ satisfying $F\left(x_0\right)\cap V\ne ⌀$, there exists a neighbourhood U of $x_0$ such that $F\left(x\right)\cap V\ne ⌀$ for all $x\in U$.

Definition 4 

([6,43]). A set-valued mapping $F:X\rightrightarrows Y$ is called continuous if it is both upper semicontinuous and lower semicontinuous.

Proposition 1 

([43]). Let X be a metric space, let Y be a compact metric space, and let $F:X\rightrightarrows Y$ have nonempty compact values. Then F is upper semicontinuous at $x_0$ if and only if, for every sequence $x_n\to x_0$ and every sequence $y_n\in F\left(x_n\right)$ such that $y_n\to y$, one has $y\in F\left(x_0\right)$.

Theorem 1 

(Berge’s maximum theorem [5,6]). Let X and Y be topological spaces, and let $K:X\rightrightarrows Y$ be a set-valued mapping with nonempty compact values. Assume that K is continuous.

Let $f:X\times Y\to \mathbb{R}$ be continuous. Define the value function $v\left(x\right)={max}_{y\in K\left(x\right)}f(x,y)$ and the solution correspondence $M\left(x\right)={arg\; max}_{y\in K\left(x\right)}f(x,y)$.

Then v is continuous, and M is nonempty-valued, compact-valued and upper semicontinuous.

Corollary 1 

(Minimum form of Berge’s maximum theorem). Under the assumptions of Berge’s maximum theorem, the function $w\left(x\right)={min}_{y\in K\left(x\right)}f(x,y)$ is continuous, and the solution correspondence $N\left(x\right)={arg\; min}_{y\in K\left(x\right)}f(x,y)$ is nonempty-valued, compact-valued and upper semicontinuous.

2.8. Kakutani–Fan–Glicksberg Fixed Point Theorem

The Brouwer fixed point theorem provides an existence result for continuous single-valued mappings defined on compact convex subsets of finite-dimensional Euclidean spaces. In optimization and equilibrium problems, however, the corresponding response operator is frequently set-valued, since an optimization problem may admit more than one maximizer. This motivates the use of fixed point principles for correspondences.

Kakutani extended Brouwer’s fixed point theorem to upper semicontinuous set-valued mappings with nonempty, compact, and convex values [7]. Fan and Glicksberg subsequently obtained extensions to compact convex subsets of locally convex topological vector spaces [8,9]. These results form one of the standard fixed point tools in mathematical economics and equilibrium theory; see also [6,10].

We recall the version, known as Kakutani–Fan–Glicksberg fixed point theorem, needed in the present paper.

Theorem 2 

([7,8,9]). Let X be a nonempty compact convex subset of a locally convex Hausdorff topological vector space, and let $F:X\rightrightarrows X$ be an upper semicontinuous set-valued mapping such that, for every $x\in X$, the set $F\left(x\right)$ is nonempty, compact, and convex. Then F has a fixed point; that is, there exists $x^{*}\in X$ such that $x^{*}\in F\left(x^{*}\right)$.

In finite-dimensional Euclidean spaces, this result is usually referred to as Kakutani’s fixed point theorem. The Fan–Glicksberg extension allows the same fixed point argument to be formulated in the more general setting of locally convex topological vector spaces. Since the strategy sets considered in the present paper are finite-dimensional, Kakutani’s theorem would be sufficient for the main existence argument. We use the Kakutani–Fan–Glicksberg formulation to emphasize the set-valued nature of the response mappings and the general fixed point structure underlying the model.

The theorem is particularly suitable for best-response correspondences. If the maximization problem defining a firm’s optimal reaction has a nonempty set of maximizers, the resulting argmax operator is naturally set-valued. Under suitable continuity and compactness assumptions, Berge’s maximum theorem provides the required regularity of the argmax correspondence, whereas the Kakutani–Fan–Glicksberg theorem yields the existence of a fixed point [5,7,8,9].

The following results of Kamburova and Marinov will be needed in the proofs of the main results below. We reproduce their original formulations and then indicate how they apply to the present model. The notation is that of [29].

Theorem 3 

([29]). Let $p:\Omega \subseteq {\mathbb{R}}_{+}\cup \left\{0\right\}\to {\mathbb{R}}_{+}$ be a decreasing, continuous, and $(-1)/n$-concave function, and let $n\ge 1$. Then

$$x⟼xp{(x+k)}^{1/n}$$

is strictly concave on the interval where it is strictly increasing, for every $k\ge 0$.

Comment. The notation $p:\Omega \subseteq {\mathbb{R}}_{+}\cup \left\{0\right\}\to {\mathbb{R}}_{+}$ is understood as $\Omega \subseteq {\mathbb{R}}_{+}\cup \left\{0\right\}$ and $p:\Omega \to {\mathbb{R}}_{+}$. Moreover, the function in the conclusion is considered for those $x\ge 0$ for which $x+k\in \Omega$. In the applications below, $\Omega ={\mathbb{R}}_{+}$ and k represents the aggregate production of the remaining firms.

Theorem 4 

([29]). Let $p:\Omega \subseteq {\mathbb{R}}_{+}\cup \left\{0\right\}\to {\mathbb{R}}_{+}$ be a $(-1)/n$-concave function, where $n\ge 1$, and let $k\ge 0$. If

$$x⟼xp{(x+k)}^{1/n}$$

has an extremum greater than 0, then this extremum is a global maximum.

Comment. For $n=1$, the two preceding results describe the shape of the revenue function $x\mapsto xp(x+k)$: it is either non-increasing, or it is strictly concave on its interval of increase and non-increasing thereafter. This structure will be used below to establish the quasi-concavity of the profit functions.

Corollary 2 

([29]). Suppose that each cost function is convex and increasing, and that $p_i$ is a positive, strictly decreasing, continuous, differentiable, and $(-1)/N$-concave function on ${\mathbb{R}}_{+}\cup \left\{0\right\}$, for every $i=1,\dots ,N$, where $N\ge 2$. Then there exists a unique equilibrium.

Comment. In the homogeneous-good model considered in the present paper, $p_i=P$ for every i. Since the market consists of three firms, $N=3$, and the required generalized concavity condition becomes $(-1)/3$-concavity. If compact strategy intervals $X_i=[0,M_i]$ are imposed, the result applies provided that the upper bounds $M_i$ are nonbinding, so that the constrained and unconstrained best responses coincide.

2.9. Diagonal Strict Concavity

Existence and uniqueness of an equilibrium are distinct problems. The fixed point arguments described above provide conditions ensuring the existence of an equilibrium, but they do not, in general, exclude the possibility of several equilibrium production profiles. A classical approach to uniqueness in concave games is based on the notion of diagonal strict concavity, introduced by Rosen [3].

Let $X=X_1\times X_2\times X_3$ and let ${\Pi }_i:X\to \mathbb{R}$ for $i=1,2,3$, denote the payoff functions of the three firms. Assume that the payoff functions are continuously differentiable with respect to the firms’ own strategy variables.

For a vector of positive weights $r=(r_1,r_2,r_3)$ for $r_i>0$, the weighted pseudogradient of the game is defined by

$$g_r(x_1,x_2,x_3)=\left(r_1{\displaystyle \frac{\partial {\Pi }_1}{\partial x_1}}(x_1,x_2,x_3),r_2{\displaystyle \frac{\partial {\Pi }_2}{\partial x_2}}(x_1,x_2,x_3),r_3{\displaystyle \frac{\partial {\Pi }_3}{\partial x_3}}(x_1,x_2,x_3)\right).$$

Definition 5 

([3]). The game is said to satisfy the diagonal strict concavity condition if there exists a weight vector $r\in {\mathbb{R}}_{++}^3$ such that, for every distinct $x,y\in X$, $〈x-y,g_r\left(x\right)-g_r\left(y\right)〉<0$, where $\langle ·,·\rangle$ denotes the standard Euclidean inner product on ${\mathbb{R}}^3$.

The diagonal strict concavity condition expresses a strict monotonicity property of the weighted pseudogradient of the game. It is stronger than the separate concavity of each payoff function in the firm’s own strategy variable and controls the strategic interaction among the players. Under the standard assumptions of a concave game, this condition provides a sufficient criterion for uniqueness of the Nash equilibrium [3,4].

In the present paper, diagonal strict concavity is not used to establish the existence of a failed-cartel equilibrium. Existence is obtained from the properties of the maximization problems and the associated set-valued response correspondences. Rather, diagonal strict concavity is introduced as an additional condition that may be used to exclude multiple equilibrium production profiles when the corresponding payoff structure is differentiable.

This distinction is important for the approach adopted below. The main existence result does not require differentiability, first-order conditions, or second-order conditions. When additional smoothness is available, diagonal strict concavity provides a classical and economically meaningful uniqueness criterion complementary to the set-valued fixed point existence argument.

3. Main Results

The failed-cartel model considered in this paper leads naturally to optimization problems whose solutions need not be uniquely determined. Even when the corresponding profit functions are continuous and maximizers exist, the optimal reaction of a firm may consist of several production levels. Consequently, the reaction structure of the market is more naturally described by set-valued mappings than by single-valued response functions.

The purpose of this section is to collect the mathematical tools required for the existence and uniqueness analysis developed below. The approach combines generalized concavity, maximum theorems, and fixed point principles for set-valued mappings. Generalized concavity is used to control the structure of the profit-maximization problems, while Berge’s maximum theorem provides existence and regularity properties of the associated argmax correspondences. The resulting set-valued response mappings are then studied by means of the Kakutani–Fan–Glicksberg fixed point theorem. When additional differentiability assumptions are available, diagonal strict concavity provides a complementary criterion for uniqueness.

Thus, the analysis does not rely on explicit reaction functions or on solving systems of first-order conditions. Instead, equilibrium is obtained through the existence of maximizers and fixed points of the correspondences generated by the underlying optimization problems.

3.1. Existence of Equilibrium

We now turn to the existence of an equilibrium in the failed-cartel market. The main difficulty arises from the asymmetric structure of the model. Firm 3 chooses its production level independently so as to maximize its own profit, whereas firms 1 and 2 preserve their coordination and optimize their joint profit while anticipating the optimal reaction of firm 3. Consequently, the equilibrium problem cannot, in general, be reduced to a classical system of single-valued reaction functions.

Moreover, the profit-maximization problem of the deviating firm may admit more than one solution. Hence, its optimal reaction is naturally described by an argmax correspondence rather than by a single-valued response function. The remaining cartel members therefore face a set of possible optimal reactions of firm 3, and their coordinated production decision must be formulated with respect to this set.

The existence result below is based directly on the optimization structure of the model. We first use the existence of maximizers and Berge’s maximum theorem to obtain the relevant properties of the argmax correspondences. These correspondences are then combined into a set-valued operator on the joint strategy space. The Kakutani–Fan–Glicksberg fixed point theorem provides a fixed point of this operator, which is interpreted as an equilibrium production profile of the failed-cartel market [5,7,8,9].

An important feature of this approach is that the existence argument does not require differentiability of the inverse demand or cost functions, the explicit construction of reaction functions, or the solution of first-order conditions. The equilibrium is obtained instead from the existence of optimal production decisions and the fixed point structure of the associated set-valued response correspondences.

Theorem 5. 

Let $X_i\subset \mathbb{R}$ for $i=1,2,3$, be nonempty, compact, and convex sets, and let ${\Pi }_i:X_1\times X_2\times X_3\to \mathbb{R}$ for $i=1,2,3$, be continuous payoff functions.

For every fixed pair $(x_1,x_2)\in X_1\times X_2$, define the follower’s reaction correspondence by

$$R_3(x_1,x_2)=\underset{x_3\in X_3}{arg\; max}{\Pi }_3(x_1,x_2,x_3).$$

Assume that, for every fixed $(x_1,x_2)\in X_1\times X_2$, the function $x_3\mapsto {\Pi }_3(x_1,x_2,x_3)$ is quasi-concave on $X_3$.

Define the best-response correspondence of leaders 1 and 2 by

$$B_1(x_2,x_3)=\underset{x_1\in X_1}{arg\; max}{\Pi }_1(x_1,x_2,x_3),B_2(x_1,x_3)=\underset{x_2\in X_2}{arg\; max}{\Pi }_2(x_1,x_2,x_3).$$

Assume that, for every fixed $(x_2,x_3)\in X_2\times X_3$, the function $x_1\mapsto {\Pi }_1(x_1,x_2,x_3)$ is quasi-concave on $X_1$, and that, for every fixed $(x_1,x_3)\in X_1\times X_3$, the function $x_2\mapsto {\Pi }_2(x_1,x_2,x_3)$ is quasi-concave on $X_2$.

Then there exists $(x_1^{*},x_2^{*},x_3^{*})\in X_1\times X_2\times X_3$ such that

$$(x_1^{*},x_2^{*},x_3^{*})\in (B_1(x_2^{*},x_3^{*}),B_2(x_1^{*},x_3^{*}),R_3(x_1^{*},x_2^{*})).$$

Equivalently,

$$(x_1^{*},x_2^{*},x_3^{*})\in (\underset{x_1\in X_1}{arg\; max}{\Pi }_1(x_1,x_2^{*},x_3^{*}),\underset{x_2\in X_2}{arg\; max}{\Pi }_2(x_1^{*},x_2,x_3^{*}),\underset{x_3\in X_3}{arg\; max}{\Pi }_3(x_1^{*},x_2^{*},x_3))$$

Thus $(x_1^{*},x_2^{*},x_3^{*})$ is an equilibrium with one follower and two Cournot leaders.

Proof. 

Since the sets $X_i$ for $i=1,2,3$, are nonempty and compact and the functions ${\Pi }_i$ for $i=1,2,3$, are continuous, each of the corresponding maximization problems has at least one solution.

We first consider the follower. For every fixed $(x_1,x_2)\in X_1\times X_2$, the function $x_3\mapsto {\Pi }_3(x_1,x_2,x_3)$ is continuous on the compact set $X_3$. Therefore, by the Weierstrass theorem, it attains its maximum on $X_3$. Hence $R_3(x_1,x_2)={arg\; max}_{x_3\in X_3}{\Pi }_3(x_1,x_2,x_3)$ is nonempty.

Moreover, the set $R_3(x_1,x_2)$ is compact.

Indeed, let $m_3(x_1,x_2)={max}_{u_3\in X_3}{\Pi }_3(x_1,x_2,u_3)$. Then

$$R_3(x_1,x_2)=\left\{x_3\in X_3:{\Pi }_3(x_1,x_2,x_3)=m_3(x_1,x_2)\right\}.$$

Since $x_3\mapsto {\Pi }_3(x_1,x_2,x_3)$ is continuous, the set on the right-hand side is closed in $X_3$. Therefore $R_3(x_1,x_2)$ is a closed subset of the compact set $X_3$, and hence it is compact.

We next prove that $R_3(x_1,x_2)$ is convex.

Let $x_3^1,x_3^2\in R_3(x_1,x_2)$ and let $\lambda \in [0,1]$. By the definition of $R_3(x_1,x_2)$, there holds the equalities

$${\Pi }_3(x_1,x_2,x_3^1)={\Pi }_3(x_1,x_2,x_3^2)=m_3(x_1,x_2).$$

Since the function $x_3\mapsto {\Pi }_3(x_1,x_2,x_3)$ is quasi-concave, we have

$${\Pi }_3\left(x_1,x_2,\lambda x_3^1+(1-\lambda )x_3^2\right)\ge min\left\{{\Pi }_3(x_1,x_2,x_3^1),{\Pi }_3(x_1,x_2,x_3^2)\right\}=m_3(x_1,x_2).$$

On the other hand, $m_3(x_1,x_2)$ is the maximal value of ${\Pi }_3(x_1,x_2,·)$ on $X_3$. Therefore,

$${\Pi }_3\left(x_1,x_2,\lambda x_3^1+(1-\lambda )x_3^2\right)=m_3(x_1,x_2).$$

Hence $\lambda x_3^1+(1-\lambda )x_3^2\in R_3(x_1,x_2),$ and consequently $R_3(x_1,x_2)$ is convex.

By Berge’s maximum theorem, the reaction correspondence $R_3:X_1\times X_2\rightrightarrows X_3$ is upper semicontinuous and compact-valued.

We now consider the two leaders. For every fixed $(x_2,x_3)\in X_2\times X_3,$ the function $x_1\mapsto {\Pi }_1(x_1,x_2,x_3)$ is continuous on the compact set $X_1$. Therefore,

$$B_1(x_2,x_3)=\underset{x_1\in X_1}{arg\; max}{\Pi }_1(x_1,x_2,x_3)$$

is nonempty and compact.

Similarly, for every fixed $(x_1,x_3)\in X_1\times X_3,$ the set

$$B_2(x_1,x_3)=\underset{x_2\in X_2}{arg\; max}{\Pi }_2(x_1,x_2,x_3)$$

is nonempty and compact.

The quasi-concavity of $x_1\mapsto {\Pi }_1(x_1,x_2,x_3)$ implies that $B_1(x_2,x_3)$ is convex. Likewise, the quasi-concavity of $x_2\mapsto {\Pi }_2(x_1,x_2,x_3)$ implies that $B_2(x_1,x_3)$ is convex.

By Berge’s maximum theorem, the correspondences $B_1:X_2\times X_3\rightrightarrows X_1$ and $B_2:X_1\times X_3\rightrightarrows X_2$ are upper semicontinuous and compact-valued.

Define the product correspondence $\mathcal{B}:X_1\times X_2\times X_3\rightrightarrows X_1\times X_2\times X_3$ by

$$\mathcal{B}(x_1,x_2,x_3)=B_1(x_2,x_3)\times B_2(x_1,x_3)\times R_3(x_1,x_2).$$

The set $X_1\times X_2\times X_3$ is nonempty, compact, and convex. The correspondence $\mathcal{B}$ is upper semicontinuous and has nonempty, compact, and convex values.

Therefore, by Theorem 2, there exists $(x_1^{*},x_2^{*},x_3^{*})\in X_1\times X_2\times X_3$ such that $(x_1^{*},x_2^{*},x_3^{*})\in \mathcal{B}(x_1^{*},x_2^{*},x_3^{*}).$

By the construction of $\mathcal{B}$, it follows $(x_1^{*},x_2^{*},x_3^{*})\in (B_1(x_2^{*},x_3^{*}),B_2(x_1^{*},x_3^{*}),R_3(x_1^{*},x_2^{*}))$.

Thus $(x_1^{*},x_2^{*},x_3^{*})$ is an equilibrium with one follower and two Cournot leaders. □

The existence theorem established in the previous section allows the follower’s reaction correspondence and the leaders’ best-response correspondences to be set-valued. One may naturally ask whether this generality is really necessary or whether these correspondences are typically single-valued under the assumptions imposed on the payoff functions. The following examples show that continuity, compactness of the strategy sets, and quasi-concavity of the payoff functions do not guarantee uniqueness of optimal responses. Consequently, the use of set-valued correspondences is not merely a technical generalization but an intrinsic feature of the optimization framework whenever several optimal strategies produce the same maximal payoff.

We illustrate why the reaction correspondence $R_3$ and the best-response correspondences $B_1$ and $B_2$ need not be single-valued, even when the relevant payoff functions are continuous and quasi-concave.

Example 1. 

Let $X_1=X_2=X_3=[0,2]$. Consider the following payoff functions:

$${\Pi }_3(x_1,x_2,z)=-dist{\left(z,[a(x_1,x_2),b(x_1,x_2)]\right)}^2,$$

where $a(x_1,x_2)={\displaystyle \frac{x_1+x_2}{4}}$, $b(x_1,x_2)=1+{\displaystyle \frac{x_1+x_2}{4}}$, and

$${\Pi }_1(x_1,x_2,z)=-dist{\left(x_1,\left[{\displaystyle \frac{x_2+z}{4}},1+{\displaystyle \frac{x_2+z}{4}}\right]\right)}^2,$$

$${\Pi }_2(x_1,x_2,z)=-dist{\left(x_2,\left[{\displaystyle \frac{x_1+z}{4}},1+{\displaystyle \frac{x_1+z}{4}}\right]\right)}^2.$$

The three payoff functions are continuous on $X_1\times X_2\times X_3$. Moreover, for every fixed values of the remaining variables, each payoff function is quasi-concave with respect to its own decision variable. We next determine the corresponding reaction and best-response correspondences.

Since $x_1,x_2\in [0,2]$, we have $0\le a(x_1,x_2)\le 1$ and $1\le b(x_1,x_2)\le 2$.

For every fixed $(x_1,x_2)\in X_1\times X_2$, the function $z\mapsto {\Pi }_3(x_1,x_2,z)$ attains its maximum value 0 at every point of the interval $[a(x_1,x_2),b(x_1,x_2)]$. Consequently,

$$R_3(x_1,x_2)=\underset{z\in X_3}{arg\; max}{\Pi }_3(x_1,x_2,z)=[a(x_1,x_2),b(x_1,x_2)].$$

Thus $R_3(x_1,x_2)$ is nonempty, compact, and convex, but it is not single-valued.

For example, if $x_1=x_2=1$, then $a(1,1)={\displaystyle \frac{1}{2}}$, $b(1,1)={\displaystyle \frac{3}{2}}$, and therefore $R_3(1,1)=\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}\right]$. Hence every quantity $z\in \left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}\right]$ is an optimal response of the follower.

For every fixed $(x_2,z)\in X_2\times X_3$, the maximum value of ${\Pi }_1(·,x_2,z)$ is 0, and it is attained at every point of the interval $\left[{\displaystyle \frac{x_2+z}{4}},1+{\displaystyle \frac{x_2+z}{4}}\right]$. Therefore,

$$B_1(x_2,z)=\left[{\displaystyle \frac{x_2+z}{4}},1+{\displaystyle \frac{x_2+z}{4}}\right].$$

Similarly,

$$B_2(x_1,z)=\left[{\displaystyle \frac{x_1+z}{4}},1+{\displaystyle \frac{x_1+z}{4}}\right].$$

Thus both $B_1$ and $B_2$ are set-valued correspondences.

Consider, for instance, the point $(x_1,x_2,z)=(1,1,1)$. We have $R_3(1,1)=\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}\right],$ while $B_1(1,1)=B_2(1,1)=\left[{\displaystyle \frac{1}{2}},{\displaystyle \frac{3}{2}}\right]$. Consequently, $1\in R_3(1,1)$, $1\in B_1(1,1)$, $1\in B_2(1,1)$. Hence $(1,1,1)\in B_1(1,1)\times B_2(1,1)\times R_3(1,1)$, and $(1,1,1)$ is a fixed point of the product best-response correspondence.

This example shows that continuity and quasi-concavity of the payoff functions do not imply uniqueness of the corresponding maximizers. The sets of maximizers may contain entire intervals. Nevertheless, these argmax sets remain nonempty, compact, and convex, which is precisely the structure required in Theorem 2.

3.2. Application to the Price-Cost Functions Model

Theorem 5 establishes an existence result for a system of three interacting optimization problems formulated solely in terms of the payoff functions ${\Pi }_i$ for $i=1,2,3$. Its assumptions involve only continuity, quasi-concavity, and compactness, without referring to any particular economic model.

We now specialize this abstract framework to the failed-cartel market introduced in Section 2.4. In this setting, the payoff functions are generated by the standard structure consisting of an inverse demand function and individual cost functions,

$${\Pi }_i(x_1,x_2,x_3)=x_{i}P(x_1+x_2+x_3)-c_i\left(x_i\right),i=1,2,3.$$

The generalized concavity assumptions imposed on the inverse demand function, together with the convexity of the cost functions, guarantee the quasi-concavity of each firm’s payoff with respect to its own decision variable. Consequently, all assumptions of Theorem 5 are satisfied, and the economic equilibrium existence result follows immediately as a direct application of the abstract optimization theorem.

Corollary 3. 

Let $X_i=[0,M_i]\subset {\mathbb{R}}_{+}$ for $i=1,2,3$, be nonempty compact intervals.

Assume that the inverse demand function $P:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ is positive, continuous, strictly decreasing and $(-1)/3$-concave. Assume further that the cost functions $c_i:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ for $i=1,2,3$, are continuous, convex and increasing.

For $i=1,2,3$, define the payoff functions by ${\Pi }_i(x_1,x_2,x_3)=x_{i}P(x_1+x_2+x_3)-c_i\left(x_i\right)$.

Then the hypotheses of Theorem 5 are satisfied. Consequently, there exists a failed-cartel equilibrium $(x_1^{*},x_2^{*},x_3^{*})\in X_1\times X_2\times X_3$.

Proof. 

Since the inverse demand function is continuous and the cost functions are continuous, each payoff function ${\Pi }_i:X_1\times X_2\times X_3\to \mathbb{R}$ is continuous.

Furthermore, since $(-1)/3$-concavity implies $(-1)$-concavity, fix $i\in \{1,2,3\}$ and denote the aggregate output of the remaining firms by $s\ge 0$. By Theorems 3 and 4, the revenue function

$$r_s\left(x\right)=xP(x+s)$$

is either non-increasing, or it is strictly concave on its interval of increase and non-increasing thereafter. In the first case, $r_s-c_i$ is non-increasing because $c_i$ is increasing. In the second case, $r_s-c_i$ is concave on the interval of increase because $c_i$ is convex, and it is non-increasing thereafter. Hence $r_s-c_i$ is quasi-concave in both cases. Consequently, the mapping $x_i\mapsto {\Pi }_i(x_1,x_2,x_3)$ is quasi-concave on $X_i$, for each $i=1,2,3$.

Therefore all assumptions of Theorem 5 are satisfied, and the conclusion follows immediately. □

Comment. If, in addition to the hypotheses of Corollary 3, the inverse demand function P is differentiable on ${\mathbb{R}}_{+}$, and the imposed upper bounds $M_i$ are nonbinding, then the equilibrium is unique. In the present application, existence is already guaranteed by Corollary 3; the cited result is used only for uniqueness. Indeed, since the market consists of three firms, the assumed $(-1)/3$-concavity is precisely the $(-1)/N$-concavity condition with $N=3$, and the conclusion follows from Corollary 2.

The following example gives set-valued best responses and more than one fixed point in a standard price–cost model. The particular inverse demand used below, however, does not satisfy the $(-1)/3$-concavity assumption of Corollary 3.

Example 2. 

Let $X_1=X_2=X_3=[0,3]$. Assume that the inverse demand function is

$$P\left(Q\right)=\left\{\begin{array}{cc}5-Q,\hfill & 0\le Q\le 3,\hfill \\ 1+{\displaystyle \frac{1}{Q-2}},\hfill & 3\le Q\le 6,\hfill \\ 1+{\displaystyle \frac{1}{2(Q-4)}},\hfill & Q\ge 6,\hfill \end{array}\right.$$

and let $c_i\left(x\right)=x$ for $i=1,2,3$.

The corresponding payoff functions are ${\Pi }_i(x_1,x_2,x_3)=x_{i}P(x_1+x_2+x_3)-c_i\left(x_i\right)$ for $i=1,2,3$.

The inverse demand function is positive, continuous, and strictly decreasing. It is not $(-1)/3$-concave on the whole domain; on $3<Q<6$, $P{\left(Q\right)}^{-1/3}={((Q-2)/(Q-1))}^{1/3}$ is concave rather than convex. Consequently, this example is not an application of Corollary 3; the equilibria below are verified directly.

We next show that the equilibrium is not unique.

Consider the production profile $(1,1,1)$. Direct computations show that $R_3(1,1)=B_1(1,1)=B_2(1,1)=[1,3]$.

Since $1\in [1,3],$ we obtain $(1,1,1)\in B_1(1,1)\times B_2(1,1)\times R_3(1,1),$ and therefore $(1,1,1)$ is an equilibrium.

Similarly, $R_3(2,2)=B_1(2,2)=B_2(2,2)=[2,3]$.

Hence $(2,2,2)$ is also an equilibrium.

The model therefore possesses at least two distinct equilibria, $(1,1,1)$ and $(2,2,2)$, both yielding strictly positive profits for every firm. Thus, the present example directly shows that its best-response correspondences and its equilibrium are not unique.

3.3. The Single-Valued Case

The previous results were formulated for general set-valued reaction and best-response correspondences. This level of generality is essential because, as illustrated by the preceding examples, multiple optimal strategies may occur even when the payoff functions satisfy all assumptions of the existence theorem.

In many economic models, however, additional assumptions imply that every optimization problem admits a unique maximizer. In this case the reaction correspondence of the follower and the best-response correspondences of the leaders become single-valued functions. Consequently, the abstract set-valued equilibrium framework reduces to the classical formulation in terms of reaction functions.

Corollary 4. 

Assume that all hypotheses of Corollary 3 are satisfied. Suppose, in addition, that for every $(x_1,x_2)\in X_1\times X_2$, the follower’s reaction set $R_3(x_1,x_2)={arg\; max}_{x_3\in X_3}{\Pi }_3(x_1,x_2,x_3)$ consists of exactly one point.

Suppose also that, for every $(x_2,x_3)\in X_2\times X_3$, the set

$$B_1(x_2,x_3)=\underset{x_1\in X_1}{arg\; max}{\Pi }_1(x_1,x_2,x_3)$$

consists of exactly one point, and that, for every $(x_1,x_3)\in X_1\times X_3$, the set

$$B_2(x_1,x_3)=\underset{x_2\in X_2}{arg\; max}{\Pi }_2(x_1,x_2,x_3)$$

consists of exactly one point.

Then there exist uniquely determined continuous reaction functions $b_3:X_1\times X_2\to X_3$, $b_1:X_2\times X_3\to X_1$, and $b_2:X_1\times X_3\to X_2$ such that $R_3(x_1,x_2)=\left\{b_3(x_1,x_2)\right\}$, $B_1(x_2,x_3)=\left\{b_1(x_2,x_3)\right\}$, and $B_2(x_1,x_3)=\left\{b_2(x_1,x_3)\right\}$.

Moreover, there exists $(x_1^{*},x_2^{*},x_3^{*})\in X_1\times X_2\times X_3$ such that $x_1^{*}=b_1(x_2^{*},x_3^{*}),$ $x_2^{*}=b_2(x_1^{*},x_3^{*}),$ and $x_3^{*}=b_3(x_1^{*},x_2^{*}).$

Equivalently, if the mapping $T:X_1\times X_2\times X_3\to X_1\times X_2\times X_3$ is defined by

$$T(x_1,x_2,x_3)=\left(b_1(x_2,x_3),b_2(x_1,x_3),b_3(x_1,x_2)\right),$$

then T has at least one fixed point $T(x_1^{*},x_2^{*},x_3^{*})=(x_1^{*},x_2^{*},x_3^{*}).$

Every fixed point of T is a failed-cartel equilibrium with one follower and two Cournot leaders.

Proof. 

By Corollary 3, the reaction correspondences $R_3:X_1\times X_2\rightrightarrows X_3$, $B_1:X_2\times X_3\rightrightarrows X_1$, and $B_2:X_1\times X_3\rightrightarrows X_2$ are nonempty, compact-valued, and upper semicontinuous.

By the additional assumptions, each of these correspondences has exactly one value at every point of its domain. Therefore, they determine single-valued functions $b_3:X_1\times X_2\to X_3$, $b_1:X_2\times X_3\to X_1$, and $b_2:X_1\times X_3\to X_2$.

We show that these functions are continuous. We present the argument for $b_3$; the proofs for $b_1$ and $b_2$ are analogous.

Let ${lim}_{n\to \infty }(x_1^n,x_2^n)=(x_1,x_2)$ and put $z_n=b_3(x_1^n,x_2^n)\in X_3$. Since $X_3$ is compact, every sequence of ${\left\{z_n\right\}}_{n=1}^{\infty }$ has a convergent subsequence. Let ${lim}_{k\to \infty }{z}_{n_k}=z$. Because $z_{n_k}\in R_3(x_1^{n_k},x_2^{n_k})$, and $R_3$ is upper semicontinuous, it follows that $z\in R_3(x_1,x_2)$. However, $R_3(x_1,x_2)$ is a singleton, namely $R_3(x_1,x_2)=\left\{b_3(x_1,x_2)\right\}.$

Therefore, $z=b_3(x_1,x_2)$.

Thus every convergent subsequence of ${\left\{z_n\right\}}_{n=1}^{\infty }$ converges to $b_3(x_1,x_2)$. Since $X_3$ is compact, this implies ${lim}_{n\to \infty }{b}_3(x_1^n,x_2^n)=b_3(x_1,x_2).$ Hence $b_3$ is continuous. By the same argument, $b_1$ and $b_2$ are continuous.

Define $T(x_1,x_2,x_3)=\left(b_1(x_2,x_3),b_2(x_1,x_3),b_3(x_1,x_2)\right)$. Since $b_1$, $b_2$, and $b_3$ are continuous, the mapping T is continuous. Moreover,

$$T\left(X_1\times X_2\times X_3\right)\subseteq X_1\times X_2\times X_3.$$

The set $X_1\times X_2\times X_3$ is nonempty, compact, and convex. Therefore, by Brouwer’s fixed point theorem, there exists $(x_1^{*},x_2^{*},x_3^{*})\in X_1\times X_2\times X_3$ such that

$$T(x_1^{*},x_2^{*},x_3^{*})=(x_1^{*},x_2^{*},x_3^{*}).$$

Consequently, $x_1^{*}=b_1(x_2^{*},x_3^{*})$, $x_2^{*}=b_2(x_1^{*},x_3^{*})$, and $x_3^{*}=b_3(x_1^{*},x_2^{*}).$ By the definitions of the reaction functions, this means that

$$(x_1^{*},x_2^{*},x_3^{*})\in (\underset{x_1\in X_1}{arg\; max}{\Pi }_1(x_1,x_2^{*},x_3^{*}),\underset{x_2\in X_2}{arg\; max}{\Pi }_2(x_1^{*},x_2,x_3^{*}),\underset{x_3\in X_3}{arg\; max}{\Pi }_3(x_1^{*},x_2^{*},x_3)).$$

Therefore, $(x_1^{*},x_2^{*},x_3^{*})$ is a failed-cartel equilibrium with one follower and two Cournot leaders. □

The following example illustrates Corollary 4. We consider a nonlinear Cournot market in which all optimization problems admit unique maximizers. Consequently, the reaction and best-response correspondences reduce to ordinary reaction functions, and the market equilibrium is characterized by the unique common fixed point of these functions.

The same market specification will be revisited later in the paper from a different perspective in order to investigate the influence of heterogeneous marginal costs on the equilibrium production levels.

Example 3. 

Let $X_1=X_2=X_3=[0,2]$, and consider the inverse demand function $P\left(Q\right)={\displaystyle \frac{1}{{(1+Q)}^5}}$. Let the cost functions be $c_1\left(x\right)=c_2\left(x\right)=0.0004201x$ and $c_3\left(x\right)=0.0017854x+{\displaystyle \frac{0.001}{2}}{x}^2$.

The corresponding profit functions are ${\Pi }_i(x_1,x_2,x_3)=x_{i}P(x_1+x_2+x_3)-c_i\left(x_i\right)$ for $i=1,2,3$.

The inverse demand function P is positive, continuous, and strictly decreasing. Moreover, $P{\left(Q\right)}^{-1/3}={(1+Q)}^{5/3}$ is convex on ${\mathbb{R}}_{+}$, and hence P is $(-1/3)$-concave. The cost functions are continuous, convex, and increasing. Therefore, all hypotheses of Corollary 3 are satisfied.

We next verify that each individual optimization problem has a unique solution. Fix $i\in \{1,2,3\}$, and denote the aggregate production of the other two firms by $s={\sum }_{j\ne i}{x}_j$. The profit of firm i, considered as a function of its own production x, is ${\phi }_{i,s}\left(x\right)={\displaystyle \frac{x}{{(1+s+x)}^5}}-c_i\left(x\right)$ for $x\in [0,2]$.

Let $A=1+s>0$. For firms 1 and 2, we have

$${\phi }_{i,s}^{\prime }\left(x\right)={\displaystyle \frac{A-4x}{{(A+x)}^6}}-0.0004201,i=1,2,$$

whereas for firm 3,

$${\phi }_{3,s}^{\prime }\left(x\right)={\displaystyle \frac{A-4x}{{(A+x)}^6}}-0.0017854-0.001x.$$

The function $h_A\left(x\right)={\displaystyle \frac{A-4x}{{(A+x)}^6}}$ satisfies $h_A^{\prime }\left(x\right)={\displaystyle \frac{20x-10A}{{(A+x)}^7}}.$ In particular, $h_A$ is strictly decreasing on $\left[0,{\displaystyle \frac{A}{4}}\right]$, which is precisely the interval on which $h_A$ may be positive. For $x\ge A/4$, we have $h_A\left(x\right)\le 0$.

It follows that each derivative ${\phi }_{i,s}^{\prime }$ can vanish at most once. Consequently, ${\phi }_{i,s}$ is either strictly decreasing on $[0,2]$, or it first increases and then strictly decreases. Thus ${\phi }_{i,s}$ is strictly quasi-concave and possesses a unique maximizer on $[0,2]$.

Therefore, for every admissible production profile of the remaining firms, the sets

$$R_3(x_1,x_2),B_1(x_2,x_3),B_2(x_1,x_3)$$

are singletons. Hence they determine single-valued reaction functions $r_3:X_1\times X_2\to X_3$, $b_1:X_2\times X_3\to X_1$, and $b_2:X_1\times X_3\to X_2$.

We can get an approximate equilibrium volumes

$$(x_1^{*},x_2^{*},x_3^{*})\approx (0.445044,0.445044,0.394932).$$

All the hypotheses of Corollary 3 are satisfied, and P is differentiable. Moreover, since $s\in [0,4]$, every positive maximizer satisfies

$$x<{\displaystyle \frac{1+s}{4}}\le {\displaystyle \frac{5}{4}}<2,$$

so the upper bound 2 is nonbinding. It therefore follows from Corollary 2 that this equilibrium is unique.

This model therefore illustrates the situation described in Corollary 4. The same specification will be considered again in Section 4.2, where it will be used to study the effect of asymmetric marginal costs on the equilibrium production levels.

3.4. Technological Asymmetry and Equilibrium Quantities

For the comparison results in this subsection, we impose an additional sequential assumption. The response $R_3(x_1,x_2)$ is single-valued, and firms 1 and 2 anticipate this response. Their induced payoffs are ${\Psi }_i(x_1,x_2)={\Pi }_i(x_1,x_2,R_3(x_1,x_2))$, $i=1,2$. We consider an interior equilibrium of this induced two-player game together with the optimal response of firm 3.

The existence result above does not, by itself, determine the relative production levels of the three firms. We first examine how differences in production technologies, expressed through marginal costs, affect the ordering of equilibrium quantities.

Corollary 5. 

Assume that P, the cost functions, and the single-valued response $R_3$ are continuously differentiable, that $P^{\prime }\left(Q^{*}\right)<0$, and that the induced equilibrium $(x_1^{*},x_2^{*},x_3^{*})$ is interior. Put $Q^{*}=x_1^{*}+x_2^{*}+x_3^{*}$ and $r_i^{*}=\partial R_3/\partial x_i(x_1^{*},x_2^{*})$. Then, for $i=1,2$,

$$P^{\prime }\left(Q^{*}\right)\left[x_i^{*}(1+r_i^{*})-x_3^{*}\right]=c_i^{\prime }\left(x_i^{*}\right)-c_3^{\prime }\left(x_3^{*}\right).$$

Consequently,

• $c_i^{\prime }\left(x_i^{*}\right)<c_3^{\prime }\left(x_3^{*}\right)$ if and only if $x_i^{*}(1+r_i^{*})>x_3^{*}$
• $c_i^{\prime }\left(x_i^{*}\right)>c_3^{\prime }\left(x_3^{*}\right)$ if and only if $x_i^{*}(1+r_i^{*})<x_3^{*}$.

Proof. 

The first-order conditions for leader i and the follower are, respectively,

$$P\left(Q^{*}\right)+x_i^{*}{P}^{\prime }\left(Q^{*}\right)(1+r_i^{*})-c_i^{\prime }\left(x_i^{*}\right)=0,P\left(Q^{*}\right)+x_3^{*}{P}^{\prime }\left(Q^{*}\right)-c_3^{\prime }\left(x_3^{*}\right)=0.$$

Subtracting the second equality from the first gives the stated identity. The comparison follows because $P^{\prime }\left(Q^{*}\right)<0$. □

The identity shows that marginal costs alone do not generally determine the ordering of the actual quantities. The follower-response term also enters through the effective quantity $x_i^{*}(1+r_i^{*})$.

Corollary 6. 

Assume that the hypotheses of Corollary 5 are satisfied. Suppose, in addition, that all three firms share the same cost function $c:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$, that is, $c_1=c_2=c_3=c$, where c is continuously differentiable and convex. Assume that P is continuously differentiable and strictly decreasing, and that the failed-cartel equilibrium $(x_1^{*},x_2^{*},x_3^{*})$ is interior.

Assume further that the follower reaction $R_3$ is single-valued and differentiable in a neighborhood of $(x_1^{*},x_2^{*})$. Then, for $i\in \{1,2\}$, the following conclusions hold:

• if ${\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})<0$ then $x_i^{*}>x_3^{*}$
• if ${\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})>0$ then $x_i^{*}<x_3^{*}$

In particular, if the equilibrium is symmetric with respect to the two leaders, so that $x_1^{*}=x_2^{*}$, then:

• if ${\displaystyle \frac{\partial R_3}{\partial x_1}}(x_1^{*},x_2^{*})<0$ and ${\displaystyle \frac{\partial R_3}{\partial x_2}}(x_1^{*},x_2^{*})<0$, then $x_1^{*}=x_2^{*}>x_3^{*}$
• if ${\displaystyle \frac{\partial R_3}{\partial x_1}}(x_1^{*},x_2^{*})>0$ and ${\displaystyle \frac{\partial R_3}{\partial x_2}}(x_1^{*},x_2^{*})>0$, then $x_1^{*}=x_2^{*}<x_3^{*}$.

Proof. 

Let $Q^{*}=x_1^{*}+x_2^{*}+x_3^{*}$. Since the equilibrium is interior, the first-order condition for the follower is

$$P\left(Q^{*}\right)+x_3^{*}{P}^{\prime }\left(Q^{*}\right)-c^{\prime }\left(x_3^{*}\right)=0.$$

For each leader $i\in \{1,2\}$, the first-order condition for the induced payoff is

$$P\left(Q^{*}\right)+x_i^{*}{P}^{\prime }\left(Q^{*}\right)\left(1+{\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})\right)-c^{\prime }\left(x_i^{*}\right)=0.$$

Subtracting the follower’s first-order condition from the corresponding condition for leader i gives

$$c^{\prime }\left(x_i^{*}\right)-c^{\prime }\left(x_3^{*}\right)=P^{\prime }\left(Q^{*}\right)\left[x_i^{*}\left(1+{\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})\right)-x_3^{*}\right].$$

(1)

We first assume that ${\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})<0$. Suppose, to the contrary, that $x_i^{*}\le x_3^{*}$. Since c is convex, $c^{\prime }$ is nondecreasing, and therefore $c^{\prime }\left(x_i^{*}\right)-c^{\prime }\left(x_3^{*}\right)\le 0$.

On the other hand, $x_i^{*}\left(1+{\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})\right)<x_i^{*}\le x_3^{*}$. Hence

$$x_i^{*}\left(1+{\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})\right)-x_3^{*}<0.$$

Since $P^{\prime }\left(Q^{*}\right)<0$, the right-hand side of (1) is strictly positive. Thus $c^{\prime }\left(x_i^{*}\right)-c^{\prime }\left(x_3^{*}\right)>0$, which is a contradiction. Therefore, $x_i^{*}>x_3^{*}$.

We next assume that ${\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})>0$. Suppose, to the contrary, that $x_i^{*}\ge x_3^{*}$. The monotonicity of $c^{\prime }$ yields $c^{\prime }\left(x_i^{*}\right)-c^{\prime }\left(x_3^{*}\right)\ge 0$. Since the equilibrium is interior, $x_i^{*}>0$, and consequently

$$x_i^{*}\left(1+{\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})\right)>x_i^{*}\ge x_3^{*}.$$

Therefore, $x_i^{*}\left(1+{\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})\right)-x_3^{*}>0$. Because $P^{\prime }\left(Q^{*}\right)<0$, the right-hand side of (1) is strictly negative. It follows that $c^{\prime }\left(x_i^{*}\right)-c^{\prime }\left(x_3^{*}\right)<0$, again a contradiction. Hence

$$x_i^{*}<x_3^{*}.$$

The two conclusions hold separately for $i=1$ and $i=2$. If, in addition, $x_1^{*}=x_2^{*}$, the asserted symmetric orderings follow immediately. □

3.5. Economic Interpretation of the Strategic Comparison Results

The preceding corollaries isolate the role of strategic asymmetry in determining the relative equilibrium quantities of the two leaders and the follower. Since the three firms have the same cost function, the inequalities between $x_1^{*}$, $x_2^{*}$, and $x_3^{*}$ cannot be attributed to differences in production technology. Instead, they are generated by the way in which the optimal reaction $R_3$ of firm 3 changes in response to the production decisions of firms 1 and 2.

If ${\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})<0$ for $i=1,2$, an increase in the production of leader i reduces the optimal output of firm 3. The leaders anticipate this response through their induced payoff functions ${\Psi }_i$. As shown in Corollary 6, this strategic effect implies $x_1^{*}>x_3^{*}$ and $x_2^{*}>x_3^{*}$. If the equilibrium is symmetric with respect to the two leaders, then $x_1^{*}=x_2^{*}>x_3^{*}$.

The opposite ordering arises in the second assumptions.

Thus, even under identical cost structures, the failed-cartel model does not imply a universal ordering of equilibrium quantities. The relative position of the leaders and the follower depends on the local response of firm 3 to changes in the leaders’ production levels. In particular, the sign of ${\displaystyle \frac{\partial R_3}{\partial x_i}}(x_1^{*},x_2^{*})$ determines whether the strategic structure favors larger equilibrium production by the leaders or by the follower.

4. Illustrative Examples and Equilibrium Configurations

The theoretical results obtained in the preceding section identify two principal mechanisms that may determine the ordering of equilibrium quantities in the failed-cartel market: technological asymmetry, reflected in differences between marginal costs, and strategic asymmetry, reflected in the response of firm 3 to changes in the production levels of firms 1 and 2.

The purpose of the following examples is to illustrate these mechanisms separately and in combination. We first consider markets in which differences in equilibrium quantities are primarily generated by heterogeneous production costs. We then examine the strategic effect associated with decreasing and increasing follower reactions under identical cost structures. Finally, we consider examples in which technological and strategic effects interact and may either reinforce or offset one another.

4.1. Strategic Dominance of the Leaders

The following example illustrates the strategic asymmetry generated by a decreasing follower reaction. Although the firms operate under the same market environment, the follower reduces its optimal production when either leader expands output. This strategic response gives the leaders an additional incentive to increase production and leads to an asymmetric equilibrium in which the leaders produce more than the follower.

Example 4. 

Let $P\left(Q\right)=100-Q$, $c_1\left(x\right)=c_2\left(x\right)=0.25x^2$, and $c_3\left(x\right)=x^2$.

On the economically relevant domain $Q<100$, the inverse demand is positive and strictly decreasing, with $P^{\prime }\left(Q\right)=-1$. The cost derivatives are $c_1^{\prime }\left(x\right)=c_2^{\prime }\left(x\right)=0.5x$ and $c_3^{\prime }\left(x\right)=2x$. The follower response is $R_3(x_1,x_2)=(100-x_1-x_2)/4$, so $\partial R_3/\partial x_i=-1/4<0$ for $i=1,2$.

The symmetric equilibrium is $x_1^{*}=x_2^{*}=300/11$ and $x_3^{*}=125/11$. At the equilibrium, $c_1^{\prime }\left(x_1^{*}\right)=c_2^{\prime }\left(x_2^{*}\right)=150/11<250/11=c_3^{\prime }\left(x_3^{*}\right)$. Thus, the lower marginal costs of firms 1 and 2 and the decreasing response of firm 3 work in the same direction, and $x_1^{*}=x_2^{*}>x_3^{*}$. Corollary 6 does not apply directly because the three cost functions are not identical.

4.2. Asymmetry Generated by Marginal-Cost Differences

Unlike the previous subsection, where the asymmetry originated from the strategic response of the follower, the present example isolates the effect of heterogeneous production costs. The follower’s reaction function remains single-valued, and the equilibrium exists by Corollary 4. The objective is to illustrate how differences in marginal costs alone affect the equilibrium allocation among the three firms.

Example 5. 

Let $P\left(Q\right)={\displaystyle \frac{1}{{(1+Q)}^5}}$, and let the cost functions be $c_1\left(x_1\right)=0.0004201x_1$, $c_2\left(x_2\right)=0.0004201x_2$, and $c_3\left(x_3\right)=0.0017854x_3+{\displaystyle \frac{0.001}{2}}{x}_3^2$.

The inverse demand function P is positive and strictly decreasing. Moreover, it is $(-1/3)$-concave, since $P{\left(Q\right)}^{-1/3}={(1+Q)}^{5/3}$ is convex.

The equilibrium quantities are $x_1^{*}=x_2^{*}=0.4$, $x_3^{*}=0.39$, and hence $Q^{*}=x_1^{*}+x_2^{*}+x_3^{*}=1.19$.

At the equilibrium aggregate output, $P\left(Q^{*}\right)\approx 0.0198509$, $P^{\prime }\left(Q^{*}\right)\approx -0.0453216$.

The marginal cost functions are $c_1^{\prime }\left(x_1\right)=c_2^{\prime }\left(x_2\right)=0.0004201$, whereas $c_3^{\prime }\left(x_3\right)=0.0017854+0.001x_3$. Therefore, $c_3^{\prime }\left(x_3^{*}\right)\approx 0.0021754$.

Consequently, $c_1^{\prime }\left(x_1^{*}\right)<c_3^{\prime }\left(x_3^{*}\right)$, $c_2^{\prime }\left(x_2^{*}\right)<c_3^{\prime }\left(x_3^{*}\right)$.

The marginal-cost inequalities alone do not determine the ordering of the actual quantities, because the follower-response term must also be taken into account.

We now examine the strategic effect generated by the follower’s reaction. The follower’s first-order condition is $P\left(Q\right)+x_3{P}^{\prime }\left(Q\right)-c_3^{\prime }\left(x_3\right)=0$, $Q=x_1+x_2+x_3$. Implicit differentiation yields

$${\displaystyle \frac{\partial R_3}{\partial x_i}}=-{\displaystyle \frac{P^{\prime }\left(Q\right)+x_3{P}^{\prime \prime }\left(Q\right)}{2P^{\prime }\left(Q\right)+x_3{P}^{\prime \prime }\left(Q\right)-c_3^{\prime \prime }\left(x_3\right)}},i=1,2.$$

At $(x_1^{*},x_2^{*},x_3^{*})=(0.4,0.4,0.39),$ we obtain ${\displaystyle \frac{\partial R_3}{\partial x_1}}(x_1^{*},x_2^{*})={\displaystyle \frac{\partial R_3}{\partial x_2}}(x_1^{*},x_2^{*})\approx 0.0718>0$.

Thus, the follower response is increasing at the equilibrium and works against larger leader quantities. Nevertheless, $x_i^{*}(1+\partial R_3/\partial x_i)-x_3^{*}\approx 0.0387>0$, in agreement with Corollary 5. Since $x_1^{*}=x_2^{*}=0.4>x_3^{*}=0.39$, the lower marginal costs of the leaders dominate the opposing strategic effect. The example does not isolate a purely technological effect, but it shows that this effect can be the stronger one.

4.3. Strategic Asymmetry with a Decreasing Follower Reaction

Corollary 6 shows that, under identical production costs, the ordering of equilibrium quantities is completely determined by the sign of the partial derivatives of the follower’s reaction function. The following example illustrates this result in the case of a decreasing follower reaction. Since the firms have identical technologies, the equilibrium asymmetry is purely strategic rather than technological.

Example 6. 

Let $P\left(Q\right)=100-Q$, $c_1\left(x\right)=c_2\left(x\right)=c_3\left(x\right)=10x$.

Since all three firms have the same cost function, technological asymmetry is absent. The follower’s profit is ${\Pi }_3(x_1,x_2,x_3)=x_3(100-x_1-x_2-x_3)-10x_3$. For fixed $x_1$ and $x_2$, the first-order condition is $90-x_1-x_2-2x_3=0$.

Hence the follower reaction is $R_3(x_1,x_2)=45-{\displaystyle \frac{x_1+x_2}{2}}$. Therefore,

$${\displaystyle \frac{\partial R_3}{\partial x_1}}={\displaystyle \frac{\partial R_3}{\partial x_2}}=-{\displaystyle \frac{1}{2}}<0.$$

The induced payoff of leader i, $i=1,2$, is

$${\Psi }_i(x_i,x_j)=x_i\left(100-x_i-x_j-R_3(x_i,x_j)\right)-10x_i,j\ne i.$$

Substituting the expression for $R_3$, we obtain ${\Psi }_i(x_i,x_j)=x_i\left(45-{\displaystyle \frac{x_i+x_j}{2}}\right)$. Thus,

$${\displaystyle \frac{\partial {\Psi }_i}{\partial x_i}}=45-x_i-{\displaystyle \frac{x_j}{2}}.$$

The equilibrium conditions for the two leaders are

$$\begin{array}{c}45-x_1-{\displaystyle \frac{x_2}{2}}=0\hfill \\ 45-x_2-{\displaystyle \frac{x_1}{2}}=0.\hfill \end{array}$$

Solving this system gives $x_1^{*}=x_2^{*}=30$. Consequently,

$$x_3^{*}=R_3(x_1^{*},x_2^{*})=45-{\displaystyle \frac{30+30}{2}}=15.$$

Therefore, $x_1^{*}=x_2^{*}=30>15=x_3^{*}$.

This example illustrates a purely strategic effect. Since the cost functions are identical, the inequality between the leaders’ quantities and the follower’s quantity cannot be explained by marginal-cost differences. It is generated by the decreasing reaction of firm 3 with respect to the production levels of firms 1 and 2.

The preceding comparison results are formulated directly at the equilibrium production profile. In applications, however, the equilibrium quantities may not be available in explicit form. The existence argument may identify only a sufficiently small neighborhood containing an equilibrium, while the marginal costs can still be compared throughout that neighborhood. The following result shows that a strict ordering of marginal costs, preserved locally around the equilibrium, is sufficient to determine the corresponding ordering of the equilibrium quantities.

4.4. Local Comparison when the Equilibrium is Not Explicitly Known

The preceding examples illustrate that, for nonlinear inverse demand and cost functions, the failed-cartel equilibrium need not be available in explicit form. In such cases, an existence result may guarantee an equilibrium in a given region of the strategy space without providing the exact values of $x_1^{*}$, $x_2^{*}$, and $x_3^{*}$. Consequently, the marginal costs $c_i^{\prime }\left(x_i^{*}\right)$ and $c_3^{\prime }\left(x_3^{*}\right)$ cannot be evaluated directly at the equilibrium.

Nevertheless, an exact computation of the equilibrium is not always necessary for comparing the equilibrium quantities. If the ordering of the marginal costs is preserved throughout a region known to contain an equilibrium, then the same comparison argument can be applied without identifying the equilibrium point explicitly. This observation leads to the following local comparison result.

Corollary 7. 

Assume the hypotheses of Corollary 5, and let $(x_1^{*},x_2^{*},x_3^{*})$ be an interior equilibrium. Suppose that, for some $i\in \{1,2\}$, there is a neighborhood U of the equilibrium such that $c_i^{\prime }\left(x_i\right)<c_3^{\prime }\left(x_3\right)$ and $\partial R_3/\partial x_i\le 0$ throughout U. Then $x_i^{*}>x_3^{*}$. Conversely, if $c_i^{\prime }\left(x_i\right)>c_3^{\prime }\left(x_3\right)$ and $\partial R_3/\partial x_i\ge 0$ throughout U, then $x_i^{*}<x_3^{*}$.

Proof. 

The assumed inequalities hold at the equilibrium. Corollary 5 gives $x_i^{*}(1+\partial R_3/\partial x_i)>x_3^{*}$ in the first case; since $\partial R_3/\partial x_i\le 0$, this implies $x_i^{*}>x_3^{*}$. The converse follows in the same way when $\partial R_3/\partial x_i\ge 0$. □

4.5. Illustration of the Combined Effect of Technology and Strategy

The preceding examples isolated two different mechanisms leading to asymmetric equilibrium production. Corollary 5 shows, however, that the equilibrium ordering is determined by the combined effect of the firms’ marginal costs and the strategic response of the follower. The following example illustrates this interaction. In contrast to the previous examples, neither technological differences nor strategic effects can be analyzed in isolation; the observed equilibrium is determined by their joint influence.

Example 7. 

Let $P\left(Q\right)=100-Q$, and let the cost functions be $c_1\left(x_1\right)=5x_1^2$, $c_2\left(x_2\right)=5x_2^2$, $\left(x_3\right)={\displaystyle \frac{1}{2}}{x}_3^2$.

The follower’s payoff is ${\Pi }_3(x_1,x_2,x_3)=x_3(100-x_1-x_2-x_3)-{\displaystyle \frac{1}{2}}{x}_3^2$.

For fixed $x_1$ and $x_2$, the first-order condition is

$${\displaystyle \frac{\partial {\Pi }_3}{\partial x_3}}=100-x_1-x_2-3x_3=0.$$

Hence the follower reaction is $R_3(x_1,x_2)={\displaystyle \frac{100-x_1-x_2}{3}}$. Therefore,

$${\displaystyle \frac{\partial R_3}{\partial x_1}}={\displaystyle \frac{\partial R_3}{\partial x_2}}=-{\displaystyle \frac{1}{3}}<0.$$

For $i,j\in \{1,2\}$, $i\ne j$, the induced payoff of leader i is

$${\Psi }_i(x_i,x_j)=x_{i}P\left(x_i+x_j+R_3(x_i,x_j)\right)-c_i\left(x_i\right).$$

Since $P\left(x_i+x_j+R_3(x_i,x_j)\right)={\displaystyle \frac{200}{3}}-{\displaystyle \frac{2}{3}}(x_i+x_j)$, we obtain

$${\Psi }_i(x_i,x_j)=x_i\left({\displaystyle \frac{200}{3}}-{\displaystyle \frac{2}{3}}(x_i+x_j)\right)-5x_i^2.$$

Thus,

$${\displaystyle \frac{\partial {\Psi }_i}{\partial x_i}}={\displaystyle \frac{200}{3}}-{\displaystyle \frac{4}{3}}{x}_i-{\displaystyle \frac{2}{3}}{x}_j-10x_i.$$

The equilibrium conditions for the two leaders are therefore

$$\begin{array}{c}200-34x_1-2x_2=0\hfill \\ 200-34x_2-2x_1=0,\hfill \end{array}$$

with a solution $x_1^{*}=x_2^{*}={\displaystyle \frac{50}{9}}$. The corresponding follower quantity is $x_3^{*}=R_3(x_1^{*},x_2^{*})={\displaystyle \frac{800}{27}}$.

Hence $x_1^{*}=x_2^{*}={\displaystyle \frac{50}{9}}<{\displaystyle \frac{800}{27}}=x_3^{*}$.

At equilibrium, the marginal costs are $c_1^{\prime }\left(x_1^{*}\right)=c_2^{\prime }\left(x_2^{*}\right)={\displaystyle \frac{500}{9}}$, whereas $c_3^{\prime }\left(x_3^{*}\right)=x_3^{*}={\displaystyle \frac{800}{27}}.$

Therefore, $c_1^{\prime }\left(x_1^{*}\right)>c_3^{\prime }\left(x_3^{*}\right)$ and $c_2^{\prime }\left(x_2^{*}\right)>c_3^{\prime }\left(x_3^{*}\right)$.

The follower reaction is decreasing,

$${\displaystyle \frac{\partial R_3}{\partial x_i}}<0,i=1,2,$$

which, in isolation, tends to increase the leaders’ equilibrium production. However, at the computed equilibrium the leaders face substantially higher marginal costs,

$$c_i^{\prime }\left(x_i^{*}\right)>c_3^{\prime }\left(x_3^{*}\right),i=1,2,$$

and this technological disadvantage dominates the strategic effect. Consequently,

$$x_1^{*}=x_2^{*}<x_3^{*}.$$

This example confirms that the equilibrium ordering is governed by the balance described in Corollary 5: neither the strategic reaction nor the cost structure alone is sufficient to determine the equilibrium allocation.

Although the previous examples employ nonlinear demand functions, the phenomenon described in Corollary 6 is not restricted to such settings. The following quadratic model shows that the same ordering of the equilibrium quantities also arises in a classical Cournot framework with linear demand and identical quadratic production costs.

Example 8. 

Let $P\left(Q\right)=100-Q$, and let the three firms have the same cost function $c_1\left(x_1\right)={\displaystyle \frac{1}{2}}{x}_1^2$, $c_2\left(x_2\right)={\displaystyle \frac{1}{2}}{x}_2^2$, $c_3\left(x_3\right)={\displaystyle \frac{1}{2}}{x}_3^2$.

Since the three firms have identical cost functions, technological asymmetry is absent. We therefore focus exclusively on the strategic effect generated by the reaction of firm 3.

The follower’s payoff is ${\Pi }_3(x_1,x_2,x_3)=x_3(100-x_1-x_2-x_3)-{\displaystyle \frac{1}{2}}{x}_3^2$.

For fixed $x_1$ and $x_2$, the first-order condition is

$${\displaystyle \frac{\partial {\Pi }_3}{\partial x_3}}=100-x_1-x_2-3x_3=0.$$

Hence the follower reaction is $R_3(x_1,x_2)={\displaystyle \frac{100-x_1-x_2}{3}}$. Consequently,

$${\displaystyle \frac{\partial R_3}{\partial x_1}}={\displaystyle \frac{\partial R_3}{\partial x_2}}=-{\displaystyle \frac{1}{3}}<0.$$

For $i,j\in \{1,2\}$, $i\ne j$, the induced payoff of leader i is

$${\Psi }_i(x_i,x_j)=x_{i}P\left(x_i+x_j+R_3(x_i,x_j)\right)-{\displaystyle \frac{1}{2}}{x}_i^2.$$

Since $P\left(x_i+x_j+R_3(x_i,x_j)\right)={\displaystyle \frac{200}{3}}-{\displaystyle \frac{2}{3}}(x_i+x_j)$, we obtain

$${\Psi }_i(x_i,x_j)=x_i\left({\displaystyle \frac{200}{3}}-{\displaystyle \frac{2}{3}}(x_i+x_j)\right)-{\displaystyle \frac{1}{2}}{x}_i^2.$$

Therefore, ${\displaystyle \frac{\partial {\Psi }_i}{\partial x_i}}={\displaystyle \frac{200}{3}}-{\displaystyle \frac{7}{3}}{x}_i-{\displaystyle \frac{2}{3}}{x}_j$.

The equilibrium conditions for the two leaders are

$$\begin{array}{c}200-7x_1-2x_2=0\hfill \\ 200-7x_2-2x_1=0,\hfill \end{array}$$

with a soution $x_1^{*}=x_2^{*}={\displaystyle \frac{200}{9}}$.

The corresponding follower quantity is $x_3^{*}=R_3(x_1^{*},x_2^{*})={\displaystyle \frac{500}{27}}$. Hence $x_1^{*}=x_2^{*}={\displaystyle \frac{200}{9}}>{\displaystyle \frac{500}{27}}=x_3^{*}$.

Thus, $x_1^{*}=x_2^{*}>x_3^{*}$, in accordance with Corollary 6.

This example isolates the strategic effect identified in Corollary 6. Since the firms have identical cost functions, the ordering of equilibrium quantities cannot be attributed to technological differences. Instead, it is generated by the decreasing reaction of firm 3:

$${\displaystyle \frac{\partial R_3}{\partial x_i}}<0,i=1,2.$$

The two leaders anticipate this response through their induced payoff functions ${\Psi }_1$ and ${\Psi }_2$, and both produce more than the follower at equilibrium.

4.6. Strategic Asymmetry with an Increasing Follower Reaction

Corollary 6 identifies two distinct strategic regimes according to the sign of the follower’s reaction. Example 8 illustrated the case of a decreasing follower reaction, leading to higher equilibrium production by the leaders. We now consider the complementary situation in which the follower’s reaction is increasing. According to Corollary 6, the equilibrium ordering is reversed. The following example confirms this prediction.

Example 9. 

Let $P\left(Q\right)={\displaystyle \frac{1}{{(1+Q)}^{3.1084}}},$ and let the three firms have the same cost function $c_i\left(x_i\right)=0.0029845x_i+{\displaystyle \frac{0.0001086}{2}}{x}_i^2$ for $i=1,2,3$.

The inverse demand function is positive and strictly decreasing. Moreover,

$$P{\left(Q\right)}^{-1/3}={(1+Q)}^{3.1084/3}$$

is convex, and hence P is $(-1/3)$-concave.

The follower’s payoff is ${\Pi }_3(x_1,x_2,x_3)=x_{3}P(x_1+x_2+x_3)-c_3\left(x_3\right)$.

For fixed $x_1$ and $x_2$, the follower reaction $R_3(x_1,x_2)$ is defined implicitly by

$$P\left(Q\right)+x_3{P}^{\prime }\left(Q\right)-c_3^{\prime }\left(x_3\right)=0,Q=x_1+x_2+x_3.$$

Solving the corresponding equilibrium system numerically gives $x_1^{*}=x_2^{*}\approx 0.939473$, $x_3^{*}\approx 0.986446$ and therefore $Q^{*}\approx 2.865392$.

At this equilibrium,

$${\displaystyle \frac{\partial R_3}{\partial x_1}}(x_1^{*},x_2^{*})={\displaystyle \frac{\partial R_3}{\partial x_2}}(x_1^{*},x_2^{*})\approx 0.050451>0.$$

Thus the follower reaction is locally increasing with respect to the production levels of both leaders.

Since the cost functions are identical, technological asymmetry is absent. The ordering of the equilibrium quantities is therefore generated by the strategic effect:

$$x_1^{*}=x_2^{*}\approx 0.939473<0.986446\approx x_3^{*}.$$

This is consistent with Corollary 6.

5. Discussion and Conclusions

The failed-cartel market considered in this paper occupies an intermediate position between the classical Cournot oligopoly and a fully coordinated cartel. Although such partially coordinated markets arise naturally when collusive agreements begin to unravel, they have received considerably less attention than the standard market organizations. The objective of this work has therefore been to develop a general equilibrium framework capable of describing these hybrid strategic interactions without relying on the restrictive assumption that firms possess uniquely determined reaction functions.

The principal contribution of the paper is methodological rather than based on new fixed-point theory. Instead of deriving equilibrium from differentiable reaction functions, we formulate the market directly through the firms’ optimization problems and represent strategic behavior by set-valued best-response correspondences. This formulation naturally accommodates situations in which firms possess several optimal production decisions. Consequently, the existence of equilibrium can be established under substantially weaker assumptions than those commonly employed in reaction-function analyses, while uniqueness is recovered only after imposing additional structural conditions.

Beyond the existence and uniqueness results, our analysis provides insight into the economic forces determining equilibrium production levels following the partial breakdown of a cartel. We identify two distinct sources of market asymmetry. The first is technological, arising from differences in firms’ marginal production costs. In this case, firms with lower marginal costs naturally produce larger equilibrium quantities. The second source is purely strategic. Even when firms possess identical technologies, the hybrid organization of the market generates unequal production levels because the coordinated firms anticipate the optimal response of the deviating firm. Depending on the behavior of this response correspondence, either the remaining cartel members or the deviating firm may become the dominant producer. Thus, differences in equilibrium market shares need not originate solely from production efficiency but may emerge from the strategic organization of the market itself.

From a broader perspective, the paper illustrates the usefulness of set-valued optimization methods in oligopoly theory. Multiple optimal responses are often regarded as technical complications to be eliminated through stronger concavity or differentiability assumptions. Our analysis instead incorporates this possibility directly into the equilibrium model. This perspective separates the existence and uniqueness questions and enlarges the class of market environments that can be analyzed within a unified mathematical framework.

Several limitations suggest directions for future research. The present analysis considers a homogeneous-good market with three firms and a single deviating cartel member. Extensions to larger markets, multiple defecting firms, differentiated products, dynamic or repeated interactions, stochastic demand, and endogenous coalition formation would provide a richer description of cartel instability while preserving the equilibrium methodology developed here. It would also be of interest to investigate welfare properties, equilibrium stability under adaptive learning dynamics, and antitrust implications of partially coordinated market structures.

In summary, this paper provides a general mathematical foundation for analyzing failed-cartel markets under non-unique optimal responses. By combining classical tools from optimization and equilibrium theory with a set-valued formulation of firms’ strategic behavior, it extends the analysis of hybrid market organizations beyond the traditional reaction-function framework. We hope that the proposed approach will provide a useful basis for future work on partially coordinated oligopolies and, more generally, on equilibrium problems in which cooperation and competition coexist.