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

We study a three-firm market following the breakdown of an initially complete cartel, taking both the breakdown and the identity of the departing firm as exogenously given. In the baseline model, the former cartel members no longer coordinate: all three firms choose quantities simultaneously and non-cooperatively. Thus, the term failed cartel describes the institutional origin of the post-cartel market rather than continued partial coordination or an endogenously profitable deviation. Because firms’ optimal quantities need not be unique, we represent their strategic behavior by set-valued best-response correspondences. We establish equilibrium existence for this simultaneous post-cartel Cournot model as a three-player specialization of the classical Debreu–Glicksberg–Fan framework, combining Berge’s maximum theorem with the Kakutani–Fan–Glicksberg fixed-point theorem. The generalized concavity of inverse demand and convexity of costs provide sufficient conditions for existence without requiring unique optimal responses. We also examine the single-valued case, distinguishing the uniqueness of individual best responses from the uniqueness of equilibrium. To enrich the benchmark analysis following the cartel breakdown, we introduce a separate anticipatory extension belonging to the class of hierarchical multi-leader–follower games. Our extension is narrowly focused on separating strategic from technological sources of production asymmetry. In our specialization, Firm 3 has a uniquely determined continuous response, while Firms 1 and 2 choose their quantities simultaneously and anticipate that response, yielding a two-leader–one-follower game. Equilibrium existence is established separately under explicit continuity and quasi-concavity assumptions on the induced payoffs. The extension shows that, even under identical technologies, the ordering of equilibrium quantities depends on the sign of Firm 3’s response to the leaders’ production decisions: depending on this response, either Firms 1 and 2 or Firm 3 may individually produce the larger equilibrium quantity. Thus, firms’ incentives to leave the cartel may differ according to the post-cartel market structure and their individual strategic positions within that structure.