Coalitional Beliefs in Cournot oligopoly TU Games

In cooperative games, due to computational complexity issues, deviant agents are not able to base their behavior on the outsiders' status but have to follow certain beliefs as to how it is in their strategic interest to act. This behavior constitutes the main interest of this paper. To this end, we quantify and characterize the set of coalitional beliefs that support cooperation of such agents. Assuming that they are engaged in a differentiated Cournot competition, for every belief of the deviants we define a TU-game, the solution to which characterizes the set of coalitional beliefs that support core nonemptiness. For this we fix the number of coalitions that deviants S will face to, say, j in number and introduce the notion of j-belief of S as the least number of coalitions into which the outsiders N\S will reorganize. We then define for every j-belief a TU-game and the j-belief core of it. We prove that the worth of S is minimized when the n – s agents split approximately equally among the j coalitions, while the worth of S is maximized when j – 1 agents have one member and one coalition has n – s – (j – 1) members. Given the above, we prove that when goods are substitutes, the j-belief core is nonempty, provided that S believe the N\S will form a sufficiently large number of coalitions, while when goods are complements, the j-belief core is nonempty irrespective of the beliefs of the agents in S. Finally, in the case of homogeneous goods we prove that the j-belief core is nonempty and depends only on the number of the outsider coalitions and not on their size.

[1]  S. Hart,et al.  On the endogenous formation of coalitions , 1983 .

[2]  X. Vives,et al.  Price and quantity competition in a differentiated duopoly , 1984 .

[3]  Jingang Zhao,et al.  The equilibria of a multiple objective game , 1991 .

[4]  W. Lucas,et al.  N‐person games in partition function form , 1963 .

[5]  Jingang Zhao,et al.  A necessary and sufficient condition for the convexity in oligopoly games , 1999 .

[6]  Philip Wolfe,et al.  Contributions to the theory of games , 1953 .

[7]  Jongsay Yong.  Horizontal Monopolization Via Alliances, or Why a Conspiracy to Monopolize is Harder than it Appears , 2004 .

[8]  Onn Shehory,et al.  Coalition structure generation with worst case guarantees , 2022 .

[9]  Jingang Zhao,et al.  Inverse Matrices and Merger Incentives from Bertrand Competition , 2007 .

[10]  Sergio Currarini,et al.  Coalition Formation in Games Without Synergies , 2004, IGTR.

[11]  A. Lardon.  The γ-core in Cournot oligopoly TU-games with capacity constraints , 2012 .

[12]  Yiannis N. Moschovakis.  Notes on set theory, Second Edition , 2006, Undergraduate texts in mathematics.

[13]  Francis Bloch.  Sequential Formation of Coalitions in Games with Externalities and Fixed Payoff Division , 1996 .

[14]  Xiaotie Deng,et al.  On the Complexity of Cooperative Solution Concepts , 1994, Math. Oper. Res..

[15]  Sang-Seung Yi.  Endogenous Formation of Economic Coalitions: A Survey on the Partition Function Approach , 2003 .

[16]  Robert J. Aumann,et al.  16. Acceptable Points in General Cooperative n-Person Games , 1959 .

[17]  E. Maskin.  Bargaining, Coalitions and Externalities , 2003 .

[18]  Sang-Seung Yi.  Stable Coalition Structures with Externalities , 1997 .

[19]  A Sequential Approach to the Characteristic Function and the Core in Games with Externalities , 2001 .

[20]  Carsten Helm.  On the existence of a cooperative solution for a coalitional game with externalities , 2001, Int. J. Game Theory.

[21]  Henry Tulkens,et al.  The core of an economy with multilateral environmental externalities , 1997, Int. J. Game Theory.

[22]  Marco A. Marini,et al.  Coalitional Approaches to Collusive Agreements in Oligopoly Games , 2011 .

[23]  Isa Emin Hafalir,et al.  Efficiency in coalition games with externalities , 2007, Games Econ. Behav..

[24]  V. Grossmann.  Firm size and diversification: Multiproduct firms in asymmetric oligopoly , 2007 .

[25]  Debraj Ray.  A Game-Theoretic Perspective on Coalition Formation , 2007 .

[26]  C. Carraro.  The Endogenous Formation of Economic Coalitions , 2003 .

[27]  Debraj Ray,et al.  Equilibrium Binding Agreements , 1997 .

[28]  Jingang Zhao.  A [beta]-Core Existence Result and Its Application to Oligopoly Markets , 1999 .

[29]  Yiannis N. Moschovakis,et al.  Notes On Set Theory , 1994 .