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 deviantsSwill face to, say,jin number and introduce the notion ofj-beliefofSas the least number of coalitions into which the outsidersN\Swill reorganize. We then define for everyj-belief a TU-game and thej-belief core of it. We prove that the worth ofSis minimized when then – sagents split approximately equally among thejcoalitions, while the worth ofSis maximized whenj – 1agents have one member and one coalition hasn – s – (j – 1)members. Given the above, we prove that when goods are substitutes, thej-belief core is nonempty, provided thatSbelieve theN\Swill form a sufficiently large number of coalitions, while when goods are complements, thej-belief core is nonempty irrespective of the beliefs of the agents inS. Finally, in the case of homogeneous goods we prove that thej-belief core is nonempty and depends only on the number of the outsider coalitions and not on their size.