The coalition concept in game theory

Abstract

One of the most difficult and major problems of game theory consists in understanding how players may choose to organize themselves in a game in order to get joint-maximizing profits. The problem raises in fact two different questions: which organizations are likely to emerge from a game (first problem) and how are members of these organizations going to share its gains (second problem)? Hence, if these organizations are coalitions of players, a solution to the game shall specify those coalitions that are likely to form (ex post stable coalitions and/or coalitions that could form during the process at an intermediary step) and the way players, in each coalition, shall share common utility they get through their coalition. In Theory of Games and Economic Behavior (TGEB) Von Neumann and Morgenstern proposed developing a theory of n-person games in the framework of the characteristic function model in order to solve the second problem. Later, Shapley (1953), Aumann and Maschler (1964), and many other authors have responded to the question by proposing various solution concepts: respectively, the Shapley value, the bargaining set, the core, the kernel, the nucleolus, etc. (Although the core is a major solution concept in cooperative games, we will not tackle it here, except in comparing it to bargaining sets, because we want to focus our attention on coalition structure. We will only give a brief definition and speak of the “core of a coalition structure.” We refer to Herve Moulin’s chapter in this book for a presentation of the core.)In cooperative game theory it seems hard to disentangle the two problems: the way players share the worth of the coalition depends on which coalitions are likely to emerge from the game. It is this link between the two problems that research on coalition structure games tries to elucidate, seeking a better understanding of how a given coalition structure affects each player’s individual payoff. Once this question is answered (and there are many ways to answer it), one can face the first problem (how players choose to organize themselves) from different viewpoints. Either one links explicitly the two problems considering that the “sharing” game within each coalition and the game between coalitions that determines stable coalitions should havecommon features,1 or one separates them. A way of avoiding arbitrary or ad hoc aspects of a specific choice of a solution concept may consist in introducing first a non-cooperative step into the game, which describes the whole set of possible links among players. Starting from a graph-theoretic extension of the Shapley value, Aumann and Myerson (1988) propose for instance justifying the existence of a cooperation structure in a cooperative game by a non-cooperative “linking” game, the issue of which is a subgame perfect Nash equilibrium.