Games With General Coalitional Structure

Abstract: This paper introduces a new solution concept for cooperative games with general coalitional structure in which only certain sets of players, including the set of all players, are able to form feasible coalitions. The solution concept takes into account the marginal contribution of players. This marginal contribution can be a joint contribution of several players and is equally divided among those players. Any set system representing a coalitional structure induces a collection of coalitional trees, whose nodes may consist of subsets of players. As solution we take the average of the marginal contribution vectors that correspond to all coalitional trees. The solution is ecient and several other properties are studied and some special cases are considered.

Structural restrictions in cooperation

Cooperative games with transferable utilities, or simply TU-games, refer to the situations where the revenues created by a coalition of players through cooperation can be freely distributed to the members of the coalition. The fundamental question in cooperative game theory deals with the problem of how much payoff every player should receive. The classical assumption for TU-games states that every coalition is able to form and earn the worth created by cooperation. In the literature, there are several different modifications of TU-games in order to cover the cases where cooperation among the players is restricted. The second chapter of this monograph provides a characterization of the average tree solution for TU-games where the restricted cooperation is represented by a connected cycle-free graph on the set of players. The third chapter considers TU-games for which the restricted cooperation is represented by a directed graph on the set of players and introduces the average covering tree solution and the dominance value for this class of games. Chapter four considers TU-games with restricted cooperation which is represented by a set system on the set of players and introduces the average coalitional tree solution for such structures. The last two chapters of this monograph belong to the social choice theory literature. Given a set of candidates and a set of an odd number of individuals with preferences on these candidates, pairwise majority comparison of the candidates yields a tournament on the set of candidates. Tournaments are special types of directed graphs which contain an arc between any pair of nodes. The Copeland solution of a tournament is the set of candidates that beat the maximum number of candidates. In chapter five, a new characterization of the Copeland solution is provided that is based on the number of steps in which candidates beat each other. Chapter six of this monograph is on preference aggregation which deals with collective decision making to obtain a social preference. A sophisticated social welfare function is defined as a mapping from profiles of individual preferences into a sophisticated social preference which is a pairwise weighted comparison of alternatives. This chapter provides a characterization of Pareto optimal and pairwise independent sophisticated social welfare functions

The Shapley value for arbitrary families of coalitions

We address the problem of finding a suitable definition of a value similar to that of Shapley's, when the games are defined on a subfamily of coalitions with no structure. We present two frameworks: one based on the familiar efficiency, linearity and null player axioms, and the other on linearity and the behavior on unanimity games. We give several properties and examples in each case, and give necessary and sufficient conditions on the family of coalitions for the approaches to coincide.Game theory Cooperative game Shapley value