The core of games on ordered structures and graphs

In cooperative games, the core is the most popular solution concept, and its properties are well known. In the classical setting of cooperative games, it is generally assumed that all coalitions can form, i.e., they are all feasible. In many situations, this assumption is too strong and one has to deal with some unfeasible coalitions. Defining a game on a subcollection of the power set of the set of players has many implications on the mathematical structure of the core, depending on the precise structure of the subcollection of feasible coalitions. Many authors have contributed to this topic, and we give a unified view of these different results

The Average Tree Solution for Multi-choice Forest Games

In this article we study cooperative multi-choice games with limited cooperation possibilities, represented by an undirected forest on the player set. Players in the game can cooperate if they are connected in the forest. We introduce a new (single-valued) solution concept which is a generalization of the average tree solution defined and characterized by Herings et al. [2008] for TU-games played on a forest. Our solution is characterized by component efficiency, component fairness and independence on the greatest activity level. It belongs to the precore of a restricted multi-choice game whenever the underlying multi-choice game is superadditive and isotone. We also link our solution with the hierarchical outcomes (Demange, 2004) of some particular TU-games played on trees. Finally, we propose two possible economic applications of our average tree solution.Average tree solution; Communication graph; (pre-)Core; Hierarchical outcomes; Multi-choice games.

Discounted Tree Solutions

This article introduces a discount parameter and a weight function in Myerson's (1977) classical model of cooperative games with restrictions on cooperation. The discount parameter aims to reflect the time preference of the agents while the weight function aims to reflect the importance of each node of a graph. We provide axiomatic characterizations of two types of solution that are inspired by the hierarchical outcomes (Demange, 2004)

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