# The core of games on distributive lattices : how to share benefits in a hierarchy

Working paper

Issue number:

2008.77

Publisher:

Maison des Sciences Économiques

Year:

2009

Finding a solution concept is one of the central problems in cooperative game theory, and the notion of core is the most popular solution concept since it is based on some rationality condition. In many real situations, not all possible coalitions can form, so that classical TU-games cannot be used. An interesting case is when possible coalitions are deﬁned through a partial ordering of the players (or hierar- chy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, it is not obvious to deﬁne a suitable notion of core, reﬂecting the team structure, and previous attempts are not satisfactory in this respect. We propose a new notion of core, which imposes eﬃciency of the allocation at each level of the hierarchy, and answers the problem of sharing beneﬁts in a hierarchy. We show that the core we deﬁned has properties very close to the classical case, with respect to marginal vectors, the Weber set, and balancedness.