Mathematical Methods of Operations Research

, Volume 73, Issue 2, pp 189–208 | Cite as

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

  • Michel GrabischEmail author
  • Lijue Xie
Original Article


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 hierarchy). Then feasible coalitions correspond to teams of players, that is, one or several players with all their subordinates. In these situations, the core in its usual formulation may be unbounded, making its use difficult in practice. We propose a new notion of core, called the restricted core, which imposes efficiency of the allocation at each level of the hierarchy, is always bounded, 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.


Cooperative game Feasible coalition Core Hierarchy 


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© Springer-Verlag 2011

Authors and Affiliations

  1. 1.Paris School of EconomicsUniversity of Paris I—Panthéon-SorbonneParisFrance
  2. 2.Centre d’Economie de la SorbonneUniversity of Paris I—Panthéon-SorbonneParisFrance

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