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

Abstract

ED EPSInternational audienceFinding 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

On the restricted cores and the bounded core of games on distributive lattices

“…Note that this generalizes a result by Grabisch and Xie (2011). As it can be observed, the proof follows the classical argument, where in the right members of (in)equalities, N is formally replaced by N ∪ N. Therefore, a strong form of the theorem can be obtained as well, which we give without proof.…”

“…A third remarkable collection is the one proposed by Grabisch and Xie (2011), and is defined as follows:…”

“…The first one is to consider restricted cores, i.e., to impose in the core further equality constraints x(S) = v(S) for all S in some collection N , which can be interpreted as binding constraints for some coalitions, in order to exclude any extremal ray in the core (Grabisch and Xie, 2011;Grabisch, 2011). Such a collection is called a normal collection.…”

The core of games on ordered structures and graphs

“…Remark 5.13 The two last theorems are shown by Grabisch and Xie (2008), but they can deduced from Derks and Gilles (1995), where they are stated for acyclic permission structures. Indeed, from Algaba et al (2004), we know that these systems are equivalent to distributive lattices of the type O(N ) (see Sect.…”

“…To avoid unboundedness, Grabisch and Xie (2008) have imposed further restrictions in the definition of the core, leading to the notion of restricted core. These additional constraints are built as follows.…”