Note On the core of a collection of coalitions

Abstract

. For a collection Ω of subsets of a finite set N we define its core to be equal to the polyhedral cone {x∈IR ^N: ∑ _i∈N x _i =0 and ∑ _i∈S x _i ≥0 for all S∈Ω}. This note describes several applications of this concept in the field of cooperative game theory. Especially collections Ω are considered with core equal to {0}. This property of a one-point core is proved to be equivalent to the non-degeneracy and balancedness of Ω. Further, the notion of exact cover is discussed and used in a second characterization of collections Ω with core equal to {0}.