Core Concepts for Share Vectors

A situation in which a finite set of players can obtain certain payoffs by cooperation can be described by a cooperative game with transferable utilities {or simply a TU-game. A value mapping for TU-games is a mapping that assigns to every game a set of vectors each representing a distribution of the payoffs. A value mapping is efficient if to every game it assigns a set of vectors which components all sum up to the worth that can be obtained by all players cooperating together. An approach to efficiently allocating the worth of the `grand coalition' is using share mappings which assign to every game a set of share vectors being vectors which components sum up to one such that every component is the corresponding players' share in the total payoff that is to be distributed among the players. In this paper we discuss a class of share mappings containing the (Shapley) share-core, the Banzhaf share-core and the Large Banzhaf share-core. We provide characterizations of this class of share mappings and show how they are related to the corresponding share functions being functions that assign to every TU-game exactly one share vector.