Potentials and Reduced Games for Share Functions

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 function for TU-games is a function that assigns to every game a distribution of the payoffs. A value function is efficient if for every game it exactly distributes 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 functions which assign to every game a vector 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 give some characterizations of a class of share functions containing the Shapley share function and the Banzhaf share function using generalizations of potentials and of Hart and Mas-Colell's reduced game property.