This discussion paper resulted in a publication in ' Discrete Mathematics', 2007, 307, 2385-2399.
A situation in which a finite set of players can obtain certainpayoffs by cooperation can be described by a cooperative game withtransferable utilities - or simply a TU-game. A valuefunction for TU-games is a function that assigns to every game adistribution of the payoffs. A value function is efficient iffor every game it exactly distributes the worth that can be obtainedby all players cooperating together.An approach to efficiently allocating the worth of the 'grandcoalition'is using share functions which assign to every gamea vector which components sum up to one such that every component isthe corresponding players' share in the total payoff that is to bedistributed among the players. In this paper we give somecharacterizations of a class of share functions containing theShapley share function and the Banzhaf share functionusing generalizations of potentials and of Hart and Mas-Colell'sreduced game property.