A value function for cooperative games with transferable utility assigns to every game a distribution of the payoffs. A value function is efficient if for every such a game it exactly distributes the worth that can be obtained by all players cooperating together. An approach to efficiently allocate the worth of the 'grand coalition' is using share functions which assign to every game a vector whose components sum up to one. Every component of this vector is the corresponding players' share in the total payoff that is to be distributed. In this paper we give 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. © 2007 Elsevier B.V. All rights reserved.