The Normalized Banzhaf Vallte and the Banzhaf Share Function

A cooperative game with transferable utilities -or simply a TU-gamedescribes a situation in which players can obtain certain payoffs by cooperation. A value function for these games is a function which assigns to every such a game a distribution of payoffs over the players in the game. A famous solution concept for TU-games is the Banzhaf value. This Banzhaf value is not efficient, i.e., in general it does not distribute the payoff that can be obtained by the 'grand coalition' consisting of all players cooperating together. In this paper we consider the normalized Banzhaf value which distributes the payoff that can be obtained by the 'grand coalition' proportional to the Banzhaf values of the players. This value does not satisfy certain axioms underlying the Banzhaf value. In this paper we discuss some characterizations of the normalized Banzhaf value and compare these with other solution concepts such as, for example, the (non-normalized) Banzhaf value and the Shapley value. Another approach to analyze efficient V'cl,lue functions is to consider share functions being functions which assign to every player in a TU-game its share in the worth of the 'grand coalition'. We discuss the characterization of a class of such share functions containing the Banzhaf and Shapley share functions. Finally, we generalize the concept of the potential function of a game as introduced by Hart and Mas-Colell to a class of potential functions and characterize any element of the class of share functions by the normalized marginal function of the corresponding potential function. 'This author is financially supported by the Netherlands Organization for Scientific Research (NWO), ESR-grant 510-01-0504