Axiomatization of a class of share functions for N-person games

The Shapley value is the unique value defined on the class of cooperative games in characteristic function form which satisfies certain intuitively reasonable axioms. Alternatively, the Banzhaf value is the unique value satisfying a different set of axioms. The main drawback of the latter value is that it does not satisfy the efficiency axiom, so that the sum of the values assigned to the players does not need to be equal to the worth of the grand coalition. By definition, the normalized Banzhaf value satisfies the efficiency axiom, but not the usual axiom of additivity. In this paper we generalize the axiom of additivity by introducing a positive real valued function sigma on the class of cooperative games in characteristic function form. The so-called axiom of sigma-additivity generalizes the classical axiom of additivity by putting the weight sigma(upsilon) on the value of the game upsilon. We show that any additive function sigma determines a unique share function satisfying the axioms of efficient shares, null player property, symmetry and sigma-additivity on the subclass of games on which sigma is positive and which contains all positively scaled unanimity games. The axiom of efficient shares means that the sum of the values equals one. Hence the share function gives the shares of the players in the worth of the grand coalition. The corresponding value function is obtained by multiplying the shares with the worth of the grand coalition. By defining the function sigma appropiately we get the share functions corresponding to the Shapley value and the Banzhaf value. So, for both values we have that the corresponding share functions belong to this class of share functions. Moreover, it shows that our approach provides an axiomatization of the normalized Banzhaf value. We also discuss some other choices of the function sigma and the corresponding share functions. Furthermore we consider the axiomatization on the subclass of monotone simple games.