A Monotonic Weighted Banzhaf Value for Voting Games

The aim of this paper is to extend the classical Banzhaf index of power to voting games in which players have weights representing different cooperation or bargaining abilities. The obtained value does not satisfy the classical total power property, which is justified by the imperfect cooperation. Nevertheless, it is monotonous in the weights. We also obtain three different characterizations of the value. Then we relate it to the Owen multilinear extension.

Solidarity Value and Solidarity Share Functions for TU Fuzzy Games

TU games under both crisp and fuzzy environments describe situations where players make full (crisp) or partial (fuzzy) binding agreements and generate worth in return. The challenge is then to decide how to distribute the profit among them in a rational manner: we call this a solution. In this paper, we introduce the notion of solidarity value and the solidarity share function as a suitable solution to TU fuzzy games. Two special classes of TU fuzzy games, namely, TU fuzzy games in Choquet integral form and in multilinear extension form, are studied and the corresponding solidarity value and the solidarity share functions are characterized.

CHOQUET EXTENSION OF COOPERATIVE GAMES

A multilinear extension of the n-person cooperative game was introduced by Owen in 1972, and a new extension method is proposed in this paper. For n-person cooperative games, any coalition can equivalently be represented by its characteristic vectors. By means of the Choquet integral, a new fuzzy extension, called the Choquet extension, is developed. Furthermore, a Shapley function in this class of fuzzy cooperative games with the Choquet extension form is defined. Axioms of the Shapley function are proposed, and an explicit formula for the Shapley function is given. Finally, an equivalent definition of this Shapley function is discussed.