A monotonic weighted Banzhaf value for voting games

Abstract

In this paper we deal with TU-games in which players possibly have different cooperation levels or different willingness to cooperate. The dividend (and thus the value) of each coalition is modified to take into account the cooperation abilities of players in that coalition. Then, we propose as point solution for these situations the Shapley value of the modified game. This allocation rule, -a new kind of weighted Shapley value- is inefficient, which is justified by the imperfect cooperation and it satisfies several interesting properties. In particular, for superadditive games, increasing the weight of a player does not decrease his value. Moreover, different characterizations for this rule can be obtained. They are parallel to those more prominent existing in the literature for the Shapley value.