Abstract
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.
Highlights
A cooperative n-person game with transferable utility or a TU game is a model for the situations in which players can obtain benefits when cooperating
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
In the proposition we prove that for convex voting games with players having different bargaining abilities, its Banzhaf value is upper-bounded by the Banzhaf value in the original voting game
Summary
A cooperative n-person game with transferable utility or a TU game is a model for the situations in which players can obtain benefits (measured by real numbers) when cooperating. Shapley [25] related cooperative and non-cooperative games by assuming that a TU game may have imperfect cooperation, that is, players can have different cooperation levels He modeled this situation by associating a positive weight to each player. The first attempt to obtain and characterize a weighted Banzhaf value was done by Radzik et al [27] who analyzed cooperative games in which some a priori importance measures were assigned to the players. In our proposal, which is the Banzhaf value of the modified game, the imperfect cooperation reduces the total power of the grand coalition, but the obtained value is monotonous in the weights. The article ends with a short section of final remarks and an extensive bibliography
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