Weighted Banzhaf power and interaction indexes through weighted approximations of games

Details Weighted Banzhaf power and interaction indexes through weighted approximations of games Weighted Banzhaf power and interaction indexes through weighted approximations of games

2010-01-18

arxiv.org/abs/1001.3052v2

European Journal of Operational Research 211 (2) (2011) 352-358

Mathematics

Optimization and Control

Scientific paper

10.1016/j.ejor.2010.11.027

The Banzhaf power index was introduced in cooperative game theory to measure the real power of players in a game. The Banzhaf interaction index was then proposed to measure the interaction degree inside coalitions of players. It was shown that the power and interaction indexes can be obtained as solutions of a standard least squares approximation problem for pseudo-Boolean functions. Considering certain weighted versions of this approximation problem, we define a class of weighted interaction indexes that generalize the Banzhaf interaction index. We show that these indexes define a subclass of the family of probabilistic interaction indexes and study their most important properties. Finally, we give an interpretation of the Banzhaf and Shapley interaction indexes as centers of mass of this subclass of interaction indexes.

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