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

Weighted Banzhaf power and interaction indexes through weighted approximations of games