Weighted Banzhaf power and interaction indexes through weighted approximations of games

TL;DR

This paper introduces weighted versions of the Banzhaf power and interaction indexes using weighted least squares approximations, generalizing existing measures and exploring their properties and interpretations.

Contribution

It defines a new class of weighted interaction indexes that extend the Banzhaf index and analyzes their properties and probabilistic interpretations.

Findings

Weighted interaction indexes generalize Banzhaf index.

These indexes are a subclass of probabilistic interaction indexes.

Banzhaf and Shapley indexes interpreted as centers of mass.

Abstract

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.