Influence and interaction indexes in cooperative games: a unified least squares approach

The classical Banzhaf power and interaction indexes used in cooperative game theory appear naturally as leading coefficients in the standard least squares approximation of the game under consideration by a set function of a specified degree. We observe that this still holds true if we consider approximations by set functions depending on specified variables. We show that the Banzhaf influence index can also be obtained from this new approximation problem. Considering certain weighted versions of this approximation, we also introduce a class of weighted Banzhaf influence indexes and analyze their most important properties.