Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach

Details Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach Influence and interaction indexes for pseudo-Boolean functions: a unified least squares approach

2012-01-17

arxiv.org/abs/1201.3543v1

Mathematics

Optimization and Control

Scientific paper

The Banzhaf power and interaction indexes for a pseudo-Boolean function (or a cooperative game) appear naturally as leading coefficients in the standard least squares approximation of the function by a pseudo-Boolean function of a specified degree. We first observe that this property still holds if we consider approximations by pseudo-Boolean functions depending only on specified variables. We then show that the Banzhaf influence index can also be obtained from the latter approximation problem. Considering certain weighted versions of this approximation problem, we introduce a class of weighted Banzhaf influence indexes, analyze their most important properties, and point out similarities between the weighted Banzhaf influence index and the corresponding weighted Banzhaf interaction index.

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