  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. We also discuss the issue of reconstructing a
pseudo-Boolean function from prescribed influences and point out very different
behaviors in the weighted and non-weighted cases.
