The inverse Banzhaf problem

Let F be a family of subsets of the ground set [n] = {1, 2, ... , n}. For each i is an element of [n] we let p(F, i) be the number of pairs of subsets that differ in the element i and exactly one of them is in F. We interpret p(F, i) as the influence of that element. The normalized Banzhaf vector of F, denoted B(F), is the vector (B(F, 1), ... , B(F, n)), where B(F, i) = p(F,i)/p(F) and p(F) is the sum of all p(F, i). The Banzhaf vector has been studied in the context of measuring voting power in voting games as well as in Boolean circuit theory. In this paper we investigate which non-negative vectors of sum 1 can be closely approximated by Banzhaf vectors of simple voting games. In particular, we show that if a vector has most of its weight concentrated in k < n coordinates, then it must be essentially the Banzhaf vector of some simple voting game with n - k dummy voters.