The inverse Banzhaf problem

Let F be a family of subsets of the ground set [n] = {1, 2, . . . , n}. For each i ∈ [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.