The inverse Banzhaf problem

Let $${\mathcal{F}}$$ be a family of subsets of the ground set [n] = {1, 2, . . . , n}. For each $${i \in [n]}$$ we let $${p(\mathcal{F},i)}$$ be the number of pairs of subsets that differ in the element i and exactly one of them is in $${\mathcal{F}}$$. We interpret $${p(\mathcal{F},i)}$$ as the influence of that element. The normalized Banzhaf vector of $${\mathcal{F}}$$, denoted $${B(\mathcal{F})}$$, is the vector $${(B(\mathcal{F},1),\dots,B(\mathcal{F},n))}$$, where $${B(\mathcal{F},i)=\frac{p(\mathcal{F},i)}{p(\mathcal{F})}}$$ and $${p(\mathcal{F})}$$ is the sum of all $${p(\mathcal{F},i)}$$. 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.