The inverse Banzhaf problem

Abstract

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.