 
Summary: The inverse Banzhaf problem
Noga Alon
Paul H. Edelman
Abstract
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 nonnegative 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.
1 Introduction
A fundamental question when analyzing a voting method is what is the distribution of power
among the voters. The most common measure of power, the Banzhaf index, quantifies the
percentage of power of a voter by its ability to alter the outcome, i.e., the probability that if
that voter were to change its vote, the outcome would change. There are numerous theorems
