The inverse Banzhaf problem Paul H. Edelman 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 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. 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 Collections: Mathematics