Inverse Banzhaf Problem
The Banzhaf vector measures the influence of a player in a monotonic simple game. I had been interested in the distribution of the Banzhaf vector is simple games and also weighted voting games (a subclass of simple games). Better knowledge of the distrbution helps to check whether an algorithm to design a game with the required Banzhaf vector has converged sufficiently. Alon and Edelman have published a recent paper where they investigate which non-negative vectors of sum 1 can be closely approximated by Banzhaf vectors of simple voting games with n players. Interestingly, they show that approximation is not necessarily easier for a larger n, and exhibit vectors in the standard n-simplex for all n that can not be well approximated.