On minimum sum representations for weighted voting games

A proposal in a weighted voting game is accepted if the sum of the (non-negative) weights of the “yea” voters is at least as large as a given quota. Several authors have considered representations of weighted voting games with minimum sum, where the weights and the quota are restricted to be integers. In [Freixas and Molinero(2009)] the authors have classified all weighted voting games without a unique minimum sum representation for up to 8 voters. Here we exhaustively classify all weighted voting games consisting of 9 voters which do not admit a unique minimum sum integer weight representation.