Bounds for the Diameter of the Weight Polytope

Abstract

A weighted game or a threshold function in general admits different weighted representations even if the sum of non-negative weights is fixed to one. Here we study bounds for the diameter of the corresponding weight polytope. It turns out that the diameter can be upper bounded in terms of the maximum weight and the quota or threshold. We apply those results to approximation results between power distributions, given by power indices, and weights.

A note on limit results for the Penrose–Banzhaf index

“…The underlying reason is that different representations of the same weighted game have to be mapped onto to same power vector, i.e., the diameter of the polytope of representations of a weighted game plays the key role, as exploited in [3].…”