TL;DR
This paper investigates bounds on the diameter of the weight polytope in weighted games, providing upper bounds based on maximum weight and threshold, with applications to approximating power distributions.
Contribution
It establishes new bounds for the diameter of weight polytopes in weighted games, linking them to maximum weights and thresholds, and applies these to power index approximations.
Findings
Diameter can be upper bounded by maximum weight and quota
Results enable better approximation of power distributions by weights
Provides theoretical bounds relevant for weighted game analysis
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.
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