On minimum sum representations for weighted voting games

TL;DR

This paper exhaustively classifies all weighted voting games with 9 voters that lack a unique minimum sum integer weight representation, extending previous classifications up to 8 voters.

Contribution

It provides a complete classification of 9-voter weighted voting games without unique minimum sum integer representations, advancing understanding of their structural properties.

Findings

Identifies all 9-voter weighted voting games without unique minimum sum representations.

Extends previous classifications from 8 to 9 voters.

Provides a comprehensive list of such games for further analysis.

Abstract

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. Freixas and Molinero 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.