Trading Transforms of Non-weighted Simple Games and Integer Weights of Weighted Simple Games
Akihiro Kawana, Tomomi Matsui

TL;DR
This paper advances the understanding of simple games by providing new proofs and bounds for trading transforms and integer-weight representations, improving theoretical limits and computational methods for weighted and roughly weighted simple games.
Contribution
It introduces a new proof using Farkas' lemma, improves bounds on trading transform size, and discusses integer-weight representations and certificates for simple games.
Findings
New proof of trading transform existence for non-weighted simple games
Improved upper bounds on trading transform size and quota in integer-weight representations
Bound on the length of certificates of non-weightedness in roughly weighted simple games
Abstract
This study investigates simple games. A fundamental research question in this field is to determine necessary and sufficient conditions for a simple game to be a weighted majority game. Taylor and Zwicker (1992) showed that a simple game is non-weighted if and only if there exists a trading transform of finite size. They also provided an upper bound on the size of such a trading transform, if it exists. Gvozdeva and Slinko (2011) improved that upper bound; their proof employed a property of linear inequalities demonstrated by Muroga (1971).In this study, we provide a new proof of the existence of a trading transform when a given simple game is non-weighted. Our proof employs Farkas' lemma (1894), and yields an improved upper bound on the size of a trading transform. We also discuss an integer-weight representation of a weighted simple game, improving the bounds obtained by Muroga…
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