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This study investigates simple games. A fundamental research question in thisfield is to determine necessary and sufficient conditions for a simple game tobe a weighted majority game. Taylor and Zwicker (1992) showed that a simplegame is non-weighted if and only if there exists a trading transform of finitesize. They also provided an upper bound on the size of such a tradingtransform, 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 tradingtransform 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 tradingtransform.  We also discuss an integer-weight representation of a weighted simple game,improving the bounds obtained by Muroga (1971). We show that our bound on thequota is tight when the number of players is less than or equal to five, basedon the computational results obtained by Kurz (2012).  Furthermore, we discuss the problem of finding an integer-weightrepresentation under the assumption that we have minimal winning coalitions andmaximal losing coalitions.In particular, we show a performance of a roundingmethod.  Lastly, we address roughly weighted simple games. Gvozdeva and Slinko (2011)showed that a given simple game is not roughly weighted if and only if thereexists a potent certificate of non-weightedness. We give an upper bound on thelength of a potent certificate of non-weightedness. We also discuss aninteger-weight representation of a roughly weighted simple game.

Trading Transforms of Non-weighted Simple Games and Integer Weights of Weighted Simple Games