  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 (1971). We show that our bound on the
quota is tight when the number of players is less than or equal to five, based
on the computational results obtained by Kurz (2012).
  Furthermore, we discuss the problem of finding an integer-weight
representation under the assumption that we have minimal winning coalitions and
maximal losing coalitions.In particular, we show a performance of a rounding
method.
  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 there
exists a potent certificate of non-weightedness. We give an upper bound on the
length of a potent certificate of non-weightedness. We also discuss an
integer-weight representation of a roughly weighted simple game.
