  This paper is concerned with simple games. One of the fundamental questions
regarding simple games is that of what makes a simple game 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 (2009) improved on that upper bound. Their proof employs a property
of linear inequalities demonstrated by Muroga (1971). 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 weights representation of a weighted simple game,
and improve on the bounds obtained by Muroga (1971). We show that our bounds
are tight when the number of players is less than or equal to five, based on
the computational results obtained by Kurz (2012).
  Lastly, we deal with the problem of finding an integer weights representation
under the assumption that we have minimal winning coalitions and maximal losing
coalitions. We discuss a performance of a rounding method.
