  A simple game $(N,v)$ is given by a set $N$ of $n$ players and a partition of
$2^N$ into a set $\mathcal{L}$ of losing coalitions $L$ with value $v(L)=0$
that is closed under taking subsets and a set $\mathcal{W}$ of winning
coalitions $W$ with $v(W)=1$. Simple games with $\alpha= \min_{p\geq
0}\max_{W\in {\cal W},L\in {\cal L}} \frac{p(L)}{p(W)}<1$ are known as weighted
voting games. Freixas and Kurz (IJGT, 2014) conjectured that $\alpha\leq
\frac{1}{4}n$ for every simple game $(N,v)$. We confirm this conjecture for two
complementary cases, namely when all minimal winning coalitions have size $3$
and when no minimal winning coalition has size $3$. As a general bound we prove
that $\alpha\leq \frac{2}{7}n$ for every simple game $(N,v)$. For complete
simple games, Freixas and Kurz conjectured that $\alpha=O(\sqrt{n})$. We prove
this conjecture up to a $\ln n$ factor. We also prove that for graphic simple
games, that is, simple games in which every minimal winning coalition has size
2, computing $\alpha$ is \NP-hard, but polynomial-time solvable if the
underlying graph is bipartite. Moreover, we show that for every graphic simple
game, deciding if $\alpha<a$ is polynomial-time solvable for every fixed $a>0$.
