Summary: 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 value \(v(W)=1\). We let \(\alpha = \min_{p\geq \mathbf{0},\, p\neq \mathbf{0}} \max_{W\in \mathcal{W}, \,L\in \mathcal{L}}\, \frac{p(L)}{p(W)}\). It is known that the subclass of simple games with \(\alpha <1\) coincides with the class of weighted voting games. Hence, \( \alpha\) can be seen as a measure of the distance between a simple game and the class of weighted voting games. We prove that \(\alpha \leqslant \frac{1}{4}n\) holds for every simple game \((N,v)\), confirming the conjecture of J. Freixas and S. Kurz [Int. J. Game Theory 43, No. 3, 659–692 (2014; Zbl 1304.91021)]. For complete simple games, Freixas and Kurz conjectured that \(\alpha =O(\sqrt{n})\). We also prove this conjecture, up to an \(\ln n\) factor. Moreover, we prove that for graphic simple games, that is, simple games in which every minimal winning coalition has size 2, the problem of computing \(\alpha\) is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Finally, we show that for every graphic simple game, deciding if \(\alpha <\alpha_0\) is polynomial-time solvable for every fixed \(\alpha_0>0\).