A simple game (N, v) is given by a set N of n players and a partition of 2N\documentclass[12pt]{minimal}
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\begin{document}$$2^N$$\end{document} into a set L\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {L}$$\end{document} of losing coalitions L with value v(L)=0\documentclass[12pt]{minimal}
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\begin{document}$$v(L)=0$$\end{document} that is closed under taking subsets and a set W\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {W}$$\end{document} of winning coalitions W with value v(W)=1\documentclass[12pt]{minimal}
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\begin{document}$$v(W)=1$$\end{document}. We let α=minp⩾0,p≠0maxW∈W,L∈Lp(L)p(W)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha = \min _{p\geqslant {\varvec{0}}, p\ne {\varvec{0}}}\max _{W\in \mathcal{W}, L\in \mathcal{L}} \frac{p(L)}{p(W)}$$\end{document}. It is known that the subclass of simple games with α<1\documentclass[12pt]{minimal}
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\begin{document}$$\alpha <1$$\end{document} coincides with the class of weighted voting games. Hence, α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} can be seen as a measure of the distance between a simple game and the class of weighted voting games. We prove that α⩽14n\documentclass[12pt]{minimal}
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\begin{document}$$\alpha \leqslant \frac{1}{4}n$$\end{document} holds for every simple game (N, v), confirming the conjecture of Freixas and Kurz (Int J Game Theory 43:659–692, 2014). For complete simple games, Freixas and Kurz conjectured that α=O(n)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =O(\sqrt{n})$$\end{document}. We also prove this conjecture, up to an lnn\documentclass[12pt]{minimal}
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\begin{document}$$\ln n$$\end{document} 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 α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Finally, we show that for every graphic simple game, deciding if α<α0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha <\alpha _0$$\end{document} is polynomial-time solvable for every fixed α0>0\documentclass[12pt]{minimal}
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\begin{document}$$\alpha _0>0$$\end{document}.
