Simple Games Versus Weighted Voting Games

Abstract

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 $\alpha = \min _{p\ge 0}\max _{W\in \mathcal{W},L\in \mathcal{L}} \frac{p(L)}{p(W)}<1$ are exactly the weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that $\alpha \le \frac{1}{4}n$ 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 $\alpha \le \frac{2}{7}n$ 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 $a>0$.