Simple Games versus Weighted Voting Games: Bounding the Critical Threshold Value

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. We show that $\alpha \le \frac{1}{4}n$ for every simple game $(N,v)$, confirming the conjecture of Freixas and Kurz (IJGT, 2014). 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 $a>0$.

10 pages; the paper is a follow-up and merge of arXiv:1805.02192 and arXiv:1806.03170