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

TL;DR

This paper investigates bounds on the critical threshold value in simple and weighted voting games, confirming a conjecture for general simple games and nearly for complete simple games, while analyzing computational complexity for graphic simple games.

Contribution

It proves an upper bound of rac{1}{4}n for simple games, nearly confirms a conjecture for complete simple games, and analyzes complexity for graphic simple games.

Findings

Bound rac{1}{4}n for simple games, confirming a conjecture.

Proves the extit{O}(\sqrt{n}) bound for complete simple games up to a extit{log} n factor.

NP-hardness for computing extit{α} in graphic simple games, polynomial-time solvable for bipartite graphs.

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 $\alpha= \min_{p\geq 0}\max_{W\in {\cal W}, L\in {\cal L}} \frac{p(L)}{p(W)}<1$ are exactly the weighted voting games. We show that $\alpha\leq \frac{1}{4}n$ 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 $a>0$.