Simple Games versus Weighted Voting Games

TL;DR

This paper investigates the relationship between simple games and weighted voting games, confirming conjectures about their bounds, analyzing special cases, and exploring computational complexity for specific subclasses.

Contribution

It proves bounds on the parameter alpha for simple games, confirms conjectures for special cases, and analyzes computational complexity for graphic simple games.

Findings

Confirmed the conjecture alpha ≤ (1/4)n for certain simple games.

Proved a general bound alpha ≤ (2/7)n for all simple games.

Established NP-hardness and polynomial-time solvability results for graphic simple 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 $\alpha= \min_{p\geq 0}\max_{W\in {\cal W},L\in {\cal L}} \frac{p(L)}{p(W)}<1$ are known as weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that $\alpha\leq \frac{1}{4}n$ 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 $\alpha\leq \frac{2}{7}n$ 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 $a>0$.