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 2N into a set L of losing coalitions L' with value v(L')=0 that is closed under taking subsets and a set W of winning coalitions W' with v(W')=1. Simple games with ∝= minp≥0max w∈W,L∈L} p(L')/p(W') <1 are known as weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that ∝≤1/4n 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 ∝≤2/7n for every simple game (N,v). For complete simple games, Freixas and Kurz conjectured that ∝=O(√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 ∝ is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Moreover, we show that for every graphic simple game, deciding if ∝< a is polynomial-time solvable for every fixed a > 0.

Locations

• SSRN Electronic Journal - View
• arXiv (Cornell University) - View - PDF
• EPub Bayreuth (University of Bayreuth) - View - PDF
• Durham Research Online (Durham University) - View - PDF