Simple Games versus Weighted Voting Games

Titelangaben

Hof, Frits ; Kern, Walter ; Kurz, Sascha ; Paulusma, Daniël:
Simple Games versus Weighted Voting Games.
Bayreuth , 2018 . - 13 S.

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Abstract

A simple game (N,v) is given by a set N of n players and a partition of 2^N 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 alpha= \min_{p>=0}\max_{W' in W,L' in L} p(L')/p(W') <1 are known as weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that alpha<=n/4 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<=2n/7 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.