Simple games versus weighted voting games: Bounding the critical threshold value

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Hof, Frits ; Kern, Walter ; Kurz, Sascha ; Pashkovich, Kanstantsin ; Paulusma, Daniël:
Simple games versus weighted voting games: Bounding the critical threshold value.
Bayreuth , 2018 . - 10 S.

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Abstract

A simple game (N,v) is given by a set N of $n$ players and a partition of the set of subsets of N into a set of losing coalitions L with value v(L)=0 that is closed under taking subsets and a set W of winning coalitions with v(W)=1. Simple games with alpha= min _{p>=0} max_{W' in W, L' in L} p(L')/p(W')<1 are exactly the weighted voting games. We show that alpha<=n/4 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.