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

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 α = min p 0 maxW ∈W,L∈L p(L) p(W ) < 1 are exactly the weighted voting games. We show that α 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 α = 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 α is NP-hard, but polynomial-time solvable if the underlying graph is bipartite. Moreover, we show that for every graphic simple game, deciding if α < a is polynomial-time solvable for every fixed a > 0.⋆ A partial answer to the conjecture of Freixas and Kurz appeared, together with some results of this paper, in an extended abstract published in the proceedings of SAGT 2018 [20].

Identically Self-blocking Clutters

“…Our proof relied on Lemma 15, which in itself has another consequence. Viewing clutters as simple games, Hof et al [16] showed that given a clutter C over n elements, its critical threshold value is always at most n 4 , and this maximum is achieved if, and only if, C has a fractional packing of value n 2 and b(C) has a fractional packing of value 2. Thus by Lemma 15, it is essentially blockers of cuboids that achieve the largest possible critical threshold value.…”