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 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 known as weighted voting games. Freixas and Kurz (IJGT, 2014) conjectured that α ≤ 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 α ≤ 2 7 n for every simple game (N, v). 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.

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

“…This is of course not always possible, that is, the core might be empty. This leads to related 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].…”

On Critical Threshold Value for Simple Games

“…In [5] the conjecture of Freixas and Kurz was verified for simple games with all minimal wining coalitions of size 3 and for simple games with no minimal winning coalitions of size 3. In [5] it was shown that α 2n/7 for general simple games.…”

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

“…⋆ 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].…”

“…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].…”