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), conrming 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 xed a > 0.

Misbehavior Detection in Ephemeral Networks: A Local Voting Game in Presence of Uncertainty

“…Finally, a potent design should provide more rewards for decisive votes. Although some authors studied weighted voting games [19], [28], they mainly based their designs on clustering heads rather than the value of a vote in the middle of a local voting games.…”

“…( 46), and eqs. ( 22), ( 23), ( 25), ( 26), (28), and (29) into eq. ( 47), and then substituting eqs.…”