On $\alpha$-roughly weighted games

TL;DR

This paper investigates the critical threshold value in simple games, exploring its possible ranges, bounds for subclasses, and its relation to the cost of stability, advancing understanding of weighted game representations.

Contribution

It provides new bounds and possible values for the critical threshold in simple games and links this concept to the cost of stability.

Findings

Bounds for the critical threshold value for different game sizes

Possible values of the critical threshold for specific game classes

Relation between critical threshold and cost of stability

Abstract

Gvozdeva, Hemaspaandra, and Slinko (2011) have introduced three hierarchies for simple games in order to measure the distance of a given simple game to the class of (roughly) weighted voting games. Their third class $\mathcal{C}_\alpha$ consists of all simple games permitting a weighted representation such that each winning coalition has a weight of at least 1 and each losing coalition a weight of at most $\alpha$. For a given game the minimal possible value of $\alpha$ is called its critical threshold value. We continue the work on the critical threshold value, initiated by Gvozdeva et al., and contribute some new results on the possible values for a given number of voters as well as some general bounds for restricted subclasses of games. A strong relation beween this concept and the cost of stability, i.e. the minimum amount of external payment to ensure stability in a coalitional game, is uncovered.