Weightedness measures from inequality systems

Abstract

A simple game is a cooperative game where some coalitions among players or voters became the (monotonic) set of winning coalitions, and the other ones form the set of losing coalitions. It is well-known that weighted voting games form a strict subclass of simple games, where each player has a voting weight so that a coalition wins if and only if the sum of weights of their members exceeds a given quota, otherwise it loses. This work studies how far away a simple game is for being representable as a weighted voting game, which allows for a more compact representation. There are several measures that determine the weightedness of a simple game, such as the dimension, the trade-robustness, the critical threshold value associated with the α-roughly weightedness property, etc. In this work we propose some new weightedness measures, all based on linear programming. In general terms, for a given simple game, a linear program is used to identify its weightedness: (i) the ϵ-roughly value (${\mu }_{ϵ}$), (ii) the $Z^{+}$-roughly value (${\mu }_{Z^{+}}$), (iii) the Δ-roughly value (${\mu }_{\Delta }$), and (iv) the outlier value (${\Psi }_M$). We show a close relation between the known critical threshold value of weightedness and the new measure ${\mu }_{\Delta }$. Finally, we also present an exhaustive comparison of weightedness measures for simple games with up to six players.