On Weights and Quotas for Weighted Majority Voting Games

Xavier Molinero,Maria Serna,Marc Taberner-Ortiz

doi:10.3390/g12040091

Xavier Molinero, Maria Serna + Show 1 more

Open Access

https://doi.org/10.3390/g12040091

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Journal: Games	Publication Date: Dec 6, 2021
Citations: 1	License type: CC BY 4.0

Affiliation: Universitat Politècnica de Catalunya

Abstract

In this paper, we analyze the frequency distributions of weights and quotas in weighted majority voting games (WMVG) up to eight players. We also show different procedures that allow us to obtain some minimum or minimum sum representations of WMVG, for any desired number of players, starting from a minimum or minimum sum representation. We also provide closed formulas for the number of WMVG with n players having a minimum representation with quota up to three, and some subclasses of this family of games. Finally, we complement these results with some upper bounds related to weights and quotas.

Highlights

Simple games are the simplest model to study decision systems in which the yes/no has to be decided cooperatively
Due to the difficulty of the problem, it is worth to analyze the optimality of representations, with many players, when only small numbers are allowed to be part of the representation. With these three goals in mind, we started analyzing the list of canonical minimum representations of Weighted majority voting games (WMVG) for up to eight players generated in Freixas and Molinero [29]
Similar as for the weights analysis, qmax denotes the maximum quota appearing in a canonical minimum n representation, and a skip is a quota value that does not appear in any representation

Summary

Introduction

Simple games are the simplest model to study decision systems in which the yes/no has to be decided cooperatively. As there are weighted majority voting games that do not admit a minimum representation [29], another notion of minimality, minimum sum, has been considered. Having access to a good approximation to the real distribution of the player’s weight might allow us to use the techniques on these papers to analyze the relation among power and weight on the complete set of weighted voting games Another natural approach to tackle the problem is to find procedures that allow the introduction of more players, while guaranteeing the optimality of the representation. With these three goals in mind, we started analyzing the list of canonical minimum representations of WMVG for up to eight players generated in Freixas and Molinero [29] Using these data, we carried on a study of the distribution of the players’ weights and the quotas in such representations.

Definitions and Preliminaries

Weights in WMVG up to Eight Players

Quotas in WMVG up to 8 Players

Generating Minimum and Minimum Sum Representations

Small Quotas in Canonical Minimum Representations of WMVG

Discussion and Conclusions