A new power index based on minimal winning coalitions without any surplus

Alonso-Meijide, J. M.; Freixas, Josep

doi:10.1016/j.dss.2010.01.003

Cited by 30 publications

(29 citation statements)

References 16 publications

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“…Bilbao et al [9] compute and apply the Shapley-Shubik and the Banzhaf-Coleman power indices for weighted majority games and study the time complexity of the corresponding algorithms. Alonso-Meijide et al [5] propose methods based in generating functions to compute the Deegan-Packel [16], the Public Good [19], and the Shift [4] power indices. Chessa [14] provides a new method also based in generating functions to compute the Public Good power index.…”

Section: Introductionmentioning

confidence: 99%

Computing Banzhaf–Coleman and Shapley–Shubik power indices with incompatible players

Alonso-Meijide

Casas-Méndez

Fiestras-Janeiro

2015

Applied Mathematics and Computation

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Section: Introductionmentioning

confidence: 99%

Computing Banzhaf–Coleman and Shapley–Shubik power indices with incompatible players

Alonso-Meijide

Casas-Méndez

Fiestras-Janeiro

2015

Applied Mathematics and Computation

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“…A prime implicant of this function is a Minimal Winning Coalition (Rushdi and Alturki, 2015), i.e., it is a winning coalition such that any defection from it negates its winning status (Steiner, 1967;Fishburn and Brams, 1996). Now, we note that the famous Banzhaf index of voting power (Banzhaf, 1964;Dubey and Shapley, 1979;Hammer and Holzman, 1992;Alonso-Meijide and Freixas, 2010;Yamamoto, 2012), is simply the weight of the Boolean derivative (Boolean difference) (Reed, 1973;Lee, 1978;Muroga, 1979;Rushdi, 1986b) of the system function with respect to the pertinent element variable…”

Section: Threshold Boolean Functionsmentioning

confidence: 99%

Weighted voting systems: A threshold- Boolean perspective

Alturki

Rushdi

2016

J Engin Res

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W ö)«d «uE «Ë WO bMN « vM « s dO W c/Ë ¡UBI « w U U «-Ëoe W Ë"u*« X uB « rE VFK WD WO{U -WI dD rEM « Ác n u W K W U b u AEW oeUB ô« WO UL ô«Ë WO UO « WO ôu « ‰«Ëb « W ôb X uB « rEM Ÿ-U n Ë WL AE…dO J « rEMK UNLOLF sJ1 W u Ë W U oeu Ë s Ë ‰«Ëb « Ác ‰u …d u *« U uKF*« ¡«d s «dO n u « «c lH ABSTRACTWeighted voting systems play a crucial role in the investigation and modeling of many engineering structures and political and socio-economic phenomena. There is an urgent need to describe these systems in a simpli ed powerful mathematical way that can be generalized to systems of any size. An elegant description of voting systems is presented in terms of threshold Boolean functions. This description bene ts considerably from the wealth of information about these functions, and of the potpourri of algebraic and map techniques for handling them. The paper demonstrates that the prime implicants of the system threshold function are its Minimal Winning Coalitions (MWC). The paper discusses the Boolean derivative (Boolean difference) of the system threshold function with respect to each of its member components. The prime implicants of this Boolean difference can be used to deduce the winning coalitions (WC) in which the pertinent member cannot be dispensed with. Each of the minterms of this Boolean difference is a winning coalition in which this member plays a pivotal role. However, the coalition ceases to be winning if the member defects from it. Hence, the number of these minterms is identi ed as the Banzhaf index of voting power. The concepts introduced are illustrated with detailed demonstrative examples that also exhibit some of the known paradoxes of voting-system theory. Finally, the paper stresses the utility of threshold Boolean functions in the understanding, study, analysis, and design of weighted voting systems irrespective of size.

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“…To define the Shift power index [4], it is necessary to introduce the desirability relation defined in [16].…”

Section: Preliminaries: Simple Games and Power Indicesmentioning

confidence: 99%

“…This is not a comprehensive list and other power indices could be listed; we just mention here one more power index, the Shift power index introduced in Alonso-Meijide and Freixas [4], since we will deal with a slight variation of it (the Shift Deegan-Packel power index) in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…The main objective of this paper is to analyze whether some modification of the generating function techniques might be used to calculate the Deegan-Packel, the Public Good, and the Shift power indices for weighted voting games. These three indices are defined on the basis of either minimal winning coalitions (in the case of the Deegan-Packel and the Public Good indices) or shift-minimal winning coalitions (in the case of the Shift power index) [4]. Specially in games with a large number of players, it is not known a priori which coalitions are minimal winning or shift-minimal winning.…”

Section: Introductionmentioning

confidence: 99%

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Computation of several power indices by generating functions

Alonso-Meijide

Freixas

Molinero

2012

Applied Mathematics and Computation

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In this paper we propose methods to compute the Deegan-Packel, the Public Good, and the Shift power indices by generating functions for the particular case of weighted voting games. Furthermore, we define a new power index which combines the ideas of the Shift and the Deegan-Packel power indices and also propose a method to compute it with generating functions. We conclude by some comments about the complexity to compute these power indices.

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A new power index based on minimal winning coalitions without any surplus

Cited by 30 publications

References 16 publications

Computing Banzhaf–Coleman and Shapley–Shubik power indices with incompatible players

Computing Banzhaf–Coleman and Shapley–Shubik power indices with incompatible players

Weighted voting systems: A threshold- Boolean perspective

Computation of several power indices by generating functions

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