Roughly weighted hierarchical simple games

Hameed, Ali; Slinko, Arkadii

doi:10.1007/s00182-014-0430-1

Cited by 3 publications

(2 citation statements)

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“…Weak separability allows the hyperplane to intersect the two sets being separated, while neat separability demands that any such intersection be the same for both sets; these notions play roles in the theory of roughly weighted simple games (see [13], [14] and [29]), and of generalized scoring rules for multicandidate elections (see [5], [30] and [31]), respectively.…”

Section: Weighted Hamming Distance and Two Characterizations Of Weighmentioning

confidence: 99%

The geometry of voting power: Weighted voting and hyper-ellipsoids

Houy

Zwicker

2014

Games and Economic Behavior

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Abstract. In cases where legislators represent districts that vary in population, the design of fair legislative voting rules requires an understanding of how the number of votes cast by a legislator is related to a measure of her influence over collective decisions. We provide three new characterizations of weighted voting, each based on the intuition that winning coalitions should be close to one another. The locally minimal and tightly packed characterizations use a weighted Hamming metric. Ellipsoidal separability employs the Euclidean metric: a separating hyperellipsoid contains all winning coalitions, and omits losing ones. The ellipsoid's proportions, and the Hamming weights, reflect the ratio of voting weight to influence, measured as Penrose-Banzhaf voting power. In particular, the spherically separable rules are those for which voting powers can serve as voting weights.

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Section: Weighted Hamming Distance and Two Characterizations Of Weighmentioning

confidence: 99%

The geometry of voting power: Weighted voting and hyper-ellipsoids

Houy

Zwicker

2014

Games and Economic Behavior

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“…This is a much broader class of games than weighted games which includes, for example, conjunctive and disjunctive hierarchical games which appear as the access structures of popular secret sharing schemes (Simmons, 1990;Tassa, 2007). Both disjunctive and conjunctive hierarchical games are seldom weighted (Gvozdeva et al, 2013) or even roughly weighted (Hameed and Slinko, 2015). Freixas and Puente (2008) studied conjunctive hierarchical games (under the name of games with a minimum) and found that their dimension grows linearly in the number of players and asked whether or not in the class of complete games the dimension can grow polynomially or even exponentially.…”

Section: Introductionmentioning

confidence: 99%

Growth of dimension in complete simple games

O'Dwyer

Slinko

2017

Mathematical Social Sciences

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The concept of dimension in simple games was introduced by Taylor and Zwicker (1993) as a measure of the remoteness of a given game from a weighted game. They demonstrated that the dimension of a simple game can grow exponentially in the number of players. However, the problem of worst-case growth of the dimension in complete games was left open. Freixas and Puente (2008) showed that complete games of arbitrary dimension exist and, in particular, their examples demonstrate that the worst-case growth of dimension in complete games is at least linear. In this paper, using a novel technique of Kurz and Napel (2015), we demonstrate that the growth of dimension in complete games can also be exponential in the number of players.

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Trading transforms of non-weighted simple games and integer weights of weighted simple games

Kawana

Matsui

2021

Theory Decis

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This paper is concerned with simple games. One of the fundamental questions regarding simple games is that of what makes a simple game a weighted majority game. Taylor and Zwicker (1992) showed that a simple game is non-weighted if and only if there exists a trading transform of finite size. They also provided an upper bound on the size of such a trading transform, if it exists. Gvozdeva and Slinko ( 2009) improved on that upper bound. Their proof employs a property of linear inequalities demonstrated by Muroga (1971). We provide a new proof of the existence of a trading transform when a given simple game is non-weighted. Our proof employs Farkas' lemma (1894), and yields an improved upper bound on the length of a trading transform. We also discuss an integer weights representation of a weighted simple game, and improve on the bounds obtained by Muroga (1971). We show that our bounds are tight when the number of players is less than or equal to five, based on the computational results obtained by Kurz (2012). We deal with the problem of finding a minimum integer weights representation under the assumption that we have minimal winning coalitions and maximal losing coalitions. We propose a polynomial time probabilistic algorithm for finding an approximate solution.

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Roughly weighted hierarchical simple games

Cited by 3 publications

References 13 publications

The geometry of voting power: Weighted voting and hyper-ellipsoids

The geometry of voting power: Weighted voting and hyper-ellipsoids

Growth of dimension in complete simple games

Trading transforms of non-weighted simple games and integer weights of weighted simple games

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