L S Penrose's limit theorem: Tests by simulation

Chang, Pao‐Li; Chua, Vincent; Machover, Moshé

doi:10.1016/j.mathsocsci.2005.06.001

Cited by 31 publications

(25 citation statements)

References 6 publications

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“…See Lindner & Machover (2004); Chang et al (2006). This phenomenon, the Penrose Limit Theorem, was originally identified in a conjecture by Penrose (1952).…”

Section: Introduction: the Weighted Voting Effect And Its Asymptotic mentioning

confidence: 89%

“…This is true not only for the PBI but also the SSI. Support for the theorem has also come from a simulation study by Chang et al (2006). Their overall conclusion is "Both real-life and randomly generated Weighted Voting Games with many voters provide much empirical evidence that the Penrose Limit Theorem holds in most cases, as a general rule".…”

Section: Proof Lindner and Machover (2004)mentioning

confidence: 98%

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Power indices in large voting bodies

Leech

2011

Public Choice

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Preliminary version: please do not quote without permission. Comments welcome.Abstract. There is no consensus on the properties of voting power indices when there are a large number of voters in a weighted voting body. On the one hand, in some real-world cases that have been studied the power indices have been found to be nearly proportional to the weights (eg the EUCM, US Electoral College). This is true for both the PenroseBanzhaf and the Shapley-Shubik indices. It has been suggested that this is a manifestation of a conjecture by Penrose (known subsequently as the Penrose limit theorem, that has been shown to hold under certain conditions). On the other hand, we have the older literature from cooperative game theory, due to Shapley and his collaborators, showing that, where there are a finite number of voters whose weights remain constant in relative terms, and where the quota remains constant in relative terms, while the total number of voters increases without limit -so called oceanic games -the powers of the voters with finite weight tend to limiting values that are, in general, not proportional to the weights. These results, too, are supported by empirical studies of large voting bodies (eg. the IMF/WB boards, corporate shareholder control). This paper proposes a restatement of the Penrose Limit theorem and shows that, for both the power indices, convergence occurs in general, in the limit as the Laakso-Taagepera index of political fragmentation increases. This new version reconciles the different theoretical and empirical results that have been found for large voting bodies.

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“…See Lindner & Machover (2004); Chang et al (2006). This phenomenon, the Penrose Limit Theorem, was originally identified in a conjecture by Penrose (1952).…”

Section: Introduction: the Weighted Voting Effect And Its Asymptotic mentioning

confidence: 89%

Section: Proof Lindner and Machover (2004)mentioning

confidence: 98%

Power indices in large voting bodies

Leech

2011

Public Choice

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“…The county of Schenectady adjoins that of Saratoga, lying just to its south. 7 On May 10, 2011, the 15-person Schenectady County Legislature voted to change from the then current "one-legislator-one-vote" rule, to a weighted rule (effective in 2012), in which the voting weights range from a low of 0.9048 to a high of 1.0799. These weights add to 15 and are chosen to be proportional, based on the 2010 Census, to the population share of each representative (equal to the district population divided by the number of representatives from that district, with 4 districts and from 3 to 5 representatives per district).…”

Section: Weight Vs Influence: Penrose-banzhaf Voting Powermentioning

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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“…p 313. 8 Les elections qui suivirent consacr erent la mont ee des lib eraux au d etriment des ultraroyalistes. Ceux-ci s'employaient pas divers moyens a tout mettre en uvre pour modi er la loi a leur pro t. 6 La chambre introuvable, selon une expression c el ebre de Louis XVIII, est issue de ces elections.…”

Section: Gen Ese Et Contexte Historique De La Loi Du 29 Juin 1820unclassified

“…7 Le mode de calcul du cens r ev ele la conception familialiste du su rage (Verjus (1996(Verjus ( , 2000(Verjus ( , 2002. 8 Rosanvallon (1992, p 273) o re une analyse tr es pouss ee de la Loi La^ n e dont le projet, r edig e par Guizot, d eveloppe deux principes fondamentaux. Le premier est que l' election doitêtre directe, c'est-a-dire que tous les citoyens qui, dans un d epartement, remplissent les conditions exig ees par la charte pourêtre electeur, doivent concourir directement, et par eux-mêmes, a la nomination des d eput es du d epartement.…”

Section: Gen Ese Et Contexte Historique De La Loi Du 29 Juin 1820unclassified

Une analyse de la loi électorale du 29 juin 1820

Breton¹,

Lepelley²

2014

Revue Économique

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Nous exploitons dans cet article la th eorie des indices de pouvoir pour evaluer les in uences respectives des deux classes d' electeurs dans le mode de scrutin instaur e par la loi electorale du 29 juin 1820, dite loi du \double vote". Nous montrons, a l'aide d'un mod ele simpli e, que le pouvoir de vote des \grands" electeurs, qui votent deux fois, est au moins trois a quatre fois sup erieur a celui des \petits" electeurs, qui ne votent qu'une fois. Le mod ele propos e constitue a certains egards une extension de celui qu'Edelman (2004) a r ecemment utilis e pour etudier les situations dans lesquelles certains electeurs appartiennent a plus d'un coll ege electoral.Classi cation JEL : D71, D72. Mots cl es : Elections, Vote, Pouvoir de vote.Les auteurs remercient particuli erement Philippe Mongin, dont la lecture attentive et les remarques ont permis d'am eliorer signi cativement la pr esentation des id ees et r esultats de cet article.Les auteurs remercient egalement deux arbitres de lecture de leurs rapports approfondis, P. Edelman d'avoir communiqu e certains de ses travaux sur des questions voisines ainsi que les participants a la conf erence de l'ADRES de leurs commentaires et suggestions. Ils remercient en n L. Vidu d'avoir conduit des simulations portant sur l' evaluation du pouvoir o u la corr elation des votes des electeurs votant deux fois est prise en compte. Elles sont consign ees dans Vidu (2011).

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L S Penrose's limit theorem: Tests by simulation

Cited by 31 publications

References 6 publications

Power indices in large voting bodies

Power indices in large voting bodies

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

Une analyse de la loi électorale du 29 juin 1820

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