L.S. Penrose's limit theorem: proof of some special cases

Lindner, Ines; Machover, Moshé

doi:10.1016/s0165-4896(03)00069-6

Cited by 63 publications

(55 citation statements)

References 6 publications

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“…However, Lindner and Machover [4] show by means of a simple counterexample that these conditions are insufficient for Penrose's approximation formula and the version of Penrose's Limit Theorem implied by it. On the other hand, they prove the approximation formula (in a somewhat improved form) as well as Penrose's Limit Theorem under more stringent conditions: both with respect to β for q = 1 2 (see [4,Theorem 3.6]); and with respect to the Shapley-Shubik index φ for arbitrary q ∈ (0, 1) (see [4,Theorem 2.3]).…”

Section: Introductionmentioning

confidence: 99%

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L S Penrose's limit theorem: Tests by simulation

Chang

Chua

Machover

2006

Mathematical Social Sciences

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This is an electronic version of an Article published in [7, p. 72] and for which he gave no rigorous proof -says that, in simple weighted voting games, if the number of voters increases indefinitely and the relative quota is pegged, then -under certain conditions -the ratio between the voting powers of any two voters converges to the ratio between their weights. Lindner and Machover [4] prove some special cases of Penrose's Limit Theorem. They give a simple counter-example showing that the theorem does not hold in general even under the conditions assumed by Penrose; but they conjecture, in effect, that under rather general conditions it holds 'almost always' -that is with probability 1 -for large classes of weighted voting games, for various values of the quota, and with respect to several measures of voting power. We use simulation to test this conjecture. It is corroborated with respect to the Penrose-Banzhaf index for a quota of 50% but not for other values; with respect to the Shapley-Shubik index the conjecture is corroborated for all values of the quota (short of 100%).

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Section: Introductionmentioning

confidence: 99%

“…On the other hand, they prove the approximation formula (in a somewhat improved form) as well as Penrose's Limit Theorem under more stringent conditions: both with respect to β for q = 1 2 (see [4,Theorem 3.6]); and with respect to the Shapley-Shubik index φ for arbitrary q ∈ (0, 1) (see [4,Theorem 2.3]). …”

Section: Introductionmentioning

confidence: 99%

L S Penrose's limit theorem: Tests by simulation

Chang

Chua

Machover

2006

Mathematical Social Sciences

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“…3.1 of this article. 34 In the limit, when the number of members goes to infinity, this optimal quota tends to 1/2, in agreement with the PLT (see Lindner and Machover 2004). 35 The Constitutional rule is now also known as the Lisbon rule as it was the one selected for the CM in the Treaty of Lisbon 2007.…”

Section: Application To the Eu Council Of Ministersmentioning

confidence: 84%

“…The latter states that under certain conditions and if the distribution of weights is not too skewed (in other words, the ratio of the largest weight to the smallest is not very high), then the relative powers of the voters tend to approximate closely to their respective relative weights. Lindner and Machover 2004;Lindner 2004;Lindner and Owen 2007 provide proofs for some special cases with respect to the Penrose-Banzhaf measure as well as the Shapley-Shubik index. In other words, when PLT holds the voters' voting weights are with close approximation proportional to their Penrose-Banzhaf measures (Shapley-Shubik indices).…”

Section: Committees Of Representativesmentioning

confidence: 99%

Annick Laruelle and Federico Valenciano: Voting and collective decision-making

Lindner

2010

Soc Choice Welf

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The book 'Voting and Collective Decision Making' by A. Laruelle and F. Valenciano provides a critical revision of the theoretical foundations of collective yes-or-no decisions. It is a study of the theory of bargaining and voting power, revolving around a fundamental question: given a committee, what voting rule should be used? Brief summary of the book and motivation of researchIn a nutshell, the theory and measurement of voting power revolves around the question to what extent a voter is able to control the outcome of a vote. This question is fundamental as it provides the foundation to the question of what voting procedure should be used for a decision-making body when we deal with criteria such as, e.g. egalitarianism. As an example take the European Union where the decision-making processes of the EU institutions should be ideally designed-from an egalitarian point of view-such that every EU citizen has equal voting power. Especially, the several enlargements of the EU have led to a renewal of interest in this topic and heated debates in the scientific community along with controversial academic papers.The main purpose of Voting and Collective Decision Making by A. Laruelle and F. Valenciano (L&V) is to provide a critical revision of the theoretical foundations of collective yes-or-no decisions as well as a revision of the recommendations when it comes to the design of decision-making procedures. The focus of their approach is to include not only the voting rule 1 as unique ingredient but to draw a clear 1 L&V define a voting rule as a monotonic and proper simple game. In other words, it is a set of winning coalitions which contains the grand coalition and all supersets of winning coalitions. It does not contain the empty set or the complement of a winning coalition (see their Definition 1, p. 5). I. Lindner (B)

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“…Conversely, when the equation = has solutions and ∈ 1 × , + = and = can be obtained relying on the Penrose theorem [9,10].…”

Section: The Equationmentioning

confidence: 99%