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L S Penrose's limit theorem: tests by simulation

Chang, Pao-Li, Chua, Vincent C. H. and Machover, Moshé (2006) L S Penrose's limit theorem: tests by simulation. Mathematical Social Sciences, 51 (1). pp. 90-106. ISSN 0165-4896

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Identification Number: 10.1016/j.mathsocsci.2005.06.001


L S Penrose’s Limit Theorem – which is implicit in Penrose [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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Additional Information: The authors gratefully acknowledge that work on this paper was partly supported by the Leverhulme Trust (Grant F/07-004m).Full appendices to the published article are included. Copyright © 2005 Elsevier B.V. LSE has developed LSE Research Online so that users may access research output of the School. Copyright and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Users may download and/or print one copy of any article(s) in LSE Research Online to facilitate their private study or for non-commercial research. You may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. You may freely distribute the URL (<>) of the LSE Research Online website.
Divisions: CPNSS
Subjects: J Political Science > JC Political theory
Date Deposited: 22 Dec 2005
Last Modified: 04 Jan 2024 18:12

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