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L S Penrose's limit theorem: Tests by simulation

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  • Chang, Pao-Li
  • Chua, Vincent C.H.
  • Machover, Moshe

Abstract

LS Penrose’s limit theorem (PLT) – which is implicit in Penrose [5, p. 72] and for which he gave no rigorous proof – says that, in simple weighted voting games, if the number of voters increases indefinitely while existing voters retain their weights 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 [3] prove some special cases of PLT; and conjecture that the theorem holds, under rather general conditions, for large classes of weighted voting games, various values of the quota, and with respect to several measures of voting power. We use simulation to test this conjecture. It is corroborated w.r.t. the Penrose–Banzhaf index for a quota of 50% but not for other values; w.r.t. the Shapley–Shubik index the conjecture is corroborated for all values of the quota (short of 100%).

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Bibliographic Info

Article provided by Elsevier in its journal Mathematical Social Sciences.

Volume (Year): 51 (2006)
Issue (Month): 1 (January)
Pages: 90-106

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Handle: RePEc:eee:matsoc:v:51:y:2006:i:1:p:90-106

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Web page: http://www.elsevier.com/locate/inca/505565

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  1. Lindner, Ines & Machover, Moshe, 2004. "L.S. Penrose's limit theorem: proof of some special cases," Mathematical Social Sciences, Elsevier, vol. 47(1), pages 37-49, January.
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Cited by:
  1. Le Breton, Michel & Lepelley, Dominique, 2012. "Une Analyse de la Loi Electorale du 29 Juin 1820," TSE Working Papers 12-312, Toulouse School of Economics (TSE).
  2. Houy, Nicolas & Zwicker, William S., 2014. "The geometry of voting power: Weighted voting and hyper-ellipsoids," Games and Economic Behavior, Elsevier, vol. 84(C), pages 7-16.
  3. Guillermo Owen & Ines Lindner & Scott Feld & Bernard Grofman & Leonard Ray, 2006. "A simple “market value” bargaining model for weighted voting games: characterization and limit theorems," International Journal of Game Theory, Springer, vol. 35(1), pages 111-128, December.
  4. Nicola Maaser & Stefan Napel, 2007. "Equal representation in two-tier voting systems," Social Choice and Welfare, Springer, vol. 28(3), pages 401-420, April.
  5. Dennis Leech, 2013. "Power indices in large voting bodies," Public Choice, Springer, vol. 155(1), pages 61-79, April.

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