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LS Penrose’s limit theorem: Tests by simulation

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  • Pao-Li Chang

    ()
    (School of Economics and Social Sciences, Singapore Management University)

  • Vincent CH Chua

    ()
    (School of Economics and Social Sciences, Singapore Management University)

  • Moshe Machover

    ()
    (CPNSS LSE, London)

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

Paper provided by Singapore Management University, School of Economics in its series Working Papers with number 26-2004.

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Length: 84 pages
Date of creation: Jul 2004
Date of revision:
Publication status: Published in SMU Economics and Statistics Working Paper Series
Handle: RePEc:siu:wpaper:26-2004

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Keywords: limit theorems; majority games; simulation; weighted voting games;

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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. Nicola Maaser & Stefan Napel, 2007. "Equal representation in two-tier voting systems," Social Choice and Welfare, Springer, vol. 28(3), pages 401-420, April.
  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. 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).

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