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Power Indices in Large Voting Bodies

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  • Leech, Dennis

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.

Suggested Citation

  • Leech, Dennis, 2010. "Power Indices in Large Voting Bodies," Economic Research Papers 270996, University of Warwick - Department of Economics.
  • Handle: RePEc:ags:uwarer:270996
    DOI: 10.22004/ag.econ.270996
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    References listed on IDEAS

    as
    1. Dennis Leech, 2003. "Computing Power Indices for Large Voting Games," Management Science, INFORMS, vol. 49(6), pages 831-837, June.
    2. Dennis Leech, 2002. "Voting Power in the Governance of the International Monetary Fund," Annals of Operations Research, Springer, vol. 109(1), pages 375-397, January.
    3. Dan S. Felsenthal & Moshé Machover, 1998. "The Measurement of Voting Power," Books, Edward Elgar Publishing, number 1489.
    4. Leech, Dennis & Aziz, Haris, 2007. "The Double Majority Voting Rule of the EU Reform Treaty as a Democratic Ideal for an Enlarging Union : an Appraisal Using Voting Power Analysis," The Warwick Economics Research Paper Series (TWERPS) 824, University of Warwick, Department of Economics.
    5. Scott L. Feld & Bernard Grofman, 2007. "The Laakso-Taagepera Index in A Mean and Variance Framework," Journal of Theoretical Politics, , vol. 19(1), pages 101-106, January.
    6. Dennis Leech, 2002. "An Empirical Comparison of the Performance of Classical Power Indices," Political Studies, Political Studies Association, vol. 50(1), pages 1-22, March.
    7. Chang, Pao-Li & Chua, Vincent C.H. & Machover, Moshe, 2006. "L S Penrose's limit theorem: Tests by simulation," Mathematical Social Sciences, Elsevier, vol. 51(1), pages 90-106, January.
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    11. Leech, Dennis, 2002. "Voting Power In The Governance Of The International Monetary Fund," Economic Research Papers 269354, University of Warwick - Department of Economics.
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