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LS Penrose’s limit theorem: Tests by simulation

Author

Listed:
  • 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%).

Suggested Citation

  • Pao-Li Chang & Vincent CH Chua & Moshe Machover, 2004. "LS Penrose’s limit theorem: Tests by simulation," Working Papers 26-2004, Singapore Management University, School of Economics.
  • Handle: RePEc:siu:wpaper:26-2004
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    File URL: https://mercury.smu.edu.sg/rsrchpubupload/5019/ccm.pdf
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    References listed on IDEAS

    as
    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.
    2. repec:cup:apsrev:v:48:y:1954:i:03:p:787-792_00 is not listed on IDEAS
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    Cited by:

    1. Le Breton, Michel & Lepelley, Dominique & Macé, Antonin & Merlin, Vincent, 2016. "Le Mécanisme Optimal de Vote au Sein du Conseil des Représentants d'un Système Fédéral," TSE Working Papers 16-617, Toulouse School of Economics (TSE), revised Dec 2016.
    2. Sascha Kurz, 2016. "The inverse problem for power distributions in committees," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 47(1), pages 65-88, June.
    3. Zaporozhets, Vera, 2015. "Power Distribution in French River Basin Committees," TSE Working Papers 15-558, Toulouse School of Economics (TSE).
    4. Michel Le Breton & Dominique Lepelley, 2014. "Une analyse de la loi électorale du 29 juin 1820," Revue économique, Presses de Sciences-Po, vol. 65(3), pages 469-518.
    5. 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.
    6. Lindner, Ines & Owen, Guillermo, 2007. "Cases where the Penrose limit theorem does not hold," Mathematical Social Sciences, Elsevier, vol. 53(3), pages 232-238, May.
    7. Artyom Jelnov & Yair Tauman, 2014. "Voting power and proportional representation of voters," International Journal of Game Theory, Springer;Game Theory Society, vol. 43(4), pages 747-766, November.
    8. Nicola Maaser & Stefan Napel, 2007. "Equal representation in two-tier voting systems," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 28(3), pages 401-420, April.
    9. 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;Game Theory Society, vol. 35(1), pages 111-128, December.
    10. Dennis Leech, 2013. "Power indices in large voting bodies," Public Choice, Springer, vol. 155(1), pages 61-79, April.
    11. Le Breton, Michel & Lepelley, Dominique & Macé, Antonin & Merlin, Vincent, 2016. "Le Mécanisme Optimal de Vote au Sein du Conseil des Représentants d'un Système Fédéral," TSE Working Papers 16-617, Toulouse School of Economics (TSE), revised Dec 2016.
    12. repec:spr:homoec:v:34:y:2017:i:2:d:10.1007_s41412-017-0042-7 is not listed on IDEAS

    More about this item

    Keywords

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

    JEL classification:

    • C71 - Mathematical and Quantitative Methods - - Game Theory and Bargaining Theory - - - Cooperative Games
    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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