IDEAS home Printed from https://ideas.repec.org/p/siu/wpaper/26-2004.html
   My bibliography  Save this paper

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
    as

    Download full text from publisher

    File URL: https://mercury.smu.edu.sg/rsrchpubupload/5019/ccm.pdf
    Download Restriction: no
    ---><---

    Other versions of this item:

    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. Dan S. Felsenthal & Moshé Machover, 1998. "The Measurement of Voting Power," Books, Edward Elgar Publishing, number 1489.
    3. Shapley, L. S. & Shubik, Martin, 1954. "A Method for Evaluating the Distribution of Power in a Committee System," American Political Science Review, Cambridge University Press, vol. 48(3), pages 787-792, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Le Breton, Michel & Lepelley, Dominique & Macé, Antonin & Merlin, Vincent, 2017. "Le mécanisme optimal de vote au sein du conseil des représentants d’un système fédéral," L'Actualité Economique, Société Canadienne de Science Economique, vol. 93(1-2), pages 203-248, Mars-Juin.
    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. 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.
    5. 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.
    6. 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.
    7. 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.
    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. Leech, Dennis, 2010. "Power Indices in Large Voting Bodies," Economic Research Papers 270996, University of Warwick - Department of Economics.
    11. Josep Freixas & Montserrat Pons, 2021. "An Appropriate Way to Extend the Banzhaf Index for Multiple Levels of Approval," Group Decision and Negotiation, Springer, vol. 30(2), pages 447-462, April.
    12. Debabrata Pal, 2021. "Does everyone have equal voting power?," Indian Economic Review, Springer, vol. 56(2), pages 515-525, December.
    13. Kurz, Sascha & Maaser, Nicola & Napel, Stefan, 2018. "Fair representation and a linear Shapley rule," Games and Economic Behavior, Elsevier, vol. 108(C), pages 152-161.
    14. Dennis Leech, 2013. "Power indices in large voting bodies," Public Choice, Springer, vol. 155(1), pages 61-79, April.
    15. Dan S. Felsenthal, 2017. "Comment on “Proposals for a Democracy of the Future” by Bruno Frey," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 34(2), pages 195-200, November.
    16. Sascha Kurz & Stefan Napel, 2014. "Heuristic and exact solutions to the inverse power index problem for small voting bodies," Annals of Operations Research, Springer, vol. 215(1), pages 137-163, April.
    17. Grimmett, Geoffrey R., 2019. "On influence and compromise in two-tier voting systems," Mathematical Social Sciences, Elsevier, vol. 100(C), pages 35-45.
    18. Fabrice Barthélémy & Mathieu Martin, 2021. "Dummy Players and the Quota in Weighted Voting Games: Some Further Results," Studies in Choice and Welfare, in: Mostapha Diss & Vincent Merlin (ed.), Evaluating Voting Systems with Probability Models, pages 299-315, Springer.
    19. Luca Alfieri & Nino Kokashvili, 2020. "Financial Safety Nets In East Asia And Europe: A Political Economy Assessment," University of Tartu - Faculty of Economics and Business Administration Working Paper Series 121, Faculty of Economics and Business Administration, University of Tartu (Estonia).
    20. de Mouzon, Olivier & Laurent, Thibault & Le Breton, Michel & Moyouwou, Issofa, 2020. "“One Man, One Vote” Part 1: Electoral Justice in the U.S. Electoral College: Banzhaf and Shapley/Shubik versus May," TSE Working Papers 20-1074, Toulouse School of Economics (TSE).
    21. Sascha Kurz, 2020. "A note on limit results for the Penrose–Banzhaf index," Theory and Decision, Springer, vol. 88(2), pages 191-203, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Le Breton, Michel & Lepelley, Dominique & Macé, Antonin & Merlin, Vincent, 2017. "Le mécanisme optimal de vote au sein du conseil des représentants d’un système fédéral," L'Actualité Economique, Société Canadienne de Science Economique, vol. 93(1-2), pages 203-248, Mars-Juin.
    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. Zaporozhets, Vera, 2015. "Power Distribution in French River Basin Committees," TSE Working Papers 15-558, Toulouse School of Economics (TSE).
    4. Kurz, Sascha & Maaser, Nicola & Napel, Stefan, 2018. "Fair representation and a linear Shapley rule," Games and Economic Behavior, Elsevier, vol. 108(C), pages 152-161.
    5. Dennis Leech, 2013. "Power indices in large voting bodies," Public Choice, Springer, vol. 155(1), pages 61-79, April.
    6. Boratyn, Daria & Kirsch, Werner & Słomczyński, Wojciech & Stolicki, Dariusz & Życzkowski, Karol, 2020. "Average weights and power in weighted voting games," Mathematical Social Sciences, Elsevier, vol. 108(C), pages 90-99.
    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. 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.
    9. Ines Lindner, 2012. "Annick Laruelle and Federico Valenciano: Voting and collective decision-making," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 161-179, January.
    10. de Mouzon, Olivier & Laurent, Thibault & Le Breton, Michel & Moyouwou, Issofa, 2020. "“One Man, One Vote” Part 1: Electoral Justice in the U.S. Electoral College: Banzhaf and Shapley/Shubik versus May," TSE Working Papers 20-1074, Toulouse School of Economics (TSE).
    11. Sascha Kurz & Stefan Napel, 2014. "Heuristic and exact solutions to the inverse power index problem for small voting bodies," Annals of Operations Research, Springer, vol. 215(1), pages 137-163, April.
    12. 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.
    13. Claus Beisbart & Luc Bovens, 2007. "Welfarist evaluations of decision rules for boards of representatives," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 29(4), pages 581-608, December.
    14. LINDNER, Ines, 2005. "Voting games with abstention : A probabilistic characterization of power and a special case of Penrose’s Limit Theorem," LIDAM Discussion Papers CORE 2005078, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    15. Matija Kovacic & Claudio Zoli, 2021. "Ethnic distribution, effective power and conflict," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 57(2), pages 257-299, August.
    16. Mikel Alvarez-Mozos & José María Alonso-Meijide & María Gloria Fiestras-Janeiro, 2016. "The Shapley-Shubik Index in the Presence of Externalities," UB School of Economics Working Papers 2016/342, University of Barcelona School of Economics.
    17. Monisankar Bishnu & Sonali Roy, 2012. "Hierarchy of players in swap robust voting games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 11-22, January.
    18. Zaporozhets, Vera & García-Valiñas, María & Kurz, Sascha, 2016. "Key drivers of EU budget allocation: Does power matter?," European Journal of Political Economy, Elsevier, vol. 43(C), pages 57-70.
    19. Silvia Fedeli & Francesco Forte, 2001. "Voting Powers and the Efficiency of the Decision-Making Process in the European Council of Ministers," European Journal of Law and Economics, Springer, vol. 12(1), pages 5-38, July.
    20. Renneboog, Luc & Szilagyi, Peter G., 2020. "How relevant is dividend policy under low shareholder protection?," Journal of International Financial Markets, Institutions and Money, Elsevier, vol. 64(C).

    More about this item

    Keywords

    limit theorems; majority games; simulation; weighted voting games;
    All these keywords.

    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

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:siu:wpaper:26-2004. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: QL THor (email available below). General contact details of provider: https://edirc.repec.org/data/sesmusg.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.