IDEAS home Printed from https://ideas.repec.org/a/eee/matsoc/v108y2020icp90-99.html
   My bibliography  Save this article

Average weights and power in weighted voting games

Author

Listed:
  • Boratyn, Daria
  • Kirsch, Werner
  • Słomczyński, Wojciech
  • Stolicki, Dariusz
  • Życzkowski, Karol

Abstract

We investigate a class of weighted voting games for which weights are randomly distributed over the standard probability simplex. We provide close-formed formulae for the expectation and density of the distribution of weight of the k-th largest player under the uniform distribution. We analyze the average voting power of the k-th largest player and its dependence on the quota, obtaining analytical and numerical results for small values of n and a general theorem about the functional form of the relation between the average Penrose–Banzhaf power index and the quota for the uniform measure on the simplex. We also analyze the power of a collectivity to act (Coleman efficiency index) of random weighted voting games, obtaining analytical upper bounds therefor.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:matsoc:v:108:y:2020:i:c:p:90-99
    DOI: 10.1016/j.mathsocsci.2020.04.002
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0165489620300408
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.mathsocsci.2020.04.002?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. William Gehrlein & Peter Fishburn, 1976. "Condorcet's paradox and anonymous preference profiles," Public Choice, Springer, vol. 26(1), pages 1-18, 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. Laruelle, Annick & Widgren, Mika, 1998. "Is the Allocation of Voting Power among EU States Fair?," Public Choice, Springer, vol. 94(3-4), pages 317-339, March.
    4. Fabrice Barthelemy & Mathieu Martin & Bertrand Tchantcho, 2011. "Some conjectures on the two main power indices," THEMA Working Papers 2011-14, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    5. Kuga, Kiyoshi & Nagatani, Hiroaki, 1974. "Voter Antagonism and the Paradox of Voting," Econometrica, Econometric Society, vol. 42(6), pages 1045-1067, November.
    6. Dan S Felsenthal & Moshé Machover, 2004. "Analysis of QM rules in the draft constitution for Europe proposed by the European Convention, 2003," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 23(1), pages 1-20, August.
    7. Owen, Guillermo, 1975. "Evaluation of a Presidential Election Game," American Political Science Review, Cambridge University Press, vol. 69(3), pages 947-953, September.
    8. Grimmett, Geoffrey R., 2019. "On influence and compromise in two-tier voting systems," Mathematical Social Sciences, Elsevier, vol. 100(C), pages 35-45.
    9. 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.
    10. Leech, Dennis, 2002. "Voting Power In The Governance Of The International Monetary Fund," Economic Research Papers 269354, University of Warwick - Department of Economics.
    11. 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.
    12. Werner Kirsch & Jessica Langner, 2010. "Power indices and minimal winning coalitions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(1), pages 33-46, January.
    13. Fabrice Barthélémy & Dominique Lepelley & Mathieu Martin, 2013. "On the likelihood of dummy players in weighted majority games," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(2), pages 263-279, July.
    14. Sascha Kurz, 2012. "On minimum sum representations for weighted voting games," Annals of Operations Research, Springer, vol. 196(1), pages 361-369, July.
    15. Sascha Kurz, 2018. "Correction to: On minimum sum representations for weighted voting games," Annals of Operations Research, Springer, vol. 271(2), pages 1087-1089, December.
    16. Rao, J. S. & Sobel, Milton, 1980. "Incomplete Dirichlet integrals with applications to ordered uniform spacings," Journal of Multivariate Analysis, Elsevier, vol. 10(4), pages 603-610, December.
    17. Dan S. Felsenthal & Moshé Machover, 1998. "The Measurement of Voting Power," Books, Edward Elgar Publishing, number 1489, December.
    18. 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.
    19. Pradeep Dubey & Lloyd S. Shapley, 1979. "Mathematical Properties of the Banzhaf Power Index," Mathematics of Operations Research, INFORMS, vol. 4(2), pages 99-131, May.
    20. 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.
    21. Leech, Dennis, 2002. "Designing the Voting System for the Council of the European Union," Public Choice, Springer, vol. 113(3-4), pages 437-464, December.
    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. Xavier Molinero & Maria Serna & Marc Taberner-Ortiz, 2021. "On Weights and Quotas for Weighted Majority Voting Games," Games, MDPI, vol. 12(4), pages 1-25, December.

    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. Sylvain Béal & Marc Deschamps & Mostapha Diss & Issofa Moyouwou, 2022. "Inconsistent weighting in weighted voting games," Public Choice, Springer, vol. 191(1), pages 75-103, April.
    2. 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.
    3. 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.
    4. Zineb Abidi & Matthieu Leprince & Vincent Merlin, 2020. "Power Inequality in Inter-communal Structures: The Simulated Impact of a Reform in the Case of the Municipalities in Western France," Post-Print halshs-02996998, HAL.
    5. Crama, Yves & Leruth, Luc, 2007. "Control and voting power in corporate networks: Concepts and computational aspects," European Journal of Operational Research, Elsevier, vol. 178(3), pages 879-893, May.
    6. Josep Freixas & Sascha Kurz, 2014. "Enumeration of weighted games with minimum and an analysis of voting power for bipartite complete games with minimum," Annals of Operations Research, Springer, vol. 222(1), pages 317-339, November.
    7. Yuto Ushioda & Masato Tanaka & Tomomi Matsui, 2022. "Monte Carlo Methods for the Shapley–Shubik Power Index," Games, MDPI, vol. 13(3), pages 1-14, June.
    8. Le Breton, Michel & Montero, Maria & Zaporozhets, Vera, 2012. "Voting power in the EU council of ministers and fair decision making in distributive politics," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 159-173.
    9. 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.
    10. N. Maaser, 2017. "Simple vs. Sophisticated Rules for the Allocation of Voting Weights," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 34(1), pages 67-78, April.
    11. 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.
    12. Matthew Gould & Matthew D. Rablen, 2016. "Equitable representation in councils: theory and an application to the United Nations Security Council," Public Choice, Springer, vol. 169(1), pages 19-51, October.
    13. Algaba, E. & Bilbao, J.M. & Fernandez, J.R., 2007. "The distribution of power in the European Constitution," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1752-1766, February.
    14. André Casajus & Frank Huettner, 2019. "The Coleman–Shapley index: being decisive within the coalition of the interested," Public Choice, Springer, vol. 181(3), pages 275-289, December.
    15. Leech, Dennis, 2010. "Power Indices in Large Voting Bodies," Economic Research Papers 270996, University of Warwick - Department of Economics.
    16. Matthew Gould & Matthew D. Rablen, 2013. "Equitable Representation in the Councils of the United Nations: Theory and Application," CEDI Discussion Paper Series 13-07, Centre for Economic Development and Institutions(CEDI), Brunel University.
    17. Stefan Napel & Mika Widgrén, 2011. "Strategic versus non-strategic voting power in the EU Council of Ministers: the consultation procedure," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(3), pages 511-541, September.
    18. Fabrice Barthelemy & Mathieu Martin & Bertrand Tchantcho, 2011. "Some conjectures on the two main power indices," THEMA Working Papers 2011-14, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    19. Julien Reynaud & Fabien Lange & Łukasz Gątarek & Christian Thimann, 2011. "Proximity in Coalition Building," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 3(3), pages 111-132, September.
    20. Alonso-Meijide, J.M. & Bilbao, J.M. & Casas-Méndez, B. & Fernández, J.R., 2009. "Weighted multiple majority games with unions: Generating functions and applications to the European Union," European Journal of Operational Research, Elsevier, vol. 198(2), pages 530-544, October.

    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:eee:matsoc:v:108:y:2020:i:c:p:90-99. 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: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/locate/inca/505565 .

    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.