IDEAS home Printed from https://ideas.repec.org/a/gam/jgames/v13y2022i3p44-d830635.html
   My bibliography  Save this article

Monte Carlo Methods for the Shapley–Shubik Power Index

Author

Listed:
  • Yuto Ushioda

    (Graduate School of Engineering, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan)

  • Masato Tanaka

    (Graduate School of Engineering, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan)

  • Tomomi Matsui

    (Graduate School of Engineering, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8552, Japan)

Abstract

This paper deals with the problem of calculating the Shapley–Shubik power index in weighted majority games. We propose an efficient Monte Carlo algorithm based on an implicit hierarchical structure of permutations of players. Our algorithm outputs a vector of power indices preserving the monotonicity, with respect to the voting weights. We show that our algorithm reduces the required number of samples, compared with the naive algorithm.

Suggested Citation

  • 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.
    • Handle: RePEc:gam:jgames:v:13:y:2022:i:3:p:44-:d:830635
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2073-4336/13/3/44/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2073-4336/13/3/44/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. 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.
    2. Moshé Machover & Dan S. Felsenthal, 2001. "The Treaty of Nice and qualified majority voting," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(3), pages 431-464.
    3. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2016. "The Shapley–Shubik power index for dichotomous multi-type games," Theory and Decision, Springer, vol. 81(3), pages 413-426, September.
    4. Prasad, K & Kelly, J S, 1990. "NP-Completeness of Some Problems Concerning Voting Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 19(1), pages 1-9.
    5. Bilbao, J. M. & Fernandez, J. R. & Jimenez, N. & Lopez, J. J., 2002. "Voting power in the European Union enlargement," European Journal of Operational Research, Elsevier, vol. 143(1), pages 181-196, November.
    6. Bolus, Stefan, 2011. "Power indices of simple games and vector-weighted majority games by means of binary decision diagrams," European Journal of Operational Research, Elsevier, vol. 210(2), pages 258-272, April.
    7. Leech, Dennis, 2002. "Voting Power In The Governance Of The International Monetary Fund," Economic Research Papers 269354, University of Warwick - Department of Economics.
    8. Xiaotie Deng & Christos H. Papadimitriou, 1994. "On the Complexity of Cooperative Solution Concepts," Mathematics of Operations Research, INFORMS, vol. 19(2), pages 257-266, May.
    9. Klinz, Bettina & Woeginger, Gerhard J., 2005. "Faster algorithms for computing power indices in weighted voting games," Mathematical Social Sciences, Elsevier, vol. 49(1), pages 111-116, January.
    10. 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.
    11. Berghammer, Rudolf & Bolus, Stefan & Rusinowska, Agnieszka & de Swart, Harrie, 2011. "A relation-algebraic approach to simple games," European Journal of Operational Research, Elsevier, vol. 210(1), pages 68-80, April.
    12. Manfred J. Holler & Florian Rupp, 2021. "Power in Networks: The Medici," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 38(1), pages 59-75, December.
    13. Owen, Guillermo, 1975. "Evaluation of a Presidential Election Game," American Political Science Review, Cambridge University Press, vol. 69(3), pages 947-953, September.
    14. Dennis Leech, 2003. "Computing Power Indices for Large Voting Games," Management Science, INFORMS, vol. 49(6), pages 831-837, June.
    15. Alonso-Meijide, J.M. & Casas-Méndez, B. & Fiestras-Janeiro, M.G., 2015. "Computing Banzhaf–Coleman and Shapley–Shubik power indices with incompatible players," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 377-387.
    16. 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.
    17. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    18. J. Bilbao & J. Fernández & A. Losada & J. López, 2000. "Generating functions for computing power indices efficiently," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 8(2), pages 191-213, December.
    19. Sébastien Courtin & Zéphirin Nganmeni & Bertrand Tchantcho, 2016. "The Shapley-Shubik power index for dichotomous multi-type games," Post-Print halshs-01545769, HAL.
    20. 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)

    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. Benati, Stefano & Rizzi, Romeo & Tovey, Craig, 2015. "The complexity of power indexes with graph restricted coalitions," Mathematical Social Sciences, Elsevier, vol. 76(C), pages 53-63.
    2. 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.
    3. 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.
    4. 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.
    5. Stefano Benati & Giuseppe Vittucci Marzetti, 2021. "Voting power on a graph connected political space with an application to decision-making in the Council of the European Union," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 57(4), pages 733-761, November.
    6. 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.
    7. Matthew Gould & Matthew D. Rablen, 2017. "Reform of the United Nations Security Council: equity and efficiency," Public Choice, Springer, vol. 173(1), pages 145-168, October.
    8. 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.
    9. 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.
    10. 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.
    11. de Keijzer, B. & Klos, T.B. & Zhang, Y., 2012. "Solving Weighted Voting Game Design Problems Optimally: Representations, Synthesis, and Enumeration," ERIM Report Series Research in Management ERS-2012-006-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    12. 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.
    13. 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.
    14. Mika Widgrén, 2008. "The Impact of Council's Internal Decision-Making Rules on the Future EU," Discussion Papers 26, Aboa Centre for Economics.
    15. Martí Jané Ballarín, 2023. "The complexity of power indices in voting games with incompatible players," UB School of Economics Working Papers 2023/441, University of Barcelona School of Economics.
    16. 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.
    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. A. Saavedra-Nieves, 2023. "On stratified sampling for estimating coalitional values," Annals of Operations Research, Springer, vol. 320(1), pages 325-353, January.
    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:gam:jgames:v:13:y:2022:i:3:p:44-:d:830635. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    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.