IDEAS home Printed from https://ideas.repec.org/a/eee/matsoc/v108y2020icp175-192.html
   My bibliography  Save this article

Dynamic programming algorithms for computing power indices in weighted multi-tier games

Author

Listed:
  • Wilms, Ingo

Abstract

In weighted games each voter has a weight assigned and “yes”-voters win if the sum of each weight is greater than or equal to the quota. In weighted multi-tier games, we have several weighted games (tiers) over the same set of voters. In this article, algorithms for calculating Banzhaf and Shapley–Shubik indices for three different type of games are proposed: weighted AND-games, weighted OR-games and games where the tiers have conjunctive normal form. The presented algorithms are generalizations of known computational methods using dynamic programming technique. Finally, some applications and experiments are carried out and these algorithms are compared to a fairly new method based on binary decision diagrams.

Suggested Citation

  • Wilms, Ingo, 2020. "Dynamic programming algorithms for computing power indices in weighted multi-tier games," Mathematical Social Sciences, Elsevier, vol. 108(C), pages 175-192.
    • Handle: RePEc:eee:matsoc:v:108:y:2020:i:c:p:175-192
    DOI: 10.1016/j.mathsocsci.2020.06.004
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0165489620300603
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.mathsocsci.2020.06.004?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. Annick Laruelle & Federico Valenciano, 2005. "Assessing success and decisiveness in voting situations," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 24(1), pages 171-197, January.
    2. Antônio Francisco Neto & Carolina Rodrigues Fonseca, 2019. "An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players," Annals of Operations Research, Springer, vol. 279(1), pages 221-249, August.
    3. Leech, Dennis, 2002. "Computing Power Indices For Large Voting Games," Economic Research Papers 269350, University of Warwick - Department of Economics.
    4. M. Musegaas & P. E. M. Borm & M. Quant, 2018. "Three-valued simple games," Theory and Decision, Springer, vol. 85(2), pages 201-224, August.
    5. 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.
    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. 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.
    8. Leech, Dennis, 2002. "Computation of Power Indices," The Warwick Economics Research Paper Series (TWERPS) 644, University of Warwick, Department of Economics.
    9. Josep Freixas & Marc Freixas & Sascha Kurz, 2017. "On the characterization of weighted simple games," Theory and Decision, Springer, vol. 83(4), pages 469-498, December.
    10. Aziz, Haris & Paterson, Mike & Leech, Dennis, 2007. "Combinatorial and computational aspects of multiple weighted voting games," Economic Research Papers 269772, University of Warwick - Department of Economics.
    11. Algaba, E. & Bilbao, J. M. & Fernandez Garcia, J. R. & Lopez, J. J., 2003. "Computing power indices in weighted multiple majority games," Mathematical Social Sciences, Elsevier, vol. 46(1), pages 63-80, August.
    12. R J Johnston, 1978. "On the Measurement of Power: Some Reactions to Laver," Environment and Planning A, , vol. 10(8), pages 907-914, August.
    13. Cheung, Wai-Shun & Ng, Tuen-Wai, 2014. "A three-dimensional voting system in Hong Kong," European Journal of Operational Research, Elsevier, vol. 236(1), pages 292-297.
    14. Guillermo Owen, 1972. "Multilinear Extensions of Games," Management Science, INFORMS, vol. 18(5-Part-2), pages 64-79, January.
    15. 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.
    16. Leech, Dennis, 2002. "Computation Of Power Indices," Economic Research Papers 269457, University of Warwick - Department of Economics.
    17. Aziz, Haris & Paterson, Mike & Leech, Dennis, 2007. "Combinatorial and computational aspects of multiple weighted voting games," The Warwick Economics Research Paper Series (TWERPS) 823, University of Warwick, Department of Economics.
    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. Bhattacherjee, Sanjay & Chakravarty, Satya R. & Sarkar, Palash, 2022. "A General Model for Multi-Parameter Weighted Voting Games," MPRA Paper 115407, University Library of Munich, Germany.

    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. Bhattacherjee, Sanjay & Chakravarty, Satya R. & Sarkar, Palash, 2022. "A General Model for Multi-Parameter Weighted Voting Games," MPRA Paper 115407, University Library of Munich, Germany.
    2. Somdeb Lahiri, 2021. "Pattanaik's axioms and the existence of winners preferred with probability at least half," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 31(2), pages 109-122.
    3. 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.
    4. Michela Chessa, 2014. "A generating functions approach for computing the Public Good index efficiently," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(2), pages 658-673, July.
    5. A. Saavedra-Nieves, 2023. "On stratified sampling for estimating coalitional values," Annals of Operations Research, Springer, vol. 320(1), pages 325-353, January.
    6. Birkmeier Olga & Käufl Andreas & Pukelsheim Friedrich, 2011. "Abstentions in the German Bundesrat and ternary decision rules in weighted voting systems," Statistics & Risk Modeling, De Gruyter, vol. 28(1), pages 1-16, March.
    7. A. Saavedra-Nieves & M. G. Fiestras-Janeiro, 2021. "Sampling methods to estimate the Banzhaf–Owen value," Annals of Operations Research, Springer, vol. 301(1), pages 199-223, June.
    8. Antônio Francisco Neto & Carolina Rodrigues Fonseca, 2019. "An approach via generating functions to compute power indices of multiple weighted voting games with incompatible players," Annals of Operations Research, Springer, vol. 279(1), pages 221-249, August.
    9. 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.
    10. Daphne Cornelisse & Thomas Rood & Mateusz Malinowski & Yoram Bachrach & Tal Kachman, 2022. "Neural Payoff Machines: Predicting Fair and Stable Payoff Allocations Among Team Members," Papers 2208.08798, arXiv.org.
    11. Josep Freixas & Montserrat Pons, 2017. "Using the Multilinear Extension to Study Some Probabilistic Power Indices," Group Decision and Negotiation, Springer, vol. 26(3), pages 437-452, May.
    12. 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.
    13. Pavel Doležel, 2011. "Optimizing the Efficiency of Weighted Voting Games," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 5(3), pages 306-323, November.
    14. 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.
    15. Macé, Antonin & Treibich, Rafael, 2012. "Computing the optimal weights in a utilitarian model of apportionment," Mathematical Social Sciences, Elsevier, vol. 63(2), pages 141-151.
    16. Leech, Dennis, 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," Economic Research Papers 269773, University of Warwick - Department of Economics.
    17. 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.
    18. 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).
    19. Fabrice Barthélémy & Mathieu Martin, 2007. "Configurations study for the Banzhaf and the Shapley-Shubik indices of power," THEMA Working Papers 2007-07, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
    20. Antônio Francisco Neto, 2019. "Generating Functions of Weighted Voting Games, MacMahon’s Partition Analysis, and Clifford Algebras," Mathematics of Operations Research, INFORMS, vol. 44(1), pages 74-101, February.

    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:175-192. 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.