IDEAS home Printed from https://ideas.repec.org/a/spr/jogath/v49y2020i1d10.1007_s00182-019-00681-5.html
   My bibliography  Save this article

Tournament solutions based on cooperative game theory

Author

Listed:
  • Aleksei Y. Kondratev

    (National Research University Higher School of Economics
    Institute for Problems of Regional Economics RAS)

  • Vladimir V. Mazalov

    (Karelian Research Center of Russian Academy of Sciences
    Qingdao University)

Abstract

A tournament can be represented as a set of candidates and the results from pairwise comparisons of the candidates. In our setting, candidates may form coalitions. The candidates can choose to fix who wins the pairwise comparisons within their coalition. A coalition is winning if it can guarantee that a candidate from this coalition will win each pairwise comparison. This approach divides all coalitions into two groups and is, hence, a simple game. We show that each minimal winning coalition consists of a certain uncovered candidate and its dominators. We then apply solution concepts developed for simple games and consider the desirability relation and the power indices which preserve this relation. The tournament solution, defined as the maximal elements of the desirability relation, is a good way to select the strongest candidates. The Shapley–Shubik index, the Penrose–Banzhaf index, and the nucleolus are used to measure the power of the candidates. We also extend this approach to the case of weak tournaments.

Suggested Citation

  • Aleksei Y. Kondratev & Vladimir V. Mazalov, 2020. "Tournament solutions based on cooperative game theory," International Journal of Game Theory, Springer;Game Theory Society, vol. 49(1), pages 119-145, March.
    • Handle: RePEc:spr:jogath:v:49:y:2020:i:1:d:10.1007_s00182-019-00681-5
    DOI: 10.1007/s00182-019-00681-5
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s00182-019-00681-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s00182-019-00681-5?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. R.J. Aumann & S. Hart (ed.), 2002. "Handbook of Game Theory with Economic Applications," Handbook of Game Theory with Economic Applications, Elsevier, edition 1, volume 3, number 3.
    2. Felix Brandt & Christian Geist & Paul Harrenstein, 2016. "A note on the McKelvey uncovered set and Pareto optimality," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 46(1), pages 81-91, January.
    3. Timothy Besley & Stephen Coate, 1997. "An Economic Model of Representative Democracy," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 112(1), pages 85-114.
    4. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    5. Vincent Anesi, 2012. "A new old solution for weak tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(4), pages 919-930, October.
    6. Dutta, Bhaskar, 1988. "Covering sets and a new condorcet choice correspondence," Journal of Economic Theory, Elsevier, vol. 44(1), pages 63-80, February.
    7. Dutta, Bhaskar & Jackson, Matthew O & Le Breton, Michel, 2001. "Strategic Candidacy and Voting Procedures," Econometrica, Econometric Society, vol. 69(4), pages 1013-1037, July.
    8. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    9. Martin J. Osborne & Al Slivinski, 1996. "A Model of Political Competition with Citizen-Candidates," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 111(1), pages 65-96.
    10. SCHMEIDLER, David, 1969. "The nucleolus of a characteristic function game," LIDAM Reprints CORE 44, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    11. Brandt, Felix & Harrenstein, Paul & Seedig, Hans Georg, 2017. "Minimal extending sets in tournaments," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 55-63.
    12. I. Good, 1971. "A note on condorcet sets," Public Choice, Springer, vol. 10(1), pages 97-101, March.
    13. Guajardo, Mario & Jörnsten, Kurt, 2015. "Common mistakes in computing the nucleolus," European Journal of Operational Research, Elsevier, vol. 241(3), pages 931-935.
    14. P. C. Fishburn, 1984. "Probabilistic Social Choice Based on Simple Voting Comparisons," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 51(4), pages 683-692.
    15. Fabien Lange & László Kóczy, 2013. "Power indices expressed in terms of minimal winning coalitions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(2), pages 281-292, July.
    16. Freixas, Josep & Pons, Montserrat, 2008. "Circumstantial power: Optimal persuadable voters," European Journal of Operational Research, Elsevier, vol. 186(3), pages 1114-1126, May.
    17. René van den Brink & Peter Borm, 2002. "Digraph Competitions and Cooperative Games," Theory and Decision, Springer, vol. 53(4), pages 327-342, December.
    18. Hannu Vartiainen, 2015. "Dynamic stable set as a tournament solution," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(2), pages 309-327, September.
    19. Dan S. Felsenthal & Moshé Machover, 1998. "The Measurement of Voting Power," Books, Edward Elgar Publishing, number 1489.
    20. Cesarino Bertini & Josep Freixas & Gianfranco Gambarelli & Izabella Stach, 2013. "Comparing Power Indices," International Game Theory Review (IGTR), World Scientific Publishing Co. Pte. Ltd., vol. 15(02), pages 1-19.
    21. Subochev, Andrey & Aleskerov, Fuad & Pislyakov, Vladimir, 2018. "Ranking journals using social choice theory methods: A novel approach in bibliometrics," Journal of Informetrics, Elsevier, vol. 12(2), pages 416-429.
    22. Laffond G. & Laslier J. F. & Le Breton M., 1993. "The Bipartisan Set of a Tournament Game," Games and Economic Behavior, Elsevier, vol. 5(1), pages 182-201, January.
    23. Felix Brandt & Maria Chudnovsky & Ilhee Kim & Gaku Liu & Sergey Norin & Alex Scott & Paul Seymour & Stephan Thomassé, 2013. "A counterexample to a conjecture of Schwartz," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 40(3), pages 739-743, March.
    24. Bordes, Georges, 1983. "On the possibility of reasonable consistent majoritarian choice: Some positive results," Journal of Economic Theory, Elsevier, vol. 31(1), pages 122-132, October.
    25. 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.
    26. Maschler, Michael, 1992. "The bargaining set, kernel, and nucleolus," Handbook of Game Theory with Economic Applications, in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 18, pages 591-667, Elsevier.
    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. Felix Brandt & Markus Brill & Hans Georg Seedig & Warut Suksompong, 2018. "On the structure of stable tournament solutions," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 65(2), pages 483-507, March.
    2. Felix Brandt, 2015. "Set-monotonicity implies Kelly-strategyproofness," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 45(4), pages 793-804, December.
    3. Felix Brandt & Markus Brill & Hans Georg Seedig & Warut Suksompong, 2020. "On the Structure of Stable Tournament Solutions," Papers 2004.01651, arXiv.org.
    4. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    5. Vicki Knoblauch, 2020. "Von Neumann–Morgenstern stable set rationalization of choice functions," Theory and Decision, Springer, vol. 89(3), pages 369-381, October.
    6. Weibin Han & Adrian Deemen, 2019. "A refinement of the uncovered set in tournaments," Theory and Decision, Springer, vol. 86(1), pages 107-121, February.
    7. Felix Brandt & Markus Brill & Felix Fischer & Paul Harrenstein, 2014. "Minimal retentive sets in tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(3), pages 551-574, March.
    8. Reiner Wolff & Yavuz Karagök, 2012. "Consistent allocation of cabinet seats: the Swiss Magic Formula," Public Choice, Springer, vol. 150(3), pages 547-559, March.
    9. Brandt, Felix & Harrenstein, Paul & Seedig, Hans Georg, 2017. "Minimal extending sets in tournaments," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 55-63.
    10. Brandt, Felix & Harrenstein, Paul, 2011. "Set-rationalizable choice and self-stability," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1721-1731, July.
    11. Mario Guajardo & Kurt Jörnsten & Mikael Rönnqvist, 2016. "Constructive and blocking power in collaborative transportation," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(1), pages 25-50, January.
    12. Bandyopadhyay, Siddhartha & Oak, Mandar P., 2008. "Coalition governments in a model of parliamentary democracy," European Journal of Political Economy, Elsevier, vol. 24(3), pages 554-561, September.
    13. Montero, M.P., 2002. "Two-Stage Bargaining with Reversible Coalitions : The Case of Apex Games," Other publications TiSEM 7dba0283-bc13-4f2c-8f5e-5, Tilburg University, School of Economics and Management.
    14. Sebastien Courtin & Boniface Mbih & Issofa Moyouwou, 2009. "Susceptibility to coalitional strategic sponsoring The case of parliamentary agendas," Post-Print hal-00914855, HAL.
    15. Borm, Peter & van den Brink, Rene & Levinsky, Rene & Slikker, Marco, 2004. "On two new social choice correspondences," Mathematical Social Sciences, Elsevier, vol. 47(1), pages 51-68, January.
    16. J. Arin & V. Feltkamp & M. Montero, 2015. "A bargaining procedure leading to the serial rule in games with veto players," Annals of Operations Research, Springer, vol. 229(1), pages 41-66, June.
    17. Vincent Anesi, 2012. "A new old solution for weak tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(4), pages 919-930, October.
    18. Slikker, Marco & Norde, Henk, 2011. "The monoclus of a coalitional game," Games and Economic Behavior, Elsevier, vol. 71(2), pages 420-435, March.
    19. Carmelo Rodríguez-Álvarez, 2006. "Candidate Stability and Voting Correspondences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 27(3), pages 545-570, December.
    20. Arnaud Dellis & Alexandre Gauthier-Belzile & Mandar Oak, 2017. "Policy Polarization and Strategic Candidacy in Elections under the Alternative-Vote Rule," Journal of Institutional and Theoretical Economics (JITE), Mohr Siebeck, Tübingen, vol. 173(4), pages 565-590, December.

    More about this item

    Keywords

    Tournament solution; Simple game; Shapley–Shubik index; Penrose–Banzhaf index; Desirability relation; Uncovered set;
    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
    • C44 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Operations Research; Statistical Decision Theory

    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:spr:jogath:v:49:y:2020:i:1:d:10.1007_s00182-019-00681-5. 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: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.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.