IDEAS home Printed from https://ideas.repec.org/a/spr/sochwe/v40y2013i3p739-743.html
   My bibliography  Save this article

A counterexample to a conjecture of Schwartz

Author

Listed:
  • Felix Brandt
  • Maria Chudnovsky
  • Ilhee Kim
  • Gaku Liu
  • Sergey Norin
  • Alex Scott
  • Paul Seymour
  • Stephan Thomassé

Abstract

In 1990, motivated by applications in the social sciences, Thomas Schwartz made a conjecture about tournaments which would have had numerous attractive consequences. In particular, it implied that there is no tournament with a partition A, B of its vertex set, such that every transitive subset of A is in the out-neighbour set of some vertex in B, and vice versa. But in fact there is such a tournament, as we show in this article, and so Schwartz’ conjecture is false. Our proof is non-constructive and uses the probabilistic method. Copyright Springer-Verlag 2013

Suggested Citation

  • 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.
    • Handle: RePEc:spr:sochwe:v:40:y:2013:i:3:p:739-743
    DOI: 10.1007/s00355-011-0638-y
    as

    Download full text from publisher

    File URL: http://hdl.handle.net/10.1007/s00355-011-0638-y
    Download Restriction: Access to full text is restricted to subscribers.
    File URL: https://libkey.io/10.1007/s00355-011-0638-y?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. Nicolas Houy, 2009. "Still more on the Tournament Equilibrium Set," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 32(1), pages 93-99, January.
    2. Dutta, Bhaskar, 1988. "Covering sets and a new condorcet choice correspondence," Journal of Economic Theory, Elsevier, vol. 44(1), pages 63-80, February.
    3. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    4. Felix Brandt & Felix Fischer & Paul Harrenstein & Maximilian Mair, 2010. "A computational analysis of the tournament equilibrium set," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(4), pages 597-609, April.
    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. 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.
    2. Felix Brandt & Markus Brill & Hans Georg Seedig & Warut Suksompong, 2020. "On the Structure of Stable Tournament Solutions," Papers 2004.01651, arXiv.org.
    3. 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.
    4. 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.
    5. Brandt, Felix & Harrenstein, Paul & Seedig, Hans Georg, 2017. "Minimal extending sets in tournaments," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 55-63.
    6. 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.

    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. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    3. Felix Brandt & Markus Brill & Hans Georg Seedig & Warut Suksompong, 2020. "On the Structure of Stable Tournament Solutions," Papers 2004.01651, arXiv.org.
    4. 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.
    5. Brandt, Felix & Harrenstein, Paul & Seedig, Hans Georg, 2017. "Minimal extending sets in tournaments," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 55-63.
    6. Berghammer, Rudolf & Rusinowska, Agnieszka & de Swart, Harrie, 2013. "Computing tournament solutions using relation algebra and RelView," European Journal of Operational Research, Elsevier, vol. 226(3), pages 636-645.
    7. Vicki Knoblauch, 2020. "Von Neumann–Morgenstern stable set rationalization of choice functions," Theory and Decision, Springer, vol. 89(3), pages 369-381, October.
    8. Josep E., Peris & Begoña, Subiza, 2015. "Rationalizable Choice and Standards of Behavior," QM&ET Working Papers 15-5, University of Alicante, D. Quantitative Methods and Economic Theory.
    9. Weibin Han & Adrian Deemen, 2019. "A refinement of the uncovered set in tournaments," Theory and Decision, Springer, vol. 86(1), pages 107-121, February.
    10. repec:hal:pseose:hal-00756696 is not listed on IDEAS
    11. repec:hal:wpaper:hal-00756696 is not listed on IDEAS
    12. Scott Moser, 2013. "A note on contestation-based tournament solutions," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 41(1), pages 133-143, June.
    13. Alex Scott & Mark Fey, 2012. "The minimal covering set in large tournaments," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 1-9, January.
    14. Fujun Hou, 2024. "A new social welfare function with a number of desirable properties," Papers 2403.16373, arXiv.org.
    15. 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.
    16. Josep E. Peris & Begoña Subiza, 2023. "Rational stability of choice functions," International Journal of Economic Theory, The International Society for Economic Theory, vol. 19(3), pages 580-598, September.
    17. Scott Moser, 2015. "Majority rule and tournament solutions," Chapters, in: Jac C. Heckelman & Nicholas R. Miller (ed.), Handbook of Social Choice and Voting, chapter 6, pages 83-101, Edward Elgar Publishing.
    18. Brandt, Felix & Harrenstein, Paul, 2011. "Set-rationalizable choice and self-stability," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1721-1731, July.
    19. De Donder, Philippe & Le Breton, Michel & Truchon, Michel, 2000. "Choosing from a weighted tournament1," Mathematical Social Sciences, Elsevier, vol. 40(1), pages 85-109, July.
    20. Thomas Demuynck, 2014. "The computational complexity of rationalizing Pareto optimal choice behavior," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(3), pages 529-549, March.
    21. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    22. Banks, Jeffrey S. & Duggan, John & Le Breton, Michel, 2002. "Bounds for Mixed Strategy Equilibria and the Spatial Model of Elections," Journal of Economic Theory, Elsevier, vol. 103(1), pages 88-105, March.

    More about this item

    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:sochwe:v:40:y:2013:i:3:p:739-743. 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.