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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
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    References listed on IDEAS

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    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.
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    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. Brandt, Felix & Harrenstein, Paul & Seedig, Hans Georg, 2017. "Minimal extending sets in tournaments," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 55-63.
    5. 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.
    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.

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