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Minimal extending sets in tournaments

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  • Brandt, Felix
  • Harrenstein, Paul
  • Seedig, Hans Georg

Abstract

Tournament solutions play an important role within social choice theory and the mathematical social sciences at large. In 2011, Brandt proposed a new tournament solution called the minimal extending set (ME) and an associated graph-theoretic conjecture. If the conjecture had been true, ME would have satisfied a number of desirable properties that are usually considered in the literature on tournament solutions. However, in 2013, the existence of an enormous counter-example to the conjecture was shown using a non-constructive proof. This left open which of the properties are actually satisfied by ME. It turns out that ME satisfies idempotency, irregularity, and inclusion in the iterated Banks set (and hence the Banks set, the uncovered set, and the top cycle). Most of the other standard properties (including monotonicity, stability, and computational tractability) are violated, but have been shown to hold for all tournaments on up to 12 alternatives and all random tournaments encountered in computer experiments.

Suggested Citation

  • Brandt, Felix & Harrenstein, Paul & Seedig, Hans Georg, 2017. "Minimal extending sets in tournaments," Mathematical Social Sciences, Elsevier, vol. 87(C), pages 55-63.
    • Handle: RePEc:eee:matsoc:v:87:y:2017:i:c:p:55-63
    DOI: 10.1016/j.mathsocsci.2016.12.007
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    References listed on IDEAS

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    4. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    5. Olivier Hudry, 2004. "A note on “Banks winners in tournaments are difficult to recognize” by G. J. Woeginger," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 23(1), pages 113-114, August.
    6. 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.
    7. 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.
    8. 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.
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    10. Duggan, John & Le Breton, Michel, 1996. "Dutta's Minimal Covering Set and Shapley's Saddles," Journal of Economic Theory, Elsevier, vol. 70(1), pages 257-265, July.
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    13. 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.
    14. Brandt, Felix & Harrenstein, Paul, 2011. "Set-rationalizable choice and self-stability," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1721-1731, July.
    15. Felix Brandt & Paul Harrenstein, 2010. "Characterization of dominance relations in finite coalitional games," Theory and Decision, Springer, vol. 69(2), pages 233-256, August.
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    Cited by:

    1. 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.
    2. Weibin Han & Adrian Deemen, 2019. "A refinement of the uncovered set in tournaments," Theory and Decision, Springer, vol. 86(1), pages 107-121, February.

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