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On the Structure of Stable Tournament Solutions

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  • Felix Brandt
  • Markus Brill
  • Hans Georg Seedig
  • Warut Suksompong

Abstract

A fundamental property of choice functions is stability, which, loosely speaking, prescribes that choice sets are invariant under adding and removing unchosen alternatives. We provide several structural insights that improve our understanding of stable choice functions. In particular, (i) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (ii) we completely characterize which simple choice functions give rise to stable choice functions, and (iii) we prove a strong relationship between stability and a new property of tournament solutions called local reversal symmetry. Based on these findings, we provide the first concrete tournament---consisting of 24 alternatives---in which the tournament equilibrium set fails to be stable. Furthermore, we prove that there is no more discriminating stable tournament solution than the bipartisan set and that the bipartisan set is the unique most discriminating tournament solution which satisfies standard properties proposed in the literature.

Suggested Citation

  • Felix Brandt & Markus Brill & Hans Georg Seedig & Warut Suksompong, 2020. "On the Structure of Stable Tournament Solutions," Papers 2004.01651, arXiv.org.
    • Handle: RePEc:arx:papers:2004.01651
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    References listed on IDEAS

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    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 & 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.
    3. Yusufcan Masatlioglu & Daisuke Nakajima & Erkut Y. Ozbay, 2012. "Revealed Attention," American Economic Review, American Economic Association, vol. 102(5), pages 2183-2205, August.
    4. Mark Fey, 2008. "Choosing from a large tournament," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 31(2), pages 301-309, August.
    5. Jac C. Heckelman & Nicholas R. Miller (ed.), 2015. "Handbook of Social Choice and Voting," Books, Edward Elgar Publishing, number 15584.
    6. Hudry, Olivier, 2009. "A survey on the complexity of tournament solutions," Mathematical Social Sciences, Elsevier, vol. 57(3), pages 292-303, May.
    7. 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.
    8. Dutta, Bhaskar, 1988. "Covering sets and a new condorcet choice correspondence," Journal of Economic Theory, Elsevier, vol. 44(1), pages 63-80, February.
    9. 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.
    10. Monjardet, B., 2008. "Statement of precedence and a comment on IIA terminology," Games and Economic Behavior, Elsevier, vol. 62(2), pages 736-738, March.
    11. Brandt, Felix, 2011. "Minimal stable sets in tournaments," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1481-1499, July.
    12. 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.
    13. Florian Brandl & Felix Brandt & Hans Georg Seedig, 2016. "Consistent Probabilistic Social Choice," Econometrica, Econometric Society, vol. 84, pages 1839-1880, September.
    14. Amartya K. Sen, 1971. "Choice Functions and Revealed Preference," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 38(3), pages 307-317.
    15. 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.
    16. Brandt, Felix & Harrenstein, Paul, 2011. "Set-rationalizable choice and self-stability," Journal of Economic Theory, Elsevier, vol. 146(4), pages 1721-1731, July.
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