Social Choice and Welfare

, Volume 42, Issue 3, pp 551–574 | Cite as

Minimal retentive sets in tournaments

  • Felix Brandt
  • Markus Brill
  • Felix Fischer
  • Paul Harrenstein
Original Paper


Tournament solutions, i.e., functions that associate with each complete and asymmetric relation on a set of alternatives a nonempty subset of the alternatives, play an important role in the mathematical social sciences at large. For any given tournament solution \(S\), there is another tournament solution  Open image in new window which returns the union of all inclusion-minimal sets that satisfy \(S\)-retentiveness, a natural stability criterion with respect to \(S\). Schwartz’s tournament equilibrium set (\({ TEQ }\)) is defined recursively as Open image in new window . In this article, we study under which circumstances a number of important and desirable properties are inherited from \(S\) to  Open image in new window . We thus obtain a hierarchy of attractive and efficiently computable tournament solutions that “approximate” \({ TEQ }\), which itself is computationally intractable. We further prove a weaker version of a recently disproved conjecture surrounding \({ TEQ }\), which establishes Open image in new window —a refinement of the top cycle—as an interesting new tournament solution.



This material is based on work supported by the Deutsche Forschungsgemeinschaft under grants BR 2312/3-3, BR 2312/6-1, BR 2312/7-1, and FI 1664/1-1. Preliminary versions of the results were presented at the Workshop on Algorithmic Aspects of Game Theory and Social Choice (Auckland, February 2010), the Dagstuhl Seminar on Computational Foundations of Social Choice (Dagstuhl, March 2010), the Doctoral School on Computational Social Choice (Estoril, April 2010), and the 9th International Joint Conference on Autonomous Agents and Multi-Agent Systems (Toronto, May 2010).


  1. Alon N (2006) Ranking tournaments. SIAM Journal on Discrete Mathematics 20(1):137–142CrossRefGoogle Scholar
  2. Arrow KJ, Raynaud H (1986) Social Choice and Multicriterion Decision-Making. MIT PressGoogle Scholar
  3. Basu K, Weibull J (1991) Strategy subsets closed under rational behavior. Economics Letters 36:141–146CrossRefGoogle Scholar
  4. Bouyssou D, Marchant T, Pirlot M, Tsoukiàs A, Vincke P (2006) Evaluation and Decision Models: Stepping Stones for the Analyst. Springer-VerlagGoogle Scholar
  5. Brandt F (2011) Minimal stable sets in tournaments. Journal of Economic Theory 146(4):1481–1499CrossRefGoogle Scholar
  6. Brandt F, Fischer F (2008) Computing the minimal covering set. Mathematical Social Sciences 56(2):254–268CrossRefGoogle Scholar
  7. Brandt F, Harrenstein P (2010) Characterization of dominance relations in finite coalitional games. Theory and Decision 69(2):233–256CrossRefGoogle Scholar
  8. Brandt F, Harrenstein P (2011) Set-rationalizable choice and self-stability. Journal of Economic Theory 146(4):1721–1731CrossRefGoogle Scholar
  9. Brandt F, Fischer F, Harrenstein P (2009) The computational complexity of choice sets. Mathematical Logic Quarterly 55(4):444–459CrossRefGoogle Scholar
  10. Brandt F, Fischer F, Harrenstein P, Mair M (2010) A computational analysis of the tournament equilibrium set. Social Choice and Welfare 34(4):597–609CrossRefGoogle Scholar
  11. Brandt F, Chudnovsky M, Kim I, Liu G, Norin S, Scott A, Seymour P, Thomassé S (2013) A counterexample to a conjecture of Schwartz. Social Choice and Welfare 40:739–743CrossRefGoogle Scholar
  12. Conitzer V (2006) Computing Slater rankings using similarities among candidates. In: Proceedings of the 21st National Conference on Artificial Intelligence (AAAI), pages 613–619. AAAI PressGoogle Scholar
  13. Duggan J, Le Breton M (1996) Dutta’s minimal covering set and Shapley’s saddles. Journal of Economic Theory 70:257–265CrossRefGoogle Scholar
  14. Dung PM (1995) On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artificial Intelligence 77:321–357CrossRefGoogle Scholar
  15. Dunne PE (2007) Computational properties of argumentation systems satisfying graph-theoretic constraints. Artificial Intelligence 171(10–15):701–729CrossRefGoogle Scholar
  16. Dutta B (1990) On the tournament equilibrium set. Social Choice and Welfare 7(4):381–383CrossRefGoogle Scholar
  17. Fisher DC, Ryan J (1995) Tournament games and positive tournaments. Journal of Graph Theory 19(2):217–236CrossRefGoogle Scholar
  18. Good IJ (1971) A note on Condorcet sets. Public Choice 10:97–101CrossRefGoogle Scholar
  19. Houy N (2009a) Still more on the tournament equilibrium set. Social Choice and Welfare 32:93–99CrossRefGoogle Scholar
  20. Houy N (2009b) A few new results on TEQ. MimeoGoogle Scholar
  21. Laffond G, Laslier J-F, Le Breton M (1993a) More on the tournament equilibrium set. Mathématiques et sciences humaines 31(123):37–44Google Scholar
  22. Laffond G, Laslier J-F, Le Breton M (1993b) The bipartisan set of a tournament game. Games and Economic Behavior 5:182–201CrossRefGoogle Scholar
  23. Laffond G, Lainé J, Laslier J-F (1996) Composition-consistent tournament solutions and social choice functions. Social Choice and Welfare 13:75–93CrossRefGoogle Scholar
  24. Laslier J-F (1997) Tournament Solutions and Majority Voting. Springer-VerlagGoogle Scholar
  25. Moulin H (1986) Choosing from a tournament. Social Choice and Welfare 3:271–291CrossRefGoogle Scholar
  26. Schwartz T (1990) Cyclic tournaments and cooperative majority voting: A solution. Social Choice and Welfare 7:19–29CrossRefGoogle Scholar
  27. Woeginger GJ (2003) Banks winners in tournaments are difficult to recognize. Social Choice and Welfare 20:523–528CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Felix Brandt
    • 1
  • Markus Brill
    • 1
  • Felix Fischer
    • 2
  • Paul Harrenstein
    • 3
  1. 1.Institut für InformatikTechnische Universität MünchenGarchingGermany
  2. 2.Statistical LaboratoryUniversity of CambridgeCambridgeUK
  3. 3.Department of Computer ScienceUniversity of OxfordOxfordUK

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