Social Choice and Welfare

, Volume 34, Issue 4, pp 597–609 | Cite as

A computational analysis of the tournament equilibrium set

  • Felix BrandtEmail author
  • Felix Fischer
  • Paul Harrenstein
  • Maximilian Mair
original paper


A recurring theme in the mathematical social sciences is how to select the “most desirable” elements given a binary dominance relation on a set of alternatives. Schwartz’s tournament equilibrium set (TEQ) ranks among the most intriguing, but also among the most enigmatic, tournament solutions proposed so far. Due to its unwieldy recursive definition, little is known about TEQ. In particular, its monotonicity remains an open problem to date. Yet, if TEQ were to satisfy monotonicity, it would be a very attractive solution concept refining both the Banks set and Dutta’s minimal covering set. We show that the problem of deciding whether a given alternative is contained in TEQ is NP-hard, and thus does not admit a polynomial-time algorithm unless P equals NP. Furthermore, we propose a heuristic that significantly outperforms the naive algorithm for computing TEQ.


Dominance Relation Maximal Element Solution Concept Condorcet Winner Naive Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Alon N (2006) Ranking tournaments. SIAM J Discret Math 20(1): 137–142CrossRefGoogle Scholar
  2. Banks JS (1985) Sophisticated voting outcomes and agenda control. Soc Choice Welf 3: 295–306CrossRefGoogle Scholar
  3. Bouyssou D, Marchant T, Pirlot M, Tsoukiàs A, Vincke P (2006) Evaluation and decision models: stepping stones for the analyst. Springer, BerlinGoogle Scholar
  4. Brandt F, Fischer F (2008) Computing the minimal covering set. Math Soc Sci 56(2): 254–268CrossRefGoogle Scholar
  5. Brandt F, Harrenstein P (2009) Characterization of dominance relations in finite coalitional games. Theory Decis (to appear)Google Scholar
  6. Brandt F, Fischer F, Harrenstein P (2009) The computational complexity of choice sets. Math Log Q 55(4): 444–459CrossRefGoogle Scholar
  7. Conitzer V (2006) Computing Slater rankings using similarities among candidates. In: Proceedings of the 21st national conference on artificial intelligence (AAAI). AAAI Press, pp 613–619Google Scholar
  8. Condorcet Marquis de Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Imprimerie Royale, 1785. Facsimile published in 1972. Chelsea, New YorkGoogle Scholar
  9. Dung PM (1995) On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games. Artif Intell 77: 321–357CrossRefGoogle Scholar
  10. Dunne PE (2007) Computational properties of argumentation systems satisfying graph-theoretic constraints. Artif Intell 171(10-15): 701–729CrossRefGoogle Scholar
  11. Dutta B (1990) On the tournament equilibrium set. Soc Choice Welf 7(4): 381–383CrossRefGoogle Scholar
  12. Dutta B, Laslier J-F (1999) Comparison functions and choice correspondences. Soc Choice Welf 16(4): 513–532CrossRefGoogle Scholar
  13. Fishburn PC (1977) Condorcet social choice functions. SIAM J Appl Math 33(3): 469–489CrossRefGoogle Scholar
  14. Fisher DC, Ryan J (1995) Tournament games and positive tournaments. J Graph Theory 19(2): 217–236CrossRefGoogle Scholar
  15. Gillies DB (1959) Solutions to general non-zero-sum games. In: Tucker AW, Luce RD (eds) Contributions to the Theory of Games IV, vol 40. Annals of Mathematics Studies. Princeton University Press, New Jersy, pp 47–85Google Scholar
  16. Houy N (2009) Still more on the tournament equilibrium set. Soc Choice Welf 32: 93–99CrossRefGoogle Scholar
  17. Laffond G, Laslier J-F, Le Breton M (1993) More on the tournament equilibrium set. Math Sci Hum 31(123): 37–44Google Scholar
  18. Laffond G, Laslier J-F, Le Breton M (1993) The bipartisan set of a tournament game. Games Econ Behav 5: 182–201CrossRefGoogle Scholar
  19. Laslier J-F (1997) Tournament solutions and majority voting. Springer, BerlinGoogle Scholar
  20. Papadimitriou CH (1994) Computational complexity. Addison-Wesley, ReadingGoogle Scholar
  21. Schwartz T (1990) Cyclic tournaments and cooperative majority voting: A solution. Soc Choice Welf 7: 19–29CrossRefGoogle Scholar
  22. Stearns R (1959) The voting problem. Am Math Mon 66(9): 761–763CrossRefGoogle Scholar
  23. Tarjan R (1972) Depth-first search and linear graph algorithms. SIAM J Comput 1(2): 146–160CrossRefGoogle Scholar
  24. Woeginger GJ (2003) Banks winners in tournaments are difficult to recognize. Soc Choice Welf 20: 523–528CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Felix Brandt
    • 1
    Email author
  • Felix Fischer
    • 1
  • Paul Harrenstein
    • 1
  • Maximilian Mair
    • 2
  1. 1.Institut für InformatikLudwig-Maximilians-Universität MünchenMunichGermany
  2. 2.Mathematisches InstitutLudwig-Maximilians-Universität MünchenMunichGermany

Personalised recommendations