Economic Theory

, Volume 65, Issue 2, pp 483–507 | Cite as

On the structure of stable tournament solutions

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


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, (1) we show that every stable choice function is generated by a unique simple choice function, which never excludes more than one alternative, (2) we completely characterize which simple choice functions give rise to stable choice functions, and (3) 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.


Choice consistency Tournament solutions Bipartisan set Tournament equilibrium set 

JEL Classification

D7 C6 



This material is based on work supported by Deutsche Forschungsgemeinschaft under Grants BR 2312/7-1 and BR 2312/7-2, by a Feodor Lynen Research Fellowship of the Alexander von Humboldt Foundation, by ERC Starting Grant 639945, by a Stanford Graduate Fellowship, and by the MIT-Germany program. The authors thank Christian Geist for insightful computer experiments and Paul Harrenstein for helpful discussions and preparing Fig. 1.


  1. Aizerman, M., Aleskerov, F.: Theory of Choice. Studies in Mathematical and Managerial Economics, vol. 38. North-Holland, Amsterdam (1995)Google Scholar
  2. Allesina, S., Levine, J.M.: A competitive network theory of species diversity. PNAS 108(14), 5638–5642 (2011)CrossRefGoogle Scholar
  3. Bordes, G.: Some more results on consistency, rationality and collective choice. In: Laffont, J.J. (ed.) Aggregation and Revelation of Preferences, Chapter 10, pp. 175–197. North-Holland, Amsterdam (1979)Google Scholar
  4. Brandl, F., Brandt, F., Seedig, H.G.: Consistent probabilistic social choice. Econometrica 84(5), 1839–1880 (2016)CrossRefGoogle Scholar
  5. Brandt, F.: Minimal stable sets in tournaments. J. Econ. Theory 146(4), 1481–1499 (2011)CrossRefGoogle Scholar
  6. Brandt, F.: Set-monotonicity implies Kelly-strategyproofness. Soc. Choice Welf. 45(4), 793–804 (2015)CrossRefGoogle Scholar
  7. Brandt, F., Harrenstein, P.: Set-rationalizable choice and self-stability. J. Econ. Theory 146(4), 1721–1731 (2011)CrossRefGoogle Scholar
  8. Brandt, F., Seedig, H.G.: On the discriminative power of tournament solutions. In: Selected Papers of the International Conference on Operations Research, OR2014, Operations Research Proceedings, pp. 53–58. Springer (2016)Google Scholar
  9. Brandt, F., Fischer, F., Harrenstein, P., Mair, M.: A computational analysis of the tournament equilibrium set. Soc. Choice Welf. 34(4), 597–609 (2010)CrossRefGoogle Scholar
  10. Brandt, F., Chudnovsky, M., Kim, I., Liu, G., Norin, S., Scott, A., Seymour, P., Thomassé, S.: A counterexample to a conjecture of Schwartz. Soc. Choice Welf. 40(3), 739–743 (2013)CrossRefGoogle Scholar
  11. Brandt, F., Brill, M., Fischer, F., Harrenstein, P.: Minimal retentive sets in tournaments. Soc. Choice Welf. 42(3), 551–574 (2014)CrossRefGoogle Scholar
  12. Brandt, F., Dau, A., Seedig, H.G.: Bounds on the disparity and separation of tournament solutions. Discrete Appl. Math. 187, 41–49 (2015)CrossRefGoogle Scholar
  13. Brandt, F., Brill, M., Harrenstein, P.: Tournament solutions. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.D. (eds.) Handbook of Computational Social Choice, chapter 3. Cambridge University Press, Cambridge (2016a)CrossRefGoogle Scholar
  14. Brandt, F., Harrenstein, P., Seedig, H.G.: Minimal extending sets in tournaments. Math. Soc. Sci. (2016b) (forthcoming)Google Scholar
  15. Chernoff, H.: Rational selection of decision functions. Econometrica 22(4), 422–443 (1954)CrossRefGoogle Scholar
  16. Duddy, C., Houy, N., Lang, J., Piggins, A., Zwicker, W.S.: Social dichotomy functions. Working paper (2014)Google Scholar
  17. Dutta, B.: Covering sets and a new Condorcet choice correspondence. J. Econ. Theory 44(1), 63–80 (1988)CrossRefGoogle Scholar
  18. Felsenthal, D.S., Machover, M.: After two centuries should Condorcet’s voting procedure be implemented? Behav. Sci. 37(4), 250–274 (1992)CrossRefGoogle Scholar
  19. Fey, M.: Choosing from a large tournament. Soc. Choice Welf. 31(2), 301–309 (2008)CrossRefGoogle Scholar
  20. Fishburn, P.C.: Probabilistic social choice based on simple voting comparisons. Rev. Econ. Stud. 51(4), 683–692 (1984)CrossRefGoogle Scholar
  21. Fisher, D.C., Reeves, R.B.: Optimal strategies for random tournament games. Linear Algebra Appl. 217, 83–85 (1995)CrossRefGoogle Scholar
  22. Fisher, D.C., Ryan, J.: Tournament games and positive tournaments. J. Graph Theory 19(2), 217–236 (1995)CrossRefGoogle Scholar
  23. Houy, N.: Still more on the tournament equilibrium set. Soc. Choice Welf. 32, 93–99 (2009a)CrossRefGoogle Scholar
  24. Houy, N.: A few new results on TEQ. Mimeo (2009b)Google Scholar
  25. Hudry, O.: A survey on the complexity of tournament solutions. Math. Soc. Sci. 57(3), 292–303 (2009)CrossRefGoogle Scholar
  26. Kreweras, G.: Aggregation of preference orderings. In: Mathematics and Social Sciences I: Proceedings of the seminars of Menthon-Saint-Bernard, France (1–27 July 1960) and of Gösing, Austria (3–27 July 1962), pp. 73–79 (1965)Google Scholar
  27. Laffond, G., Laslier, J.-F., Le Breton, M.: More on the tournament equilibrium set. Math. sci. hum. 31(123), 37–44 (1993a)Google Scholar
  28. Laffond, G., Laslier, J.-F., Le Breton, M.: The bipartisan set of a tournament game. Games Econ. Behav. 5(1), 182–201 (1993b)CrossRefGoogle Scholar
  29. Laffond, G., Laslier, J.-F., Le Breton, M.: The Copeland measure of Condorcet choice functions. Discrete Appl. Math. 55(3), 273–279 (1994)CrossRefGoogle Scholar
  30. Landau, H.G.: On dominance relations and the structure of animal societies: I. Effect of inherent characteristics. Bull. Math. Biophys. 13(1), 1–19 (1951)CrossRefGoogle Scholar
  31. Laslier, J.-F.: Tournament Solutions and Majority Voting. Springer, Berlin (1997)CrossRefGoogle Scholar
  32. Masatlioglu, Y., Nakajima, D., Ozbay, E.Y.: Revealed attention. Am. Econ. Rev. 102(5), 2183–2205 (2012)CrossRefGoogle Scholar
  33. McGarvey, D.C.: A theorem on the construction of voting paradoxes. Econometrica 21(4), 608–610 (1953)CrossRefGoogle Scholar
  34. Mnich, M., Shrestha, Y.R., Yang, Y.: When does Schwartz conjecture hold? In: Proceedings of the 24th International Joint Conference on Artificial Intelligence (IJCAI), pp. 603–609. AAAI Press (2015)Google Scholar
  35. Monjardet, B.: Statement of precedence and a comment on IIA terminology. Games Econ. Behav. 62, 736–738 (2008)CrossRefGoogle Scholar
  36. Moser, S.: Majority rule and tournament solutions. In: Heckelman, J.C., Miller, N.R. (eds.) Handbook of Social Choice and Voting, chapter 6, pp. 83–101. Edward Elgar, Cheltenham (2015)CrossRefGoogle Scholar
  37. Saari, D.G., Barney, S.: Consequences of reversing preferences. Math. Intell. 25, 17–31 (2003)CrossRefGoogle Scholar
  38. Schjelderup-Ebbe, T.: Beiträge zur Sozialpsychologie des Haushuhns. Z. Psychol. 88, 225–252 (1922)Google Scholar
  39. Schwartz, T.: Cyclic tournaments and cooperative majority voting: a solution. Soc. Choice Welf. 7(1), 19–29 (1990)CrossRefGoogle Scholar
  40. Scott, A., Fey, M.: The minimal covering set in large tournaments. Soc. Choice Welf. 38(1), 1–9 (2012)CrossRefGoogle Scholar
  41. Sen, A.K.: Choice functions and revealed preference. Rev. Econ. Stud. 38(3), 307–317 (1971)CrossRefGoogle Scholar
  42. Slater, P.: Inconsistencies in a schedule of paired comparisons. Biometrika 48(3–4), 303–312 (1961)CrossRefGoogle Scholar
  43. Yang, Y.: A further step towards an understanding of the tournament equilibrium set. Technical report. arXiv:1611.03991. (2016)

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Technical University of MunichMunichGermany
  2. 2.Oxford UniversityOxfordUK
  3. 3.Stanford UniversityStanfordUSA

Personalised recommendations