# On the structure of stable tournament solutions

- 205 Downloads
- 1 Citations

## 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, (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.

## Keywords

Choice consistency Tournament solutions Bipartisan set Tournament equilibrium set## JEL Classification

D7 C6## Notes

### Acknowledgements

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.

## References

- Aizerman, M., Aleskerov, F.: Theory of Choice. Studies in Mathematical and Managerial Economics, vol. 38. North-Holland, Amsterdam (1995)Google Scholar
- Allesina, S., Levine, J.M.: A competitive network theory of species diversity. PNAS
**108**(14), 5638–5642 (2011)CrossRefGoogle Scholar - 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
- Brandl, F., Brandt, F., Seedig, H.G.: Consistent probabilistic social choice. Econometrica
**84**(5), 1839–1880 (2016)CrossRefGoogle Scholar - Brandt, F.: Minimal stable sets in tournaments. J. Econ. Theory
**146**(4), 1481–1499 (2011)CrossRefGoogle Scholar - Brandt, F.: Set-monotonicity implies Kelly-strategyproofness. Soc. Choice Welf.
**45**(4), 793–804 (2015)CrossRefGoogle Scholar - Brandt, F., Harrenstein, P.: Set-rationalizable choice and self-stability. J. Econ. Theory
**146**(4), 1721–1731 (2011)CrossRefGoogle Scholar - 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
- 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 - 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 - Brandt, F., Brill, M., Fischer, F., Harrenstein, P.: Minimal retentive sets in tournaments. Soc. Choice Welf.
**42**(3), 551–574 (2014)CrossRefGoogle Scholar - 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 - 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
- Brandt, F., Harrenstein, P., Seedig, H.G.: Minimal extending sets in tournaments. Math. Soc. Sci. (2016b) (forthcoming)Google Scholar
- Chernoff, H.: Rational selection of decision functions. Econometrica
**22**(4), 422–443 (1954)CrossRefGoogle Scholar - Duddy, C., Houy, N., Lang, J., Piggins, A., Zwicker, W.S.: Social dichotomy functions. Working paper (2014)Google Scholar
- Dutta, B.: Covering sets and a new Condorcet choice correspondence. J. Econ. Theory
**44**(1), 63–80 (1988)CrossRefGoogle Scholar - Felsenthal, D.S., Machover, M.: After two centuries should Condorcet’s voting procedure be implemented? Behav. Sci.
**37**(4), 250–274 (1992)CrossRefGoogle Scholar - Fey, M.: Choosing from a large tournament. Soc. Choice Welf.
**31**(2), 301–309 (2008)CrossRefGoogle Scholar - Fishburn, P.C.: Probabilistic social choice based on simple voting comparisons. Rev. Econ. Stud.
**51**(4), 683–692 (1984)CrossRefGoogle Scholar - Fisher, D.C., Reeves, R.B.: Optimal strategies for random tournament games. Linear Algebra Appl.
**217**, 83–85 (1995)CrossRefGoogle Scholar - Fisher, D.C., Ryan, J.: Tournament games and positive tournaments. J. Graph Theory
**19**(2), 217–236 (1995)CrossRefGoogle Scholar - Houy, N.: Still more on the tournament equilibrium set. Soc. Choice Welf.
**32**, 93–99 (2009a)CrossRefGoogle Scholar - Houy, N.: A few new results on TEQ. Mimeo (2009b)Google Scholar
- Hudry, O.: A survey on the complexity of tournament solutions. Math. Soc. Sci.
**57**(3), 292–303 (2009)CrossRefGoogle Scholar - 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
- Laffond, G., Laslier, J.-F., Le Breton, M.: More on the tournament equilibrium set. Math. sci. hum.
**31**(123), 37–44 (1993a)Google Scholar - 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 - 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 - 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 - Laslier, J.-F.: Tournament Solutions and Majority Voting. Springer, Berlin (1997)CrossRefGoogle Scholar
- Masatlioglu, Y., Nakajima, D., Ozbay, E.Y.: Revealed attention. Am. Econ. Rev.
**102**(5), 2183–2205 (2012)CrossRefGoogle Scholar - McGarvey, D.C.: A theorem on the construction of voting paradoxes. Econometrica
**21**(4), 608–610 (1953)CrossRefGoogle Scholar - 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
- Monjardet, B.: Statement of precedence and a comment on IIA terminology. Games Econ. Behav.
**62**, 736–738 (2008)CrossRefGoogle Scholar - 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
- Saari, D.G., Barney, S.: Consequences of reversing preferences. Math. Intell.
**25**, 17–31 (2003)CrossRefGoogle Scholar - Schjelderup-Ebbe, T.: Beiträge zur Sozialpsychologie des Haushuhns. Z. Psychol.
**88**, 225–252 (1922)Google Scholar - Schwartz, T.: Cyclic tournaments and cooperative majority voting: a solution. Soc. Choice Welf.
**7**(1), 19–29 (1990)CrossRefGoogle Scholar - Scott, A., Fey, M.: The minimal covering set in large tournaments. Soc. Choice Welf.
**38**(1), 1–9 (2012)CrossRefGoogle Scholar - Sen, A.K.: Choice functions and revealed preference. Rev. Econ. Stud.
**38**(3), 307–317 (1971)CrossRefGoogle Scholar - Slater, P.: Inconsistencies in a schedule of paired comparisons. Biometrika
**48**(3–4), 303–312 (1961)CrossRefGoogle Scholar - Yang, Y.: A further step towards an understanding of the tournament equilibrium set. Technical report. arXiv:1611.03991. (2016)