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Robust Bounds on Choosing from Large Tournaments

  • Christian Saile
  • Warut SuksompongEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

Abstract

Tournament solutions provide methods for selecting the “best” alternatives from a tournament and have found applications in a wide range of areas. Previous work has shown that several well-known tournament solutions almost never rule out any alternative in large random tournaments. Nevertheless, all analytical results thus far have assumed a rigid probabilistic model, in which either a tournament is chosen uniformly at random, or there is a linear order of alternatives and the orientation of all edges in the tournament is chosen with the same probabilities according to the linear order. In this work, we consider a significantly more general model where the orientation of different edges can be chosen with different probabilities. We show that a number of common tournament solutions, including the top cycle and the uncovered set, are still unlikely to rule out any alternative under this model. This corresponds to natural graph-theoretic conditions such as irreducibility of the tournament. In addition, we provide tight asymptotic bounds on the boundary of the probability range for which the tournament solutions select all alternatives with high probability.

Notes

Acknowledgments

This material is based upon work supported by the Deutsche Forschungsgemeinschaft under grant BR 2312/11-1 and by a Stanford Graduate Fellowship. The authors thank Felix Brandt, Pasin Manurangsi, and Fedor Petrov for helpful discussions and the anonymous reviewers for helpful comments.

References

  1. 1.
    Allesina, S., Levine, J.M.: A competitive network theory of species diversity. Proc. Natl. Acad. Sci. (PNAS) 108(14), 5638–5642 (2011)CrossRefGoogle Scholar
  2. 2.
    Arrow, K.J., Raynaud, H.: Social Choice and Multicriterion Decision-Making. MIT Press, Cambridge (1986)zbMATHGoogle Scholar
  3. 3.
    Bell, C.E.: A random voting graph almost surely has a Hamiltonian cycle when the number of alternatives is large. Econometrica 49(6), 1597–1603 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bouyssou, D.: Monotonicity of ‘ranking by choosing’: a progress report. Soc. Choice Welf. 23(2), 249–273 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    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. Cambridge University Press, Cambridge (2016). Chapter 3Google Scholar
  6. 6.
    Brandt, F., Brill, M., Seedig, H.G., Suksompong, W.: On the structure of stable tournament solutions. Econ. Theory 65(2), 483–507 (2018)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brandt, F., Seedig, H.G.: On the Discriminative power of tournament solutions. In: Lübbecke, M., Koster, A., Letmathe, P., Madlener, R., Peis, B., Walther, G. (eds.) Operations Research Proceedings 2014. ORP, pp. 53–58. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-28697-6_8CrossRefGoogle Scholar
  8. 8.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)zbMATHGoogle Scholar
  9. 9.
    Fey, M.: Choosing from a large tournament. Soc. Choice Welf. 31(2), 301–309 (2008)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Fisher, D.C., Ryan, J.: Tournament games and positive tournaments. J. Graph Theory 19(2), 217–236 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Frank, O.: Stochastic competition graphs. Rev. Int. Stat. Inst. 36(3), 319–326 (1968)CrossRefGoogle Scholar
  12. 12.
    Good, I.J.: A note on condorcet sets. Public Choice 10(1), 97–101 (1971)CrossRefGoogle Scholar
  13. 13.
    Hudry, O.: A survey on the complexity of tournament solutions. Math. Soc. Sci. 57(3), 292–303 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Kim, M.P., Suksompong, W., Vassilevska Williams, V.: Who can win a single-elimination tournament? SIAM J. Discret. Math. 31(3), 1751–1764 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Landau, H.G.: On dominance relations and the structure of animal societies: III. The condition for a score structure. Bull. Math. Biophys. 15(2), 143–148 (1953)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Laslier, J.-F.: Tournament Solutions and Majority Voting. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  17. 17.
    Laslier, J.-F.: In silico voting experiments. In: Laslier, J.-F., Sanver, M.R. (eds.) Handbook on Approval Voting. Studies in Choice and Welfare, pp. 311–335. Springer, Heidelberg (2010).  https://doi.org/10.1007/978-3-642-02839-7_13. Chapter 13CrossRefzbMATHGoogle Scholar
  18. 18.
    Łuczak, T., Ruciński, A., Gruszka, J.: On the evolution of a random tournament. Discret. Math. 148(1–3), 311–316 (1996)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Maurer, S.B.: The king chicken theorems. Math. Mag. 53, 67–80 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Miller, N.R.: Graph-theoretic approaches to the theory of voting. Am. J. Polit. Sci. 21(4), 769–803 (1977)CrossRefGoogle Scholar
  21. 21.
    Miller, N.R.: A new solution set for tournaments and majority voting: further graph-theoretical approaches to the theory of voting. Am. J. Polit. Sci. 24(1), 68–96 (1980)CrossRefGoogle Scholar
  22. 22.
    Moon, J.W.: Topics on Tournaments. Holt, Reinhard and Winston, New York (1968)zbMATHGoogle Scholar
  23. 23.
    Moon, J.W., Moser, L.: Almost all tournaments are irreducible. Can. Math. Bull. 5, 61–65 (1962)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Moulin, H.: Choosing from a tournament. Soc. Choice Welf. 3(4), 271–291 (1986)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Reid, K.B.: Every vertex a king. Discret. Math. 38(1), 93–98 (1982)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Saile, C., Suksompong, W.: Robust bounds on choosing from large tournaments. CoRR, abs/1804.02743 (2018)Google Scholar
  27. 27.
    Schjelderup-Ebbe, T.: Beiträge zur Sozialpsychologie des Haushuhns. Z. für Psychol. 88, 225–252 (1922)Google Scholar
  28. 28.
    Schwartz, T.: Rationality and the myth of the maximum. Noûs 6(2), 97–117 (1972)CrossRefGoogle Scholar
  29. 29.
    Scott, A., Fey, M.: The minimal covering set in large tournaments. Soc. Choice Welf. 38(1), 1–9 (2012)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Slater, P.: Inconsistencies in a schedule of paired comparisons. Biometrika 48(3–4), 303–312 (1961)CrossRefGoogle Scholar
  31. 31.
    Ushakov, I.A.: The problem of choosing the preferred element: an application to sport games. In: Machol, R.E., Ladany, S.P., Morrison, D.G. (eds.) Management Science in Sports, pp. 153–161. North-Holland, Amsterdam (1976)Google Scholar
  32. 32.
    Vassilevska Williams, V.: Fixing a tournament. In: Proceedings of the 24th AAAI Conference on Artificial Intelligence (AAAI), pp. 895–900. AAAI Press (2010)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of InformaticsTechnical University of MunichMunichGermany
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA

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