Advertisement

Permutation-Based Randomised Tournament Solutions

  • Justin Kruger
  • Stéphane Airiau
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10767)

Abstract

Voting rules that are based on the majority graph typically output large sets of winners. In this full original paper our goal is to investigate a general method which leads to randomized version of such rules. We use the idea of parallel universes, where each universe is connected with a permutation over alternatives. The permutation allows us to construct resolute voting rules (i.e. rules that always choose unique winners). Such resolute rules can be constructed in a variety of ways: we consider using binary voting trees to select a single alternative. In turn this permits the construction of neutral rules that output the set the possible winners of every parallel universe. The question of which rules can be constructed in this way has already been partially studied under the heading of agenda implementability. We further propose a randomised version in which the probability of being the winner is the ratio of universes in which the alternative wins. We also briefly consider (typically novel) rules that elect the alternatives that have maximal winning probability. These rules typically output small sets of winners, thus provide refinements of known tournament solutions.

Keywords

Tournament Probabilistic rules Refinements Condorcet consistency 

Notes

Acknowledgement

Justin Kruger and Stéphane Airiau are supported by the ANR project CoCoRICo-CoDec.

Supplementary material

References

  1. 1.
    Altman, A., Kleinberg, R.: Nonmanipulable randomized tournament selections. In: AAAI (2010)Google Scholar
  2. 2.
    Aziz, H.: Maximal recursive rule: a new social decision scheme. In Proceedings of IJCAI 2013, pp. 34–40. AAAI Press (2013)Google Scholar
  3. 3.
    Aziz, H., Stursberg, P.: A generalization of probabilistic serial to randomized social choice. In: Proceedings of the Twenty-Eighth AAAI Conference on Artificial Intelligence, 27–31 July 2014, Québec City, Québec, Canada, pp. 559–565 (2014)Google Scholar
  4. 4.
    Banks, S.J.: Sophisticated voting outcomes and agenda control. Soc. Choice Welfare 1(4), 295–306 (1985)CrossRefGoogle Scholar
  5. 5.
    Barberà, S.: Majority and positional voting in a probabilistic framework. Rev. Econ. Stud. 46(2), 379–389 (1979)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bogomolnaia, A., Moulin, H., Stong, R.: Collective choice under dichotomous preferences. J. Econ. Theory 122(2), 165–184 (2005)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Brandt, F., Brill, M., Harrenstein, P.: Tournament solutions. In: Handbook of Computational Social Choice, chap. 3. Cambridge University Press, Cambridge (2016)Google Scholar
  8. 8.
    Brill, M., Fischer, F.: The price of neutrality for the ranked pairs method. In: Proceedings of AAAI-2012, pp. 1299–1305 (2012)Google Scholar
  9. 9.
    Conitzer, V., Rognlie, M., Xia, L.: Preference functions that score rankings and maximum likelihood estimation. In: Proceedings of IJCAI-2009, pp. 109–115 (2009)Google Scholar
  10. 10.
    Fischer, F., Procaccia, A.D., Samorodnitsky, A.: A new perspective on implementation by voting trees. In: Proceedings of the 10th ACM Conference on Electronic Commerce, pp. 31–40. ACM (2009)Google Scholar
  11. 11.
    Freeman, R., Brill, M., Conitzer, V.: General tiebreaking schemes for computational social choice. In: Proceedings of AAMAS-2015 (2015)Google Scholar
  12. 12.
    Gibbard, A.: Manipulation of schemes that mix voting with chance. Econometrica 45, 665–681 (1977)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Horan, S.: Implementation of majority voting rules. Preprint (2013)Google Scholar
  14. 14.
    Hudry, O.: A note on “Banks winners in tournaments are difficult to recognize” by G. J. Woeginger. Soc. Choice Welfare 23(1), 113–114 (2004)MathSciNetCrossRefGoogle Scholar
  15. 15.
    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, Gösing, Austria, 3–27 July 1962, pp. 73–79 (1965)Google Scholar
  16. 16.
    Lang, J., Pini, M.S., Rossi, F., Venable, K.B., Walsh, T.: Winner determination in sequential majority voting. In: IJCAI 2007, vol. 7, pp. 1372–1377 (2007)Google Scholar
  17. 17.
    Laslier, J.-F.: Tournament Solutions and Majority Voting. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  18. 18.
    Moon, J.W.: Topics on Tournaments in Graph Theory. Holt, Rinehart and Winston (1968)Google Scholar
  19. 19.
    Moulin, H.: Choosing from a tournament. Soc. Choice Welfare 3(4), 271–291 (1986)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Nicolaus, T.: Independence of clones as a criterion for voting rules. Soc. Choice Welfare 4(3), 185–206 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Woeginger, G.J.: Banks winners in tournaments are difficult to recognize. Soc. Choice Welfare 20(3), 523–528 (2003)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Université Paris-Dauphine, PSL Research University, CNRS, LAMSADEParisFrance

Personalised recommendations