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Social Choice and Welfare

, Volume 51, Issue 2, pp 193–222 | Cite as

Extending tournament solutions

  • Felix Brandt
  • Markus Brill
  • Paul Harrenstein
Original Paper
  • 144 Downloads

Abstract

An important subclass of social choice functions, so-called majoritarian (or C1) functions, only take into account the pairwise majority relation between alternatives. In the absence of majority ties—e.g., when there is an odd number of agents with linear preferences—the majority relation is antisymmetric and complete and can thus conveniently be represented by a tournament. Tournaments have a rich mathematical theory and many formal results for majoritarian functions assume that the majority relation constitutes a tournament. Moreover, most majoritarian functions have only been defined for tournaments and allow for a variety of generalizations to unrestricted preference profiles, none of which can be seen as the unequivocal extension of the original function. In this paper, we argue that restricting attention to tournaments is justified by the existence of a conservative extension, which inherits most of the commonly considered properties from its underlying tournament solution.

Notes

Acknowledgements

This work was supported by the Deutsche Forschungsgemeinschaft under Grant BR 2312/7-2, by a Feodor Lynen research fellowship of the Alexander von Humboldt Foundation, and by the ERC under Advanced Grant 291528 (“RACE”) and Starting Grant 639945 (“ACCORD”). We thank Vincent Conitzer, Christian Geist, and Hans Georg Seedig for helpful discussions and the anonymous reviewers and an associate editor for valuable feedback. Preliminary results of this paper have been presented at the 13th International Symposium on Artificial Intelligence and Mathematics (Fort Lauderdale, January 2014) and at the 28th AAAI Conference on Artificial Intelligence (Québec City, July 2014).

References

  1. Aizerman M, Aleskerov F (1995) Theory of choice, Studies in mathematical and managerial economics, vol 38. North-Holland, AmsterdamGoogle Scholar
  2. Arrow KJ, Raynaud H (1986) Social choice and multicriterion decision-making. MIT Press, CambridgeGoogle Scholar
  3. Aziz H, Brill M, Fischer F, Harrenstein P, Lang J, Seedig HG (2015) Possible and necessary winners of partial tournaments. J Artif Intell Res 54:493–534Google Scholar
  4. Banks JS, Bordes GA (1988) Voting games, indifference, and consistent sequential choice rules. Soc Choice Welf 5:31–44CrossRefGoogle Scholar
  5. Bordes G (1979) Some more results on consistency, rationality and collective choice. In: Laffont JJ (ed) Aggregation and revelation of preferences, chapter 10. North-Holland, Amsterdam, pp 175–197Google Scholar
  6. Bouyssou D, Marchant T, Pirlot M, Tsoukiàs A, Vincke P (2006) Evaluation and decision models: stepping stones for the analyst. Springer, BerlinGoogle Scholar
  7. Brandt F(2009) Tournament solutions—extensions of maximality and their applications to decision-making. Habilitation Thesis, Faculty for Mathematics, Computer Science, and Statistics, University of MunichGoogle Scholar
  8. Brandt F (2015) Set-monotonicity implies Kelly-strategyproofness. Soc Choice Welf 45(4):793–804CrossRefGoogle Scholar
  9. Brandt F, Harrenstein P (2010) Characterization of dominance relations in finite coalitional games. Theory Decis 69(2):233–256CrossRefGoogle Scholar
  10. Brandt F, Harrenstein P (2011) Set-rationalizable choice and self-stability. J Econ Theory 146(4):1721–1731CrossRefGoogle Scholar
  11. Brandt F, Fischer F, Harrenstein P (2009) The computational complexity of choice sets. Math Log Q 55(4):444–459 (Special Issue on Computational Social Choice)CrossRefGoogle Scholar
  12. Brandt F, Brill M, Harrenstein P (2016) Tournament solutions. In: Brandt F, Conitzer V, Endriss U, Lang J, Procaccia AD (eds) Handbook of computational social choice, Chapter 3. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  13. Brandt F, Brill M, Seedig HG, Suksompong W (2018) On the structure of stable tournament solutions. Econ Theory (forthcoming).  https://doi.org/10.1007/s00199-016-1024-x Google Scholar
  14. Brill M, Fischer F (2012) The price of neutrality for the ranked pairs method. In: Proceedings of the 26th AAAI Conference on Artificial Intelligence (AAAI). AAAI Press, Palo Alto, pp 1299–1305Google Scholar
  15. Brill M, Freeman R, Conitzer V (2016) Computing possible and necessary equilibrium actions (and bipartisan set winners). In: Proceedings of the 30th AAAI Conference on Artificial Intelligence (AAAI). AAAI Press, Palo Alto, pp 418–424Google Scholar
  16. Chernoff H (1954) Rational selection of decision functions. Econometrica 22(4):422–443CrossRefGoogle Scholar
  17. Conitzer V, Rognlie M, Xia L (2009) Preference functions that score rankings and maximum likelihood estimation. In: Proceedings of the 21st International Joint Conference on Artificial Intelligence (IJCAI). AAAI Press, Palo Alto, pp 109–115Google Scholar
  18. Cook WJ, Cunningham WH, Pulleyblank WR, Schrijver A (1998) Combinatorial optimization. Wiley, New YorkGoogle Scholar
  19. Duggan J (2013) Uncovered sets. Soc Choice Welf 41(3):489–535CrossRefGoogle Scholar
  20. Duggan J, Le Breton M (1996) Dutta’s minimal covering set and Shapley’s saddles. J Econ Theory 70(1):257–265CrossRefGoogle Scholar
  21. Duggan J, Le Breton M (2001) Mixed refinements of Shapley’s saddles and weak tournaments. Soc Choice Welf 18(1):65–78CrossRefGoogle Scholar
  22. Dutta B, Laslier J-F (1999) Comparison functions and choice correspondences. Soc Choice Welf 16(4):513–532CrossRefGoogle Scholar
  23. Faliszewski P, Hemaspaandra E, Hemaspaandra L, Rothe J (2009) Llull and Copeland voting computationally resist bribery and constructive control. J Artif Intell Res 35:275–341Google Scholar
  24. Fisher DC, Ryan J (1995) Tournament games and positive tournaments. J Graph Theory 19(2):217–236CrossRefGoogle Scholar
  25. Freeman R, Brill M, Conitzer V (2015) General tiebreaking schemes for computational social choice. In: Proceedings of the 14th International Conference on Autonomous Agents and Multiagent Systems (AAMAS). IFAAMAS, pp 1401–1409Google Scholar
  26. Henriet D (1985) The Copeland choice function: an axiomatic characterization. Soc Choice Welf 2(1):49–63CrossRefGoogle Scholar
  27. Laffond G, Laslier J-F, Le Breton M (1993) The bipartisan set of a tournament game. Games Econ Behav 5(1):182–201CrossRefGoogle Scholar
  28. Lang J, Pini MS, Rossi F, Salvagnin D, Venable KB, Walsh T (2012) Winner determination in voting trees with incomplete preferences and weighted votes. J Auton Agents Multi Agent Syst 25(1):130–157CrossRefGoogle Scholar
  29. Laslier J-F (1997) Tournament solutions and majority voting. Springer, BerlinCrossRefGoogle Scholar
  30. Masatlioglu Y, Nakajima D, Ozbay EY (2012) Revealed attention. Am Econ Rev 102(5):2183–2205CrossRefGoogle Scholar
  31. May K (1952) A set of independent, necessary and sufficient conditions for simple majority decisions. Econometrica 20(4):680–684CrossRefGoogle Scholar
  32. Monjardet B (2008) Statement of precedence and a comment on IIA terminology. Games Econ Behav 62:736–738CrossRefGoogle Scholar
  33. Moulin H (1986) Choosing from a tournament. Soc Choice Welf 3(4):271–291CrossRefGoogle Scholar
  34. Peris JE, Subiza B (1999) Condorcet choice correspondences for weak tournaments. Soc Choice Welf 16(2):217–231CrossRefGoogle Scholar
  35. Schwartz T (1972) Rationality and the myth of the maximum. Noûs 6(2):97–117CrossRefGoogle Scholar
  36. Schwartz T (1986) The logic of collective choice. Columbia University Press, New YorkGoogle Scholar
  37. Schwartz T (1990) Cyclic tournaments and cooperative majority voting: a solution. Soc Choice Welf 7(1):19–29CrossRefGoogle Scholar
  38. Sen AK (1986) Social choice theory. In: Arrow KJ, Intriligator MD (eds) Handbook of mathematical economics, vol 3, Chapter 22. Elsevier, Amsterdam, pp 1073–1181CrossRefGoogle Scholar
  39. Woeginger GJ (2003) Banks winners in tournaments are difficult to recognize. Soc Choice Welf 20(3):523–528CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für InformatikTechnische Universität MünchenGarchingGermany
  2. 2.Institut für Softwaretechnik und Theoretische InformatikTechnische Universität BerlinBerlinGermany
  3. 3.Department of Computer ScienceUniversity of OxfordOxfordUK

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