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A refinement of the uncovered set in tournaments

  • Weibin Han
  • Adrian Van Deemen
Article

Abstract

We introduce a new solution for tournaments called the unsurpassed set. This solution lies between the uncovered set and the Copeland winner set. We show that this solution is more decisive than the uncovered set in discriminating among alternatives, and avoids a deficiency of the Copeland winner set. Moreover, the unsurpassed set is more sensitive than the uncovered set but less sensitive than the Copeland winner set to the reinforcement of the chosen alternatives. Besides, it turns out that this solution violates the other standard properties including independence of unchosen alternatives, stability, composition consistency and indempotency.

Keywords

Uncovered set Unsurpassed set Copeland winner set Monotonicity 

Notes

Acknowledgements

We would like to thank the anonymous referees and the editor in charge for their excellent comments and insightful remarks. The authors are also grateful to Dr. Jean Derks whose advices and comments improved the paper substantially. Preliminary results of this paper were presented at the 13th Meeting of the Society for Social Choice and Welfare (Lund, Sweden, June 2016). All mistakes are our responsibility. This research was supported by SCNU under the Project No. 508/8S0253.

References

  1. Banks, J. S. (1985). Sophisticated voting outcomes and agenda control. Social Choice and Welfare, 1(4), 295–306.CrossRefGoogle Scholar
  2. Bordes, G. (1979). Some more results on consistency, rationality and collective choice. In J. J. Laffont (Ed.), Aggregation and revelation of preferences, Chap 10, (pp. 175–197). North-Hollabd.Google Scholar
  3. Brandt, F. (2011). Minimal stable sets in tournaments. Journal of Economic Theory, 146(4), 1481–1499.CrossRefGoogle Scholar
  4. Brandt, F., Conitzer, V., Endriss, U., & Procaccia, A. D. (2016a). Handbook of computational social choice. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  5. Brandt, F., Geist, C., & Harrenstein, P. (2016b). A note on the mckelvey uncovered set and pareto optimality. Social Choice and Welfare, 46(1), 81–91.CrossRefGoogle Scholar
  6. Brandt, F., Harrenstein, P., & Seedig, H. G. (2017). Minimal extending sets in tournaments. Mathematical Social Sciences, 87, 55–63.CrossRefGoogle Scholar
  7. Chernoff, H. (1954). Rational selection of decision functions. Econometrica, 22(4), 422–443.CrossRefGoogle Scholar
  8. Copeland, A. H. (1951). A ‘reasonable’ social welfare function. Mimeographed, University of Michigan Seminar on Applications of Mathematics to the Social Sciences.Google Scholar
  9. Deb, R. (1977). On schwartz’s rule. Journal of Economic Theory, 16(1), 103–110.CrossRefGoogle Scholar
  10. Dutta, B. (1988). Covering sets and a new condorcet choice correspondence. Journal of Economic Theory, 44(1), 63–80.CrossRefGoogle Scholar
  11. Fishburn, P. C. (1977). Condorcet social choice functions. SIAM Journal on applied Mathematics, 33(3), 469–489.CrossRefGoogle Scholar
  12. Henriet, D. (1985). The copeland choice function an axiomatic characterization. Social Choice and Welfare, 2(1), 49–63.CrossRefGoogle Scholar
  13. Kalai, E., & Schmeidler, D. (1977). An admissible set occurring in various bargaining situations. Journal of Economic Theory, 14(2), 402–411.CrossRefGoogle Scholar
  14. Laslier, J. F. (1997). Tournament solutions and majority voting. Berlin: Springer.CrossRefGoogle Scholar
  15. Masatlioglu, Y., Nakajima, D., & Ozbay, E. Y. (2012). Revealed attention. American Economic Review, 102(5), 2183–2205.CrossRefGoogle Scholar
  16. Miller, N. R. (1980). A new solution set for tournaments and majority voting: Further graph-theoretical approaches to the theory of voting. American Journal of Political Science, 24(1), 68–96.CrossRefGoogle Scholar
  17. Moulin, H. (1986). Choosing from a tournament. Social Choice and Welfare, 3(4), 271–291.CrossRefGoogle Scholar
  18. Özkal-Sanver, I., & Sanver, M. R. (2010). A new monotonicity condition for tournament solutions. Theory and Decision, 69(3), 439–452.CrossRefGoogle Scholar
  19. Schwartz, T. (1972). Rationality and the myth of the maximum. Nous, 6(2), 97–117.CrossRefGoogle Scholar
  20. Schwartz, T. (1986). The logic of collective choice. New York: Columbia University Press.Google Scholar
  21. Schwartz, T. (1990). Cyclic tournaments and cooperative majority voting: A solution. Social Choice and Welfare, 7(1), 19–29.CrossRefGoogle Scholar
  22. Von Neumann, J., & Morgenstern, O. (1944). Theory of games and economic behavior. Princeton: Princeton University Prress.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Economics and ManagementSouth China Normal University, Guangzhou Higher Education Mega CenterGuangzhouPeople’s Republic of China
  2. 2.Institute for Management ResearchRadboud University NijmegenNijmegenThe Netherlands

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