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The Dominance, Covering and Supercovering Relations in the Context of Multiple Partially Specified Reciprocal Relations

  • Raúl Pérez-FernándezEmail author
  • Irene Díaz
  • Susana Montes
  • Bernard De Baets
Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 1000)

Abstract

The problem of ranking different candidates or alternatives according to the preferences of different voters or experts is a common study subject in the fields of social choice theory and preference modelling. Whereas the former field normally restricts its attention to preferences given in the form of rankings (with ties), the latter field embraces the use of (partially specified) reciprocal relations. In this contribution, we study the notions of dominance relation, covering relation and supercovering relation, which are widely studied in the setting in which we are dealing with rankings (with ties), and adapt them to the setting in which we are dealing with partially specified reciprocal relations by using a tool similar to stochastic dominance.

Keywords

Reciprocal relation Dominance Covering Supercovering 

Notes

Acknowledgements

This research has been partially supported by Spanish MINECO (TIN2017-87600-P). Raúl Pérez-Fernández acknowledges the support of the Research Foundation of Flanders (FWO17/PDO/160).

References

  1. 1.
    Arrow, K.J.: A difficulty in the concept of social welfare. J. Polit. Econ. 58(4), 328–346 (1950)CrossRefGoogle Scholar
  2. 2.
    Arrow, K.J.: Social Choice and Individual Values, 2nd edn. Yale University Press, New Haven (1963)zbMATHGoogle Scholar
  3. 3.
    Balinski, M., Laraki, R.: A theory of measuring, electing and ranking. PNAS 104(21), 8720–8725 (2007)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Balinski, M., Laraki, R.: Majority Judgment: Measuring, Ranking, and Electing. MIT Press, Cambridge (2010)zbMATHGoogle Scholar
  5. 5.
    Barthelemy, J.P., Monjardet, B.: The median procedure in cluster analysis and social choice theory. Math. Soc. Sci. 1, 235–267 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Borda, J.C.: Mémoire sur les Élections au Scrutin. Histoire de l’Académie Royale des Sciences, Paris (1781)Google Scholar
  7. 7.
    Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations. Fuzzy Sets Syst. 97(1), 33–48 (1998)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating multiplicative preference relations in a multipurpose decision-making model based on fuzzy preference relations. Fuzzy Sets Syst. 122, 277–291 (2001)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Condorcet, M.: Essai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix. De l’Imprimerie Royale, Paris (1785)Google Scholar
  10. 10.
    De Baets, B., De Meyer, H., De Schuymer, B., Jenei, S.: Cyclic evaluation of transitivity of reciprocal relations. Soc. Choice Welfare 26(2), 217–238 (2006)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Fishburn, P.C.: Conditions for simple majority decision functions with intransitive individual indifference. J. Econ. Theory 2, 354–367 (1970)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fishburn, P.C.: Paradoxes of voting. Am. Polit. Sci. Rev. 68(2), 537–546 (1974)CrossRefGoogle Scholar
  13. 13.
    Fodor, J.C., Roubens, M.R.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht (1994)CrossRefGoogle Scholar
  14. 14.
    García-Lapresta, J.L.: A general class of simple majority decision rules based on linguistic opinions. Inf. Sci. 176, 352–365 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    García-Lapresta, J.L., Martínez-Panero, M.: Borda counts versus approval voting: a fuzzy approach. Public Choice 112, 167–184 (2002)CrossRefGoogle Scholar
  16. 16.
    García-Lapresta, J.L., Martínez-Panero, M., Meneses, L.C.: Defining the Borda count in a linguistic decision making context. Inf. Sci. 179, 2309–2316 (2009)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Inada, K.: The simple majority decision rule. Econometrica 37(3), 490–506 (1969)CrossRefGoogle Scholar
  18. 18.
    Kacprzyk, J., Nurmi, H., Fedrizzi, M.: Consensus Under Fuzziness, vol. 10. Kluwer Academic, Boston (1996)zbMATHGoogle Scholar
  19. 19.
    Kemeny, J.G.: Mathematics without numbers. Daedalus 88(4), 577–591 (1959)Google Scholar
  20. 20.
    Levy, H.: Stochastic Dominance: Investment Decision Making under Uncertainty, 3rd edn. Springer, Berlin (2016)CrossRefGoogle Scholar
  21. 21.
    May, K.O.: A set of independent necessary and sufficient conditions for simple majority decision. Econometrica 20, 680–684 (1952)MathSciNetCrossRefGoogle Scholar
  22. 22.
    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
  23. 23.
    Nash, J.F.: The bargaining problem. Econometrica 28, 155–162 (1950)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Nguyen, H.T., Kosheleva, O., Kreinovich, V.: Decision making beyond arrow’s “impossibility theorem," with the analysis of effects of collusion and mutual attraction. Int. J. Intell. Syst. 24, 27–47 (2008)CrossRefGoogle Scholar
  25. 25.
    Pérez-Fernández, R., De Baets, B.: The supercovering relation, the pairwise winner, and more missing links between Borda and Condorcet. Soc. Choice and Welfare 50, 329–352 (2018)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Pérez-Fernández, R., De Baets, B.: The superdominance relation, the positional winner, and more missing links between Borda and Condorcet. J. Theor. Polit. 31(1), 46–65 (2019)CrossRefGoogle Scholar
  27. 27.
    Pérez-Fernández, R., Rademaker, M., Alonso, P., Díaz, I., Montes, S., De Baets, B.: Monotonicity-based ranking on the basis of multiple partially specified reciprocal relations. Fuzzy Sets Syst. 325, 69–96 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Roubens, M.: Choice procedures in fuzzy multicriteria decision analysis based on pairwise comparisons. Fuzzy Sets Syst. 84, 135–142 (1996)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Roubens, M., Vincke, P.: Preference Modelling. Springer, Berlin (1985)CrossRefGoogle Scholar
  30. 30.
    Schulze, M.: A new monotonic, clone-independent, reversal symmetric, and Condorcet-consistent single-winner election method. Soc. Choice Welfare 36, 267–303 (2011)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sen, A.K.: A possibility theorem on majority decisions. Econometrica 34(2), 491–499 (1966)CrossRefGoogle Scholar
  32. 32.
    Sen, A.K.: Collective Choice and Social Welfare. Holden-Day, San Francisco (1970)zbMATHGoogle Scholar
  33. 33.
    Tideman, T.N.: Independence of clones as a criterion for voting rules. Soc. Choice Welfare 4(3), 185–206 (1987)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Young, H.P.: Condorcet’s theory of voting. Am. Polit. Sci. Rev. 82(4), 1231–1244 (1988)CrossRefGoogle Scholar
  35. 35.
    Zavist, T.M., Tideman, T.N.: Complete independence of clones in the ranked pairs rule. Soc. Choice Welfare 6(2), 167–173 (1989)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Raúl Pérez-Fernández
    • 1
    • 2
    Email author
  • Irene Díaz
    • 3
  • Susana Montes
    • 1
  • Bernard De Baets
    • 2
  1. 1.Department of Statistics and O.R. and Mathematics DidacticsUniversity of OviedoOviedoSpain
  2. 2.KERMIT, Department of Data Analysis and Mathematical ModellingGhent UniversityGhentBelgium
  3. 3.Department of InformaticsUniversity of OviedoOviedoSpain

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