On Some Voting Paradoxes: A Fuzzy Preference and a Fuzzy Majority Perspective

  • Janusz Kacprzyk
  • Sławomir Zadrożny
  • Hannu Nurmi
  • Mario Fedrizzi
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 305)


Group decision making, as meant in this paper, is the following choice problem which proceeds in a multiperson setting. There is a group of individuals (decision makers, experts, …) who provide their testimonies concerning an issue in question as individual preference relations over some set of option (alternatives, variants, …). The problem is to find a solution, i.e., an alternative or a set of alternatives which best reflects the preferences of the group of individuals as a whole. First, we survey main developments in group decision making under fuzziness and outline some basic inconsistencies and negative results in group decision making and social choice, and show how they can be alleviated by some plausible modifications of underlying assumptions, mainly by introducing fuzzy preference relations and a fuzzy majority. We concentrate on how to derive solutions under individual fuzzy preference relations, and a fuzzy majority equated with a fuzzy linguistic quantifier (e.g., most, almost all, …), and discuss a related issue of how to define a “soft” degree of consensus in the group. Finally, we show how fuzzy preferences can help alleviate some known voting paradoxes.


fuzzy logic linguistic quantifier fuzzy preference relation fuzzy majority group decision making social choice consensus 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arrow, K.J.: Social Choice and Individual Values, 2nd edn. Wiley, New York (1963)Google Scholar
  2. 2.
    Black, D.: Theory of Committees and Elections. Cambridge University Press, Cambridge (1958)MATHGoogle Scholar
  3. 3.
    Bordogna, G., Fedrizzi, M., Pasi, G.: A linguistic modelling of consensus in group decision making based on OWA operators. IEEE Trans. on Systems, Man and Cybernetics SMC-27, 126–132 (1997)Google Scholar
  4. 4.
    Chiclana, F., Herrera, F., Herrera-Viedma, E.: Integrating multiplicative preference relations in a multipurpose decision making model based on fuzzy preference relations. Fuzzy Sets and Systems 122, 277–291 (2001)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    DeGrazia, A.: Mathematical Derivation of an Election System. Isis 44, 42–51 (1953)CrossRefGoogle Scholar
  6. 6.
    Fedrizzi, M., Kacprzyk, J., Nurmi, H.: Consensus degrees under fuzzy majorities and fuzzy preferences using OWA (ordered weighted average) operators. Control and Cybernetics 22, 71–80 (1993)MathSciNetGoogle Scholar
  7. 7.
    Fedrizzi, M., Kacprzyk, J., Nurmi, H.: How different are social choice functions: a rough sets approach. Quality and Quantity 30, 87–99 (1996)Google Scholar
  8. 8.
    Fedrizzi, M., Kacprzyk, J., Zadrożny, S.: An interactive multi-user decision support system for consensus reaching processes using fuzzy logic with linguistic quantifiers. Decision Support Systems 4, 313–327 (1988)CrossRefGoogle Scholar
  9. 9.
    Fodor, J., Roubens, M.: Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer, Dordrecht (1994)MATHCrossRefGoogle Scholar
  10. 10.
    García-Lapresta, J.L., Llamazares, B.: Aggregation of fuzzy preferences: Some rules of the mean. Social Choice and Welfare 17, 673–690 (2000)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Herrera, F., Herrera-Viedma, E., Verdegay, J.L.: A rational consensus model in group decision making using linguistic assessments. Fuzzy Sets and Systems 88, 31–49 (1997a)CrossRefGoogle Scholar
  12. 12.
    Herrera, F., Martínez, L.: An approach for combining numerical and linguistic information based on the 2-tuple fuzzy linguistic representation model in decision making. International J. of Uncertainty, Fuzziness and Knowledge-Based Systems 8, 539–562 (2000)MATHCrossRefGoogle Scholar
  13. 13.
    Intrilligator, M.D.: A probabilistic model of social choice. Review of Economic Studies 40, 553–560 (1973)CrossRefGoogle Scholar
  14. 14.
    Intrilligator, M.D.: Probabilistic models of choice. Mathematical Social Sciences 2, 157–166 (1982)CrossRefGoogle Scholar
  15. 15.
    Kacprzyk, J.: Group decision-making with a fuzzy majority via linguistic quantifiers. Part I: A consensory-like pooling; Part II: A competitive-like pooling. Cybernetics and Systems: an International J 16, 119–129 (Part I), 131–144 (Part II) (1985)Google Scholar
  16. 16.
    Kacprzyk, J.: Group decision making with a fuzzy linguistic majority. Fuzzy Sets and Systems 18, 105–118 (1986)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kacprzyk, J., Fedrizzi, M.: “Soft” consensus measures for monitoring real consensus reaching processes under fuzzy preferences. Control and Cybernetics 15, 309–323 (1986)MathSciNetGoogle Scholar
  18. 18.
    Kacprzyk, J., Fedrizzi, M.: A “soft” measure of consensus in the setting of partial (fuzzy) preferences. European Journal of Operational Research 34, 315–325 (1988)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kacprzyk, J., Fedrizzi, M.: A ’human-consistent‘ degree of consensus based on fuzzy logic with linguistic quantifiers. Mathematical Social Sciences 18, 275–290 (1989)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Kacprzyk, J., Fedrizzi, M. (eds.): Multiperson Decision Making Models Using Fuzzy Sets and Possibility Theory. Kluwer, Dordrecht (1990)MATHGoogle Scholar
  21. 21.
    Kacprzyk, J., Fedrizzi, M., Nurmi, H.: Group decision making and consensus under fuzzy preferences and fuzzy majority. Fuzzy Sets and Systems 49, 21–31 (1992)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kacprzyk, J., Nurmi, H.: Group decision making under fuzziness, in R. Słowiński (Ed.). In: Fuzzy Sets in Decision Analysis, Operations Research and Statistics, pp. 103–136. Kluwer, Boston (1998)CrossRefGoogle Scholar
  23. 23.
    Kacprzyk, J., Nurmi, H., Fedrizzi, M. (eds.): Consensus under Fuzziness. Kluwer, Boston (1996)Google Scholar
  24. 24.
    Kacprzyk, J., Nurmi, H., Fedrizzi, M.: Group decision making and a measure of consensus under fuzzy preferences and a fuzzy linguistic majority. In: Zadeh, L.A., Kacprzyk, J. (eds.) Computing with Words in Information/Intelligent Systems. Part 2. Foundations, pp. 233–243. Physica–Verlag, Springer, Heidelberg, New York (1999)Google Scholar
  25. 25.
    Kacprzyk, J., Zadrożny, S.: Collective choice rules in group decision making under fuzzy preferences and fuzzy majority: a unified OWA operator based approach. Control and Cybernetics 31, 937–948 (2002)MATHGoogle Scholar
  26. 26.
    Kacprzyk, J., Zadrożny, S.: Dealing with imprecise knowledge on preferences and majority in group decision making: towards a unified characterization of individual and collective choice functions. Bulletin of the Polish Academy of Sciences. Tech. Sci. 3, 286–302 (2003)Google Scholar
  27. 27.
    Kacprzyk, J., Zadrony, S., Fedrizzi, M., Nurmi, H.: On group decision making, consensus reaching, voting and voting paradoxes under fuzzy preferences and a fuzzy majority: a survey and a granulation perspective. In: Pedrycz, W., Skowron, A., Kreinovich, V. (eds.) Handbook of Granular Computing, pp. 906–929. Wiley, Chichester (2008a)Google Scholar
  28. 28.
    Kacprzyk, J., Zadrony, S., Fedrizzi, M., Nurmi, H.: On group decision making, consensus reaching, voting and voting paradoxes under fuzzy preferences and a fuzzy majority: a survey and some perspectives. In: Bustince, H., Herrera, F., Montero, J. (eds.) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models, pp. 263–295. Springer, Heidelberg (2008b)CrossRefGoogle Scholar
  29. 29.
    Kacprzyk, J., Zadrony, S., Nurmi, H., Fedrizzi, M.: Fuzzy preferences as a convenient tool in group decision making and a remedy for voting paradoxes. In: Seising, R. (ed.) Views on Fuzzy Sets and Systems from Different Perspectives: Philosophy and Logic, Criticisms and Applications, pp. 345–360. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  30. 30.
    Kelly, J.S.: Arrow Impossibility Theorems. Academic Press, New York (1978)MATHGoogle Scholar
  31. 31.
    Kelly, J.S.: Social Choice Theory: An Introduction. Academic Press, New York (1988)MATHCrossRefGoogle Scholar
  32. 32.
    Lagerspetz, E.: Paradoxes and representation. Electoral Studies 15, 83–92 (1995)CrossRefGoogle Scholar
  33. 33.
    Loewer, B., Laddaga, R.: Destroying the consensus. Special Issue on Consensus, Synthese 62(1), 79–96 (1985)Google Scholar
  34. 34.
    Montero, J.: Arrow‘s theorem under fuzzy rationality. Behavioral Science 32, 267–273 (1987)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Montero, J., Tejada, J., Cutello, V.: A general model for deriving preference structures from data. European Journal of Operational Research 98, 98–110 (1997)MATHCrossRefGoogle Scholar
  36. 36.
    Nurmi, H.: Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets and Systems 6, 249–259 (1981)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Nurmi, H.: Imprecise notions in individual and group decision theory: resolution of Allais paradox and related problems. Stochastica VI, 283–303 (1982)Google Scholar
  38. 38.
    Nurmi, H.: Voting procedures: a summary analysis. British Journal of Political Science 13, 181–208 (1983)CrossRefGoogle Scholar
  39. 39.
    Nurmi, H.: Probabilistic voting. Political Methodology 10, 81–95 (1984)Google Scholar
  40. 40.
    Nurmi, H.: Comparing Voting Systems. Reidel, Dordrecht (1987)CrossRefGoogle Scholar
  41. 41.
    Nurmi, H.: Referendum design: an exercise in applied social choice theory. Scandinavian Political Studies 20, 33–52 (1997)CrossRefGoogle Scholar
  42. 42.
    Nurmi, H.: Voting paradoxes and referenda. Social Choice and Welfare 15, 333–350 (1998)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Nurmi, H.: Voting Paradoxes and How to Deal with Them. Springer, Heidelberg (1999)MATHCrossRefGoogle Scholar
  44. 44.
    Nurmi, H.: Voting Procedures under Uncertainty. Springer, Heidelberg (2002)MATHCrossRefGoogle Scholar
  45. 45.
    Nurmi, H., Kacprzyk, J.: On fuzzy tournaments and their solution concepts in group decision making. European Journal of Operational Research 51, 223–232 (1991)MATHCrossRefGoogle Scholar
  46. 46.
    Nurmi, H., Kacprzyk, J.: Social choice under fuzziness: a perspective. In: Fodor, J., De Baets, B., Perny, P. (eds.) Preferences and Decisions under Incomplete Knowledge, pp. 107–130. Physica–Verlag, Springer, Heidelberg, New York (2000)Google Scholar
  47. 47.
    Nurmi, H., Kacprzyk, J., Fedrizzi, M.: Probabilistic, fuzzy and rough concepts in social choice. European Journal of Operational Research 95, 264–277 (1996)MATHCrossRefGoogle Scholar
  48. 48.
    Nurmi, H., Meskanen, T.: Voting paradoxes and MCDM. Group Decision and Negotiation 9(4), 297–313 (2000)CrossRefGoogle Scholar
  49. 49.
    Yager, R.R.: On ordered weighted averaging aggregation operators in multicriteria decision making. IEEE Trans. on Systems, Man and Cybernetics SMC-18, 183–190 (1988)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Yager, R.R., Kacprzyk, J. (eds.): The Ordered Weighted Averaging Operators: Theory and Applications. Kluwer, Boston (1997)Google Scholar
  51. 51.
    Yager, R.R., Kacprzyk, J., Beliakov, G.: Recent Developments in the Ordered Weighted Averaging Operators: Theory and Practice. Springer, Berlin (2011)CrossRefGoogle Scholar
  52. 52.
    Zadeh, L.A.: A computational approach to fuzzy quantifiers in natural languages. Computers and Maths. with Appls. 9, 149–184 (1983)MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Zadrożny, S.: An approach to the consensus reaching support in fuzzy environment. In: Kacprzyk, J., Nurmi, H., Fedrizzi, M. (eds.) Consensus under Fuzziness, pp. 83–109. Kluwer, Boston (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Janusz Kacprzyk
    • 1
  • Sławomir Zadrożny
    • 1
  • Hannu Nurmi
    • 2
  • Mario Fedrizzi
    • 3
  1. 1.Systems Research InstitutePolish Academy of SciencesWarsawPoland
  2. 2.Department of Political ScienceUniversity of TurkuTurkuFinland
  3. 3.Dipartimento di Informatica e Studi AziendaliUniversitá degli Studi di Trento via Inama 5TrentoItaly

Personalised recommendations