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Extension of Some Voting Systems to the Field of Gradual Preferences

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Fuzzy Sets and Their Extensions: Representation, Aggregation and Models

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 220))

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Abstract

In the classical theory of social choice, there exist many voting procedures for determining a collective preference on a set of alternatives. The simplest situation happens when a group of individuals has to choose between two alternatives. In this context, some voting procedures such as simple and absolute special majorities are frequently used. However, these voting procedures do not take into account the intensity with which individuals prefer one alternative to the other. In order to consider this situation, one possibility is to allow individuals showing their preferences through values located between 0 and 1. In this case, the collective preference can be obtained by means of an aggregation operator. One of the most important matter in this context is how to choose such aggregation operator. When we consider the class of OWA operators, it is necessary to determine the associated weights. In this contribution we survey several methods for obtaining the OWA operator weights. We pay special attention to the way the weights are chosen, regarding the concrete voting system we want to obtain when individuals do not grade their preferences between the alternatives.

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References

  1. Ašan G, Sanver MR (2002) Another characterization of the majority rule. Economics Letters 75: 409–413.

    Article  MathSciNet  Google Scholar 

  2. Beliakov G, Calvo T (2006) Identification of weights in aggregation operators. In this book.

    Google Scholar 

  3. Beliakov G, Calvo T (2006) Interpolarity type construction of general aggregation operators. In this book.

    Google Scholar 

  4. Bezdek JC, Spillman B, Spillman R (1978) A fuzzy relation space for group decision theory. Fuzzy Sets and Systems 1: 255–268.

    Article  MATH  MathSciNet  Google Scholar 

  5. Bordogna G, Fedrizzi M, Pasi G (1997) A linguistic modelling of consensus in group decision making based on OWA operators. IEEE Transactions on Systems, Man and Cybernetics 27: 126–132.

    Article  Google Scholar 

  6. Calvo T, Kolesárová A, Komorníková M, Mesiar R (2002) Aggregation operators: Properties, classes and construction methods. In Calvo et al. [7], 3–104.

    Google Scholar 

  7. Calvo T, Mayor G, Mesiar R (eds) (2002) Aggregation Operators: New Trends and Applications. Physica-Verlag, Heidelberg.

    MATH  Google Scholar 

  8. Campbell DE (1982) On the derivation of majority rule. Theory and Decision 14: 133–140.

    Article  MATH  MathSciNet  Google Scholar 

  9. Campbell DE (1988) A characterization of simple majority rule for restricted domains. Economics Letters 28: 307–310.

    Article  MathSciNet  Google Scholar 

  10. Campbell DE, Kelly JS (2000) A simple characterization of majority rule. Economic Theory 15: 689–700.

    Article  MATH  MathSciNet  Google Scholar 

  11. Campbell DE, Kelly JS (2003) A strategy-proofness characterization of majority rule. Economic Theory 22: 557–568.

    Article  MATH  MathSciNet  Google Scholar 

  12. De Baets B, Fodor J (1997) Twenty years of fuzzy preference structures. Rivista di Matematica per le Scienze Economiche e Sociali 20: 45–66.

    MATH  MathSciNet  Google Scholar 

  13. De Baets B, De Meyer H, De Schuymer B, Jenei S (2006) Cyclic evaluation of transitivity of reciprocal relations. Social Choice and Welfare 26: 217–238.

    Article  MATH  MathSciNet  Google Scholar 

  14. Delgado M, Verdegay JL, Vila MA (1993) Linguistic decision making models. International Journal of Intelligent Systems 7: 479–492.

    Article  Google Scholar 

  15. Delgado M, Verdegay JL, Vila MA (1993) On aggregation operations of linguistic labels. International Journal of Intelligent Systems 8: 352–370.

    Article  Google Scholar 

  16. Dubois D, Koning JL (1991) Social choice axioms for fuzzy set agregation. Fuzzy Sets and Systems 43: 257–274.

    Article  MATH  MathSciNet  Google Scholar 

  17. Dubois D, Prade H (1985) A review of fuzzy set aggregation connectives. Information Sciences 36: 85–121.

    Article  MATH  MathSciNet  Google Scholar 

  18. Ferejohn JA, Grether DM (1974) On a class of rational social decisions procedures. Journal of Economic Theory 8: 471–482.

    Article  MathSciNet  Google Scholar 

  19. Filev D, Yager RR (1995) Analytic properties of maximal entropy OWA operators. Information Sciences 85: 11–27.

    Article  MATH  Google Scholar 

  20. Filev D, Yager RR (1998) On the issue of obtaining OWA operator weights. Fuzzy Sets and Systems 94: 157–169.

    Article  MathSciNet  Google Scholar 

  21. Fishburn PC (1973) The Theory of Social Choice. Princeton University Press, Princeton.

    MATH  Google Scholar 

  22. Fishburn PC (1983) A new characterization of simple majority. Economics Letters 13: 31–35.

    Article  MathSciNet  Google Scholar 

  23. Fodor J, De Baets, B (2006) Fuzzy preference modelling: Fundamentals and recent advances. In this book.

    Google Scholar 

  24. Fodor J, Marichal JL, Roubens M (1995) Characterization of the ordered weighted averaging operators. IEEE Transactions on Fuzzy Systems 3: 236–240.

    Article  Google Scholar 

  25. Fodor J, Roubens M (1994) Fuzzy Preference Modelling and Multicriteria Decision Support. Kluwer Academic Publishers, Dordrecht.

    MATH  Google Scholar 

  26. Fullér R, Majlender P (2001) An analytic approach for obtaining maximal entropy OWA operator weights. Fuzzy Sets and Systems 124: 53–57.

    Article  MATH  MathSciNet  Google Scholar 

  27. Fullér R, Majlender P (2003) On obtaining minimal variability OWA operator weights. Fuzzy Sets and Systems 136: 203–215.

    Article  MATH  MathSciNet  Google Scholar 

  28. Garcí a-Lapresta JL (2006) A general class of simple majority decision rules based on linguistic opinions. Information Sciences 176: 352–365.

    Article  MathSciNet  Google Scholar 

  29. García-Lapresta JL, Llamazares B (2000) Aggregation of fuzzy preferences: Some rules of the mean. Social Choice and Welfare 17: 673–690.

    Article  MATH  MathSciNet  Google Scholar 

  30. García-Lapresta JL, Llamazares B (2001) Majority decisions based on difference of votes. Journal of Mathematical Economics 35: 463–481.

    Article  MATH  MathSciNet  Google Scholar 

  31. Garcí a-Lapresta JL, Martí nez-Panero M, Meneses LC (To appear) Defining the Borda count in a linguistic decision making context. Information Sciences.

    Google Scholar 

  32. Grabisch M (1995) Fuzzy integral in multicriteria decision making. Fuzzy Sets and Systems 69: 279–298.

    Article  MATH  MathSciNet  Google Scholar 

  33. Grabisch M (1996) The application of fuzzy integrals in multicriteria decision making. European Journal of Operational Research 89: 445–456.

    Article  MATH  Google Scholar 

  34. Grabisch M, Orlovski SA, Yager RR (1998) Fuzzy aggregation of numerical preferences. In: Słowiński R (ed) Fuzzy Sets in Decision Analysis, Operations Research and Statistics. Kluwer Academic Publishers, Dordrecht, 31–68.

    Google Scholar 

  35. Herrera F, Herrera-Viedma E (2000) Linguistic decision analysis: Steps for solving decision problems under linguistic information. Fuzzy Sets and Systems 115: 67–82.

    Article  MATH  MathSciNet  Google Scholar 

  36. Herrera F, Herrera-Viedma E, Verdegay JL (1995) A sequential selection process in group decision making with linguistic assessment. Information Sciences 85: 223–239.

    Article  MATH  Google Scholar 

  37. Herrera F, Herrera-Viedma E, Verdegay JL (1996) A linguistic decision process in group decision making. Group Decision and Negotiation 5: 165–176

    Article  Google Scholar 

  38. Herrera F, Herrera-Viedma E, Verdegay JL (1996) Direct approach processes in group decision making using linguistic OWA operators. Fuzzy Sets and Systems 79:175–190.

    Article  MATH  MathSciNet  Google Scholar 

  39. Kandel A, Byatt WJ (1978) Fuzzy sets, fuzzy algebra and fuzzy statistics. Proceedings of IEEE 66: 1619–1639.

    Article  Google Scholar 

  40. Liu X, Chen L (2004) On the properties of parametric geometric OWA operator. International Journal of Approximate Reasoning 35: 163–178.

    Article  MATH  MathSciNet  Google Scholar 

  41. Llamazares B (2004) Simple and absolute special majorities generated by OWA operators. European Journal of Operational Research 158: 707–720.

    Article  MATH  MathSciNet  Google Scholar 

  42. Llamazares B (2006) The forgotten decision rules: Majority rules based on difference of votes. Mathematical Social Sciences 51: 311–326.

    Article  MATH  MathSciNet  Google Scholar 

  43. Llamazares B (2006) Choosing OWA operator weights in the field of Social Choice. Mimeo.

    Google Scholar 

  44. Llamazares B, García-Lapresta JL (2003) Voting Systems generated by quasiarithmetic means and OWA operators. In: De Baets B, Fodor J (eds) Principles of Fuzzy Preference Modelling and Decision Making. Academia Press, Gent, 195–213.

    Google Scholar 

  45. Majlender P (2005) OWA operators with maximal Rényi entropy. Fuzzy Sets and Systems 155: 340–360.

    Article  MATH  MathSciNet  Google Scholar 

  46. Marichal JL (1998) Aggregation Operators for Multicriteria Decision Aid. MA Thesis, Liège University, Liège.

    Google Scholar 

  47. Marichal JL, Mathonet P (1999) A characterization of the ordered weighted averaging functions based on the ordered bisymmetry property. IEEE Transactions on Fuzzy Systems 7: 93–96.

    Article  Google Scholar 

  48. Maskin ES (1995) Majority rule, social welfare functions and game forms. In: Basu K, Pattanaik PK, Suzumura K (eds) Choice, Welfare and Development. The Clarendon Press, Oxford, 100–109.

    Google Scholar 

  49. May KO (1952) A set of independent necessary and sufficient conditions for simple majority decision. Econometrica 20: 680–684.

    Article  MATH  MathSciNet  Google Scholar 

  50. Mesiar R, Komorníková M, Kolesárová A, Calvo T, (2006) Aggregation functions: A revision. In this book.

    Google Scholar 

  51. Miroiu A (2004) Characterizing majority rule: from profiles to societies. Economics Letters 85: 359–363.

    Article  MathSciNet  Google Scholar 

  52. Mizumoto M (1989) Pictorial representations of fuzzy connectives, part I: cases of t-norms, t-conorms and averaging operators. Fuzzy Sets and Systems 31: 217–242.

    Article  MathSciNet  Google Scholar 

  53. Mizumoto M (1989) Pictorial representations of fuzzy connectives, part II: cases of compensatory operators and self-dual operators. Fuzzy Sets and Systems 32: 45–79.

    Article  MATH  MathSciNet  Google Scholar 

  54. Murofushi T, Sugeno M (1993) Some quantities represented by the Choquet integral. Fuzzy Sets and Systems 56: 229–235.

    Article  MATH  MathSciNet  Google Scholar 

  55. Nakamura K (1992) On the nature of intransitivity in human preferential judgements. In: Novák V, Ramík J et al. (eds) Fuzzy Approach to Reasoning and Decision-Making. Kluwer Academic Publishers, Dordrecht, 147–162.

    Google Scholar 

  56. Nettleton D, Torra V (2001) A comparison of active set method and genetic algorithm approaches for learning weighting vectors in some aggregation operators. International Journal of Intelligent Systems 16: 1069–1083.

    Article  MATH  Google Scholar 

  57. Nurmi H (1981) Approaches to collective decision making with fuzzy preference relations. Fuzzy Sets and Systems 6: 249–259.

    Article  MATH  MathSciNet  Google Scholar 

  58. O’Hagan M (1988) Aggregating template or rule antecedents in real-time expert systems with fuzzy set logic. Proc. 22nd Annual IEEE Asilomar Conf. on Signals, Systems and Computers, Pacific Grove, California, 681–689.

    Google Scholar 

  59. Ovchinnikov SV (1998) An analytic characterization of some aggregation operators. International Journal of Intelligent Systems 13: 3–10.

    Article  MathSciNet  Google Scholar 

  60. Sen AK (1970) Collective Choice and Social Welfare. Holden-Day, San Francisco.

    MATH  Google Scholar 

  61. Świtalski Z (1999) Rationality of fuzzy reciprocal preference relations. Fuzzy Sets and Systems 107: 187–190.

    Article  MathSciNet  Google Scholar 

  62. Świtalski Z (2003) General transitivity conditions for fuzzy reciprocal preference matrices. Fuzzy Sets and Systems 137: 85–100.

    Article  MathSciNet  Google Scholar 

  63. Tanino T (1984) Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems 12: 117–131.

    Article  MATH  MathSciNet  Google Scholar 

  64. Torra V (1997) The weighted OWA operator. International Journal of Intelligent Systems 12: 153–166.

    Article  MATH  Google Scholar 

  65. Torra V (1999) On the learning of weights in some aggregation operators: The weighted mean and OWA operators. Mathware and Soft Computing 6: 249–265.

    MathSciNet  Google Scholar 

  66. Torra V (2006) On the construction of models for decision using aggregation operators. In this book.

    Google Scholar 

  67. Wang YM, Parkan C (2005) A minimax disparity approach for obtaining OWA operator weights. Information Sciences 175: 20–29.

    Article  MATH  MathSciNet  Google Scholar 

  68. Woeginger GJ (2003) A new characterization of the majority rule. Economics Letters 81: 89–94.

    Article  MathSciNet  Google Scholar 

  69. Xu ZS (2005) An overview of methods for determining OWA weights. International Journal of Intelligent Systems 20: 843–865.

    Article  MATH  Google Scholar 

  70. Xu ZS (2006) Dependent OWA operators. In: Torra V, Narukawa Y, Valls A, Domingo-Ferrer J (eds) Modeling Decisions for Artificial Intelligence: Third International Conference MDAI 2006, Lecture Notes in Artificial Intelligence 3885. Springer, Berlin, 172–178.

    Google Scholar 

  71. Xu ZS, Da QL (2002) The uncertain OWA operator. International Journal of Intelligent Systems 17: 569–575.

    Article  MATH  Google Scholar 

  72. Xu ZS, Da QL (2003) An overview of operators for aggregating information. International Journal of Intelligent Systems 18: 953–969.

    Article  MATH  Google Scholar 

  73. Yager RR (1988) On ordered weighted averaging operators in multicriteria decision making. IEEE Transactions on Systems, Man and Cybernetics 18: 183–190.

    Article  MATH  MathSciNet  Google Scholar 

  74. Yager RR (1991) Connectives and quantifiers in fuzzy sets. Fuzzy Sets and Systems 40: 39–75.

    Article  MATH  MathSciNet  Google Scholar 

  75. Yager RR (1993) Families of OWA operators. Fuzzy Sets and Systems 59: 125–148.

    Article  MATH  MathSciNet  Google Scholar 

  76. Yager RR (1995) An approach to ordinal decision making. International Journal of Approximate Reasoning 12: 237–261.

    Article  MATH  MathSciNet  Google Scholar 

  77. Yager RR (1996) Quantifier guided aggregation using OWA operators. International Journal of Intelligent Systems 11: 49–73.

    Article  Google Scholar 

  78. Yager RR (2002) Heavy OWA operators. Fuzzy Optimization and Decision Making 1:379–397.

    Article  MATH  MathSciNet  Google Scholar 

  79. Yager RR (2003) E-Z OWA weights. Proc. 10th International Fuzzy Systems Association World Congress, Istanbul, 39–42.

    Google Scholar 

  80. Yager RR, Filev D (1999) Induced ordered weighted averaging operators. IEEE Transactions on Systems, Man and Cybernetics – Part B: Cybernetics 29: 141–150.

    Article  Google Scholar 

  81. Yager RR, Kacprzyk J (eds) (1997) The Ordered Weighted Averaging Operators: Theory and Applications. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  82. Yi J (2005) A complete characterization of majority rules. Economics Letters 87: 109–112.

    Article  MathSciNet  Google Scholar 

  83. Zadeh LA (1975) The concept of a linguistic variable and its applications to approximate reasoning. Information Sciences. Part I 8: 199–249. Part II 8: 301–357. Part III 9: 43–80.

    Google Scholar 

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Authors
  1. Bonifacio Llamazares
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  2. José Luis García-Lapresta
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Editors and Affiliations

  1. Department of Automatics and Computation, Universidad Pública de Navarra, Campus Arrosadia s/n, 31006 Pamplona, Spain

    Humberto Bustince

  2. Department of Computer Science and Artificial Intelligence (DECSAI), University of Granada, Periodista Daniel Saucedo Aranda s/n, 18071 Granada, Spain

    Francisco Herrera

  3. Faculty of Mathematics, Universidad Complutense, Plaza de las Ciencias 3, 28040 Madrid, Spain

    Javier Montero

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Llamazares, B., García-Lapresta, J.L. (2008). Extension of Some Voting Systems to the Field of Gradual Preferences. In: Bustince, H., Herrera, F., Montero, J. (eds) Fuzzy Sets and Their Extensions: Representation, Aggregation and Models. Studies in Fuzziness and Soft Computing, vol 220. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73723-0_15

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  • DOI: https://doi.org/10.1007/978-3-540-73723-0_15

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