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The structure of fuzzy preferences: Social choice implications


  • Gregory Richardson

    (Shorter College, Box 393, Rome, GA 30165, USA)


It has been shown that, with an alternative factorization of fuzzy weak preferences into symmetric and antisymmetric components, one can prove a fuzzy analogue of Arrow's Impossibility Theorem even when the transitivity requirements on individual and social preferences are very weak. It is demonstrated here that the use of this specification of strict preference, however, requires preferences to also be strongly connected. In the absence of strong connectedness, another factorization of fuzzy weak preferences is indicated, for which nondictatorial fuzzy aggregation rules satisfying the weak transitivity requirement can still be found. On the other hand, if strong connectedness is assumed, the fuzzy version of Arrow's Theorem still holds for a variety of weak preference factorizations, even if the transitivity condition is weakened to its absolute minimum. Since Arrow's Impossibility Theorem appeared nearly half a century ago, researchers have been attempting to avoid Arrow's negative result by relaxing various of his original assumptions. One approach has been to allow preferences - those of individuals and society or just those of society alone - to be "fuzzy." In particular, Dutta [4] has shown that, to a limited extent, one can avoid the impossibility result (or, more precisely, the dictatorship result) by using fuzzy preferences, employing a particularly weak version of transitivity among the many plausible (but still distinct) definitions of transitivity that are available for fuzzy preferences. Another aspect of exact preferences for which the extension to the more general realm of fuzzy preferences is ambiguous is the factorization of a weak preference relation into a symmetric component (indifference) and an antisymmetric component (strict preference). There are several ways to do this for fuzzy weak preferences, all of them equivalent to the traditional factorization in the special case when preferences are exact, but quite different from each other when preferences are fuzzy (see, for example, [3]). A recent paper in this journal [1], by A. Banerjee, argues that the choice of definitions for indifference and strict preference, given a fuzzy weak preference, can also have "Arrovian" implications. In particular, [1] claims that Dutta's version of strict preference presents certain intuitive difficulties and recommends a different version, with its own axiomatic derivation, for which the dictatorship results reappear even with Dutta's weak version of transitivity. However, the conditions used to derive [1]'s version of strict preference imply a restriction on how fuzzy the original weak preference can be, namely, that the fuzzy weak preference relation must be strongly connected. Without this restriction, I will show that the rest of [1]'s conditions imply yet a third version of strict preference, for which Dutta's possibility result under weak transitivity still holds. On the other hand, if one accepts the strong connectedness required in order for it to be valid, I show that [1]'s dictatorship theorem can in fact be strengthened to cover any version of transitivity for fuzzy preferences, no matter how weak, and further, that this dictatorship result holds for any "regular" formulation of strict preference, including the one originally used by Dutta.

Suggested Citation

  • Gregory Richardson, 1998. "The structure of fuzzy preferences: Social choice implications," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 15(3), pages 359-369.
  • Handle: RePEc:spr:sochwe:v:15:y:1998:i:3:p:359-369
    Note: Received: 13 May 1996 / Accepted: 13 January 1997

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    Cited by:

    1. repec:spr:grdene:v:10:y:2001:i:2:d:10.1023_a:1008747918121 is not listed on IDEAS
    2. Robert Draeseke & David E. A. Giles, 1999. "A Fuzzy Logic Approach to Modelling the Underground Economy," Econometrics Working Papers 9909, Department of Economics, University of Victoria.
    3. Draeseke, Robert & Giles, David E.A., 2002. "A fuzzy logic approach to modelling the New Zealand underground economy," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 59(1), pages 115-123.
    4. Piggins, Ashley & Duddy, Conal, 2016. "Oligarchy and soft incompleteness," MPRA Paper 72392, University Library of Munich, Germany.
    5. Dinko Dimitrov, 2001. "Fuzzy Preferences, Liberalism and Non-discrimination," Economic Thought journal, Bulgarian Academy of Sciences - Economic Research Institute, issue 2, pages 63-76.
    6. Peter Casey & Mark Wierman & Michael Gibilisco & John Mordeson & Terry Clark, 2012. "Assessing policy stability in Iraq: a fuzzy approach to modeling preferences," Public Choice, Springer, vol. 151(3), pages 409-423, June.
    7. David E. A. Giles & Robert Draeseke, 2001. "Econometric Modelling based on Pattern recognition via the Fuzzy c-Means Clustering Algorithm," Econometrics Working Papers 0101, Department of Economics, University of Victoria.
    8. Yu, Tiffany Hui-Kuang & Wang, David Han-Min & Chen, Su-Jane, 2006. "A fuzzy logic approach to modeling the underground economy in Taiwan," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 471-479.
    9. Louis Fono & Maurice Salles, 2011. "Continuity of utility functions representing fuzzy preferences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 37(4), pages 669-682, October.
    10. Subramanian, S., 2009. "The Arrow paradox with fuzzy preferences," Mathematical Social Sciences, Elsevier, vol. 58(2), pages 265-271, September.
    11. Espin, Rafael & Fernandez, Eduardo & Mazcorro, Gustavo & Lecich, Maria Ines, 2007. "A fuzzy approach to cooperative n-person games," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1735-1751, February.
    12. Perote-Pena, Juan & Piggins, Ashley, 2007. "Strategy-proof fuzzy aggregation rules," Journal of Mathematical Economics, Elsevier, vol. 43(5), pages 564-580, June.

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