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Arrow’s theorem and max-star transitivity

Author

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  • Conal Duddy

    ()

  • Juan Perote-Peña

    ()

  • Ashley Piggins

    ()

Abstract

In the literature on social choice and fuzzy preferences, a central question is how to represent the transitivity of a fuzzy binary relation. Arguably the most general way of doing this is to assume a form of transitivity called max-star transitivity. The star operator in this formulation is commonly taken to be a triangular norm. The familiar max-min transitivity condition is a member of this family, but there are infinitely many others. Restricting attention to fuzzy aggregation rules that satisfy counterparts of unanimity and independence of irrelevant alternatives, we characterise the set of max-star transitive relations that permit preference aggregation to be non-dictatorial. This set contains all and only those triangular norms that contain a zero divisor.
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Suggested Citation

  • Conal Duddy & Juan Perote-Peña & Ashley Piggins, 2011. "Arrow’s theorem and max-star transitivity," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 36(1), pages 25-34, January.
  • Handle: RePEc:spr:sochwe:v:36:y:2011:i:1:p:25-34
    DOI: 10.1007/s00355-010-0461-x
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    References listed on IDEAS

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    1. Sen, Amartya, 1970. "Interpersonal Aggregation and Partial Comparability," Econometrica, Econometric Society, vol. 38(3), pages 393-409, May.
    2. Richard Barrett & Maurice Salles, 2006. "Social Choice With Fuzzy Preferences," Economics Working Paper Archive (University of Rennes 1 & University of Caen) 200615, Center for Research in Economics and Management (CREM), University of Rennes 1, University of Caen and CNRS.
    3. Rajat Deb & Manabendra Dasgupta, 1996. "Transitivity and fuzzy preferences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 13(3), pages 305-318.
    4. Asley Piggins & Maurice Salles, 2007. "Instances of Indeterminacy," Post-Print halshs-00337772, HAL.
    5. Basu, Kaushik, 1984. "Fuzzy revealed preference theory," Journal of Economic Theory, Elsevier, vol. 32(2), pages 212-227, April.
    6. Dutta, Bhaskan, 1987. "Fuzzy preferences and social choice," Mathematical Social Sciences, Elsevier, vol. 13(3), pages 215-229, June.
    7. Perote-Pena, Juan & Piggins, Ashley, 2007. "Strategy-proof fuzzy aggregation rules," Journal of Mathematical Economics, Elsevier, vol. 43(5), pages 564-580, June.
    8. Barrett, C.R. & Pattanaik, P.K. & Salles, M., 1990. "Rationality and Aggregation of Preferences in an Ordinally Fuzzy Framework," Institut des Mathématiques Economiques – Document de travail de l’I.M.E. (1974-1993) 9006, Institut des Mathématiques Economiques. LATEC, Laboratoire d'Analyse et des Techniques EConomiques, CNRS, Université de Bourgogne.
    9. Dutta, Bhaskar & Panda, Santosh C. & Pattanaik, Prasanta K., 1986. "Exact choice and fuzzy preferences," Mathematical Social Sciences, Elsevier, vol. 11(1), pages 53-68, February.
    10. Duddy, Conal & Piggins, Ashley, 2013. "Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary," Journal of Economic Theory, Elsevier, vol. 148(2), pages 793-805.
    11. Maurice Salles, 2005. "Social Choice," Post-Print halshs-00337075, HAL.
    12. Conal Duddy & Juan Perote-Peña & Ashley Piggins, 2010. "Manipulating an aggregation rule under ordinally fuzzy preferences," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 34(3), pages 411-428, March.
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    Cited by:

    1. Piggins, Ashley & Duddy, Conal, 2016. "Oligarchy and soft incompleteness," MPRA Paper 72392, University Library of Munich, Germany.
    2. Duddy, Conal & Piggins, Ashley, 2013. "Many-valued judgment aggregation: Characterizing the possibility/impossibility boundary," Journal of Economic Theory, Elsevier, vol. 148(2), pages 793-805.
    3. Maurice Salles, 2014. "‘Social choice and welfare’ at 30: its role in the development of social choice theory and welfare economics," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 42(1), pages 1-16, January.
    4. Conal Duddy & Ashley Piggins, 2012. "The proximity condition," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 39(2), pages 353-369, July.

    More about this item

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations

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