Abstract
In the literature on social choice with 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 triangular norms that permit preference aggregation to be non-dictatorial. This set contains all and only those norms that contain a zero divisor.
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References
Arrow KJ (1951) Social choice and individual values. Wiley, New York
Banerjee A (1993) Rational choice under fuzzy preferences: the Orlovsky choice function. Fuzzy Sets Syst 53: 295–299
Banerjee A (1994) Fuzzy preferences and arrow-type problems in social choice. Soc Choice Welf 11: 121–130
Barrett CR, Salles M (2006) Social choice with fuzzy preferences, Working paper, Centre for Research in Economics and Management, UMR CNRS 6211, University of Caen
Barrett CR, Pattanaik PK, Salles M (1986) On the structure of fuzzy social welfare functions. Fuzzy Sets Syst 19: 1–10
Barrett CR, Pattanaik PK, Salles M (1992) Rationality and aggregation of preferences in an ordinally fuzzy framework. Fuzzy Sets Syst 49: 9–13
Basu K (1984) Fuzzy revealed preference. J Econ Theory 32: 212–227
Basu K, Deb R, Pattanaik PK (1992) Soft sets: an ordinal reformulation of vagueness with some applications to the theory of choice. Fuzzy Sets Syst 45: 45–58
Billot A (1995) Economic theory of fuzzy equilibria. Springer, Berlin
Dasgupta M, Deb R (1991) Fuzzy choice functions. Soc Choice Welf 8: 171–182
Dasgupta M, Deb R (1996) Transitivity and fuzzy preferences. Soc Choice Welf 13: 305–318
Dasgupta M, Deb R (1999) An impossibility theorem with fuzzy preferences. In: de Swart H (ed) Logic, game theory and social choice: proceedings of the international conference, LGS ’99, May 13–16, 1999, Tilburg University Press
Dasgupta M, Deb R (2001) Factoring fuzzy transitivity. Fuzzy Sets Syst 118: 489–502
Dietrich F, List C (2009) The aggregation of propositional attitudes: towards a general theory, forthcoming in Oxford Studies in Epistemology
Duddy C, Piggins A (2009) Many-valued judgment aggregation: characterizing the possibility/impossibility boundary for an important class of agendas, Working paper, Department of Economics, National University of Ireland, Galway
Duddy C, Perote-Peña J, Piggins A (2010) Manipulating an aggregation rule under ordinally fuzzy preferences. Soc Choice Welf 34: 411–428
Dutta B (1987) Fuzzy preferences and social choice. Math Soc Sci 13: 215–229
Dutta B, Panda SC, Pattanaik PK (1986) Exact choice and fuzzy preferences. Math Soc Sci 11: 53–68
Fono LA, Andjiga NG (2005) Fuzzy strict preference and social choice. Fuzzy Sets Syst 155: 372–389
Goguen JA (1967) L-fuzzy sets. J Math Anal Appl 18: 145–174
Jain N (1990) Transitivity of fuzzy relations and rational choice. Ann Oper Res 23: 265–278
Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, Dordrecht
Leclerc B (1984) Efficient and binary consensus functions on transitively valued relations. Math Soc Sci 8: 45–61
Leclerc B (1991) Aggregation of fuzzy preferences: a theoretic Arrow-like approach. Fuzzy Sets Syst 43: 291–309
Leclerc B, Monjardet B (1995) Lattical theory of consensus. In: Barnett W, Moulin H, Salles M, Schofield N (eds) Social choice, welfare and ethics. Cambridge University Press, Cambridge
Orlovsky SA (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets Syst 1: 155–167
Ovchinnikov SV (1981) Structure of fuzzy binary relations. Fuzzy Sets Syst 6: 169–195
Ovchinnikov SV (1991) Social choice and Lukasiewicz logic. Fuzzy Sets Syst 43: 275–289
Ovchinnikov SV, Roubens M (1991) On strict preference relations. Fuzzy Sets Syst 43: 319–326
Ovchinnikov SV, Roubens M (1992) On fuzzy strict preference, indifference and incomparability relations. Fuzzy Sets Syst 49: 15–20
Perote-Peña J, Piggins A (2007) Strategy-proof fuzzy aggregation rules. J Math Econ 43: 564–580
Perote-Peña J, Piggins A (2009a) Non-manipulable social welfare functions when preferences are fuzzy. J Log Comput 19: 503–515
Perote-Peña J, Piggins A (2009b) Social choice, fuzzy preferences and manipulation. In: Boylan T, Gekker R (eds) Economics, rational choice and normative philosophy. Routledge, London
Piggins A, Salles M (2007) Instances of indeterminacy. Analyse und Kritik 29: 311–328
Ponsard C (1990) Some dissenting views on the transitivity of individual preference. Ann Oper Res 23: 279–288
Richardson G (1998) The structure of fuzzy preferences: social choice implications. Soc Choice Welf 15: 359–369
Salles M (1998) Fuzzy utility. In: Barbera S, Hammond PJ, Seidl C (eds) Handbook of utility theory, vol 1: principles. Kluwer, Dordrecht
Sen AK (1970) Interpersonal aggregation and partial comparability. Econometrica 38: 393–409
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This article was first presented at the conference “New Developments in Social Choice and Welfare Theories: A Tribute to Maurice Salles”, which was held at the Université Caen in June 2009. It was later presented at the PET 09 conference in NUI Galway.
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Duddy, C., Perote-Peña, J. & Piggins, A. Arrow’s theorem and max-star transitivity. Soc Choice Welf 36, 25–34 (2011). https://doi.org/10.1007/s00355-010-0461-x
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DOI: https://doi.org/10.1007/s00355-010-0461-x