Arrow’s theorem and max-star transitivity

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.

This is a preview of subscription content, access via your institution.

References

  1. Arrow KJ (1951) Social choice and individual values. Wiley, New York

    Google Scholar 

  2. Banerjee A (1993) Rational choice under fuzzy preferences: the Orlovsky choice function. Fuzzy Sets Syst 53: 295–299

    Article  Google Scholar 

  3. Banerjee A (1994) Fuzzy preferences and arrow-type problems in social choice. Soc Choice Welf 11: 121–130

    Article  Google Scholar 

  4. 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

  5. Barrett CR, Pattanaik PK, Salles M (1986) On the structure of fuzzy social welfare functions. Fuzzy Sets Syst 19: 1–10

    Article  Google Scholar 

  6. Barrett CR, Pattanaik PK, Salles M (1992) Rationality and aggregation of preferences in an ordinally fuzzy framework. Fuzzy Sets Syst 49: 9–13

    Article  Google Scholar 

  7. Basu K (1984) Fuzzy revealed preference. J Econ Theory 32: 212–227

    Article  Google Scholar 

  8. 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

    Article  Google Scholar 

  9. Billot A (1995) Economic theory of fuzzy equilibria. Springer, Berlin

    Google Scholar 

  10. Dasgupta M, Deb R (1991) Fuzzy choice functions. Soc Choice Welf 8: 171–182

    Article  Google Scholar 

  11. Dasgupta M, Deb R (1996) Transitivity and fuzzy preferences. Soc Choice Welf 13: 305–318

    Article  Google Scholar 

  12. 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

  13. Dasgupta M, Deb R (2001) Factoring fuzzy transitivity. Fuzzy Sets Syst 118: 489–502

    Article  Google Scholar 

  14. Dietrich F, List C (2009) The aggregation of propositional attitudes: towards a general theory, forthcoming in Oxford Studies in Epistemology

  15. 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

  16. Duddy C, Perote-Peña J, Piggins A (2010) Manipulating an aggregation rule under ordinally fuzzy preferences. Soc Choice Welf 34: 411–428

    Article  Google Scholar 

  17. Dutta B (1987) Fuzzy preferences and social choice. Math Soc Sci 13: 215–229

    Article  Google Scholar 

  18. Dutta B, Panda SC, Pattanaik PK (1986) Exact choice and fuzzy preferences. Math Soc Sci 11: 53–68

    Article  Google Scholar 

  19. Fono LA, Andjiga NG (2005) Fuzzy strict preference and social choice. Fuzzy Sets Syst 155: 372–389

    Article  Google Scholar 

  20. Goguen JA (1967) L-fuzzy sets. J Math Anal Appl 18: 145–174

    Article  Google Scholar 

  21. Jain N (1990) Transitivity of fuzzy relations and rational choice. Ann Oper Res 23: 265–278

    Article  Google Scholar 

  22. Klement EP, Mesiar R, Pap E (2000) Triangular norms. Kluwer Academic Publishers, Dordrecht

    Google Scholar 

  23. Leclerc B (1984) Efficient and binary consensus functions on transitively valued relations. Math Soc Sci 8: 45–61

    Article  Google Scholar 

  24. Leclerc B (1991) Aggregation of fuzzy preferences: a theoretic Arrow-like approach. Fuzzy Sets Syst 43: 291–309

    Article  Google Scholar 

  25. 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

    Google Scholar 

  26. Orlovsky SA (1978) Decision-making with a fuzzy preference relation. Fuzzy Sets Syst 1: 155–167

    Article  Google Scholar 

  27. Ovchinnikov SV (1981) Structure of fuzzy binary relations. Fuzzy Sets Syst 6: 169–195

    Article  Google Scholar 

  28. Ovchinnikov SV (1991) Social choice and Lukasiewicz logic. Fuzzy Sets Syst 43: 275–289

    Article  Google Scholar 

  29. Ovchinnikov SV, Roubens M (1991) On strict preference relations. Fuzzy Sets Syst 43: 319–326

    Article  Google Scholar 

  30. Ovchinnikov SV, Roubens M (1992) On fuzzy strict preference, indifference and incomparability relations. Fuzzy Sets Syst 49: 15–20

    Article  Google Scholar 

  31. Perote-Peña J, Piggins A (2007) Strategy-proof fuzzy aggregation rules. J Math Econ 43: 564–580

    Article  Google Scholar 

  32. Perote-Peña J, Piggins A (2009a) Non-manipulable social welfare functions when preferences are fuzzy. J Log Comput 19: 503–515

    Article  Google Scholar 

  33. 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

    Google Scholar 

  34. Piggins A, Salles M (2007) Instances of indeterminacy. Analyse und Kritik 29: 311–328

    Google Scholar 

  35. Ponsard C (1990) Some dissenting views on the transitivity of individual preference. Ann Oper Res 23: 279–288

    Article  Google Scholar 

  36. Richardson G (1998) The structure of fuzzy preferences: social choice implications. Soc Choice Welf 15: 359–369

    Article  Google Scholar 

  37. Salles M (1998) Fuzzy utility. In: Barbera S, Hammond PJ, Seidl C (eds) Handbook of utility theory, vol 1: principles. Kluwer, Dordrecht

    Google Scholar 

  38. Sen AK (1970) Interpersonal aggregation and partial comparability. Econometrica 38: 393–409

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ashley Piggins.

Additional information

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.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

Keywords

  • Social Choice
  • Social Welfare Function
  • Transitivity Condition
  • Zero Divisor
  • Judgment Aggregation