Fuzzy Arrovian theorems when preferences are strongly-connected

Document Type : Research Paper

Author

Inarbe (Institute for Advanced Research in Business and Economics) and Departamento de Estad´ıstica, Inform´atica y Matem´aticas. Universidad P´ublica de Navarra. 31006 Pamplona, Spain

Abstract

In this paper we study the aggregation of fuzzy preferences on non-necessarily finite societies. We characterize in terms of possibility and impossibility a family of models of strongly-connected preferences in which the transitivity is defined for any t-norm. For that purpose, we have described each model by means of some crisp binary relations and we have applied the results obtained by Kirman and Sondermann about ultrafilters and Arrovian models.

Keywords


[1] J. C. R. Alcantud, S. Díaz, S. Montes, Liberalism and dictatorship in the problem of fuzzy classification, International Journal of Approximate Reasoning, 110 (2019), 82-95.
[2] K. J. Arrow, Social choice and individual values, Cowles Commission Monograph, No. 12, John Wiley and Sons, Inc., New York, N. Y.; Chapman and Hall, Ltd., London, 1951.
[3] A. Banerjee, Fuzzy preferences and arrow-type problems in social choice, Social Choice and Welfare, 11(2) (1994), 121-130.
[4] A. Basile, V. Scalzo, A new view on Arrovian dictatorship in a fuzzy setting, Fuzzy Sets and Systems, 349 (2018), 99-106.
[5] G. Beliakov, A. Pradera, T. Calvo, Aggregation functions: A guide for practitioners, Springer, Berlin, Heidelberg, 2007.
[6] P. Bevilacqua, G. Bosi, M. Zuanon, Existence of order-preserving functions for nontotal fuzzy preference relations under decisiveness, Axioms, 6(4) (2017), 29.
[7] A. Billot, Economic theory of fuzzy equilibria: An axiomatic analysis, Springer Berlin Heidelberg, Berlin, Heidelberg, 2 Edition, 1995.
[8] D. S. Bridges, G. B. Mehta, Representations of preferences orderings, Volume 422 of Lecture Notes in Economics and Mathematical Systems. Springer Berlin Heidelberg, Berlin, Heidelberg, 1995.
[9] M. J. Campión, J. C. Candeal, E. Induráin, Representability of binary relations through fuzzy numbers, Fuzzy Sets and Systems, 157(1) (2006), 1-19.
[10] G. Chichilnisky, Social choice and the topology of spaces of preferences, Advances in Mathematics, 37(2) (1980), 165-176.
[11] W. J. Cho, C. W. Park, Fractional group identification, Journal of Mathematical Economics, 77 (2018), 66-75.
[12] C. Duddy, J. Perote-Peña, A. Piggins, Arrow’s theorem and max-star transitivity, Social Choice and Welfare, 36(1) (2011), 25-34.
[13] C. Duddy, A. Piggins, On some oligarchy results when social preference is fuzzy, Social Choice and Welfare, 51 (2018), 717-735.
[14] B. Dutta, Fuzzy preferences and social choice, Mathematical Social Sciences, 13(3) (1987), 215-229.
[15] B. Dutta, S. C. Panda, P. K. Pattanaik, Exact choice and fuzzy preferences, Mathematical Social Sciences, 11(1) (1986), 53-68.
[16] F. Fioravanti, F. Tohmé, Fuzzy group identification problems, Fuzzy Sets and Systems, 434 (2021), 159-171.
[17] L. A. Fono, N. G. Andjiga, Fuzzy strict preference and social choice, Fuzzy Sets and Systems, 155(3) (2005), 372-389.
[18] L. W. Fung, K. S. Fu, An axiomatic approach to rational decision making in a fuzzy environment, In L. A. Zadeh, K. S. Fu, K. Tanaka, and M. Shimura, editors, Fuzzy Sets and their Applications to Cognitive and Decision Processes, pages 227–256. Academic Press, 1975.
[19] I. Georgescu, Fuzzy choice functions: A revealed preference approach, Springer-Verlag Berlin Heidelberg, 2007.
[20] A. Gibbard, Manipulation of voting schemes: A general result, Econometrica, 41(4) (1973), 587-601.
[21] M. B. Gibilisco, A. M. Gowen, K. E. Albert, J. N. Mordeson, M. J. Wierman, T. D. Clark, Fuzzy social choice theory, Springer International Publishing, Cham, 2014.
[22] B. Hansson, The existence of group preference functions, Public Choice, 28(1) (1976), 89-98.
[23] J. S. Kelly, Arrow impossibility theorems, Academic Press, New York, NY, USA, 1978.
[24] J. S. Kelly, Social choice theory: An introduction, Springer-Verlag Berlin Heidelberg, 1988.
[25] A. P. Kirman, D. Sondermann, Arrow’s theorem, many agents, and invisible dictators, Journal of Economic Theory, 5(2) (1972), 267-277.
[26] J. N. Mordeson, M. B. Gibilisco, T. D. Clark, Independence of irrelevant alternatives and fuzzy Arrow’s theorem, New Mathematics and Natural Computation, 8(2) (2012), 219-237.
[27] S. V. Ovchinnikov, Structure of fuzzy binary relations, Fuzzy Sets and Systems, 6(2) (1981), 169-195.
[28] A. Raventós-Pujol, M. J. Campión, E. Induráin, Arrow theorems in the fuzzy setting, Iranian Journal of Fuzzy Systems, 17(5) (2020), 29-41.
[29] A. Raventós-Pujol, M. J. Campión, E. Induráin, Decomposition and arrow-like aggregation of fuzzy preferences, Mathematics, 8(3) (2020), 436.
[30] G. Richardson, The structure of fuzzy preferences: Social choice implications, Social Choice and Welfare, 15(3) (1998), 359-369.
[31] M. A. Satterthwaite, Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions, Journal of Economic Theory, 10(2) (1975), 187-217.
[32] S. Willard, General topology, Dover Publications, Mineola, N.Y, 2004.