Abstract
Choices among alternatives in a set can be expressed in three different ways: by means of choice functions, by means of preference relations or using choice probabilities. The connection between the two first formalizations has been widely studied in the literature, both in the crisp or classical context and in the setting of fuzzy relations. However, the connection between probabilistic choice functions and fuzzy choice functions seems to have been forgotten and as far as we know, no literature can be found about it.
In this contribution we focus on the comparison of both types of choice functions. We provide a way to obtain the fuzzy choice function from the probabilistic choice function and the other way around. Moreover,we can prove that under Luce’s Choice Axiom the fuzzy choice function derived from the probabilistic choice function is G-normal.
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References
Arrow, K.J.: Rational choice functions and orderings. Economica 26, 121–127 (1959)
Banerjee, A.: Fuzzy choice functions, revealed preference and rationality. Fuzzy Sets and Systems 70, 31–43 (1995)
Barrett, R., Pattanaik, P.K., Salles, M.: On choosing rationally when preferences are fuzzy. Fuzzy Sets and Systems 34, 197–212 (1990)
Barrett, R., Pattanaik, P.K., Salles, M.: Rationality and aggregation of preferences in an ordinally fuzzy framework. Fuzzy Sets and Systems 49, 9–13 (1992)
De Baets, B., De Meyer, H., De Loof, K.: On the cycle-transitivity of the mutual rank probability relation of a poset. Fuzzy Sets and Systems 161, 2695–2708 (2010)
De Schuymer, B., De Meyer, H., De Baets, B.: Cycle-transitive comparison of independent random variables. Journal of Multivariate Analysis 96, 352–373 (2005)
Fishburn, P.C.: Choice probabilities and choice functions. Journal of Mathematical Psychology 18, 205–219 (1978)
Georgescu, I.: Fuzzy choice functions, A revealed Preference Approach. Springer, Berlin (2007)
Georgescu, I.: Arrow’s axiom and full rationality for fuzzy choice functions. Social Choice and Welfare 28, 303–319 (2007)
Georgescu, I.: Acyclic rationality indicators of fuzzy choice functions. Fuzzy Sets and Systems 160, 2673–2685 (2009)
Luce, R.D.: Individual Choice Behavior: A Theoretical Analysis. Wiley, New York (1959)
Luce, R.D., Suppes, P.: Preferences utility and subject probability. In: Luce, R.D., Bush, R.R., Galanter, E. (eds.) Handbook of Mathematical Psychology III, pp. 249–410. Wiley (1965)
Martinetti, D., De Baets, B., Díaz, S. and Montes, S.: On the role of acyclicity in the study of rationality of fuzzy choice functions. In: International Conference on Intelligent Systems Design and Applications, pp. 350–355 (2011)
Martinetti, D., Montes, I., Díaz, S.: From Preference Relations to Fuzzy Choice Functions. In: Liu, W. (ed.) ECSQARU 2011. LNCS, vol. 6717, pp. 594–605. Springer, Heidelberg (2011)
Martinetti, D., Montes, I., Díaz, S., Montes, S.: A study on the transitivity of probabilistic and fuzzy relations. Fuzzy Sets and Systems 184, 156–170 (2011)
Martinetti, D., Montes, S., Díaz, S., De Baets, B.: Some comments to the fuzzy version of the Arrow-Sen theorem. In: Greco, S., Bouchon-Meunier, B., Coletti, G., Fedrizzi, M., Matarazzo, B., Yager, R.R. (eds.) IPMU 2012, Part IV. CCIS, vol. 300, pp. 286–295. Springer, Heidelberg (2012)
Martinetti, D., De Baets, B., Díaz, S., Montes, S.: On the role of acyclicity in the study of rationality of fuzzy choice functions (under revision)
Richardson, G.: The structure of fuzzy preference: Social choice implications. Social Choice and Welfare 15, 359–369 (1998)
Richter, M.K.: Revealed preference theory. Econometrica 34, 635–645 (1996)
Samuelson, P.A.: Foundation of Economic Analysis, 604 pages. Harvard University Press, Cambridge (1983)
Sen, A.K.: Choice functions and revealed preference. Review of Economic Studies 38, 307–317 (1971)
Sen, A.K.: Collective Choice and Social Welfare. Holden-Day, San Francisco (1970)
Suzumura, K.: Rational choice and revealed preference. Review of Economic Studies 46, 149–158 (1976)
Uzawa, I.: Preference and Rational Choice in the Theory of Consumption. In: Arrow, K.J., Karlin, S., Suppes, P. (eds.) Mathematical Methods in the Social Sciences. Stanford University Press, Stanford (1959)
Wang, X.: A note on congruence conditions of fuzzy choice functions. Fuzzy Sets and Systems 145, 355–358 (2004)
Wang, X., Wu, C., Wu, X.: Choice Functions in Fuzzy Environment: An Overview. In: Cornelis, C., Deschrijver, G., Nachtegael, M., Schockaert, S., Shi, Y. (eds.) 35 Years of Fuzzy Set Theory. STUDFUZZ, vol. 261, pp. 149–169. Springer, Heidelberg (2010)
Wu, C., Wang, X., Hao, Y.: A further study on rationality conditions of fuzzy choice functions. Fuzzy Sets and Systems 176, 1–19 (2011)
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Martinetti, D., Montes, S., Díaz, S., De Baets, B. (2013). Uncertain Choices: A Comparison of Fuzzy and Probabilistic Approaches. In: Bustince, H., Fernandez, J., Mesiar, R., Calvo, T. (eds) Aggregation Functions in Theory and in Practise. Advances in Intelligent Systems and Computing, vol 228. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39165-1_28
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DOI: https://doi.org/10.1007/978-3-642-39165-1_28
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