# zbMATH — the first resource for mathematics

Possibilistic risk aversion. (English) Zbl 1186.91120
Summary: Risk theory is usually developed within probability theory. The aim of this paper is to propose an approach of the risk aversion by possibility theory, initiated by Zadeh in 1978. The main notion studied in this paper is the possibilistic risk premium associated with a fuzzy number $$A$$ and a utility function $$u$$. Under the hypothesis that the utility function $$u$$ verifies certain hypotheses, one proves a formula to evaluate the possibilistic risk premium in terms of $$u$$ and of some possibilistic indicators.

##### MSC:
 91B30 Risk theory, insurance (MSC2010) 60A86 Fuzzy probability
##### Keywords:
fuzzy number; risk premium; possibility theory; risk aversion
Full Text:
##### References:
 [1] Arrow, K.J., Essays in the theory of risk bearing, (1970), North-Holland Amsterdam · Zbl 0215.58602 [2] Billot, A., An existence theorem for fuzzy utility functions: a new elementary proof, Fuzzy sets and systems, 74, 271-276, (1995) · Zbl 0858.90014 [3] Campos, L.; Gonzalez, A., Further contributions to the study of average value for ranking fuzzy numbers, International journal of approximate reasoning, 10, 135-163, (1994) · Zbl 0798.90003 [4] Carlsson, C.; Fullér, R., Fuzzy reasoning in decision making and optimization, () · Zbl 1016.68111 [5] Carlsson, C.; Fullér, R., On possiblistic Mean value and variance of fuzzy numbers, Fuzzy sets and systems, 122, 315-326, (2001) · Zbl 1016.94047 [6] Carlsson, C.; Fullér, R.; Majlender, P., On possibilistic correlations, Fuzzy sets and systems, 155, 425-445, (2005) · Zbl 1085.94028 [7] C. Carlsson, R. Fullér, P. Majlender, Some normative properties of possibility distributions, in: Proc. Third Internat. Symp. of Hungarian Researchers in Computational Intelligence, Budapest, 2002, pp. 61-71. [8] I. Couso, D. Dubois, S. Montes, L. Sanchez, On various definitions of the variance of a fuzzy random variable, in: G. De Cooman, J. Vejnarova, M. Zaffalon (Eds.), Dans: Internat. Symp. on Imprecise Probability: Theories and Applications (ISIPTA 2007), Prague, Czech Republic, Sipta, 2007, pp. 135-144. [9] De Campos Ibane˜z, L.; Gonzalez-Muñoz, M., A subjective approach for ranking fuzzy numbers, Fuzzy sets and systems, 29, 145-154, (1989) · Zbl 0672.90001 [10] Djemane, K.; Gourlay, I.; Padgett, J.; Birkenheuer, G.; Novestadt, M.; Kao, O.; Voß, K., Introducing risk management into the grid, () [11] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049 [12] Dubois, D.; Prade, H., The Mean value of a fuzzy number, Fuzzy sets and systems, 24, 279-300, (1987) · Zbl 0634.94026 [13] D. Dubois, H. Fargier, J. Fortin, The empirical variance of a set of fuzzy intervals, in: Proc. of the 2005 IEEE Internat. Conf. on Fuzzy Systems (FUZZ-IEEE 2005), Reno, Nevada, 22-25 mai, IEEE Press, New York, pp. 885-890. [14] Dubois, D.; Prade, H., Possibility theory: an approach to computerized processing of uncertainty, (1988), Plenum Press New York [15] Fullér, R.; Majlenderm, P., On weighted possibilistic Mean and variance of fuzzy numbers, Fuzzy sets and systems, 136, 363-374, (2003) · Zbl 1022.94032 [16] Fullér, R., Introduction to neuro-fuzzy systems, advances in soft computing, (2000), Springer Berlin, Heidelberg [17] Gonzalez, A., A study of the ranking function approach through Mean values, Fuzzy sets and systems, 35, 29-43, (1990) · Zbl 0733.90003 [18] Kreinovich, V.; Xiang, G.; Ferson, S., Computing Mean and variance under dempster – shafer uncertainty: towards faster algorithms, International journal of approximate reasoning, 42, 3, 212-227, (2006) · Zbl 1103.68121 [19] Laffont, J.J., The economics of uncertainty and information, (1993), MIT Press Cambridge [20] Liu, B.; Liu, Y.-K., Expected value of fuzzy variable and fuzzy expected models, IEEE transactions on fuzzy systems, 10, (2002) [21] P. Majlender, A normative approach to possibility theory and decision support, Ph.D. Thesis, Turku Centre for Computer Science, 2004. [22] Mares, M., Fuzzy cooperative games, (2001), Physica Verlag Heidelberg · Zbl 1005.91013 [23] Pratt, J., Risk aversion in the small and in the large, Econometrica, 32, 122-130, (1964) · Zbl 0132.13906 [24] Quiggin, J., Generalized expected utility theory/the rank-dependent expected utility model, (1993), Kluwer-Nijhoff Amsterdam [25] Rothschild, M.; Stiglitz, J., Increasing risk: I. A definition, Journal of economic theory, 2, 225-243, (1970) [26] Zadeh, L.A., Fuzzy sets as a basis for a theory of possibility, Fuzzy sets and systems, 1, 3-28, (1978) · Zbl 0377.04002 [27] Zadeh, L.A., Fuzzy sets, Information and control, 8, 335-338, (1965) · Zbl 0139.24606
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.