ORIGINAL PAPER A possibilistic approach to risk aversion Irina Georgescu Published online: 10 July 2010 Ó Springer-Verlag 2010 Abstract In this paper a possibilistic model of risk aversion based on the lower and upper possibilistic expected values of a fuzzy number is studied. Three notions of possibilistic risk premium are deﬁned for which calculation formulae in terms of Arrow–Pratt index and a possibilistic variance are established. A possibilistic ver- sion of Pratt theorem is proved. Keywords Possibilistic indicators  Possibilistic risk premium  Possibilistic Pratt theorem 1 Introduction Possibility theory was initiated by Zadeh (1965) as an alternative to probability theory. Probabilistic methods are not efﬁcient in the study of those uncertain situations in which phenomena occur with a small frequency. In such cases it is preferable to apply techniques offered by pos- sibility theory. The development of this domain was done by contributions of several authors, especially Dubois and Prade (1980, 1988). In foundation of possibility theory one started on a paralel line with probability theory. Random variables were replaced by possibilistic distributions and for probabilistic indicators (mean value, variance, and covariance) corre- spondents in possibilistic context were looked for. The transition for probabilistic to possibilistic indicators proved not to be easy. Mostly the indicators of fuzzy numbers, the most impostant class of possibilistic distri- butions were studied (see, e.g. Carlsson and Fulle ´r 2001; Dubois and Prade 1987; Fulle ´r and Majlender 2003; Majlender 2004; Thavaneswaran et al. 2009; Zhang and Nie 2003; Zhang and Whang 2007). Fuzzy numbers are a natural generalization of real numbers. Using Zadeh’s (1965) extension principle, operations with real numbers extend to fuzzy numbers. Properties of operations with fuzzy numbers lead to a rich algebraic structure which helps deﬁne some notions, prove results required in the study of uncertain phenomena. First an adequate concept of possibilistic mean value was looked for. Dubois and Prade (1987) introduced the interval-valued expectation of a fuzzy number, and Carlsson et al. (2005) deﬁned the notions of lower and upper mean values and the possibilistic mean value of a fuzzy number. The weighted possibilistic mean value of a fuzzy number was introduced by Fulle ´r and Majlender (2003). In the context of credibility theory (Liu and Liu 2002; Liu 2007), we ﬁnd a notion of expected value of an arbitrary possi- bilistic distribution. Secondly, starting from these possibilistic mean values one obtained more notions of possibilistic variances. In paper (Carlsson and Fulle ´r 2001), two concepts of possi- bilistic variances were deﬁned. These have been general- ized in case of a weighting function (Fulle ´r and Majlender 2003). The lower and upper possibilistic variances of a fuzzy number were studied in Fulle ´r and Majlender (2003). Other types of possibilistic variances are in Zhang and Nie (2003) and Zhang and Whang (2007). Risk aversion of an agent faced with an uncertain situ- ation is a topic treated especially with probabilistic meth- ods. In Georgescu (2009), an approach to risk aversion was tried in a possibilistic context, mathematically speciﬁed by a triple (A, u, f), formed by a fuzzy number A, a utility I. Georgescu (&) Department of Economic Cybernetics, Academy of Economic Studies, P. O. Box 15-432, Piat ¸a Romana No 6 R 70167, Oﬁciul Postal 22, 014700 Bucharest, Romania e-mail: irina.georgescu@csie.ase.ro 123 Soft Comput (2011) 15:795–801 DOI 10.1007/s00500-010-0634-7