Uncertain risk aversion

Transcription

1 J Intell Manuf (7) 8:65 64 DOI.7/s Uncertain risk aversion Jian Zhou Yuanyuan Liu Xiaoxia Zhang Xin Gu Di Wang Received: 5 August 4 / Accepted: 8 November 4 / Published online: 7 December 4 Springer Science+Business Media New York 4 Abstract This paper discusses the risk aversion within the framework of the uncertainty theory (Liu in Uncertainty theory: A branch of mathematics for modeling human uncertainty. Springer, Berlin, b), and introduces the notions of uncertain expected utility and uncertain risk premium. In terms of the Arrow Pratt index, an uncertain version of Pratt s theorem is proved, which offers an effective way to make comparisons between different individuals risk-averse attitudes. We suggest that uncertain risk aversion can be used to measure human s risk-averse attitudes when uncertainty exists due to lack of the observed data, just as probabilistic risk aversion when sufficient data can be obtained. Uncertain risk aversion provides an alternative method to compare the risk aversions between individuals under uncertain situations. Keywords Uncertainty theory Risk aversion Risk premium Pratt s theorem J. Zhou Y. Liu (B) School of Management, Shanghai University, Shanghai 444, China liuyuanyuan@shu.edu.cn X. Zhang Business School, University of Sydney, Sydney, NSW 6, Australia X. Gu School of Law and Economics, University of Mannheim, Mannheim 683, Germany D. Wang Department of Mathematics, Rutgers, The State University of New Jersey, Camden, NJ 8854, USA Introduction In economics, finance, and psychology, risk aversion is a widely used concept that describes an attitude that individuals are always attempting to accept an amount of certain income rather than an even higher expected income associated with uncertain risks. Probability measure is the standard measure in risk aversion, risk premium and relative risk premium, because they were initiated through this measure by Arrow (97) and Pratt (964). Then Laffont (989) provided a generalized idea and made a fundamental contribution for risk aversion. Afterwards many scholars and researchers have extensively applied risk aversion to theoretical research, thanks to which some relevant anomalies were found (see, e.g., Rabin and Thaler ). Up to now, phenomena of risk aversion have been widely considered in the fields of lottery choice (Holt and Laury ), optimal portfolio choice (Dow and Werlang 99), asset return and consumption (Hansen and Singleton 983), scale transformation (Rabin and Thaler ), product development (Yang et al. 4), etc. Based on the possibility theory, established by Zadeh (978) and serving as an alternative theory of the probability theory to model indeterminacy (phenomena where outcomes cannot be exactly predicted in advance), concepts of the possibilistic risk premium and the relative possibilistic risk premium were put forward by Georgescu (9, ). Following that, some further investigations on multidimensional possibilistic risk aversion and risk aversion with many parameters were made in the literature (see, e.g., Georgescu and Kinnunen a, b). Risk aversion aims to evaluate individual s attitudes towards future risks and uncertain incomes. It is usually difficult to estimate the future income via statistics, because human behavior and unpredictable risks always lead to a lack 3

2 66 J Intell Manuf (7) 8:65 64 of the observed data in practice. In this case, the probability theory may not be applicable to measure such uncertain risks although it has been used in risk aversion for a long time. Besides, a lot of surveys showed that the subjective uncertainty cannot be modeled by fuzziness (Liu ). In order to deal with this problem, an uncertainty theory was founded by Liu (7, b) based on normality, duality, subadditivity and product axioms. In this paper, the risk aversion within the framework of the uncertainty theory is discussed. The main target of this paper is to present a framework of uncertain risk aversion on the basis of the uncertainty theory in Liu (b). Firstly, we introduce the uncertain expected utility (Yao and Ji 4) as the starting point of uncertain risk aversion. Then the concept of uncertain risk premium concerning the Arrow Pratt index is defined. Subsequently, the uncertain version of Pratt s theorem is proved. In conclusion, under the circumstances of uncertainty that the observed data is insufficient or lacking, uncertain risk aversion can be used to measure individual s riskaverse attitudes. And in such cases, it is more reasonable to use the uncertain risk aversion rather than the probabilistic risk aversion to make comparisons between individuals. The rest of this paper is organized as follows. Section Preliminaries reviews the fundamental knowledge on probabilistic risk aversion including the definition of risk premium, the Arrow Pratt index and Pratt s theorem as well as some related definitions of uncertain variables. Section Uncertain expected utility, concepts of uncertain expected utility together with some properties which will be employed in the following sections are discussed. Section Uncertain risk premium establishes the foundation of the theory of uncertain risk aversion. The concept of uncertain risk premium is defined associating with the utility function and uncertain variance. According to the Arrow Pratt index of utilities and uncertain variance, the computation of the uncertain risk premium is provided. Section An uncertain version of Pratt s theorem obtains the main result of this paper, an uncertain version of Pratt s theorem. With this theorem, the uncertain risk aversions between two individuals are compared. Probabilistic risk premium Let Ω be a nonempty set, A a σ -algebra over Ω, and Pr a probability measure. Then the triplet (Ω, A, Pr) is a probability space. The sets in A represent the events. A random variable η is a measurable function from a probability space (Ω, A, Pr) to the set of real numbers. Assume that Ω =R and u :R Ris a continuous function representing the utility of people. Then u(η) is also a random variable. Denote the expected value of η and the expected utility of η by E[η] and E[u(η)], respectively. We have E[u(η)] = + u(x)dφ(x) provided that the integral + u(x)dφ(x) is finite, where Φ :R [,] is the probability distribution of η. Definition (Pratt 964)Let η be a random variable defined on (R, A, Pr) and u :R Ra utility function. The risk premium ρ η,u is defined by the following equality E[u(η)] =u(e[η] ρ η,u ). () It should be noted that as an indicator of risk aversion, the risk premium ρ η,u here is in fact the probabilistic risk premium. Theorem (Pratt 964) Assume that u is twice differentiable, strictly concave and strictly increasing. Then ρ η,u V [η]u (E[η]) u (E[η]) where V [η] is the variance of η. Definition (Arrow 97; Pratt 964) The Arrow Pratt index associated with the utility function u is a function r u :R Rdefined by r u (x) = u (x) u (x) for any x R. The Arrow Pratt index r u is also termed as the absolute risk aversion, the coefficient of absolute risk aversion, or risk aversion. Furthermore, it follows immediately from Theorem that () (3) ρ η,u V [η]r u(e[η]). (4) Preliminaries Arrow (97) and Pratt (964) proposed risk aversion in the context of probability theory, and later Laffont (989) put forward a summarized version of risk aversion. In the first part of this section, we will review some notions and results corresponding to the probabilistic means. The second part will introduce the uncertainty theory which provides the foundation of uncertain risk aversion. 3 Theorem (Pratt 964, Pratt s Theorem) Let r ui, ρ i be the Arrow Pratt indices, risk premiums corresponding to the utility functions u i,i =,, respectively. Then the following assertions are equivalent: (a) r u (x) r u (x) for each x R; (b) u u is concave; (c) ρ ρ.

3 J Intell Manuf (7) 8: Besides the risk premium ρ η,u, there is another probabilistic indicator of risk aversion, i.e., the relative risk premium ˆρ η,u, which was defined in Pratt (964) by the equality E[u(η)] =u(e[η]( ˆρ η,u )). (5) Moreover, in Pratt (964), it was shown that ρ η,u = E[η]ˆρ η,u. (6) Uncertainty theory Uncertainty theory is a branch of axiomatic mathematics to deal with human uncertainty. It has been applied to a wide range of areas, such as uncertain programming (Liu 9b; Zhong et al. 4; Zhou et al. 4b), uncertain statistics (Chen and Ralescu ; Liu b), uncertain differential equations (Yao 3a, b), uncertain process (Yao and Li ), uncertain control (Liu a; Zhu ), uncertain graph (Zhang et al. 3; Zhou et al. 4a), uncertain network (Liu 4; Sheng and Gao 4), uncertain game (Yang and Gao 3, 4), uncertain finance (Liu 9a, 3; Peng and Yao ), etc. The emphasis in this section is mainly related to definitions of uncertain variables. Definition 3 (Liu 7)Let L be a σ -algebra on a nonempty set Γ.AsetfunctionM : L [,] is called an uncertain measure if it satisfies the following axioms: Axiom (Normality Axiom) M{Γ }= for the universal set Γ ; Axiom (Duality Axiom) M{Λ}+M{Λ c }= for any event Λ; Axiom 3 (Subadditivity Axiom) For every countable sequence of events Λ,Λ,, we have { } M Λ i M{Λ i }. (7) i= i= Moreover, let (Γ k, L k, M k ) be uncertainty spaces for k =,,. Denote Γ = Γ Γ, L = L L. (8) Then the product uncertain measure M on the product σ - algebra L is defined by the following axiom (Liu 9a). Axiom 4 (Product Axiom) Let (Γ k, L k, M k ) be uncertainty spaces for k =,,... The product uncertain measure M is an uncertain measure satisfying { } M Λ k = M k {Λ k } (9) k= k= where Λ k are arbitrarily chosen events from L k for k =,,..., respectively. Definition 4 (Liu 7) An uncertain variable ξ is a measurable function from an uncertainty space (Γ, L, M) to the set of real numbers, i.e., for any Borel set B of real numbers, the set {ξ B} ={γ Γ ξ(γ) B} () is an event. Definition 5 (Liu 7) The uncertainty distribution Φ of an uncertain variable ξ is defined by Φ(x) = M{ξ x} () for any real number x. Definition 6 (Liu b) An uncertainty distribution Φ(x) is said to be regular if it is a continuous and strictly increasing function with respect to x at which <Φ(x) <, and lim Φ(x) =, lim x Φ(x) =. () x + The inverse function Φ (α) is called the inverse uncertainty distribution of an uncertain variable ξ if it exists and is unique for each α (,). The inverse uncertainty distribution plays a crucial role in operations of independent uncertain variables. It should be noted that according to the product axiom (Axiom 4) and the definition of independence for uncertain variables, the operational law of uncertain variables is absolutely different from that of random variables by means of inverse uncertainty distributions. Definition 7 (Liu 9a) The uncertain variables ξ, ξ,..., ξ n are said to be independent if { n } M {ξ i B i } = i= n M{ξ i B i } (3) i= for any Borel sets B, B,...,B n of real numbers. Theorem 3 (Liu b) Let ξ, ξ,..., ξ n be independent uncertain variables with regular uncertainty distributions Φ, Φ,..., Φ n, respectively. If the function f (x, x,...,x n ) is strictly increasing with respect to x, x,...,x m and strictly decreasing with respect to x m+, x m+,, x n, then the uncertain ξ = f (ξ,ξ,...,ξ n ) (4) has an inverse uncertainty distribution ( Ψ (α) = f (α),..., Φ Φ m (α), Φ m+ ( α),..., Φ n ( α) ). (5) 3

4 68 J Intell Manuf (7) 8:65 64 Uncertain expected utility As seen in section Preliminaries, the risk premium proposed by Arrow (97) and Pratt (964) used the expected utility on the basis of the probability theory. With the help of the expected utility theory which serves as a foundation in the theory of risk aversion (see, e.g., Von Neumann and Morgenstern 953; Quiggin 993; Rabin ), risk aversion provides a way of comparing risk-averse attitudes of people. In this section, concerning to define the uncertain risk premium, we present the uncertain expected utility first. Let ξ be an uncertain variable defined on the uncertainty space (Γ, L, M). Then the expected value of ξ is defined by Liu (7) as + E[ξ] = M{ξ r}dr M{ξ r}dr (6) provided that at least one of the two integrals is finite. Besides, if ξ is an uncertain variable with regular uncertainty distribution Φ, Liu (b) showed that E[ξ] = Φ (α)dα. (7) Furthermore, the variance of ξ is defined by Liu (7) as V [ξ] =E [ (ξ E[ξ]) ]. (8) As to an uncertain variable ξ with regular uncertainty distribution Φ and finite expected value E[ξ], Yao (4) proved that ( V [ξ] = Φ (α) E[ξ]) dα. (9) In addition, Liu (b) has also proved the linearity of expected value operator. That is, E[aξ + bη] = ae[ξ]+be[η] () holds for any real numbers a and b, and any independent uncertain variables ξ and η with finite expected values. Now let us discuss a continuous utility u. It is clear that u(ξ) is also an uncertain variable for any uncertain variable ξ. Thus the uncertain expected utility of ξ is defined as follows. Definition 8 (Yao and Ji 4) Let u be a continuous utility function for any uncertain variable ξ. Then the uncertain expected utility of ξ, i.e., the expected value E[u(ξ)],is defined by E[u(ξ)] = + M{u(ξ) r}dr M{u(ξ) r}dr. () Theorem 4 Let u :R R + be a nonnegative continuous function. Then E[u(ξ)] holds for any uncertain variable ξ. Proof From the definition of E[ξ], wehavee[ξ] = + M{ξ r}dr M {ξ r}dr. Ifξ, M{ξ r}dr equals to zero and + M{ξ r}dr. Thus we have E[ξ]. Since u is a nonnegative continuous function and also an uncertain variable for any ξ, we obtain E[u(ξ)]. The result from Theorem 4 will be directly applied in the proof of Theorem 5. Theorem 5 Let g :R Rand h :R Rbe two continuous strictly monotone functions such that g(x) h(x) for any real number x R. Then E[g(ξ)] E[h(ξ)] holds for any uncertain variable ξ with regular uncertainty distribution. Proof Let us denote the uncertainty distributions of ξ, g(ξ) and h(ξ) by Φ, Ψ and Ψ, respectively. Without loss of generality, assume g is strictly increasing, and h is strictly decreasing. Then it follows from Theorem 3 that g(ξ) and h(ξ) are uncertain variables with regular uncertainty distributions, and their inverse uncertainty distributions are ( ) Ψ (α) = g Ψ (α) = h Φ (α) ( Φ ( α), () ), (3) respectively, where Φ is the inverse uncertainty distribution of ξ. Based upon Eq. (7), we have E[h(ξ)] E[g(ξ)] = = = = Ψ (α)dα Ψ h ( Φ ( α) ) dα h ( Φ (α) ) dα (α)dα g ( Φ (α) ) dα g ( Φ (α) ) dα ( h ( Φ (α) ) g ( Φ (α) )) dα. (4) Besides, since g(x) h(x) holds for any real number x R, then for any α [,],wehaveh(φ (α)) g(φ (α)), which may deduce based on Theorem 4 that E[h(ξ)] E[g(ξ)]. (5) As seen below, Theorem 6 will use the result from Theorem 5 in the process of deduction. 3

5 J Intell Manuf (7) 8: Theorem 6 Let u :R Rbe a convex, continuous and strictly monotone function, and v :R Ra continuous and strictly monotone function. Then u(e[v(ξ)]) E[u v(ξ)] holds for any uncertain variable ξ with regular uncertainty distribution. Proof From real analysis we know that if u is convex, then there must exist two sequences of real numbers {a n } and {b n } such that u(x) = sup(a n x + b n ) (6) n holds for all x R. For each natural number n, consider the function w n :R Rdefined by w n (x) = a n v(x) + b n. (7) According to Eqs. (6) and (7), it is easy to deduce that w n (x) = a n v(x) + b n sup(a n v(x) + b n ) = u(v(x)) n = u v(x). (8) Since u(x) and v(x) are two strictly monotone functions, it is clear that a n v(x)+b n and u v(x) are also strictly monotone. Hence for any uncertain variable ξ with regular uncertainty distribution, it follows from Theorem 5 that E[w n (ξ)] E[u v(ξ)]. (9) Owing to the linearity of expected value operator [see Eq. ()], we get a n E[v(ξ)]+b n E[u v(ξ)]. (3) Since the inequality (3) holds for any natural number n, it follows that Then we have u(e[v(ξ)]) E[u v(ξ)], and hence u(e[v(ξ)]) E[u v(ξ)]. Theorem 7 clarifies the relationship between the expected value operator and the utility function, which plays an essential role in the demonstration of the uncertain version of Pratt s theorem in section An uncertain version of Pratt s theorem. Uncertain risk premium In this section, we launch the uncertain risk aversion represented by the uncertain risk premium and the relative uncertain risk premium as well as the calculation methods. Let us start from an example of the uncertain risk aversion as shown in Fig.. Suppose that a risk-averse agent with a starting wealth $, plans to make an investment with this amount of money. In order to simplify the discussion, assume there are only two investment choices for the agent, i.e., Treasure bills (T-bills) with a -month return rate of 5 % and a portfolio of new issued Mutual fund (although actually there are plenty of investment products that people can consider). T-bills can be regarded as a risk-free investment, while the Mutual fund is likely to bring him a higher return after -month holding period, say $,, i.e., the return rate of months is %. However, due to the lack of historical data of the new launched Mutual fund, it is accompanied with uncertain risks. Here we consider the risks of u(e[v(ξ)]) = sup(a n E[v(ξ)]+b n ) E[u v(ξ)]. (3) n Theorem 6 will be immediately applied to obtain the following conclusion in Theorem 7. Theorem 7 Let u :R Rbe a concave, continuous and strictly monotone function, and v :R Ra continuous and strictly monotone function. Then E[u v(ξ)] u(e[v(ξ)]) holds for any uncertain variable ξ with regular uncertainty distribution. Proof Since u is concave, then u is a convex function. Besides, both u and v are strictly monotone functions. It follows from Theorem 6 that for any uncertain variable ξ with regular uncertainty distribution, u(e[v(ξ)]) E[ u v(ξ)]. Fig. Background example 3

6 6 J Intell Manuf (7) 8:65 64 receiving variable returns to be uncertain (instead of random risks) due to the absence of historical data about the future uncertain returns (thus the probability distribution of the risks in obtaining varied returns cannot be estimated in this situation). Naturally, we suppose the uncertain income is an uncertain variable. Moreover, this agent considers two choices of such similar degrees of satisfaction that he feels indifferent to choose either one. In the example, the degree of the agent s satisfaction is represented by the value of u, a twice differentiable, strictly concave and strictly increasing utility function. The conditions of u ensure the uncertain version of the Arrow Pratt index to be defined. The higher but uncertain income coming from the mutual fund is denoted by an uncertain variable ξ associated with an uncertain risk ξ E[ξ]. That is to say, the uncertain income ξ is constituted by the uncertain part (the uncertain risk ξ E[ξ]) and the certain part (the expected value of ξ, i.e., E[ξ]). What s more, it is easy to deduce that E [ξ E[ξ]] = E[ξ] E[ξ] =, (3) which implies that the expected value of the uncertain risk is zero. In other words, the risk of losing and earning the money is equivalent. It should be noted that the uncertain income ξ is assumed to have a regular uncertainty distribution in our paper. Starting from Pratt s definition of probabilistic risk aversion (see Definition ), we consider the current example of two equivalent satisfaction choices the agent is facing. Suppose that a utility function u is given. By replacing the random variable η and the expected utility E[u(η)] with an uncertain variable ξ and an uncertain expected utility E[u(ξ)], respectively, we give the definition of the uncertain risk premium. Definition 9 The uncertain risk premium ρ(ξ, u) of ξ with respect to u is defined by means of the following equality u(e[ξ] ρ(ξ,u)) = E[u(ξ)]. (33) It follows from the injectivity of u that the uncertain risk premium ρ(ξ,u) is uniquely defined. In the above example, the uncertain risk premium can be regarded as the compensation for undertaking the uncertain risk, in other words, the price paid for bearing the risk. By means of the expected utility E[u(ξ)], the uncertain risk premium ρ(ξ,u) implies the agent s degree of risk aversion. The bigger ρ(ξ,u) is, the more risk-averse the agent is. In order to get the computational formula of the uncertain riskpremiumρ(ξ,u) that can be approximately indicated by a linear function of the Arrow Pratt index r u for a fixed ξ, our attention is restricted to small values of uncertain risk ξ E[ξ]. Upon the definition of the uncertain risk premium, the following proposition will establish the computational formula of ρ(ξ, u) within the framework of the uncertainty theory. Proposition Assume that the utility function u is twice differentiable, strictly concave and strictly increasing. Then ρ(ξ,u) V u (E[ξ]) r u (E[ξ]) where V r is the variance of the uncertain income ξ. (34) Proof Since u is twice differentiable, therefore upon the application of Taylor s formula of the second degree, expanding u around E[ξ],wehave u(x) = u(e[ξ]) + u (E[ξ])(x E[ξ]) + u (E[ξ]) (x E[ξ]) + o(x E[ξ]). (35) Evaluating Eq. (35) atx = ξ, we obtain u(ξ) = u(e[ξ]) + u (E[ξ])(ξ E[ξ]) + u (E[ξ]) (ξ E[ξ]) + o(ξ E[ξ]). (36) Since μ is strictly increasing and ξ is an uncertain variable with regular uncertainty distribution, there exist inverse uncertainty distributions of μ(ξ) and ξ, denoted as Ψ Φξ u(ξ) and, respectively. Based on Theorem 3 and Eq. (36), for any α [,], wehave Ψ u(ξ) (α) = u(φ ξ (α)) = u(e[ξ]) + u (E[ξ])(Φξ (α) E[ξ]) + u (E[ξ]) (Φ ξ (α) E[ξ]) + o(φξ (α) E[ξ]). (37) Subsequently, it can be derived immediately that Ψ u(ξ) (α)dα = u(e[ξ])+u (E[ξ]) + u (E[ξ]) + (Φξ (Φξ (α) E[ξ])dα (α) E[ξ]) dα o(φ ξ (α) E[ξ]) dα. (38) According to Eqs. (7), (3) and (9), we can derive that E[μ(ξ)] =μ(e[ξ]) + u (E[ξ]) V r + o(v r ). (39) Since o(v r ) is negligible compared to V r, we eliminate the error rest o(v r ) in Eq. (39) and obtain E[u(ξ)] u(e[ξ]) + u (E[ξ]) V r. (4) Furthermore, by writing Taylor s formula for u(e[ξ] ρ(ξ,u)), considering V r is sufficiently small, we retain only the first two terms and then get u(e[ξ] ρ(ξ,u)) u(e[ξ]) u (E[ξ])ρ(ξ, u). 3

7 J Intell Manuf (7) 8: It follows from the definition of the uncertain risk premium (see Definition 9) that E[u(ξ)] =u(e[ξ] ρ(ξ,u)) u(e[ξ]) u (E[ξ])ρ(ξ, u). (4) We can identify the right hand sides of Eqs. (4) and (4), and then obtain ρ(ξ,u) V r u (E[ξ]) u (E[ξ]). Remark Pratt (964) proposed and proved risk aversion function with an error rest, and then Laffont (989) used the equation ignoring the error rest in the deduction. In this paper, we treat the error rest in the same way that Laffont adopted, and then utilize the approximation equality where the approximation in itself does not have a meaningful effect on the uncertain risk aversion. Recall the Arrow Pratt index defined in Definition which serves as a measure of risk aversion, and take it into the formula of Proposition. Then we obtain the following proposition. Proposition Assume that a utility function u satisfying the conditions in Proposition is given. Then ρ(ξ,u) V rr u (E[ξ]) (4) where r u is the Arrow Pratt index defined in Definition. Equation (4) implies that the agent will be indifferent between bearing and refusing an uncertain risk when the compensation of undertaking the risk, i.e., the uncertain risk premium, is approximately r u (E[ξ]) times half the variance of the uncertain risk ξ E[ξ]. Hence r u (E[ξ]) can be interpreted as twice the compensation the agent requires per unit of variance for the uncertain risk. It should be noted that although the Arrow Pratt index r u was initiated as an indicator of probabilistic risk aversion, it can also serve as the measure of uncertain risk aversion and be used to compare uncertain risk aversions between different people. The following notion of the relative uncertain risk premium is inspired from the relative probabilistic risk premium. Definition The relative uncertain risk premium ˆρ(ξ,u) of ξ with respect to u is defined by means of the following equation u(e[ξ]( ˆρ(ξ,u))) = E[u(ξ)]. (43) The relative uncertain risk premium corresponds to a relative income. That is, an agent represented by a utility function u would be indifferent between receiving an uncertain income ξ with the relative uncertain risk ξ E[ξ] E[ξ] and receiving the certain amount of money E[ξ] ˆρ(ξ,u)E[ξ]. Proposition 3 Assume that a utility function u satisfying the conditions in Proposition is given. Then ˆρ(ξ,u) V r E[ξ]u (E[ξ]) u (E[ξ]) (44) where Vr ξ E[ξ] is the variance of the relative uncertain risk E[ξ]. Proof Since Vr is the variance of the relative uncertain risk ξ E[ξ] E[ξ], and V r is the variance of the uncertain risk ξ E[ξ] from Proposition, it follows immediately that V r = E [ξ]v r. (45) From the definition of the uncertain risk premium (see Definition 9) and the relative uncertain risk premium (see Definition ), we have E[ξ]ˆρ(ξ,u) = ρ(ξ,u) V u (E[ξ]) r u (E[ξ]) = V r E [ξ] u (E[ξ]) u (E[ξ]), (46) which can arrive at ˆρ(ξ,u) V r (E[ξ]) E[ξ]u u (E[ξ]). An uncertain version of Pratt s theorem In the field of risk aversion, Pratt s theorem characterizes the way we can compare risk aversions of two agents. To make comparison of risk-averse attitudes between agents under uncertainty, we introduce an uncertain version of Pratt s theorem which reveals the relations among the Arrow Pratt indices, utility functions and uncertain risk premiums. In this section, the uncertain version of Pratt s theorem is established and proved. Theorem 8 Let r ui, ρ(ξ,u i ) be the Arrow Pratt indices, uncertain risk premiums corresponding to the utility functions u i,i =,, respectively, where u i,i =,, are twice differentiable, strictly concave and strictly increasing. Then the following assertions are equivalent: (i) r u (x) r u (x) for any x R; (ii) u u is concave; (iii) ρ(ξ,u ) ρ(ξ,u ) for any uncertain variable ξ. Proof (i) (ii). It follows immediately from Pratt s theorem. (ii) (iii). For any given uncertain variable ξ, let us denote ρ i = ρ(ξ,u i ), i =,. According to the definition of the uncertain risk premium (see Definition 9), we get u (E[ξ] ρ ) = E[u (ξ)] (47) 3

8 6 J Intell Manuf (7) 8:65 64 and u (E[ξ] ρ ) = E[u (ξ)]. (48) Since u is a strictly increasing and continuous function, we can get the inverse of u, i.e., u. By operating the inverse u of u to both sides of the Eq. (47), we have u (E[u (ξ)]) = u u (E[ξ] ρ ) = E[ξ] ρ, (49) which subsequently infers ρ = E[ξ] u (E[u (ξ)]). (5) Similarly, we have ρ = E[ξ] u (E[u (ξ)]). (5) By subtracting the Eqs. (5) and (5), we obtain ρ ρ = u (E[u (ξ)]) u (E[u (ξ)]). (5) Because u and u are strictly increasing, it is ascertained that the inverse u as well as u u is also strictly increasing. In addition, since u u is concave, Theorem 7 is employed to demonstrate that E[u (ξ)] =E[u u u (ξ)] u u (E[u (ξ)]). (53) Similarly, because u is strictly increasing, it is easy to know that the inverse u is also strictly increasing. Thus we attain the equation ( ) u (E[u (ξ)]) u u u (E[u (ξ)]) = u (E[u (ξ)]). (54) Back to Eq. (5), ρ ρ is verified. (iii) (i). For any given real number x R,itiseasy to prove that there exists an uncertain variable ξ such that x = E[ξ]. According to the computational formula (4) of the uncertain risk premium, we have ρ(ξ,u ) /V r r u (x) and ρ(ξ,u ) /V r r u (x). Since V r and ρ(ξ,u ) ρ (ξ, u ),wehaver u (x) r u (x). Remark According to Pratt s theorem, the assertion (ii) is equivalent to the statement that u is more concave than u, or u = g u with g >, g <, which gives an overall description of utility functions. Recall that in the probabilistic risk aversion, the binary relation Pr is defined on the set of utility functions, which is generally used to compare probabilistic risk aversions between two agents. If condition (c) of Theorem holds, meaning that agent is more risk-averse than agent, then we denote u Pr u. Similarly, as for the uncertain risk aversion, let us define the symbol M which represents a way of comparing uncertain risk aversions between two agents. We say that the relation u M u exists if condition (iii) of Theorem 8 holds, meaning that the agent with the utility function u is more risk-averse than the agent with the utility function u. Pratt s theorem is of great importance in comparing risk aversions on the basis of the probability theory when data can be obtained with big frequency. In reality, however, people cannot always get the probability distribution of future variable income associated with uncertain risks due to the lack of observed data. Under such circumstances, the uncertain version of Pratt s theorem provides an alternative for people to make those comparisons between different risk-averse attitudes. The following example is presented to illustrate the application of Theorem 8. Example Assume that there are two risk-averse agents. The objective is to compare the different risk-averse attitudes between them. It should be noted that Pratt s theorem (see Theorem ) and the uncertain version of Pratt s theorem (see Theorem 8) have common conditions (a), (b), and (i), (ii), respectively. Therefore condition (c) of Pratt s theorem has the similar implication with condition (iii) of Theorem 8, however, under different assumptions. Therefore, the utility functions together with the corresponding Arrow Pratt indices may be utilized. Suppose that agent and agent have the utility functions u and u, respectively. For example, take u (x) = e c x and (55) u (x) = e c x, (56) where c > and c >. From the definition of the Arrow Pratt index (see Definition ), we have r u (x) = u (x) u (x) = c e cx c e c x and r u (x) = u (x) u (x) = c e cx c e c x = c (57) = c, (58) 3

9 J Intell Manuf (7) 8: respectively. Without loss of generality, we assume that c > c. Then according to Eqs. (57) and (58), we obtain the following relation r u (x) >r u (x). (59) Now let us discuss this example with different assumptions. On the one hand, if we have enough data to derive the probability distributions of future incomes of the two agents, Pratt s theorem is used to conclude that agent is more riskaverse than agent, denoted by u Pr u. On the other hand, if there is not enough data for us to obtain the probability distributions, the experts knowledge may be employed to obtain the uncertainty distributions of future incomes of agent and agent. In this situation, Pratt s theorem is no more effective, and alternatively the uncertain version of Pratt s theorem, i.e., Theorem 8, can be used to draw a conclusion that agent is more risk-averse than agent, denoted by u M u. Although the conclusions are analogous, they have different interpretations referring to different frameworks. Conclusion The risk aversion was measured in the settings of uncertainty in this paper. Under the uncertainty theory, we firstly defined the uncertain expected utility, the uncertain risk premium, and the relative uncertain risk premium. Furthermore, we put forward an uncertain version of Pratt s theorem. Finally, the associations between Pratt s theorem under uncertainty and probability had been discussed and illustrated with a practical example. It was shown that when people have enough data to obtain the probability distribution of future income, we should use the probabilistic version of Pratt s theorem. However, when people do not have access to the data needed for the probabilistic estimation, it is more appropriate to use the uncertain version of Pratt s theorem to compare the risk aversions of different people. 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