Theoretical Foundation of Uncertain Dominance

Theoretical Foundation of Uncertain Dominance Yang Zuo, Xiaoyu Ji 2 Uncertainty Theory Laboratory, Department of Mathematical Sciences Tsinghua University, Beijing 84, China 2 School of Business, Renmin University of China, Beijing 872, China zuoy7@mails.tsinghua.edu.cn, xyji@ruc.edu.cn Abstract: In this paper, uncertain dominance is introduced, which is a new ranking criterion for uncertain variables. Some properties of uncertain dominance are investigated. Necessary and sufficient conditions for lower order uncertain dominance are given, and for higher order uncertain dominance, we give two necessary conditions. And we give an example of portfolio for its application. Keywords: uncertain variable, uncertain dominance, uncertainty theory, uncertainty distribution function. Introduction There are various types of uncertainty. Randomness reflects objective uncertainty, and probability theory is a branch of mathematics for studying random phenomena. In another perspective, fuzziness decribes subjective uncertainty, and Zadeh [22] initiated fuzzy theory via membership function in 965. As further development in fuzzy theory, Liu and Liu [] introduced credibility measure in 22. Then based on this measure, Liu [2] proposed credibility theory in 24, which is considered as a branch of mathematics for studying fuzzy phenomena. In order to give a better characterization of subjective uncertainty, Liu [3] proposed uncertainty theory in 27, which is a branch of mathematics based on normality, monotonicity, self-duality, countable subadditivity, and product measure axioms. From then on, uncertainty theory entered a period of rapid development. Gao [], Liu [6, 7], You [6] and Peng [6] investigated on the properties of uncertain variables and uncertainty measure, besides Chen [5] studied on the application of operational law. Furthermore, some new sub-disciplines of uncertainty theory appear. Li and Liu [9] proposed uncertain logic. Then Gao [2] made some research on inference rule, and Chen [3] studied on the truth value in uncertain logic. The concept and foundations of uncertain process were given by Liu [4], and Peng [8] applied it to finance. In 94s, stochastic dominance was introduced in decision theory, and Fishburn and Vickson [7] founded the theorical foundation of stochastic dominance in 979. In Proceedings of the Seventh International Conference on Information and Management Sciences, Urumchi, China, August 2-9, 29, pp. 827-832. another paper of Fishburn [9], he studied the relation between stochastic dominance and moments. In 982, Bawa [8] wrote a research bibliography of stochastic dominance including about 4 papers. In 25, Peng [2] introduced fuzzy dominance. Then Jiang and Peng [2] used second order fuzzy dominance to order trapezoidal fuzzy variables. Inspired by stochastic dominance and fuzzy dominance, we propose the concept of uncertain dominance, and investigate some related properties. The paper proceeds as follows: In Section 2, we recall some basic concepts and theorems in uncertainty theory. The concept of uncertain dominance along with corresponding properties are given in Section 3. In Section 4, we present necessary and sufficient conditions for lower order uncertain dominance and necessary conditions for higher order uncertain dominance. In Section 5, we use uncertain dominance to solve portfolio problem. 2 Preinaries In this section, we introduce some foundamental concepts of uncertainty theory, which is used throughout this paper. Definition 2.. Liu [3] Let Γ be a nonempty set, and let L be a σ-algebra over Γ. Each element Λ L is called an event. A function M: L R is called an uncertain measure if it satisfies the following four axioms:. Normality M{Γ}. 2.Monotonicity M{Λ } M{Λ 2 } whenever Λ Λ 2. 3. Self-Duality M{Λ} + M{Λ c } for any event Λ. 4. Countable Subaditivity For every countable sequence of events {Λ i }, we have { } M Λ i M{Λ i }. i i Definition 2.2. Liu [3] An uncertain variable 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 { B} {γ Γ γ B} is an event. Its uncertainty distribution Φx : R [, ] is defined by Φx M{γ Γ γ x}.

828 YANG ZUO, XIAOYU JI Its expected value is defined by E[] M{ r} dr M{ r} dr provided that at least one of the two integrals is finite. Definition 2.3. Liu [5] The uncertain variables, 2,..., m are said to be independent if { m } M { i B i } min M{ i B i } i m i for any Borel sets B, B 2,..., B m of real numbers. Theorem 2.4. Liu [7] Let be an uncertain variable whose uncertainty distribution Φ satisfies Φx, x Φx. If fx is a monotone function such that the Lebesgue- Stieltjies integral is finite, then we have E[f] fx dφx fx dφx. Definition 2.5. Liu [3] Let be an uncertain variable, and α, ]. Then sup α sup{r M{ r} α} is called the α-optimistic value to, and inf α inf{r M{ r} α} is called the α-pessimistic value to. 3 Definition and Properties of Uncertain Dominance 3. Definition of Uncertain Dominance Let be an uncertain variable with uncertainty distribution function Φx. Then let Φ k+ x Φ x Φx, Φ k tdt for k, 2, 3,.... In this paper, we simply focus on the uncertain variables with uncertainty distribution function Φx satisfying Φx, x Φx and the Lebesgue-Stieltjies integral x dφx 2 is finite. So the following conclusions all base on Equation and Equation 2. Definition 3.. Let L k be a space of uncertain variables such that L k { E[ k ] < } for k, 2, 3,.... The norm in L k is defined as k E[ k ] /k. 3 Definition 3.2. Let [ be an uncertain variable in L k. Then the E[] ] k expected value E + is called the kth semideviation of, and the k-th root E + [ k E[] ] k is called the kth standard semideviation of denoted by δ k. Definition 3.3. Let and η be two uncertain variables with uncertainty distribution functions Φx and Ψx, respectively. If Φ k x Ψ k x for all x R, we say that uncertainly dominates η in the k-th order, denoted by k η. If k η and η k, we say strictly uncertainly dominates η in the k-th order, denoted by k η. 3.2 Some Basic Properties of Uncertain Dominance Property 3.4. Let, η and ζ be uncertain variables in L k. Then we have a Reflexivity i for any i, 2,..., k +. b Transitivity i η and η i ζ implies i ζ for any i, 2,..., k +. c If i η, then i+ η for any i, 2,..., k. Proof. These are simple results of the definition of uncertain dominance. Property 3.5. The function Φ k is nondecreasing for k and strictly convex for k 2. And the function k Φ k+ is also a strictly convex function. Proof. These are simple results of the definition of Φ k. Theorem 3.6. Let k and let be an uncertain variable with an uncertainty distribution function Φx satisfying Φ k+ x < + for all x R. Then we have Φ k+ x E[x + k ] x + k k.

THEORETICAL FOUNDATION OF UNCERTAIN DOMINANCE 829 Proof. We proceed by induction, and the case k is proved in the following Assuming that we have Φ 2 x Φ k+ x xφx Φt dt t dφt x t dφt x t + dφt E[x + ] x +. Φ k x k! E[x + k ] k! x + k k, k! k! k! Φ k t dt k! E[t + k ] dt t s t s + k dφs dt t s + k dt dφs k x s + k dφs E[x + k ] x + k k. Then the theorem is proved. Theorem 3.7. Liu [3] Let p be a real number with p, and let and η be independent uncertain variables with E[ p ] < and E[ η p ] <. Then we have p E[ + η p ] p E[ p ] + p E[ η p ]. Corollary 3.8. Let be an uncertain variable in L k. Then we have Φ k+ x < for all x R. Proof. From Theorem 3.6 and Theorem 3.7, we have Φ k+ x E[x k +] E[ x k ] x + k E[ k ] k <. 4 Necessary and Sufficient Conditions In this part, the necessary and sufficient conditions of first and second order uncertain dominance are presented. Theorem 4.. Let and η be two uncertain variables and α, ]. Let inf α and η inf α denote the α-pessimistic values of and η, respectively. Then η if and only if inf α η inf α for all α, ]. Proof. Let Φx and Ψx be uncertainty distribution functions of and η, respectively. Since η, i.e., Φx Ψx for all x R, then for all α, ], we have inf α inf{ r Φr α} inf{ r Ψr α} η inf α. Conversely, assume that there exists some α, ] such that Φx > α > Ψx for some x, ]. From the definition of critical values, we have x < η inf α and x inf α. Then inf α < η inf α which contradicts to inf α η inf α. Therefore, Φx Ψx for all x R. That is, η. Let U and U 2 be spaces of functions as following U {u u, u are continuous and bounded, and u > }, U 2 {u u, u, u are continuous and bounded, and u >, u < }. Theorem 4.2. η if and only if E[u] E[uη] for all u U. 2 η if and only if E[u] E[uη] for all u U 2. Proof. Because the function u is continuous and u > for every x R, we have E[u] Then we can get M{u r} dr M{ u r} dr Φ u r dr E[u] E[uη] Φ u r dr M{u r} dr Ψ u r dr Ψ u r Φ u r dr M{ u r} dr Φ u r dr. 4 Φ u r dr Ψ u r dr Ψr Φr u r dr. 5

83 YANG ZUO, XIAOYU JI Since Φx Ψx and u x > for every x R, we have E[u] E[uη]. Therefore we proved the sufficiency part of Theorem 4.2. Conversely, we assume that there exists some x that Φx > Ψx. Because Φx and Ψx are right continuous, we can find a continuity point of both Φx and Ψx denoted by x, such that Φx > Ψx. Now we get Φx Ψx [ Φx ] [ Ψx ] E[f x ] E[f x η] <, 6 where f x is the step-function, if x > x f x x /2, if x x, if x < x. We can find a series of functions u n U with u n x f x x for all x R. For example, let u n x 2 + 2 x x x x 2 + n 2. 7 2 Thus we have E[u n] E[f x ], E[u nη] E[f x η]. Therefore there exists some n satisfying E[u n ] E[u n η] <, which contradicts to E[u] E[uη] for all u U. Then the assumption is wrong. From Equation 5, we have where and E[u] E[uη] D xu x dx u x d D 2 x D 2 + u + u xd 2 x dx, 8 D k x Ψ k x Φ k x for k, 2, D 2 + D2 x, u + u x. Since D 2 x, u x and u x, we have E[u] E[uη]. Conversely, we assume that there exists some x that D 2 x <. For Φx and Ψx right continuous, we can find a continuity point of both Φx and Ψx denoted by x, such that D 2 x <. Now D 2 x Ψx Φx dx x x Ψx Φx + x dψx dφx x x dψx dφx E[g x ] E[g x η], 9 where g x is g x x { x x, if x x, if x > x. We can find a series of functions u n U 2 with u nx f x x for all x R. Assuming that u n x 2 x x + 2 we have E[u n] E[g x ], x x 2 + n 2 2, E[u nη] E[g x η]. Therefore there exists some n satisfying E[u n ] E[u n η] <, which contradicts to E[u] E[uη] for all u U 2. Then the theorem is proved. Theorem 4.3. Liu [3] Let be an uncertain variable, and t a positive number. If E[ t ] <, then x xt M{ x} Conversely, if Equation holds for some positive number t, then E[ s ] < for any s < t. Theorem 4.4. Let and η be two nonnegative uncertain variables in L k. If k+ η, then we have E[] E[η]. Proof. According to Theorem 3.6, we have Φ k+ x i i x t k dφt k x i t i dφx x k i i k i i k i i t i dφx x k i t i dφx x k i i x k i E[ k ] x k i i i k + i t i dφx x k i. i x i From Theorem 4.3, we have xk Φx x xk M{ x}. 2 For i k, we have x t i dφt x i k t i dφt ox i k. 3 x

THEORETICAL FOUNDATION OF UNCERTAIN DOMINANCE 83 Then we have Φk+ x i i i k i E[ k ] x k i i t i dφx x k i k i i x k i E[ k ] x k i. i Expressing Ψk+ x in the same form, we obtain Φ k+ x Ψ k+ x i i k i E[ i ] E[η i ] x k i. Since Φx Ψx for every x R, then we have Φk+ x Ψ k+ x. Then from the Equation 4, we have E[] E[η]. The theorem is proved. Corollary 4.5. Let and η be two nonnegative uncertain variables in L k. If k+ η, then E[] E[η] and E[] E[η] δ k δ k η where the last inequality is strict whenever E[] > E[η]. Proof. According to Theorem 4.4, we have E[] E[η] and Φ k+ E[] Φ k+ E[η] + Ψ k+ E[η] + E[] E[η] E[] E[η] Φ k x dx Φ k x dx. 4 k From Theorem 3.5, we know that F k+ is also a convex function. Hence we have E[] E[η] Φ k x dx E[] E[η] E[] E[η] + t Φ k E[η] k k dt E[] E[η] k! Φ k te[] + te[η] dt t Φ k E[] k + t E[η] η + k k k E[] E[η] k! +t E[η] η + k k dt E[] E[η] k! E[] E[η] k! δ k t E[] + k k k dt t E[] + k tδ k k δ k δ k η δ k η k + tδ η k dt k. k Substitution into Equation 4 and simplification with the use of Theorem 3.6 yield E[] E[η] δ k δ k η. Since k F k+ is strictly convex, if E[] > E[η], then we have E[] E[η] > δ k δ η k. The theorem is proved. 5 Application of Uncertain Dominance As a new ranking criterion, uncertain dominance can be used in uncertain programming, decision making, finance and so on. In this paper, we simply investigate its application in portfolio problem. We consider the problem of constructing a portfolio of finitely many assets whose return rates are described by uncertain variables R, R 2,..., R n. Then the return rate of the portfolio is Rx R x + R 2 x 2 +... + R n x n, 5 where x, x 2,..., x n are the fractions of the capital invested in the assets. We propose a new portfolio optimization model involving uncertain dominance constraints on the portfolio return rate. The starting point for our model is the assumption

832 YANG ZUO, XIAOYU JI that a benchmark random return rate R having a finite expected value is available. It may be an index or our current portfolio. Our intention is to have the return rate of the new portfolio, Rx, preferable over R. Therefore, we introduce the following optimization model max E[RX] subject to: Rx k R 6 x + x 2 +... + x n x i, i, 2,..., n. From Theorem 4.4 and Corollar 4.5, we have that a solution x is feasible, if E[Rx] E[R ], and E[Rx] λδ k Rx E[R ] λδ k R, where λ. So our new model consists with expected value model and E-V model, which have been widely used in finance and decision making. 6 Conclusion In this paper we presented a new way to compare uncertain variables, which was called uncertain dominance. Some basic properties of uncertain dominance was investigated. The necessary and sufficient conditions for uncertain dominance is founded. And we proposed a model to optimize portfolio based on uncertain dominance to explain its application. Acknowledgments This work was supported by National Natural Science Foundation of ChinaNSFC Grant No. 687467 and No. 786. References [2] Liu, B., Uncertainty therory: An introduction to its axiomatic foundations, Springer-Verlag, Berlin, 24. [3] Liu, B., Uncertainty therory, 2nd ed., Springer-Verlag, Berlin, 27. [4] Liu, B., Fuzzy Process, hybrid process and uncertain process, Journal of Uncertain Systems, Vol.2, No., 3-6, 28. [5] Liu, B., Some research problems in uncertainty theory, Journal of Uncertain Systems, Vol.3, No., 3-, 29. [6] Liu, Y., How to generate uncertain measures, Proceedings of Tenth National Youth Conference on Information and Management Sciences, Luoyang, pp.23-26, 28. [7] Liu, Y., Expected value of function of uncertain variables, http://orsc.edu.cn/online 922.pdf. [8] Peng, J., A stock model for uncertain markets, http://orsc.edu.cn/online/929.pdf. [9] Li, X., and Liu, B., Hybrid logic and uncertain logic, Journal of Uncertain Systems, Vol.3, No.2, 83-94, 29. [] You, C., Some convergence theorems of uncertain sequences, Mathematical and Computer Modelling, Vol.49, Nos.3-4, 482-487, 29. [] Gao, X., Some properties of continuous uncertain measure, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, to be published. [2] Gao, X., On Liu s inference Rule for uncertain systems, http://orsc.edu.cn/online/92.pdf. [3] Chen, X., A note on truth value in uncertain logic, http://orsc.edu.cn/online/92.pdf. [4] Chen, X., and Liu, B., Existence and uniqueness theorem for uncertain differential equations, http://orsc.edu.cn/online/823.pdf. [5] Chen, X., Some formulas of operational law on uncertain variables, http://orsc.edu.cn/online/ 92.pdf. [6] Peng, Z., Some properties of product uncertain measure, http://orsc.edu.cn/online/8228.pdf. [7] Fishburn P. C., and Vickson R. G., Theoretical foundations of stochastic dominance, in Whitmore, G. A. and M. C. Findlay, eds., Stochastic Dominance Lexington, MA: Lexington Books, 978. [8] Bawa V.S., Stochastic dominance: a research bibliography, Management Science, Vol.28, No.6, 698-72, 982. [9] Fishburn, P.C., Stochastic dominance and moments of distributions, Mathematics of Operations Research, Vol.5, No., 94-, 98. [2] Peng J., Henry M.K. Mok, and Tse W.M., Fuzzy dominance based on credibility distributions, Lecture Notes in Artificial Intelligent 363, 292-33, 25. [2] Jiang, Q., and Peng, J., Ranking trapezoidal fuzzy variables by secondorder fuzzy dominance based on credibility distribution, International Conference on Robotics and Biomimetics December 5-8, Sanya, China, 2278-2282, 27. [22] Zadeh, L.A., Fuzzy set, Information Sciences Vol.8, 338-353, 965. [] Liu, B., and Liu, Y.K., Expected value of fuzzy variable and fuzzy expedted value model, IEEE Transactions on Fuzzy Systems, Vol., 445-45, 22.