Multidimensional possibilistic risk aversion

Multidimensional possibilistic risk aversion

Mathematical and Computer Modelling 54 (2011) 689–696 Contents lists available at ScienceDirect Mathematical and Computer Modelling journal homepage...

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Mathematical and Computer Modelling 54 (2011) 689–696

Contents lists available at ScienceDirect

Mathematical and Computer Modelling journal homepage: www.elsevier.com/locate/mcm

Multidimensional possibilistic risk aversion Irina Georgescu a,1 , Jani Kinnunen b,∗ a

Academy of Economic Studies, Department of Economic Cybernetics, Piaţa Romana No 6 R 70167, Oficiul Postal 22, Bucharest, Romania

b

Institute for Advanced Management Systems Research, Åbo Akademi University, Joukahaisenkatu 3-5 A 4th floor, 20520, Turku, Finland

article

info

Article history: Received 2 September 2010 Received in revised form 3 March 2011 Accepted 4 March 2011 Keywords: Fuzzy number Risk premium Possibility theory Risk aversion

abstract This paper deals with the analysis of risk aversion of an agent faced with a situation of uncertainty with several risk parameters. These risk parameters are represented by fuzzy numbers and the attitude of the agent to the risk situation by a multidimensional utility function. Risk aversion is measured by the notion of generalized possibilistic risk premium. The main result of the paper is an approximate calculation formula of generalized possibilistic risk premium in terms of the utility function and of possibilistic indicators (mean value and covariance). © 2011 Elsevier Ltd. All rights reserved.

1. Introduction Uncertainty is one of the main features of the social and economic life. A phenomenon subject to uncertainty may have several possible outcomes. An agent’s decisions should take into consideration the risk, which appears in all these outcomes. The mathematical modeling of the uncertainty is mainly realized by probability theory [1–5]. Therefore, risk theory is traditionally developed by using probabilistic methods. More precisely, the notions, which describe risk are defined and studied in the framework of expected utility theory (=EU theory) (see [1,2,6,3–5]). Risk aversion is a main theme in risk theory. According to [4], p. 20, ‘‘the central behavioral concept in EU theory is that of risk aversion’’. The probabilistic framework in which risk theory is treated consists of two components: a random variable which models the experience in which risk appears and a utility function which represents the attitude of an agent with respect to various outcomes of this experience. The concepts of probabilistic risk are defined in terms of probabilistic indicators (expected value, variance, etc.). Possibility theory initiated by Zadeh in [7] is an alternative to probability theory in the treatment of uncertainty. It models those situations of uncertainty in which the frequency of events is not big, therefore for them we do not have a database large enough for credible probabilistic inference. Possibility theory has been developed by Dubois and Prade [8,9] and by several authors and has been in decision-making problems in conditions of uncertainty (see e.g. [10–12]). The transition from probabilistic to possibilistic models is done by replacing the random variables with possibilistic distributions (particularly fuzzy numbers) and replacing probabilistic indicators (expected value, variance, covariance) with possibilistic indicators. In paper [13], a possibilistic model for risk aversion has been proposed. The framework for the treatment of possibilistic risk has two components: a possibilistic distribution which models the phenomenon subject to risk and a utility function which represents the behavior of an agent with respect to different outcomes. For the possibilistic model of [13] we restricted to fuzzy numbers possibilistic distributions for which we have a well-developed mathematical theory [2]. In [13] the



Corresponding author. Tel.: +358 452388050. E-mail addresses: [email protected] (I. Georgescu), [email protected] (J. Kinnunen).

1 Tel.: +40 217456082. 0895-7177/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2011.03.011

690

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possibilistic risk premium associated with a utility function u and a fuzzy number A is introduced. The notion of possibilistic risk premium measures the risk aversion of an agent represented by u with respect to the possibilistic phenomenon described by a fuzzy number A. In this paper we propose a model of risk aversion of an agent faced with a situation with several risk components. The risk situation is mathematically modeled by a possibilistic vector (A1 , . . . , An ). One component was a fuzzy number in paper [13]. The idea came from a grid computing where the fuzzy numbers A1 , . . . , An describe the components in which risk appears. The attitude of the agent with respect to the n components of possibilistic risk A1 , . . . , An is expressed by a utility function of the form u : Rn → R. Such a utility function is a multiattribute linear utility [8], p. 20. The main concept of the paper is the generalized possibilistic risk premium. It generalizes the notion of possibilistic risk premium from the unidimensional case studied in [13] and measures the risk aversion of an agent with respect to several risk components. The paper is organized as follows. Section 2 recalls some basic knowledge on the possibilistic indicators of fuzzy numbers studied in [14–22], etc. Definitions of fuzzy numbers, weighted possibilistic expected value, variance and covariance are recalled. In Section 3 the possibilistic expected utility E (f , g (A1 , . . . , An )) of a possibilistic vector (A1 , . . . , An ) w.r.t. a weighting function f : [0, 1] → R and a multidimensional utility g : Rn → R is introduced. Several properties of E (f , g (A1 , . . . , An )) are established. They will be used in the next section to prove a formula of calculation for the generalized possibilistic risk aversion. In Section 4 the generalized possibilistic risk premium ρA1 ,...,An ,f ,u associated with a possibilistic vector (A1 , . . . , An ), a utility function u : Rn → R and a weighting function f : [0, 1] → R is defined. If u has the class C 2 then an analytical formula for the calculation of ρA1 ,...,An ,f ,u is proved. The formula expresses ρA1 ,...,An ,f ,u with help of the weighted possibilistic expected value, the possibilistic variance and the possibilistic covariance. For the unidimensional case (n = 1) we find the definition of [13] of the possibilistic risk premium and its calculation formula. In Section 5 a possibilistic model regarding the risk aversion in grid computing is proposed. The functioning of a grid composed of n nodes is described by a possibilistic vector (A1 , . . . , An ). A1 (x1 ), . . . , An (xn ) represent the possibilities that the n nodes have the states x1 , . . . , xn . A utility function u : Rn → R expresses the attitude of an agent with respect to various values of (A1 , . . . , An ). Then we can apply the possibilistic model of Section 5: the generalized possibilistic risk premium will evaluate the risk aversion of the agent with respect to (A1 , . . . , An ). 2. Possibilistic indicators of fuzzy numbers Possibility theory has been initiated by Zadeh in [7] as an instrument for the treatment of those phenomena of uncertainty in which theory cannot be applied. We talk about events which do not have a big frequency and for which several data do not exist. In the center of possibility theory there are the notions of possibility measure and necessity measure. The random variables from probability theory are replaced here by possibilistic distributions. A possibilistic distribution is usually interpreted as a fuzzy set. The fuzzy numbers are an important class of possibilistic distributions. They generalize real numbers and replace them when describing the uncertainty situations. By using Zadeh’s extension principle [23], the operations with real numbers are extended to fuzzy numbers [8,11]. In papers [14,24,17–19,25–27,20–22] possibilistic versions of the mean value, variance and covariance of random variables were defined and studied. In this section we repeat from the above mentioned papers some definitions and basic properties of some possibilistic indicators of fuzzy numbers (expected value, variance and covariance). Definition 2.1. Let X be a set of states. A fuzzy subset of X (=fuzzyset) is a function A : X → [0, 1]. For any x ∈ X the number A(x) is the degree of membership of x to A. Definition 2.2. Let A be a fuzzy set in X . A is normal if there exists x ∈ X such that A(x) = 1. The support of A is defined by supp(A) = {x ∈ X |A(x) > 0}. Definition 2.3. In the following we consider that X is the set R of real numbers. For any γ ∈ [0, 1], the γ -level set of a fuzzy set A in R is defined by γ

[A] =



{x ∈ R|A(x) ≥ γ } cl(supp(A))

if γ > 0 if γ = 0

(cl(supp(A)) is the topological closure of the set supp(A) ⊆ R). A fuzzy number is a fuzzy set of R normal, fuzzy convex, continuous, and with bounded support. Let A be a fuzzy number and γ ∈ [0, 1]. Then [A]γ is a closed and convex subset of R. The 1-level set [A]1 = {x ∈ R | A(x) = 1} is called the core or peak of A. We denote a1 (γ ) = min[A]γ and a2 (γ ) = max[A]γ . Hence [A]γ = [a1 (γ ), a2 (γ )] for all γ ∈ [0, 1]. A non-negative and monotone increasing function f : [0, 1] → R is a weighting function if it satisfies the normality 1 condition 0 f (γ )dγ = 1. We fix a fuzzy number A and a weighting function f . Assume that [A]γ = [a1 (γ ), a2 (γ )] for all γ ∈ [0, 1].

I. Georgescu, J. Kinnunen / Mathematical and Computer Modelling 54 (2011) 689–696

Definition 2.4. The f -weighted possibilistic expected value of was defined in [2] by E (f , A) = Definition 2.5. The f -weighted possibilistic variance of A is defined by Var (f , A) = E (f , A))2 ]f (γ )dγ .

1 1 2

0

1 2

691

1 0

(a1 (γ ) + a2 (γ ))f (γ )dγ .

[(a1 (γ ) − E (f , A))2 + (a2 (γ ) −

If f (γ ) = 2γ for any γ ∈ [0, 1] then E (f , A) is the crisp possibilistic mean value introduced in [19], p. 318 and Var (f , A) is the second possibilistic variance defined in [15], p. 324. Definition 2.6. Let A and B be two fuzzy numbers and f a weighting function. Assume that [A]γ = [a1 (γ ), a2 (γ )] and [B]γ = [b1 (γ ), b2 (γ )] for any γ ∈ [0, 1]. The f -weighted possibilistic covariance cov(f , A, B) of A and B is defined in [22], p. 261 by Cov(f , A, B) =

1 2

1 0

[(a1 (γ ) − E (f , A))(b1 (γ ) − E (f , B)) + (a2 (γ ) − E (f , A))(b2 (γ ) − E (f , B))]f (γ )dγ .

If f (γ ) = 2γ for any γ ∈ [0, 1] then cov(f , A, B) is the second possibilistic covariance defined in [15] p. 324. Remark 2.7. In papers [14,15,19,20,22] various possibilistic covariances were introduced by formulas which use only γ level sets of fuzzy numbers. In other papers [17,18,25,26] covariances were defined using the joint possibility distribution of fuzzy numbers and the formulas proposed in [14,15,19,22] appeared as particular forms. This paper uses the form of the possibilistic covariance from Definition 2.6. 3. Possibilistic expected utility A possibilistic vector has the form (A1 , . . . , An ) where A1 , . . . , An are fuzzy numbers. Let g : Rn → R be a continuous function. Then, the sup-min extension of g is defined by g (A1 , . . . , An )(y) =

sup

g (x1 ,...,xn )=y

min{A1 (x1 ), . . . , An (xn )},

∀y ∈ R.

The construction above called Zadeh’s extension principle [23] plays an important role in fuzzy set theory. The operations with real numbers can be extended to operations with fuzzy numbers by this construction [8]. Definition 3.1. Let f : [0, 1] → R be a weighting function and g : Rn → R a continuous function. We consider a possibilistic vector (A1, . . . , An ) where [Ai ]γ = [ai (γ ), bi (γ )] for any i = 1, . . . , n and γ ∈ [0, 1]. We define the f -possibilistic expected utility of (A1 , . . . , An ) w.r.t. g by E (f , g (A1 , . . . , An )) =

1

1



2

[g (a1 (γ ), . . . , an (γ )) + g (b1 (γ ), . . . , bn (γ ))]f (γ )dγ . 0

If n = 1 we obtain the notion of f -weighted possibilistic expected utility of [13]. Remark 3.2. Let n = 2 and g (x, y) = (x − E (f , A1 ))(y − E (f , A2 )) for any x, y ∈ R. Then E (f , g (A1 , A2 )) = cov(f , A1 , A2 ). In this section we fix a possibilistic vector (A1 , . . . , An ) and a weighting function f : [0, 1] → R. Assume that [Ai ]γ = [ai (γ ), bi (γ )] for any γ ∈ [0, 1]. The following propositions describe the behavior of f -weighted possibilistic utility with respect to some operations of utility functions. They will be used in the next section to obtain approximate calculation formulas of generalized possibilistic risk premium. Proposition 3.3. Let g : Rn → R, h : Rn → R be two continuous functions and a, b ∈ R. We consider the function u : Rn → R defined by u(x1 , . . . , xn ) = ag (x1 , . . . , xn ) + bg (x1 , . . . , xn ) for any (x1 , . . . , xn ) ∈ Rn . Then E (f , u(A1 , . . . , An )) = aE (f , g (A1 , . . . , An )) + bE (f , h(A1 , . . . , An )). Proof. According to the Definition 3.1 of possibilistic expected utility E (f , u(A1 , . . . , An )) =

=

1 2 a 2

1



[u(a1 (γ ), . . . , an (γ )) + u(b1 (γ ), . . . , bn (γ ))]f (γ )dγ 0 1



[g (a1 (γ ), . . . , an (γ )) + g (b1 (γ ), . . . , bn (γ ))]f (γ )dγ 0

b



1

[h(a1 (γ ), . . . , an (γ )) + h(b1 (γ ), . . . , bn (γ ))]f (γ )dγ 2 0 = aE (f , g (A1 , . . . , An )) + bE (f , h(A1 , . . . , An )).  +

Proposition 3.4. Let g : Rn → R, h : Rn → R be two continuous functions such that g (x1 , . . . , xn ) ≤ h(x1 , . . . , xn ) for any (x1 , . . . , xn ) ∈ Rn . Then E (f , g (A1 , . . . , An )) ≤ E (f , h(A1 , . . . , An )).

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Proof. For any γ ∈ [0, 1] we have g (a1 (γ ), . . . , an (γ )) ≤ h(a1 (γ ), . . . , an (γ )) g (b1 (γ ), . . . , bn (γ )) ≤ h(b1 (γ ), . . . , bn (γ )). Therefore according to the definition of possibilistic expected utility and monotony of the integral we have E (f , g (A1 , . . . , An )) =

1 2 1

1



[g (a1 (γ ), . . . , an (γ )) + g (b1 (γ ), . . . , bn (γ ))]f (γ )dγ 0 1



[h(a1 (γ ), . . . , an (γ )) + h(b1 (γ ), . . . , bn (γ ))]f (γ )dγ 2 0 = E (f , h(A1 , . . . , An )).  ≤

n Proposition 3.5. Let n continuous ∑n functions gi : R → R, i = 1, . . . , n and a1 , . . . , an ∈ R. We consider ∑n the function g : R → R defined by g (x1 , . . . , xn ) = i=1 ai gi (xi ) for any (x1 , . . . , xn ) ∈ Rn . Then E (f , g (A1 , . . . , An )) = i=1 ai E (f , gi (Ai )).

Proof. By applying the definition of the possibilistic expected utility it follows that E (f , g (A1 , . . . , An )) =

=

1

1



2

[g (a1 (γ ), . . . , an (γ )) + g (b1 (γ ), . . . , bn (γ ))]f (γ )dγ 0

∫ n − ai 2

i =1

=

n −

1

[gi (ai (γ )) + gi (bi (γ ))]f (γ )dγ 0

ai E (f , gi (Ai )).



i =1

Proposition 3.6. Let n2 continuous functions gij ∑ : R2 → R, i, j = 1, . . . , n and aij ∈ R, i, j = 1, . . . , n. We consider the n n n function g : R → R defined by g (x1 , . . . , xn ) = i=1 aij gij (xi , xj ) for any (x1 , . . . , xn ) ∈ R . Then E (f , g (A1 , . . . , An )) =

n −

aij E (f , gij (Ai , Aj )).

i=1

Proof. By Definition 3.1 one gets E (f , g (A1 , . . . , An )) =

=

1

1



2

f (γ )dγ 0

∫ n − aij i,j=1

=

n −

2

1

[gij (ai (γ ), aj (γ )) + gij (bi (γ ), bj (γ ))]f (γ )dγ 0

aij E (f , gij (Ai , Aj )).



i,j=1

4. Generalized possibilistic risk aversion This section emphasizes the situations with several components of possibilistic risk, each represented by a fuzzy number. For the evaluation of the risk aversion of an agent with respect to several components of possibilistic risk, we shall propose the notion of generalized risk premium and we shall give a formula for its calculation. We consider an agent with respect to a possibilistic vector (A1 , . . . , An ) in which the fuzzy number Ai describes the possibilistic component i (1 ≤ i ≤ n). Instead of the utility function u : R → R from [6], we shall consider a function u : Rn → R. For the unidimensional case the utility function represents the attitude of the agent towards a fuzzy number. In the n-dimensional case the function u : Rn → R will express the attitude of the agent with respect to the set of the n fuzzy numbers A1 , . . . , An . In the following one assumes that an n-dimensional utility function has the class C 2 and is increasing in each argument (see [2,4]). One fixes a weighting function f and a utility function u : Rn → R. Definition 4.1. Let (A1 , . . . , An ) be a possibilistic vector, where each Ai is a fuzzy number. A generalized possibilistic risk premium ρ = ρA1 ,...,An ,f ,u (associated with the possibilistic risk vector (A1 , . . . , An ), the weighting function f and the utility function u) is defined by the following inequality: E (f , u(A1 , . . . , An )) = u(E (f , A1 ) − ρ, . . . , E (f , An ) − ρ).

(1)

I. Georgescu, J. Kinnunen / Mathematical and Computer Modelling 54 (2011) 689–696

693

Remark 4.2. (1) is an equation in ρ and it can have several solutions. To see it let us consider the bidimensional possibilistic vector (A1 , A2 ) and a utility function u(x1 , x2 ) = x1 x2 . Then (1) becomes (E (f , A1 ) − ρ)(E (f , A2 ) − ρ) = E (f , A1 , A2 ). Taking f (γ ) = 2γ for γ ∈ [0, 1] and A1 = (r1 , α1 ), A2 = (r2 , α2 ) one has E (f , A1 ) = r1 , E (f , A2 ) = r2 and E (f , u(A1 , A2 )) =

1



(r1 − (1 − γ )α1 )(r2 − (1 − γ )α2 ) + (r1 + (1 − γ )α1 )(r2 + (1 − γ )α2 )f (γ )dγ . 0

The (1) takes the form (r1 − ρ)(r2 − ρ) = E (f , u(A1 , A2 )) and one can determine distinct solutions. In the unidimensional case if the utility function u : R → R is injective, then the solution of (1) is unique. Remark 4.3. The notion of the generalized possibilistic risk premium from Definition 4.1 measures the risk aversion of the agent u with respect to the possibilistic vector (A1 , . . . , An ). The bigger the generalized possibilistic risk premium is the higher the agent’s risk aversion is. If n = 1 then one obtains the notion of possibilistic risk premium studied in [13]. Now a formula for the calculation of the generalized possibilistic risk premium in terms of the possibilistic indicators will be established. Theorem 4.4. Let (A1 , . . . , An ) be a possibilistic vector in which Ai is a fuzzy number for any i = 1, . . . , n. Then an approximate solution of Eq. (1) is given by n ∑

ρ≈−

1 i,j=1 2

cov(f , Ai, Aj ) n ∑ i=1

∂ 2 (E (f ,A1 ),...,E (f ,An )) ∂ xi ∂ xj

∂ u(E (f ,A1 ),...,E (f ,An )) ∂ xi

.

(2)

(One supposes that the denominator of (2) is non-zero.) Proof. Let us denote mi = E (f , Ai ) for i = 1, . . . , n. By applying the Taylor formula for the function u : Rn → R and by neglecting the Taylor remainder of the second order, one obtains u(x1 , . . . , xn ) ≈ u(m1 , . . . , mn ) −

=

n 1−

n − ∂ u(m1 , . . . , mn ) (xi − mi ) ∂ xi i =1

(xi − mi )(xj − mj )

2 i,j=1

∂ 2 u(m1 , . . . , mn ) . ∂ xi ∂ xj

Consider the function g : Rn → R and h : Rn → R defined by g (x1 , . . . , xn ) =

n −

(xi − mi )

i=1

h(x1 , . . . , xn ) =

n −

∂ u(m1 , . . . , mn ) ∂ xi

(xi − mi )(xj − mj )

i,j=1

∂ 2 u(m1 , . . . , mn ) . ∂ xi ∂ xj

According to Proposition 3.3 one has E (f , u(A1 , . . . , An )) ≈ u(m1 , . . . , mn ) − E (f , g (A1 , . . . , An )) +

1 2

E (f , h(A1 , . . . , An )).

(3)

One considers the functions gi : R → R defined by gi (x) = x − mi for any x ∈ R. ∑n ∂ u(m1 ,...,mn ) Then g (x1 , . . . , xn ) = gi (xi ) for any (x1 , . . . , xn ) ∈ Rn . i =1 ∂x i

∂ u(m ,...,m )

n 1 According to Proposition 3.5 E (f , g (A1 , . . . , An )) = E (f , gi (Ai )). i=1 ∂ xi By applying Proposition 3.3 it follows that E (f , gi (Ai )) = E (f , Ai ) − mi = 0, i = 1, . . . , n, therefore E (f , g (A1 , . . . , An )) = 0. A straightforward application of Proposition 3.6 shows that

E (f , h(A1 , . . . , An )) =

∑n

n − ∂ 2 u(m1 , . . . , mn ) cov(f , Ai , Aj ). ∂ xi ∂ xj i ,j = 1

Replacing in (3) one obtains E (f , u(A1 , . . . , An )) = u(m1 , . . . , mn ) +

n 1 − ∂ 2 u(m1 , . . . , mn )

2 i ,j = 1

∂ xi ∂ xj

cov(f , Ai , Aj ).

(4)

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By applying again the Taylor formula and by omitting the Taylor remainder of first order, it follows that u(E (f , A1 ) − ∑ mn ) . ρ), . . . , u(E (f , An ) − ρ) = u(m1 − ρ, . . . , mn − ρ) ≈ u(m1 , . . . , mn ) − ρ ni=1 ∂ u(m1∂,..., xi Then taking into account Definition 4.1 it follows that E (f , u(A1 , . . . , An )) ≈ u(m1 , . . . , mn ) − ρ

n − ∂ u(m1 , . . . , mn ) i =1

∂ xi

.

(5)

By (4) and (5) it follows that n ∑

ρ≈−

1 i,j=1

∂ 2 u(m1 ,...,mn ) cov(f , Ai, Aj ) ∂ xi ∂ xj n ∑

2

i =1

u(m1 ,...,mn ) ∂ xi

. 

Remark 4.5. One considers the case n = 1, A1 = A. According to Theorem 4.4 one obtains: 1

u′′ (E (f , A))

2

u′ (E (f , A))

ρ ≈ − Var (f , A)

.

(6)

According to Remark 4.5, one has found the formula from [13] for the calculation of the possibilistic risk premium associated with the fuzzy number A, the weighting function f and the utility function u : R → R. Remark 4.6. One shall consider the particular case of a situation with n components of possibilistic risk A1 , . . . , An and in which the agent has the same attitude towards A1 , . . . , An with respect to risk aversion. The attitude of risk towards each of components Ai will be expressed by a common utility function u : R → R. One supposes that u has the class C 2 . One intends to show how this situation can be framed in the model of possibilistic risk with several components studied in this section. One considers the function v : Rn → R defined by

v(x1 , . . . , xn ) = u(x1 ) + · · · + u(xn ) for any x1 , . . . , xn ∈ R.

(7)

This construction (called in [2] p. 20 multiattribute linear utility) allows us to set ourselves in the Definition 4.1 and to consider the possibilistic risk premium ρ associated with (A1 , . . . , An ), f and v . One notices that

∂v(E (f , A1 ), . . . , E (f , An )) = u′ (E (f , Ai )) ∂ xi  ′′ ∂ 2 v(E (f , A1 ), . . . , E (f , An )) u (E (f , Ai )) = 0 ∂ xi ∂ xj

if i = j if i ̸= j.

Then by applying formula (2) for (A1 , . . . , An ) and for the utility function v defined by (7), one will obtain the following form of ρ : n ∑

ρ≈−

1 i =1 2

Var (f , Ai )u′′ (E (f , Ai )) n ∑

.

(8)

u′ (E (f , Ai ))

i=1

5. Possibilistic risk aversion in grid computing Grid Computing is one of the main themes in computer science area. Grid technologies assure a better distribution and leadership of the computational and informational resources. This makes the technique of grid computing be more and more interesting for commercial applications, which motivates the study of the risk phenomenon in the context of grid computing. In this section, one will propose a possibilistic method by which to evaluate the risk aversion of an agent with respect to grid computing. The model is based on the concept of generalized possibilistic risk premium introduced in the previous section. One considers a grid formed of n nodes N1 , . . . , Nn . One denotes by Si the set of the states in which the node Ni , i = 1, . . . , n can exist. For the functioning of the grid, both overall and for each node in part, situations of uncertainty might appear. To know the situation in which the node Ni is can be described in terms of probability theory or possibility theory. In the first case the functioning of the node Ni is described by a random variable Xi . If x ∈ Si , then P (Xi = x) is the probability that the node Ni is in state x.

I. Georgescu, J. Kinnunen / Mathematical and Computer Modelling 54 (2011) 689–696

695

In the second case, the functioning of the node Ni is described by a possibilistic distribution Ai . If x ∈ Si , then Ai (x) is the possibility that Ni is in state x. (One agrees that the states are represented by real numbers, and the possibilistic distributions are fuzzy numbers.) In the following the second case will be considered. The functioning of each node is subject to risk. An agent takes into consideration the risk with respect to the entire grid. A way of evaluating the risk aversion of the agent with respect to the grid will be presented. A utility function having the form u : Rn → R will describe the attitude of the agent with respect to various states x1 , . . . , xn in which the nodes N1 , . . . , Nn may be. One faces a situation in which one can apply the possibilistic risk model from the previous section: a possibilistic vector (A1 , . . . , An ) representing the functioning of the nodes N1 , . . . , Nn and a utility function u : Rn → R. By computing the generalized possibilistic risk premium ρA1 ,...,An ,f ,u (f is a weighting function, conveniently chosen) one obtains an evaluation of the risk aversion of the agent with respect to (A1 , . . . , An ). Based on these facts one can appreciate whether the existent grid satisfies some conditions initially imposed (e.g., ρA1 ,...,An ,f ,u should be inferior to a threshold α ). In order to reach a conveniently chosen level of the risk aversion one will modify the possibilistic vector (A1 , . . . , An ) either by adding new nodes, or by improving the functioning of the existing nodes. Example 5.1. One considers a grid with n nodes N1 , . . . , Nn whose functioning is described by triangular fuzzy numbers Ai = (ai , αi , βi ), i = 1, . . . , n.

 ai − t   1 − α  i t − ai Ai (t ) = 1 −   βi   0

if ai − αi ≤ t ≤ αi (9)

if ai ≤ t ≤ αi + βi otherwise.

Assume that the utility function u : Rn → R and the weighting function f : [0, 1] → R have the form: u(x1 , . . . , xn ) = −e−2(x1 +···+xn ) , f (γ ) = 2γ ,

x1 , . . . , xn ∈ R;

γ ∈ [0, 1].

(10) (11)

According to a simple calculation, for any i, j = 0, . . . , n one obtains the following expression of the covariance cov(f , Ai , Aj ): Cov(f , Ai , Aj ) =

αi αj + βi βj + (αi + βi )(αj + βj ) 36

.

(12)

Applying the formula from Theorem 4.4 for the utility function (10) it follows:

ρ≈

n 1−

n i,j=1

cov(f , Ai , Aj ).

(13)

Replacing in (13) cov(f , Ai , Aj ) with the value given by (12) one obtains:

ρ≈

n 1 −

36n i,j=1

(αi αj + βi βj + (αi + βi )(αj + βj )).

(14)

6. Conclusions This paper is a proposal to treat risk aversion when several risk parameters exist. The main contributions of the paper are: (i) the introduction of a notion of possibilistic expected utility and the study of some of its properties, (ii) the definition of the generalized possibilistic risk premium as a measure of risk aversion of an agent faced with a situation of possibilistic risk with several parameters, and (iii) the proof of an approximate calculation formula for generalized possibilistic risk premium. The application of these formulas in the evaluation of risk, which might appear in the functioning of a grid, suggests the possibility of ulterior developments of the topic of this paper. Acknowledgement The work of Irina Georgescu was supported by CNCSIS-UEFISCSU project number PN II-RU 651/2010. References [1] K.J. Arrow, Essays in the Theory of Risk Bearing, North–Holland, Amsterdam, 1970. [2] P.C. Fishburn, Nonlinear Preference and Utility Theory, John Hopkins University Press, Baltimore, MD, 1988.

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