[ieee 2010 11th international symposium on computational intelligence and informatics (cinti) -...

Multidimensional Possibilistic Risk Aversion

Irina Georgescu* and Jani Kinnunen** * Academy of Economic Studies, Department of Economic Cybernetics, Bucharest, Romania

** Institute for Advanced Management Systems Research, Åbo Akademi University, Turku, Finland E-mails: [email protected], [email protected]

Abstract—In this paper the notion of generalized possibilistic risk premium is introduced as a measure of the risk aversion of an agent faced with several components of possibilistic risk. The main result of the paper is a formula for the calculation of the generalized possibilistic risk premium expressed in terms of a utility function and of some possibilistic indicators.

I. 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 modelling of the uncertainty is mainly realized by probability theory [1], [9], [12], [15], [17], [18]. 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], [9], [15], [17], [18], [19]).

Risk aversion is a main theme in risk theory. According to [18], 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 [20] 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 [6], [7] and by several authors and has been in decision–making problems in conditions of uncertainty (see, e.g., [3]).

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 behaviour 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 [6]. In [13] the 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 possibilistic 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) in which one component is a fuzzy number. 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 . Such a utility function is a multiattribute linear utility [9], 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 [2], [3], [4], [5], [10], [11], [13], [17], 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 0, 1 and a multidimensional utility 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 ,…, , , associated with a possibilistic vector (A1,...,An), a utility function and a weighting function 0, 1 is defined. If u has the class C2 then an analytical formula for the calculation of ,…, , , is proved. The formula expresses ,…, , , 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). , . . . , represent the possibilities that the n nodes have the states , . . . , . A utility function 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).

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II. POSSIBILISTIC INDICATORS OF FUZZY NUMBERS Possibility theory has been initiated by Zadeh in [21] 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 centre 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, the operations with real numbers are extended to fuzzy numbers [7]. In papers [2], [3], [4], [5], [10], [11], [16] 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 (= fuzzy set) is a function 0, 1 . For any the number A(x) is the degree of membership of to A.

Definition 2.2 Let A be a fuzzy set in X. A is normal if there exists such that A(x) = 1. The support of A is defined by

| 0 . Definition 2.3 In the following we consider that X is the

set of real numbers. For any 0, 1 , the –level set of a fuzzy set A in is defined by

| 0 0

(cl(supp(A)) is the topological closure of the set

.) A fuzzy number is a fuzzy set of normal, fuzzyconvex, continuous, and with bounded support.

Let A be a fuzzy number and 0, 1 . Then is a closed and convex subset of . We denote

and . Hence , for all 0, 1 . A non–negative and monotone increasing function 0, 1 is a weighting function if it satisfies the

normality condition 1. We fix a fuzzy number A and a weighting function f.

Assume that , for all 0, 1 . Definition 2.4 The possibilistic expected value of A

w.r.t. f was defined in [11] by , 12 .

Definition 2.5 The possibilistic variance of A w.r.t. f is defined by

, ,, .

If 2 for any 0, 1 then , is the crisp possibilistic mean value introduced in [11], p. 318 and , is the second possibilistic variance defined in [2], p. 324.

Definition 2.6 Let A and B be two fuzzy numbers and f a weighting function. Assume that , and , for any 0, 1 . The possibilistic covariance , , of A and B w.r.t. f is defined in [22], p. 261 by

, , ,, , , . If 2 for any 0, 1 then , , is

the second possibilistic covariance defined in [2], p. 324. A triangular fuzzy number , , is defined by

the function 0,1 : 11 0 .

Recall from [6] that 1 ,1 for all 0,1 . Hence 1 , 1 . Assume that 2 for all 0,1 . Then , .

III. POSSIBILISTIC EXPECTED UTILITY A possibilistic vector has the form (A1,...,An) where

A1,...,An are fuzzy numbers. Definition 3.1 Let 0, 1 be a weighting

function and a continuous function. We consider a possibilistic vector (A1,...,An) where , for any = 1, ..., n and 0, 1 . We define the possibilistic expected utility of (A1,...,An) w.r.t. f and g by

, , . . . , , … ,, … , .

If n = 1 we obtain the notion of possibilistic expected utility of [13].

Remark 3.2 Let n = 2 and , , , for any , . Then

I. Georgescu and J. Kinnunen • Multidimensional Possibilistic Risk Aversion

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, , , , .

In this section we fix a possibilistic vector (A1,...,An) and a weighting function 0, 1 . Assume that , for any 0, 1 .

Proposition 3.3 Let , be two continuous functions and , . We consider the function defined by

, … , , … , , … , for any , … , . Then , , . . . , , , . . . ,, , . . . , . Proof. According to Definition 3.1 of possibilistic

expected utility , , . . . , 12 … ,, … ,2 , … ,, … ,2 , … ,, … ,, , . . . ,, , . . . , .

Proposition 3.4 Let , be two

continuous functions such that , … , , … , for any , … , . Then , , . . . , , , . . . , . Proof. We apply the definition of possibilisitic

expected utility and the monotony of the integral. Proposition 3.5 Let n continuous functions

, i = 1,….,n and , … , . We consider the function defined by , … , ∑ for any , … , . Then

, , . . . , , . Proof. By applying the definition of the possibilistic

expected utility it follows

, , . . . , , … ,, … , ∑∑ , .

Proposition 3.6 Let n2 continuous functions , , 1, … , and , , 1, … , . We consider the function defined by

, … , ,

for any , … , . Then , , . . . , , , … , . Proof. Similar with the previous proof. Example 3.7 Let (A1,...,An) be a possibilistic vector

such that , for each 0,1 . We assume that 2 for each 0,1 . We consider the n-dimensional utility function : defined by

, … , Then the possibilistic expected value has the form: , , … ,∑ ∑ .

We compute it for the case , , , 1, … , .

In this case for any 0,1 1 and 1 , 1, … , . We denote ∑ , ∑ , ∑ . Then ∑ 1 and ∑ 1

, and thus the possibilistic expected value becomes: , , … ,

. After we compute the integrals we obtain: , , … , .

IV. GENERALIZED POSSIBILISTIC RISK AVERSION Usually risk aversion is treated in a probabilistic

context. Risk aversion of an agent represented by a utility function is studied with respect to a situation subject to risk, represented by a random variable. An instrument for evaluation of the risk aversion is the notion of risk premium ([1], [15], [17]). If we intend to study the risk aversion of an agent with respect to a situation of possibilistic uncertainty, then we have to replace the random variable with a possibilistic distribution (particularly, with a fuzzy number). In [13] the risk


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premium associated with a utility function u and a fuzzy number was introduced, as a possibilistic version of the notion of probabilistic risk premium. Instead of the mean value and the probabilistic dispersion, used in the definition and the description of the probabilistic risk premium, in [13] we used the notions of possibilistic mean value and the possibilistic variance. In the real life, an agent can be set in front of some complex phenomena of uncertainty, in which several components subject to risk exist. The multidimensional risk aversion has been treated by several authors by probabilistic mehods (see, e.g., [8], [14]).

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 each calculation.

We consider an agent with respect to a possibilistic vector (A1,...,An) in which the fuzzy number Ai describes the possibilistic component 1 . Instead of the utility function from [13], we shall consider a function . For the unidimensional case the utility function represents the attitude of the agent towards a fuzzy number; in this case the function will express the attitude of the agent with respect to the set of the n fuzzy numbers A1,...,An. We shall fix a weighting function f and a utility function of class C2.

Definition 4.1 Let (A1,...,An) be a possibilistic vector, where each Ai is a fuzzy number. A generalized possibilistic risk premium ,…, , , (associated with the possibilistic risk vector (A1,...,An), the weighting function f and the utility function u) is defined by the following equality:

, , . . . , , , … , , . (1)

Remark 4.2 The notion of the generalized possibilistic

risk premium from the preceding definition measures the risk aversion of the agent u with respect to the possibilistic vector (A1,...,An). If n = 1 then we obtain the notion of the possibilistic risk premium studied in [13].

Now we shall establish a formula for the calculation of an approximate value of the generalized possibilistic risk premium in terms of the possibilistic indicators.

Remark 4.3 (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 , . Then (1) becomes

, , , , . Taking 2 for 0,1 and , , , , we have , , , , and , , 1 1 1 1 .

Then (1) takes the form , , and we can determine distinct solutions. In the unidimensional case if the utility function is injective then the solution of (1) is unique. Now we shall establish a formula for the calculation of

an approximate value for the generalized possibilistic risk premium in terms of the possibilistic indicators.

Theorem 4.4 Let (A1,...,An) be a possibilistic vector in which Ai is a fuzzy number for any i = 1, ..., n. Then (We suppose that the denominator of (2) is non–zero) 12 ∑ , , , , … , ,, ∑ , , … , , . (2)

Proof. Let us denote , for i = 1,...,n. By

applying the Taylor formula for the function and by neglecting the Taylor remainder of the second order, one obtains

, . . . , , . . . , ∑,..., ∑ , ,..., . Consider the function and

defined by , . . . , ∑ ,..., ; , . . . ,∑ , ,..., .

According to Proposition 3.3 we have

, , . . . , , . . . , , , . . . , , , . . . , . (3)

We consider the functions defined by

for any . Then , . . . , , . . . ,

for any , . . . , . According with Proposition

3.5 , , . . . , ∑ ,..., , . By applying Proposition 3.3 it follows that , , 0, i = 1,…,n, therefore , , . . . , 0.


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A straightforward application of Proposition 3.6 shows that , , . . . , , . . . ,, , , .

Replacing in (3) we obtain

, , . . . , , . . . ,∑ ,...,, , , . (4)

By applying again the Taylor formula and by omitting

the Taylor remainder of first order, it follows that , , … , ,, … , , … , ∑ ,..., . Then taking into account Definition 4.1 it follows that

, , . . . , , . . . ,∑ ,..., . (5)

According to (4) and (5) it follows that

12 ∑ , . . . , , ,, ∑ , . . . ,

Remark 4.5 We consider the case n = 1, A1 = A.

According to Theorem 4.4 we obtain: 12 , ′′ ,

′ , . (6)

According to Remark 4.5, we have 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 .

Remark 4.6 We 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 .

We suppose that u has the class C2. We intend to show how this situation can be framed in the model of possibilistic risk with several components studied in this section.

We consider the function defined by , . . . ,

for any , . . . , . (7)

This construction (called in [9], 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. We notice that , , … , , ′ , ; , , … , , ′′ , 0 .

Then by applying formula (2) for (A1 ..., An) and for the utility function v defined by (7), we will obtain the following form of ρ : 12 ∑ , ′′ ,∑ ′ , . (8)

V. 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, we shall 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.

We consider a grid formed of n nodes N1,...,Nn. We denote 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 , then P(Xi = x) is the probability that the node Ni is in state x.

In the second case, the functioning of the node Ni is described by a possibilistic distribution Ai. If , then Ai(x) is the possibility that Ni is in state x. (We agree that the states are represented by real numbers, and the possibilistic distributions are fuzzy numbers.)

In the following we shall consider the second case. An agent takes into consideration the risk with respect

to the entire grid. We shall present a way of evaluating the risk aversion of the agent with respect to the grid.

We face a situation in which we 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 .

By computing the generalized possibilistic risk premium ,…, , , (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


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satisfies some conditions initially imposed (e.g., ,…, , , 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 We consider a grid made on n nodes N1,...,Nn whose functioning is described by triangular fuzzy numbers , , , 1, … , , for which

1 1 0 . (9)

Assume that the utility function and the

weighting function : 0,1 have the form: , … , , , … , ; (10) 2 , 0,1 . (11) According to a simple calculation for any , 1, … ,

we have the following expression of the covariance: , , . (12)

Applying the formula from Theorem 4.4 for the utility

function it follows that 1 , , , . (13)

By replacing in (13) the value from (12) we obtain: 136 , . (14)

With the approximate value of ρ from (14) we measure

the risk aversion of the agent represented by the utility function (10) w.r.t. a grid in which the functioning of the n nodes is described by the triangular fuzzy numbers A1,...,An from (9).

VI. CONCLUSIONS The notion of generalized possibilistic risk premium

introduced in this paper is a measure of risk aversion of an agent faced with a phenomenon of uncertainty with several risk parameters. Its definition is based on the possibilistic expected utility associated with a possibilistic vector whose components are fuzzy numbers. The formula for the calculation of generalized possibilistic risk

premium established in the paper can be used in evaluating risk aversion in situations of uncertainty concerning the functioning of a grid.

ACKNOWLEDGMENT The work of Irina Georgescu was supported by

CNCSIS-UEFISCSU project number PN II-RU 651/2010.

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