Abstract This paper attempts to treat some topics of risk theory by means of cred-ibility theory. We study the risk aversion of an agent faced with a situation of un-certainty represented by a discrete fuzzy variable, the relationship between stochasticdominance and credibilistic dominance, and an index of riskiness of discrete credi-bilistic gambles. In the framework of an optimal saving credibilistic model, the waythe presence of risk modifies the level of optimal saving is studied. The main tool ofour investigation is an operator defined by B. Liu and Y. K. Liu by which to a discretefuzzy variable one associates a discrete random variable with the same expected valueas the former.
Keywords Credibility theory · Possibility theory · Credibilistic risk aversion · Cred-ibilistic risk premium · Credibilistic dominance · Credibilistic index of riskiness
1. Introduction
The mathematical modeling of phenomena of uncertainty is traditionally done byprobability theory [1, 2]. Zadeh’s possibility theory [3, 4] and Liu and Liu’s credibil-ity theory [5, 6] offer other ways for modeling uncertainty.
The transitions from probabilistic models to possibilistic and credibilistic modelsrequire that:
– random variables are replaced by fuzzy variables, and– probabilistic indicators (expected value, variance, etc.) are replaced with appro-
priate possibilistic or credibilistic indicators.
Irina Georgescu (�)Academy of Economic Studies, Department of Economic Cybernetics Piata Romana No 6 R 70167, OficiulPostal 22, Bucharest, Romaniaemail: [email protected] Kinnunen (�)Institute for Advanced Management Systems Research, Abo Akademi University Jouhakaisenkatu 3-5 B6th floor, 20520 Turku, Finlandemail: [email protected]
400 Irina Georgescu · Jani Kinnunen (2013)
For the probabilistic approach to risk two main directions can be distinguished ([7,2]):
(a) How much riskier is a situation and how can one evaluate if a situation is riskierthan another?
(b) How is the evaluation done if an agent is more risk averse than another?The fundamental contributions of Arrow [8, 9] and Pratt [10] to the issue (b) cre-
ated the theory of risk aversion (see also [1, 2]). With respect to the issue (a), werecall the results of [11, 7] on stochastic dominance and those of [12, 13] on theindex of riskiness. A possibilistic approach to risk aversion is found in papers [14,15].
The authors in this paper make an attempt to study the risk by credibility theory.As we know, probability theory of risk is built in the framework of expected utilitytheory (=EU-theory). Similarly, a credibilistic theory of risk requires the develop-ment of a credibilistic EU-theory. We will focus on the case when risk situations aredescribed by discrete fuzzy variables. A procedure from [5, 6] lies at the basis ofour investigations by which a random variable Xζ is associated with a discrete fuzzyvariable ζ. The expected value of Xζ coincides with the credibilistic expected valueof ζ.
The paper is structured by nine sections as follows. In Section 2, some basicnotions (of possibilistic measure, possibilistic distribution, and credibilistic measure)and their relations are recalled.
Section 3 deals with the membership function of a fuzzy variable w.r.t. a credibilitymeasure and the credibilistic expected value Q(ζ) of a fuzzy variable ζ. The way adiscrete random variable Xζ associated with a discrete fuzzy variable ζ is presented,such that Q(ζ) coincides with the probabilistic mean value M(Xζ) ([5, 6]).
In Section 4, the way the operator ζ �→ Xζ behaves w.r.t. a utility function isstudied. The notion of credibility expected utility Q(u(ζ)) of a fuzzy variable ζ w.r.t.a utility function u is introduced. If ζ is a discrete fuzzy variable and u is strictlyincreasing, then Q(u(ζ)) coincides with the probabilistic expected value M(u(Xζ)).By this, for the discrete case one obtains a credibilistic EU-theory isomorphic toa probabilistic EU-theory. The building of the risk models in the next section isfounded by this thesis.
Section 5 is concerned with the credibilistic risk aversion. The notion of credibilis-tic risk premium associated with a fuzzy variable and a utility function is introduced.This indicator measures the risk aversion of an agent (represented by a utility func-tion u) in the face of a risk described by a fuzzy variable ζ. By means of the operatorζ �→ Xζ , an approximate calculation formula for the credibilistic risk premium in caseof discrete fuzzy variable is obtained.
In Section 6, one proves that the credibilistic dominance is turned into stochasticdominance by the operator ζ �→ Xζ .
Section 7 introduces an index of riskiness of type Aumann-Serrano [12] for anydiscrete credibilistic gamble. The main result of the section is a credibilistic versionof Hart theorem [13].
Section 8 studies an optimal saving model in case when the risk is represented bya fuzzy variable ζ. The notion of credibilistic precautionary saving is defined as a
Fuzzy Inf. Eng. (2013) 4: 399-416 401
measure of the effect of credibilistic risk on the level of optimal saving. In case whenζ is discrete, the study of optimal saving comes down to a probabilistic model.
The paper is concluded in Section 9.
2. Possibility and Credibility
In this section, we recall, by [16-18, 5, 6], some notions and results on possibilitymeasure and credibility measure.
LetΩ be a non-empty set andP(Ω) its power set. The elements ofΩ are interpretedas states and the elements of P(Ω) as events. For any D ∈ P(Ω), we denote Dc =
Ω − D.
Definition 1 [18] A possibility measure on Ω is a function Π : P(Ω) → [0, 1] suchthat the following conditions hold:
(Pos 1) P(∅) = 0;P(Ω) = 1;(Pos 2) Π(
⋃i∈I
Di) = supi∈IΠ(Di) for any family (Di)i∈I , of subsets of Ω.
If A ∈ P(Ω), then Π(A) is called the possibility of the event A.A possibility distribution on Ω is a function μ : Ω→ [0, 1] such that sup
x∈Ωμ(x) = 1;
μ is said to be normalized if μ(x) = 1 for some x ∈ Ω.Let Π be a possibility measure on Ω and μ a possibility distribution on Ω. Let us
consider the functions μΠ : Ω→ [0, 1] and Posμ : P(Ω)→ [0, 1] defined by
μΠ(x) = Π({x}) for any x ∈ Ω; (1)
Posμ(D) = supx∈Dμ(x) for any D ⊆ Ω. (2)
Proposition 1 [18](i) μΠ is a possibility distribution on Ω and Posμ is a possibility measure on Ω;(ii) μPosμ = μ and PosμΠ = Π.
The above proposition establishes a bijective correspondence between the possi-bility measures on Ω and possibility distributions on Ω.
Definition 2 [6] A credibility measure on Ω is a function Cr : P(Ω) → [0, 1] suchthat the following conditions hold:
(Cred 1) Cr(Ω) = 1;(Cred 2) If A, B ∈ P(Ω), then A ⊆ B implies Cr(A) � Cr(B);(Cred 3) For any A ∈ P(Ω),Cr(A) +Cr(AC) = 1;(Cred 4) For any family (Ai)i∈I of subsets of Ω such that sup Cr(Ai) < 1/2, the
following equality holds: Cr(⋃i∈I
Ai) = supi∈I
Cr(Ai).
If A ∈ P(Ω), then Cr(A) is called the credibility of the event A.If a, b ∈ R, then we denote a ∨ b = sup(a, b) and a ∧ b = inf(a, b).Let Π be a possibility measure on Ω and Cr a credibility measure on Ω. Let us
consider the functions ΠCr : P(Ω)→ [0, 1], and CrΠ : P(Ω)→ [0, 1], defined by
ΠCr(A) = 2Cr(A) ∧ 1 for any A ⊆ Ω; (3)
402 Irina Georgescu · Jani Kinnunen (2013)
CrΠ = 1/2[Π(A) + 1 − Π(AC)] for any A ⊆ Ω. (4)
Proposition 2 (i) ΠCr is a possibility measure on Ω and CrΠ is a credibility measureon Ω;
(ii) ΠCrΠ = Π and CrΠCr = Cr.
The above proposition shows that there exists a bijective correspondence betweenthe possibility measures on Ω and credibility measures on Ω.
Let μ : Ω → [0, 1] be a possibility distribution and Posμ the possibility mea-sure defined by (2). We denote by Crμ the credibility measure associated with Posμaccording to (4):
Crμ(A) =12
[supx∈Aμ(x) + 1 − sup
x�Aμ(x)]. (5)
3. Fuzzy Variables and Credibilistic Indicators
In this section, we will present some credibilistic indicators associated with fuzzyvariables (the credibilistic expected value and the credibilistic variance).
From now on in this paper, we will assume that the set of states Ω is the set R ofreal numbers. A fuzzy variable is an arbitrary function ζ : R→ R.
Let Cr : P(R) → [0, 1] be a credibility measure and ζ a fuzzy variable. Weconsider the function μ : R→ [0, 1] defined by
μ(x) = 2Cr(ζ = x) ∧ 1 for any x ∈ R. (6)
μ is called the membership function of ζ w.r.t. Cr.
Proposition 3 [6](i) μ is a possibility distribution;(ii) Cr = Crμ.
Proposition 4 [6] If A ⊆ R, then
Cr(ζ ∈ A) =12
[supx∈Aμ(x) + 1 − sup
x�Aμ(x)].
Corollary 1 For any r ∈ R, we have:
(a) Cr(ζ � r) =12
[supx�rμ(x) + 1 − sup
x>rμ(x)];
(b) Cr(ζ > r) =12
[supx>rμ(x) + 1 − sup
x�rμ(x)].
Let Cr be a credibility measure, ζ : R→ R a fuzzy variable, and μ the membershipfunction of ζ w.r.t. Cr.
Definition 3 [5] The credibilistic expected value Q(ζ) of ζ, provided that the twointegrals are finite, is defined by
Q(ζ) =∫ ∞
0Cr(ζ � r)dr −
∫ 0
−∞Cr(ζ � r)dr. (7)
Fuzzy Inf. Eng. (2013) 4: 399-416 403
Definition 4 [6] Assume that the credibilistic expected value Q(ζ) = e exists, thenthe credibilistic variance Var(ζ) of ζ is defined by
Var(ζ) = Q[(ζ − e)2]. (8)
Proposition 5 [6] If a, b ∈ R, then Q(aζ + b) = aQ(ζ) + b.
A fuzzy variable ζ is discrete if it takes a finite number of values. Let a1, a2, · · · , an
be the distinct values of a discrete fuzzy variable ζ. Then the membership function μof ζ has the form:
μ(x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
μ1, if x = a1;μ2, if x = a2;...
...
μn, if x = an.
We write it in the matrix:
μ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
μ1 μ2 · · · μn
⎤⎥⎥⎥⎥⎥⎦ . (9)
(9) represents the discrete fuzzy variable ζ.Let us consider the following real numbers:
pi =12
[max1� j�n{μ j|a j � ai} − max
1� j�n{μ j|aj < ai}]
+12
[max1� j�n{μ j|aj ≥ ai} − max
1� j�n{μ j|a j > ai}]. (10)
Proposition 6 [5] The real numbers p1, p2, · · · , pn verify the following conditions:(i) pi ≥ 0, i = 1, · · · , n;
(ii)n∑
i=1
pi = 1.
Then one can consider the following discrete random variable:
Xζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
p1 p2 · · · pn
⎤⎥⎥⎥⎥⎥⎦ . (11)
Proposition 7 [5] The credibilistic expected value Q(ζ) of ζ coincides with the prob-abilistic expected value M(Xζ) of Xζ .
Remark 1 [5] If a1 < a2 < · · · < an, then the probabilities p1, · · · , pn get the form:
pi =12
[i∨
j=1
μ j −i−1∨j=0
μ j] +12
[n∨
j=i
μ j −n+1∨
j=i+1
μ j], i = 1, · · · , n,
where μ0 = μn+1 = 0.
404 Irina Georgescu · Jani Kinnunen (2013)
4. The Operator ζ �→ Xζ
In the previous sections, we associated a discrete random variable Xζ with a discretefuzzy variable ζ. This way we define an operator from the set of discrete fuzzyvariables to the set of discrete random variables. In this section, we will study theway the operator ζ �→ Xζ behaves w.r.t. the expected utility.
A utility function is a function u : R→ R of class C2, strictly concave and strictlyincreasing.
Let (Ω,K , P) be a probability space and X : Ω → R a random variable. If u is autility function, then the function u(X) : u ◦ X : Ω→ R is a random variable.
The probabilistic expected value M(u(X)) is called probabilistic expected utility ofX, w.r.t. u.
Let Cr : P(R) → [0, 1] be a credibility measure, ζ a fuzzy variable, and μ themembership function of ζ. If u serves as a utility function, then u(ζ) = u ◦ ζ : Ω→ Ris a fuzzy variable. The credibilistic mean value Q(u(ζ)) of u(ζ) is called credibilisticexpected utility of ζ, w.r.t. u.
We consider now the discrete fuzzy variable given by:
ζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
μ1 μ2 · · · μn
⎤⎥⎥⎥⎥⎥⎦ , (12)
and the random variable Xζ associated with ζ:
Xζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
p1 p2 · · · pn
⎤⎥⎥⎥⎥⎥⎦ . (13)
Proposition 8 The discrete fuzzy variable u(ζ) is represented by:
u(ζ) :
⎡⎢⎢⎢⎢⎢⎣u(a1) u(a2) · · · u(an)μ1 μ2 · · · μn
⎤⎥⎥⎥⎥⎥⎦ . (14)
Proof Since u is injective, u(a1), · · · , u(an) are distinct from each other. Due to theinjectivity of u, we have for any i = 1, · · · , n:
We recall that μ is the membership function of ζ and that μi = μ(ai), i = 1, · · · , n.Let η be the membership function of u(ζ). Then, by the relation (6) of Section 3:
ηi = (2Cr(u(ζ) = u(ai))) ∧ 1
= (2Cr(ζ = ai)) ∧ 1 = μi, i = 1, · · · , n.This shows that (14) represents the fuzzy variable u(ζ).
By (13), the discrete random variable u(Xζ) has the distribution:
u(Xζ) :
⎡⎢⎢⎢⎢⎢⎣u(a1) u(a2) · · · u(an)p1 p2 · · · pn
⎤⎥⎥⎥⎥⎥⎦ .
Fuzzy Inf. Eng. (2013) 4: 399-416 405
Remark 2 By Proposition 8, we have u(Xζ) = Xu(ζ).
Proposition 9 Q(u(ζ)) = M(u(Xζ)).
Proof By Proposition 7 and Remark 2, it follows that:
Q(u(ζ)) = M(Xu(ζ)) = M(u(Xζ)).
5. Credibilistic Risk Aversion
In this section, we will introduce the notion of credibilistic risk premium associatedwith a fuzzy variable and a utility function. Using the operator ζ �→ Xζ one will studythe credibilistic risk premium for discrete fuzzy variables.
We recall first the notion of probabilistic risk premium and one of its approximatecalculation formulas.
Let X be a random variable and u a utility function. In this section, assume that uhas the class C2, strictly concave and strictly increasing.
Definition 5 [8, 10] The probabilistic risk premium Π = Π(X, u) associated with Xand u is the unique solution to the equation:
M(u(X)) = u(M(X) − Π). (15)
The probabilistic risk premium Π is an indicator of the risk aversion of the agentu in the face of probabilistic risk. If we interpret X as a lottery, then M(X) is theexpected return associated with the lottery X. Then Π is the sum the agent is willingto pay to achieve the sure return M(u(X)).
Proposition 10 [8, 10] An approximate solution of Equation (15) is given by
Π ≈ −12
Var(X)u′′(M(X))u′(M(X))
. (16)
Now, let Cr : P(R) → [0, 1] be a credibility measure, ζ a fuzzy variable, and u autility function.
Definition 6 The credibilistic risk premium λ = λ(ζ, u) associated with ζ and u is theunique solution to the equation:
Q(u(ζ)) = u(Q(ζ) − λ). (17)
Condition (17) by which the credibilistic risk premium is defined is similar inform to (15). Therefore the interpretation of λ will be similar to Π, with the essentialdifference that it will be done in terms of credibilistic EU-theory and not in terms ofprobabilistic EU-theory.
We don’t know an approximate calculation formula for λ analogous to Formula(16). We can prove only the following weaker result.
406 Irina Georgescu · Jani Kinnunen (2013)
Proposition 11 An approximate solution of Equation (17) is given by
λ ≈ u(Q(ζ)) − Q(u(ζ))u′(Q(ζ))
. (18)
Proof We develop in Taylor series the expression u(Q(ζ) − λ) and we omit theremainder of the first order: u(Q(ζ) − λ) ≈ u(Q(ζ)) − λu′(Q(ζ)). By (17), we obtainQ(u(ζ)) ≈ u(Q(ζ)) − λu′(Q(ζ)) from where (18) follows.
Let ζ be a discrete fuzzy variable and Xζ the associated discrete random variable:
ζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
μ1 μ2 · · · μn
⎤⎥⎥⎥⎥⎥⎦ ; Xζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
p1 p2 · · · pn
⎤⎥⎥⎥⎥⎥⎦ . (19)
Let u be a utility function.
Proposition 12 The credibilistic risk premium λ = λ(ζ, u) associated with ζ and ucoincides with the probabilistic risk premium Π = Π(Xζ , u) associated with Xζ and u.
Proof By (15) and (17), M(u(Xζ)) = u(M(Xζ) − Π) and Q(u(ζ)) = u(Q(ζ) − λ). ByProposition 9, M(u(Xζ)) = Q(u(ζ)), hence u(M(Xζ)−Π) = u(Q(ζ)−λ). u is injective,hence M(Xζ)−Π = Q(ζ)−λ. By Proposition 7, M(Xζ) = Q(ζ), and therefore Π = λ.
By the previous proposition, the evaluation of risk aversion of an agent in frontof a risk situation represented by a discrete fuzzy variable reduces to the evaluationof probabilistic risk aversion corresponding to the random variable Xζ . This thesis issustained by the approximate evaluation formula from the following proposition.
Proposition 13 We consider a discrete fuzzy variable ζ and the discrete random vari-able Xζ from (19). Then, an approximate value of credibilistic risk aversion λ(ζ, u)is:
λ(ζ, u) ≈ −12
u′′(Q(ζ))u′(Q(ζ))
n∑i=1
(ai − Q(ζ))2 pi. (20)
Proof By Propositions 12 and 10, the following relations hold:
λ(ζ, u) = Π(Xζ , u) ≈ −12
Var(Xζ)u′′(M(Xζ))u′(M(Xζ))
= −12
u′′(M(Xζ))u′(M(Xζ))
n∑i=1
pi(ai − M(Xζ))2.
By Proposition 7, M(Xζ) = Q(ζ) , and hence (20) is obtained.
Example 1 Let us consider the discrete fuzzy variable
ζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 a3
μ1 μ2 μ3
⎤⎥⎥⎥⎥⎥⎦ .Then the discrete random variable Xζ associated with ζ will have the distribution
Fuzzy Inf. Eng. (2013) 4: 399-416 407
Xζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 a3
p1 p2 p3
⎤⎥⎥⎥⎥⎥⎦ ,where according to Remark 1, the probabilities p1, p2, p3 will be given by:
If u is a utility function, then by Proposition 8,
Q(u(ζ)) = M(u(Xζ)) = p1u(a1) + p2u(a2) + p3u(a3).
We consider the following numerical example:
ζ :
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣1 3 514
13
1
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ .
Applying p1, p2, p3, one obtains:
p1 =12× 1
4+
12
[1 − 13∨ 1] =
18
;
p2 =12
[14∨ 1
5− 1
4] +
12
[15∨ 1 − 1] =
124
;
p3 =12
[1 − 14× 1
5] +
12× 1 =
56.
Then
Q(ζ) =18× 1 +
124× 3 +
56× 5 =
5312.
Consider the utility function u(x) = −e−x for any x ∈ R. It is easy to notice thatu′′(x)u′(x)
= −1 for any x ∈ R.
By applying Proposition 13, we can compute the approximate value of credibilisticrisk premium λ(ζ, u):
λ(ζ, u) ≈ 12
3∑i=1
pi(ai − Q(ζ))2 = 0.913.
6. Credibilistic and Stochastic Dominance
408 Irina Georgescu · Jani Kinnunen (2013)
Credibilistic dominance was introduced in [11, 7] as a way of ranking the fuzzy vari-ables. It is similar to stochastic dominance, a notion intensively studied in proba-bilistic models of risk in [1, 19, 11, 2, 7]. In this section, we will establish a relationbetween the credibilisitic dominance of discrete fuzzy variables and the stochasticdominance of discrete random variables associated with them by the operator ζ �→ Xζ .
We recall from [19] the notion of stochastic dominance of order k. Let (Ω,K , P)be a probability space with Ω ⊆ R and X : Ω → R a random variable. We recall thedefinition of the distribution function FX : R→ [0, 1] associated with X:
FX(x) = P(X � x), x ∈ R. (21)
For any x ∈ R and k � 0, we will define by induction the stochastic dominanceindex of order k:
F[0]X (x) = FX(x); F[k+1]
X (x) =∫ x
−∞F[k]
X (t)dt. (22)
Definition 7 Let X and Y be two random variables and k ∈ N. We define X �(k)prob Y
iff F[k]X (x) � F[k]
Y (x), for any x ∈ R. If X �(k)prob Y, then we say that X dominates
stochastically Y in order k.
Now, we present, following [20, 21], the notion of credibilistic dominance of orderk. Let Cr : P(R)→ [0, 1] be a credibility measure and ζ a fuzzy variable. By [6], thecredibility distribution φζ : R→ [0, 1] associated with ζ is defined by:
φζ(x) = Cr(ζ � x), x ∈ R. (23)
For any x ∈ R and k � 0, we will define by induction the credibilistic dominanceindex of order k:
φ[0]ζ (x) = φζ(x); φ[k+1]
ζ (x) =∫ x
−∞φ[k]ζ (t)dt. (24)
Definition 8 Let ζ and ε be two fuzzy variables and k ∈ N. We define ζ �(k)cred ε
iff φ[k]ζ (x) � φ[k]
ε (x), for any x ∈ R. If ζ �(k)cred ε, then we say that ζ dominates
credibilistically ε in order k.
�(k)prob is a preorder on the set of random variables X : Ω→ R and �(k)
cred is a preorderon the set of fuzzy variables ζ : R→ R.
Next, we study the restriction of the relation �(k)cred to the set of discrete fuzzy
variables. Let ζ be a discrete fuzzy variable defined by:
ζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
μ1 μ2 · · · μn
⎤⎥⎥⎥⎥⎥⎦ , a1 < a2 < · · · < an. (25)
Proposition 14 If φζ is the credibility distribution of ζ, then
φζ(ai) =12
[max1� j�iμ j + 1 − max
1< j�nμ j], i = 1, · · · , n.
Fuzzy Inf. Eng. (2013) 4: 399-416 409
Proof By Proposition 4, for any i = 1, · · · , n, we have
Cr(ζ � ai) =12
[max1� j�iμ j + 1 − max
1< j�nμ j].
Remark 1 is applied then.
Proposition 15 The values of φζ are given by
φζ(x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0, if x < a1,
φζ(a1), if a1 � x � a2,...
...
φζ(an−1), if an−1 � x � an,
1, if x � an.
We consider now the random variable Xζ associated with ζ:
Xζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
p1 p2 · · · pn
⎤⎥⎥⎥⎥⎥⎦ . (26)
Proposition 16 The distribution function FXζ of Xζ coincides with ζ’s credibility dis-tribution φζ .
Proof The form of FXζ is well known:
FXζ (x) =
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
0, if x < a1,
p1, if a1 � x < a2,
p1 + p2 if a2 � x < a3,...
...
p1 + ... + pn−1, if an−1 � x < an,
1, if x � an.
(27)
By Remark 1, one gets, for i = 1, · · · , n − 1 immediately
pi = φζ(ai) − φζ(ai−1). (28)
Replacing in (27) the probabilities pi with their values from (28), it follows thatFXζ (ai) = φζ(ai) for any i = 1, · · · , n. Applying then the Proposition 15, we obtainFXζ (x) = φζ(x), for any x ∈ R.
Proposition 17 For any x ∈ R and k ∈ N, we have φ[k]ζ (x) = F[k]
Xζ(x).
Proof Similarly, by induction on k, the case k = 0 is treated in Proposition 16, andthe step k → k + 1 follows by (22) and (24).
Proposition 18 Let ζ and ε be two discrete fuzzy variables. For any k ∈ N, thefollowing equivalence holds: ζ �(k)
cred ε iff Xζ �(k)prob Xε.
410 Irina Georgescu · Jani Kinnunen (2013)
Proof By Proposition 17.
7. Credibilistic Index of Riskiness
In this section, we define an index of riskiness associated with every discrete credi-bilistic gamble. This indicator, similar to the Aumann-Serrano index of probabilisticgambles [12], expresses the riskiness of a credibilistic gamble. We will prove a cred-ibilistic version of Hart theorem [13].
We introduce the following terminology:– A probabilistic gamble is a random variable, and– A credibilistic gamble is a fuzzy variable.
Proposition 19 [12] For any probabilistic gamble X, there exists a unique positivereal number R(X) such that
M(e−X
R(X) ) = 1. (29)
R(X) is called the Aumann-Serrano index of riskiness of X.One naturally raises the question of defining an index of riskiness associated with
a credibilistic gamble. We will answer this problem for discrete credibilistic gambles.Let us consider a discrete credibilistic gamble
ζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
μ1 μ2 · · · μn
⎤⎥⎥⎥⎥⎥⎦ , (30)
and the associated probabilistic gamble
Xζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 · · · an
p1 p2 · · · pn
⎤⎥⎥⎥⎥⎥⎦ . (31)
Proposition 20 If ζ is the discrete credibilistic gamble (30), then there exists a uniquepositive real number R(ζ) such that
Q(e−ζ
R(ζ) ) = 1. (32)
Proof We consider the function u : R → R defined by u(x) = e−x/R(Xζ ), for anyx ∈ R , and u is injective. Hence, by Proposition 9, we have
Q(e− ζ
R(Xζ ) ) = Q(u(ζ)) = M(u(Xζ)) = M(e− Xζ
R(Xζ ) ) = 1.
It follows that R(ζ) = R(Xζ) is the unique solution to Equation (32).
R(ζ) will be called the credibilistic index of riskiness of ζ. We consider an agentrepresented by a utility function of class C2, strictly increasing and strictly concave.
Let w be a real number interpreted as a wealth level.
Definition 9 [12] Let X be a probabilistic gamble. We say that the agent u acceptsX at wealth level w if M(u(w + X)) > u(w); otherwise u rejects X at level w.
The following definition introduces similar notions for credibilistic gambles.
Fuzzy Inf. Eng. (2013) 4: 399-416 411
Definition 10 Let ζ be a credibilistic gamble. We say that the agent u accepts ζ atwealth level w if Q(u(w + X)) > u(w) ; otherwise u rejects ζ at wealth level w.
Proposition 21 Let ζ be the discrete credibilistic gamble (30) and Xζ the probabilis-tic gamble (31). The following assertions are equivalent:
(i) u accepts ζ at wealth level w;(ii) u accepts Xζ at wealth level w.
Proof Let the function v : R → R defined by v(x) = u(w + x) for any x ∈ R.Since u is injective, it follows that v is injective too. By Proposition 9, one obtains:Q(u(w + ζ)) = Q(v(ζ)) = M(v(Xζ)) = u(v + Xζ). The equivalence of (i) and (ii) isimmediate.
We say that an agent u totally rejects a probabilistic gamble X (respectively, acredibilistic gamble ζ) if u rejects X (respectively, ζ) at all wealth level w.
Corollary 2 In conditions of Proposition 21, the following assertions are equivalent:(i) u totally rejects ζ;(ii) u totally rejects Xζ .
We recall the definition of Arrow-Pratt index associated with the utility function ufor any w ∈ R,
ru(w) = −u′′(w)u′(w)
.
We denote by U the set of utility functions u for which ru is non-increasing:ru(w′) � ru(w) for all w′ > w.
Definition 11 [13] Let X and Y be two probabilistic gambles. We say that X is riskierthan Y (X �prob Y) if any agent u ∈ U that totally rejects Y also totally rejects X.
We introduce a similar concept for credibilistic gambles.
Definition 12 Let ζ and ε be two credibilistic gambles. We say that ζ is riskier thanε (ζ �cred ε) if any agent u ∈ U that totally rejects ε also totally rejects ζ.
Proposition 22 Let ζ and ε be two discrete credibilistic gambles and Xζ and Xε theassociated probabilistic gambles. Then ζ �cred ε iff Xζ �prob Xε .
Proof By Definitions 11 and 12 and Corollary 2.
We recall following Hart theorem [13].
Theorem 1 [13] For any probabilistic gambles ζ and ε, X �prob Y iff R(X) � R(Y).
For discrete credibilistic gambles, a similar result holds.
Proposition 23 For any discrete credibilistic gambles associated with ζ and ε, thefollowing equivalence holds: ζ �cred ε iff R(ζ) � R(ε).
412 Irina Georgescu · Jani Kinnunen (2013)
Proof Let Xζ and Xε be the discrete probabilistic gambles associated with ζ and ε.From the proof of Proposition 20, we have R(ζ) = R(Xζ) and R(ε) = R(Xε). Then
ζ �cred ε i f f Xζ �prob Xε (by Proposition 22)
i f f R(Xζ) � R(Xε) (by Theorem 1)
i f f R(ζ) � R(ε).
8. A Credibilistic Model of Precautionary Saving
The way the presence of risk influences a choice of optimal saving by a consumerwas studied for the first time by Leland [22] and Sandmo [23]. They introducedthe precautionary saving as a measure of the risk effect on saving. In [24], Kimballconnected precautionary saving and prudence.
In this section, we will define a credibilistic model of optimal saving. In such amodel, the risk is represented by a fuzzy variable and the expected lifetime utility isdefined using the credibilistic expected utility of Section 4. When the risk is a discretefuzzy variable, we can convert our credibilistic model into a probabilistic one by theoperator ζ �→ Xζ . The credibilistic precautionary saving of the first model will equalto the probabilistic precautionary saving of the second one. This way the results onthe probabilistic model can be transferred to the credibilistic one.
The starting point of our discussion is the probabilistic two-period model of pre-cautionary saving of [25] (p. 95). We briefly present the components of this model:
– u(y) and v(y) are the utility functions of the consumer in periods 0 and 1;– In period 0 there is a sure income y0 and in period 1 there is an uncertain income
modeled by the random variable X;– s is the level of saving (in period 0); and– r is the rate of interest for saving.Assume that the utility functions u and v have the class C2 and u′ > 0, v′ > 0,
u′′ < 0, v′′ < 0. The expected lifetime utility of the model is:
V(s) = u(y0 − s) + M(v((1 + r)s + X)) (33)
and the associated optimization problem is:
maxs
V(s). (34)
Consider the case when the uncertain income X is replaced with M(X), and weobtain a model without uncertainty in which the lifetime utility is:
V1(s) = u(y0 − s) + v((1 + r)s + M(X)), (35)
and the optimization problem is:
maxs
V1(s). (36)
The functions V and V1 are strictly concave. If s∗(X) is the optimal solution of(34) and s∗1(M(X)) is the optimal solution of (36), then s∗(X) − s∗1(M(X)) is called
Fuzzy Inf. Eng. (2013) 4: 399-416 413
precautionary saving. This indicator measures the change of optimal saving inducedby the presence of possibilistic risk X.
Assume now that the uncertain income of period 1 is modeled by a fuzzy variableζ. The corresponding credibilistic model will have the following expected lifetimeutility:
W(s) = u(y0 − s) + Q(v((1 + r)s + ζ)). (37)
The other components of (37) have the same significance as in the probabilisticmodel.
The optimization problem of this model will be:
maxs
W(s). (38)
As in the probabilistic case, we can consider an uncertainty model by replacing ζwith Q(ζ). Then the lifetime utility will be:
W1(s) = u(y0 − s) + v((1 + r)s + Q(ζ)), (39)
and the associated optimization problem will be:
maxs
W1(s). (40)
Assume there exist the optimal solution s◦(ζ) of (38) and the optimal solutions◦1(Q(ζ)) of (40). The difference s◦(ζ) − s◦1(Q(ζ)) will be called credibilistic precau-tionary saving and will evaluate the change of optimal saving induced by the presenceof credibilistic risk ζ.
Next we assume that ζ is a discrete fuzzy variable defined by:
ζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 . . . an
μ1 μ2 . . . μn
⎤⎥⎥⎥⎥⎥⎦ , a1 < a2 · · · < an. (41)
Let Xζ be the discrete random variable associated with ζ. Then
Xζ :
⎡⎢⎢⎢⎢⎢⎣a1 a2 . . . an
p1 p2 . . . pn
⎤⎥⎥⎥⎥⎥⎦ (42)
with p1, · · · , pn given by Remark 1.We consider the lifetime utility function associated with Xζ ,
V(s) = u(y0 − s) + M(v((1 + r)s + Xζ)). (43)
This way we obtain an optimal model of optimal saving. We will also consider themodel without uncertainty with the lifetime utility:
V1(s) = u(y0 − s) + v((1 + r)s + M(Xζ)). (44)
Since ζ is now the fuzzy variable (41), according to Proposition 9, W(s) gets theform:
W(s) = u(y0 − s) + M(v((1 + r)s + Xζ))
= u(y0 − s) +n∑
i=1
piv((1 + r)s + ai).
414 Irina Georgescu · Jani Kinnunen (2013)
By derivation, we obtain:
W ′(s) = −u′(y0 − s) + (1 + r)n∑
i=1
piv′((1 + r)s + ai), (45)
W′′(s) = u′′(y0 − s) + (1 + r)2n∑
i=1
piv′′((1 + r)s + ai). (46)
Since u′′ < 0, v′′ < 0, it follows that W ′′(s) < 0 for any s, thus W is strictlyconcave. Similarly, W1 is strictly concave. Then the optimal solutions s◦(ζ) ands◦1(Q(ζ)) of Problems (38) and (40) exist and they can be computed from W ′(s◦(ζ)) =0, resp. W ′1(s◦1(Q(ζ))) = 0.
Taking into account (45), s◦(ζ) is a solution of the equation:
u′(y0 − s) = (1 + r)n∑
i=1
piv′((1 + r)s + ai). (47)
Similarly, s◦1(Q(ζ)) is a solution of the equation:
u′(y0 − s) = (1 + r)v′((1 + r)s + Q(ζ)). (48)
Proposition 24 The credibilistic precautionary saving s◦(ζ) − s◦1(Q(ζ)) is identicalto the probabilistic precautionary saving s∗(Xζ) − s∗1(M(Xζ)).
Proof One notices that V(s) = W(s) and V1(s) = W1(s) for any s, thus s◦(ζ) =s∗(Xζ) and s◦1(Q(ζ)) = s∗1(M(Xζ)).
The previous proposition enables some results on probabilistic model (defined byV) to be transferred to the initial credibilistic model (defined by W).
Proposition 25 If the utility function v has the class C3 and v′′′ > 0, then s◦(ζ) −s◦1(Q(ζ)) > 0 for any discrete fuzzy variable ζ.
Proof By [25] (p. 96), if v′′′ > 0, then s∗(Xζ)− s∗1(M(Xζ)) > 0 and then Proposition24 is applied.
Remark 3 Condition v′′′ > 0 represents the notion of “prudence” of consumer v[24]. Then Proposition 25 has the following interpretation: in case of a prudentconsumer (v′′′ > 0), the presence of credibilistic risk leads to the increase of theoptimal saving (s∗(Xζ) > s∗1(M(Xζ)) > 0).
Open question: Is the converse of Proposition 25 true: if s∗(Xζ) > s∗1(M(Xζ)) > 0for any discrete fuzzy variable ζ, then v′′′ > 0?
9. Conclusion
In this paper, three themes of risk theory were approached by credibility theory: riskaversion, credibilistic dominance, and an index of riskiness.
The main contributions of the paper are:
Fuzzy Inf. Eng. (2013) 4: 399-416 415
(i) The introduction of the notion of credibilistic risk premium as an indicator ofrisk aversion and the proofs of formulas for its evaluation;
(ii) The establishment of a relation between the credibilistic dominance of discretefuzzy variables and the stochastic dominance of associated discrete random variables;
(iii) The definition and the study of an index of riskiness of Aumann-Serrano typefor discrete credibilistic gambles;
(iv) The study of discrete credibilistic optimal saving models.The three credibilistic models of the paper focus on risk situations represented by
discrete fuzzy variables. They can be useful to an agent in a decision-making processof credibilistic risk problems. An open problem is to elaborate similar models for thegeneral case of risk situations described by arbitrary fuzzy variables.
In the study of the four themes of this paper, the following common scheme ap-pears: With a discrete credibilistic model, one canonically associates a discrete prob-abilistic model. The indicators of the probabilistic model coincide with the indicatorsof the initial one making both discrete credibilistic models be eventually studied byprobabilistic techniques.
It is expected that in case of non-discrete fuzzy variables, credibilistic models dif-fer significantly from probabilistic ones.
We mention that some results of this paper were published without a proof in [26].
Acknowledgments
The authors would like to thank the referees for their helpful comments on a previousversion of the paper.
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