A Probabilistic Approach to Possibilistic Risk Aversion Irina Georgescu
Jani Kinnunen
Department of Economic Cybernetics Academy of Economic Studies Bucharest, Romania and Universidad Loyola-Andalucia Department of Quantitative Methods, Cordoba, Spain
[email protected]
Institute for Advanced Management Systems Research
Åbo Akademi University Turku, Finland
[email protected]
Abstract—This paper proposes an approach to possibilistic risk aversion by means of probabilistic concepts. A notion of probabilistic risk premium is defined as the probabilistic risk premia associated with the uniform distributions on the level sets of a fuzzy number. We prove an approximate computation formula for this indicator and a Pratt-type theorem to compare two agents’ possibilistic risk aversions. IndexTerms—Possibilistic probability theory
risk
aversion,
Pratt
theorem,
I. INTRODUCTION Probabilistic theory of risk aversion was founded by Arrow [1] and Pratt [13]. An indicator of an agent’s risk aversion in front of a situation of uncertainty is probabilistic risk premium. In [1] and [13] this indicator is evaluated in terms of the Arrow–Pratt index. Pratt theorem proved in [13] shows that to compare two agents’ risk aversions reduces to comparing the corresponding Arrow-Pratt indices. [8] and [10] propose an approach to risk aversion by Zadeh’s possibility theory [14]. The risk situations are described by fuzzy numbers and the notions and results on possibilistic risk aversion are expressed by possibilistic indicators. The possibilistic expected value and possibilistic variance of a fuzzy number have been introduced in [2] and [7] and their theory has been developed in several papers (e.g. [3], [4], [9], [11], [12]). In particular, in [2], [4], [11] and [12] the two possibilistic indicators are characterized probabilistically using the so–called ”principle of average value” (see [3], p. 27). Following this line, this paper aims to possibilistically treat risk aversion by probabilistic methods. We summarize the content of the paper. Section 2 presents the probabilistic definitions of possibilistic expected value, possibilistic variance and possibilistic expected utility. In Section 3 the possibilistic risk premium π(A, u, f) associated with a fuzzy number A, a utility function u and a weighting function f is introduced. The fuzzy number A represents the risk situation, the utility function u characterizes
the risk situation of the agent in front of risk and f assures that a weighting process w.r.t. the level sets. π(A, u, f) is a new indicator of risk aversion w.r.t. the risk situation A and it is defined as the f–weighted average of the probabilistic risk premia associated with the uniform distributions on the level sets of A. The section has two propositions. The first one refers to the exact computation of π(A, u, f) and the second one establishes a relationship between π(A, u, f) and the possibilistic risk premium ρ(A, u, f) of [8]. Section 4 contains an approximate computation formula for π(A, u, f)analogous to the Arrow–Pratt approximation formula of probabilistic risk theory (see e.g. [6], p. 11). It expresses π(A, u, f) according to u’s Arrow–Pratt index [1], [13]. Another result of the section is a Pratt–type theorem [13]. One proves that to compare two agents’ aversions to the possibilistic risk is equivalent to comparing the Arrow–Pratt indices associated with their utility functions. II. INDICATORS OF FUZZY NUMBERS We recall from [3], [4], [12], the probabilistic definitions of the following indicators of fuzzy numbers: possibilistic expected value, possibilistic variance and possibilistic expected utility. Let A be a fuzzy subset of the set R of real numbers. Then A is a fuzzy number if it is continuous, fuzzy convex and its support supp(A) = {x∈ R|A(x) > 0} is bounded. If γ ∈ [0, 1] γ then the γ–level set [A] of A is defined by [𝐴]𝛾 =
𝑥 ∈ 𝑅 𝐴(𝑥) ≥ 𝛾 𝑐𝑙 𝑠𝑢𝑝𝑝 𝐴
𝑖𝑓 𝛾 > 0 𝑖𝑓 𝛾 = 0
(1)
(cl(supp(A)) is the topological closure of supp(A)). γ γ By [5], the γ–level set [A] has the form [A] = [a1(γ), a2(γ)] with a1(γ) ≤a2(γ). A non–negative and monotone increasing function f: [0,1] → R is a weighting function if it satisfies the 1 normality condition 0 𝑓 𝛾 𝑑𝛾 = 1. Let A be a fuzzy number and f a weighting function. We
γ assume that [A] = [a1(γ), a2(γ)] for any γ ∈ [0, 1]. For any γ ∈ [0, 1] we denote by Xγ the uniform probability γ distribution on [A] . We recall that the probability density function of Xγ has the form:
M(u(X)) = u(M(X) − ρ(X, u)).
1
𝑔 𝑥 =
𝑖𝑓 𝑎1 𝛾 ≤ 𝑎2 𝛾
𝑎 2 𝛾 −𝑎 1 𝛾
0 𝑖𝑓 𝑜𝑡𝑒𝑟𝑤𝑖𝑠𝑒
(2)
andthe probabilistic expected value M(Xγ) probabilistic variance Var(Xγ) are given by
and
M(Xγ)= (a1(γ)+a2(γ))/2; Var(Xγ)= (a2(γ)−a1(γ))2 /12.
the
(3)
Let u: R → R be a continuous utility function. Then u(Xγ) is a random variable and M(u(Xγ)) is called probabilistic expected utility of Xγ w.r.t. u. The f–weighted possibilistic expected value Ef(A) and the f– weighted possibilistic variance Varf (A) are defined by 𝐸𝑓 𝐴 =
1 𝑀(𝑋𝛾 )𝑓 0
𝑉𝑎𝑟𝑓 𝐴 =
2 ′ ′′ class C with the properties u > 0 andu < 0. If the risk situation is described by the random variable X, then by [1], [13] the probabilistic risk premium ρ(X, u) associated with (X, u) is defined by the equality:
𝛾 𝑑𝛾 =
1 𝑉𝑎𝑟(𝑋𝛾 )𝑓 0
1 𝑎 1 𝛾 +𝑎 2 𝛾 0 2
𝛾 𝑑𝛾 =
(9)
Assume that the risk situation is described by a fuzzy γ number A whose γ–level sets are [A] = [a1(γ), a2(γ)], γ ∈ γ [0,1]. We recall that Xγ is a uniform distribution on [A] . We fix a weighting function f. A notion of possibilistic risk premium ρ(A, u, f) associated with the triple (A, u, f) has been introduced in [8] by the equality: Ef(u(A)) = u(Ef (A) − ρ(A, u, f)).(10) We will define a second notion of possibilistic risk premium using the family of uniform distributions (Xγ), γ ∈ [0,1]. For any γ ∈ [0,1] we consider the probabilistic risk premium ρ(Xγ, u) associated with (Xγ, u) by the equality
𝑓 𝛾 𝑑𝛾;(4)
1 (𝑎 2 𝛾 −𝑎 1 𝛾 )2 0 12
M(u(Xγ)) = u(M(Xγ)) − ρ(Xγ, u)). 𝑑𝛾.(5)
Let u: R → R be a continuous utility function. For any γ ∈ [0,1] we consider the probabilistic expected utility M(u(Xγ)). The possibilistic expected utility Ef(u(A)) of the fuzzy number A w.r.t u and f is defined by
(11)
Taking into account that Xγ is uniformly distributed on the interval [a1(γ), a2(γ)] we have 𝑀 𝑢 𝑋𝛾
=
𝑎 2 (𝛾) 1 𝑢 𝑎 2 𝛾 −𝑎 1 (𝛾) 𝑎 1 (𝛾)
𝑥 𝑑𝑥 . (12)
The equality (11) gets the form: 𝐸𝑓 𝑢 𝐴
=
1 𝑀 0
𝑢 𝑋𝛾
𝑓 𝛾 𝑑𝛾 .
(6)
A simple computation shows that 𝐸𝑓 𝑢 𝐴
=
1 1 0 𝑎 2 𝛾 −𝑎 1 𝛾
𝑎2 𝛾 𝑎1 𝛾
𝑢 𝑥 𝑑𝑥 𝑓 𝛾 𝑑𝛾.(7)
If u is the identity of R then Ef(u(A)) = Ef(A). If u(x) = 2 (x−Ef(A)) , then Ef(u(A)) is another possibilistic variance ∗ Varf (A) introduced in [8]. The relation between the ∗ possibilistic variances Varf (A) and Varf (A) is given by the relation from [8]: 𝑉𝑎𝑟𝑓∗ 𝐴 = 4𝑉𝑎𝑟𝑓 𝐴 − 𝐸𝑓2 𝐴 +
1 𝑎 0 1
𝛾 𝑎2 𝛾 𝑓 𝛾 𝑑𝛾.(8)
III. A PROBABILISTICALLY INTRODUCED NOTION OF POSSIBILISTIC RISK PREMIUM In this section we will define a notion of possibilistic risk premium in a probabilistic context, using the uniform distributions associated with the γ–level sets of a fuzzy number. We consider an agent represented by a utility function u of
𝑎2 𝛾 1 𝑎 2 𝛾 −𝑎 1 (𝛾) 𝑎 1 (𝛾)
𝑢 𝑥 𝑑𝑥 = 𝑢
𝑎 1 𝛾 +𝑎 2 𝛾 2
− 𝜌 𝑋𝛾 , 𝑢 .(13)
Definition 3.1 The possibilistic risk premium π(A, u, f) associated with (A, u, f) is defined by equality: 𝜋 𝐴, 𝑢, 𝑓 =
1 𝜌 0
𝑋𝛾 , 𝑢 𝑓 𝛾 𝑑𝛾. (14)
The above definition falls into the principle of average value: π(A, u, f) is the f–weighted average of the family (ρ(Xγ, u)), γ ∈ [0,1]. The function u is strictly increasing therefore we can take its −1 inverse u (defined on the image of u). Proposition 1 3.2𝜋 𝐴, 𝑢, 𝑓 = 𝐸𝑓 𝐴 − 0 𝑢−1 (𝑀 𝑢 𝑋𝛾 )𝑓 𝛾 𝑑𝛾. Proof. From (11) it follows that ρ(Xγ, u) = M(Xγ) −1 =u (M(u(Xγ))). Replacing in (14) one gets: 𝜋 𝐴, 𝑢, 𝑓 = 1 𝑀 0
𝑋𝛾 𝑓 𝛾 𝑑𝛾 − = 𝐸𝑓 𝐴 −
1 −1 𝑢 0 1 −1
𝑢
0
Proposition 3.3
𝑀 𝑢 𝑋𝛾
𝑓 𝛾 𝑑𝛾
𝑀 𝑢 𝑋𝛾
𝑓 𝛾 𝑑𝛾 .
γ Let A be a fuzzy number with [A] = [a1(γ), a2(γ)] for γ ∈ [0, 1]. Applying Proposition 4.1 for u1and u2we obtain
𝜋 𝐴, 𝑢, 𝑓 − 𝜌 𝐴, 𝑢, 𝑓 = 𝑢−1 𝐸𝑓 𝑢 𝐴
1 −1 𝑢 0
−
𝑀 𝑢 𝑋𝛾
𝑓 𝛾 𝑑𝛾.
Proof. From (10) it follows that ρ(A, u, f) = −1 Ef(A)−u (Ef(u(A))), which together with Proposition 3.2 lead to the desired equality. −2x Example 3.4 We consider the utility function u(x) = −e . 2 ′ ′′ One notices that uhas the class C and u > 0, u < 0. u’s inverse −1 has the form u (y) =−(1/2)ln(−y). By applying in this case (12) of Section 3 we obtain: 𝑀 𝑢 𝑋𝛾
=
𝑎 2 (𝛾) 𝑢 𝑎 2 𝛾 −𝑎 1 (𝛾) 𝑎 1 (𝛾) 1
𝑥 𝑑𝑥 =
𝑒 −2𝑎 2 (𝛾 ) −𝑒 −2𝑎 1 (𝛾 ) 2(𝑎 2 𝛾 −𝑎 1 𝛾 )
.
Here we deduce 𝑢
−1
𝑀 𝑢 𝑋𝛾
1
𝑒 −2𝑎 2 𝛾 −𝑒 −2𝑎 1 𝛾
2
2 𝑎 2 𝛾 −𝑎 1 𝛾
= − ln[−
].
By Proposition 3.2 it follows that𝜋 𝐴, 𝑢, 𝑓 = 𝐸𝑓 𝐴 − 1 −1 𝑢 0
𝑀 𝑢 𝑋𝛾
𝑓 𝛾 𝑑𝛾 =
1 ln 2 0
𝑒 −2𝑎 2 𝛾 −𝑒 −2𝑎 1 𝛾
𝐸𝑓 𝐴 +
1
−
2 𝑎 2 𝛾 −𝑎 1 𝛾
𝑓 𝛾 𝑑𝛾.
IV. THE APPROXIMATE CALCULATION OF POSSIBILISTIC RISK PREMIUM In this section we will prove an approximate calculation formula for possibilistic risk premium π(A, u, f) and a Pratttype theorem [13] to compare the possibilistic risk aversions. 2 ′ ′′ Let u be a utility function of class C with u > 0 and u < 0. By [1], [13] the Arrow–Pratt index of u is defined by 𝑟𝑢 𝑥 = −
𝑢 ′′ (𝑥) 𝑢 ′ (𝑥)
for any x∈ R.
(15)
1 1 𝑉𝑎𝑟 𝑋𝛾 𝑟𝑢 𝑀 𝑋𝛾 24 0 𝑎 𝛾 +𝑎 2 𝛾 − 𝑎1 𝛾 )2 𝑟𝑢 1 2
1 (𝑎2 24 0 1
𝛾
𝑓 𝛾 𝑑𝛾 = 𝑓 𝛾 𝑑𝛾.
−2x Example 4.2 We consider the utility function u(x) = −e . One notices that ru(x) = 2 for any x∈ R. Applying Proposition 4.1 in this case one obtains 𝜋(𝐴, 𝑢, 𝑓) ≈ 2
1 24 0
𝑎2 𝛾 − 𝑎1 𝛾
𝜋 𝐴, 𝑢1 , 𝑓 𝑎 𝛾 +𝑎 2 𝑟1 1
𝛾
≈
1 24 0
𝜋 𝐴, 𝑢2 , 𝑓 𝑎 𝛾 +𝑎 2 𝑎2 𝛾 − 𝑎1 𝛾 𝑟2 1
𝛾
1
1
1 1 12 0
𝑎2 𝛾 −
𝑎1 𝛾 𝑓 𝛾 𝑑𝛾 = 𝑉𝑎𝑟𝑓 𝐴 . Consider two agents with the utility functions u1and u2. We denote by r1(x) = ru1(x) and r2(x) = ru2(x) the Arrow–Pratt indices of u1and u2.
2
2
2
𝑓 𝛾 𝑑𝛾;(16)
2
𝑓 𝛾 𝑑𝛾.(17)
The following result is a Pratt–type theorem [13]. Proposition 4.3 The following assertions are equivalent: (a) r1(x) ≥ r2(x) for any x∈ R;
(b) π(A, u1, f) ≥ π(A, u2, f) for any fuzzy number A. Proof. (a) ⇒ (b): Let A be an arbitrary fuzzy number. By hypothesis, for anyγ ∈ [0, 1] the following inequality holds: r1((a1(γ)+a2(γ))/2) ≥ r2((a1(γ)+a2(γ))/2). Then, by (16) and (17) we obtain π(A, u1, f) ≥ π(A, u2, f). (b) ⇒ (a): We consider the symmetrical triangular fuzzy number A = (x, α). γ For any γ ∈ [0,1], [A] =[a1(γ), a2(γ)]=[x−α, x+α], therefore (a1(γ)+a2(γ))/2=x. Then (16) and (17) are written: 𝜋 𝐴, 𝑢1 , 𝑓 1 1 ≈ 𝑎2 𝛾 − 𝑎1 𝛾 0
2
𝜋 𝐴, 𝑢2 , 𝑓 1 1 ≈ 𝑎2 𝛾 − 𝑎1 𝛾 0
2
24
24
We fix a weighting function f. Let A be a fuzzy number γ whose γ–level sets are [A] = [a1(γ), a2(γ)] for any γ ∈ [0, 1]. Proposition 4.1𝜋(𝐴, 𝑢, 𝑓) ≈ 2 1 1 𝑎 𝛾 +𝑎 2 𝛾 𝑎2 𝛾 − 𝑎1 𝛾 𝑟𝑢 1 𝑓 𝛾 𝑑𝛾. 24 0 2 Proof. Let γ ∈ [0,1]. By applying the Arrow–Pratt approximation formula ([1], [13] or [6], p. 11) to the probabilistic risk premium ρ(Xγ, u) one obtains ρ(Xγ, u) ≈ (1/2)Var(Xγ)ru(M(Xγ)). By Definition 3.1 and (3) of Section 2 it follows that 𝜋(𝐴, 𝑢, 𝑓) ≈
≈
𝑟1 𝛾 𝑓 𝛾 𝑑𝛾 =
𝑟2 𝛾 𝑓 𝛾 𝑑𝛾 =
𝑟1 𝑥 2
𝑟2 𝑥 2
𝑉𝑎𝑟𝑓 𝐴 ;(18)
𝑉𝑎𝑟𝑓 𝐴 .(19)
Taking into account that π(A, u1, f) ≥ π(A, u2, f) and Varf(A) ≥ 0 it follows that r1(x) ≥ r2(x). In the risk aversion theory corresponding to possibilistic risk premium ρ(A, u, f) a Pratt–type theorem also holds (see [9], Theorem 4.3.7). Putting together Pratt theorem from [13], Theorem 4.3.7 from [9] and Proposition 4.3 from above the following result is obtained. Theorem 4.4 The following assertions are equivalent: (i) r1(x) ≥ r2(x) for any x∈ R; −1 (ii) u1 ◦ u2 is concave; (iii) For any random variable X, ρ(X, u1) ≥ ρ(X, u2); (iv) For any fuzzy number A, ρ(A, u1, f) ≥ ρ(A, u2, f); (v) For any fuzzy number A, π(A, u1, f) ≥ π(A, u2, f). Proof. The equivalences (i) ⇔ (ii) ⇔ (iii) are exactly Pratt theorem from [13]; (i) ⇔(iv) is Theorem 4.7 from [9]; and (i) ⇔(v) is Proposition 4.3. V. CONCLUSIONS The paper proposes a new treatment of risk aversion founded on the notion of possibilistic risk premium π(A, u, f). This approach is based on the idea: probability theory of risk aversion from [1], [13] is applied for uniform distributions associated with the level sets of a fuzzy number A (representing the risk situation), and then, by the principle of average value,
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