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Exact capacities and star-shaped distorted probabilities
Mathematical Social Sciences 56 (2008) 185–194 www.elsevier.com/locate/econbase
Exact capacities and star-shaped distorted probabilities Zaier Aouani a,∗ , Alain Chateauneuf b a University of Kansas, Economics Department, 415 Snow Hall, Lawrence, KS 66045, USA b Universit´e Paris I, PSE-CES, 106-112 boulevard de l’Hopital, 75647 Paris Cedex 13, France
Received 22 February 2006; received in revised form 18 January 2008; accepted 18 January 2008 Available online 13 February 2008
Abstract We are interested in capacities which are deformations of probability, i.e. v = f ◦ P. We characterize balanced, totally balanced, and exact capacities by properties concerning the probability transformation function, f . These results allow us to obtain simple new characterizations of a large pattern of risk aversions relevant to Yaari’s dual theory of choice under risk. c 2008 Elsevier B.V. All rights reserved.
JEL classification: C71; D80 Keywords: Capacity; Exact; Balanced; Star-shaped function; Risk aversion
1. Introduction Set functions, which are not necessarily additive, are frequently used in decision theory. Indeed, they have different interpretations: on the one hand, they represent transferable utility or the characteristic function of a cooperative game; while on the other hand, they represent non-additive probabilities. In this work, we are interested in capacities which are distorted probabilities, i.e. deformations of a given probability, v = f ◦ P where P is a non-atomic additive probability on a measurable space (Ω , A) where Ω is a non-empty set, A a σ -algebra of subsets of Ω , and f : [0, 1] → [0, 1] is a non-decreasing function, satisfying f (0) = 0 and f (1) = 1. Because of their special form, ∗ Corresponding author.
E-mail addresses: [email protected], [email protected] (Z. Aouani), [email protected] (A. Chateauneuf). c 2008 Elsevier B.V. All rights reserved. 0165-4896/$ - see front matter doi:10.1016/j.mathsocsci.2008.01.006
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distorted probabilities play an important role in mathematical economics, specifically in risk theory and game theory. The main purpose of this work is to characterize a wide classical range of monotone normalized games, i.e. capacities, by properties concerning the probability transformation f . An important motivation for such characterizations is that, building upon (Chateauneuf et al., 2004), this will allow us to directly obtain simple new characterizations of a large pattern of risk attitudes relevant to Yaari’s dual theory of choice under risk (Yaari, 1987), in terms of simple specific properties of the capacity arising in Yaari’s model. The paper is organized as follows: in Section 2, we introduce our notation, definitions and preliminary results. Section 3 is devoted to the characterization of balanced, totally balanced, and exact capacities by properties related to f . The most salient result – related to Wallner (2003) who dealt with robust statistics – concerns exact capacities which are proved to be obtained through star-shaped distortions. In Section 4, results of Section 3 are straightforwardly applied to derive new characterizations of several kinds of risk aversion for decision makers a` la Yaari. Section 5 concludes the paper. 2. Definitions, notation and preliminary results A game is a triple (Ω , A, v) where Ω is a non-empty set, A is an algebra of subsets of Ω and v is a function v : A → R+ such that v(∅) = 0. The game v is called a (normalized) capacity if (i) v is normalized i.e. v(Ω ) = 1, and (ii) v is monotone, i.e. A, B ∈ A and A ⊂ B implies v(A) ≤ ¯ for all A ∈ A. Elements v(B). Each capacity v has a dual capacity v¯ defined by v(A) ¯ = 1 − v( A) of Ω are usually called players or states of nature and the elements of A are called coalitions or events, depending on whether the context is game theory or decision under uncertainty. For each A ∈ A, v(A) is interpreted as a measurement of the power of coalition A or as the subjective likelihood of the event A. If A is an algebra of subsets of Ω and A ∈ A, the family A ∩ A := {B ∩ A, B ∈ A} is an algebra of subsets of A. If A ∈ A, then the game (A, A ∩ A, v|A∩A ) is called the restriction of v to the coalition A. The game v|A∩A will be simply denoted v|A . The core of a game (Ω , A, v) is the set C(v) = {λ : A → R+ , λ is a finitely additive measure, λ(Ω ) = v(Ω ) and λ ≥ v}. A game v is said to be: (a) balanced if C(v) 6= ∅. (b) totally balanced if for all A ∈ A, its restriction to A is balanced. (c) exact if it is balanced and for all A ∈ A, v(A) = min{µ(A) : µ ∈ C(v)}. The minimum in the preceding expression is reached since C(v) is compact under the weak? topology in the space of additive and bounded measures. (d) convex if A1 and A2 ∈ A implies v(A1 ) + v(A2 ) ≤ v(A1 ∪ A2 ) + v(A1 ∩ A2 ). The following implications are well-known (see for example Schmeidler (1972)): v is convex ⇒ v is exact ⇒ v is totally balanced ⇒ v is balanced. Next, we overview some earlier classical characterizations of exact, balanced, totally balanced and convex games. For all A ⊆ Ω , let A∗ denote the characteristic function of A, i.e. the function defined on Ω , being equal to 1 on A and 0 elsewhere.
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Theorem 2.1. Let A be a σ -algebra of subsets of Ω and v a game. The game v is exact if and only if for all n ∈ N∗ , a ∈ R, a1 , . . . , an ∈ R+ , A, A1 , . . . , An ∈ A, A∗ + aΩ ∗ ≥
n X
ai Ai∗ ⇒ v(A) + av(Ω ) ≥
i=1
n X
ai v(Ai ).
i=1
To prove Theorem 2.1, Kannai (1969) used a fundamental result, due to Fan (1956), on linear systems of inequalities in a real normed vector space. Schmeidler (1972) proved the same result using a separation theorem. ∗ Corollary 2.1. The game P v is balanced if and Pnonly if ∀n ∈ N , a1 , . . . , an ≥ 0 and n ∗ ∗ A1 , . . . , An ∈ A we have: i=1 ai Ai ≤ Ω ⇒ i=1 ai v(Ai ) ≤ v(Ω ). ∗ Corollary 2.2. The game v isPtotally balanced if and Pn only if ∀n ∈ N , ∀a1 , . . . , an ≥ 0 and n ∗ ∗ A, A1 , . . . , An ∈ A we have: i=1 ai Ai ≤ A ⇒ i=1 ai v(Ai ) ≤ v(A).
The following theorem characterizes convex games (see e.g. Delbaen (1974)). Theorem 2.2. The game v is convex if and only if for any finite sequence ∅ = A0 ⊆ A1 ⊆ · · · ⊆ An = Ω in A, there exists λ ∈ C(v) such that λ(A0 ) = v(A0 ), . . . , λ(An ) = v(An ). Our Proposition 3.4 provides, when v is a convex distorted probability, an explicit expression for a probability λ satisfying the above condition in Theorem 2.2. The anticore of a game v, introduced by Gilboa and Lehrer (1991), is the set of all finitely additive measures that are setwise dominated by v, that is, AC(v) = {λ : A → R+ , λ is a finitely additive measure with λ(Ω ) = v(Ω ) and λ ≤ v}. The game v is said to be antibalanced if AC(v) 6= ∅. The game v is said to be totally antibalanced if for all A ∈ A, v|A is antibalanced. Because v and v¯ play dual roles, it is obvious that C(v) = AC(v) ¯ and therefore a game is balanced if and only if its dual is antibalanced. As the following example shows, this duality relationship is no longer valid in the case of total balancedness since the restriction of a dual is not the dual of the restriction i.e. (v|A ) 6= v¯|A . Example 2.1. Let Ω = {1, 2, 3, 4} and for A a subset of Ω ; 1 if 1 ∈ A and |A| ≥ 3, v(A) = 0 otherwise. The game v is totally balanced. From monotonicity of v, Corollary 2.2 is trivially satisfied if v(A) = 0. If v(A) = 1, either |A| = 3 and without loss of generality we may assume A = {1, 2, hence if there exists i 0 ∈ {1, 2, . . . , n} such that Ai0 = A, clearly ai0 ≤ 1 and P3}, n ai v(Ai ) ≤ ai0 ≤ v(A), or A = Ω and it suffices to check that the probability therefore i=1 P defined by P({1}) = 1 belongs to C(v). However, v¯ is not totally antibalanced because AC(v¯|{2,3} ) = ∅ since v({2}) ¯ = v({3}) ¯ = 0 and v({2, ¯ 3}) = 1. The same techniques used to prove Theorem 2.1 lead to the following characterization of antibalanced games. Proposition 2.1. The game v is antibalanced if and ∀a1 , . . . , an ≥ 0 and Pn only if∗ ∀n ∗∈ N,P n A1 , . . . , An ∈ A, the following implication holds: i=1 ai Ai ≥ Ω ⇒ i=1 ai v(Ai ) ≥ v(Ω ). From Proposition 2.1 we may state:
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Corollary 2.3. The game v is totally antibalanced if and only if ∀n ∈ N, P∀a1 , .∗. . , an ∗≥ 0, ∀A ∈ A and ∀A , . . . , A ∈ A ∩ A, the following implication holds: n 1 i ai Ai ≥ A ⇒ P a v(A ) ≥ v(A). i i i Notice however that now we require the Ai ’s to be subsets of A. This condition was omitted in Corollary 2.2 because it would be superfluous. 3. Distorted pobabilities As pointed out in the introduction, an important motivation for this paper is to detect in the framework of Yaari’s dual theory of choice under risk, simple specific properties of the underlying capacity, characterizing a large known pattern of risk attitudes. This will be achieved as soon as known characterizations of risk aversion in terms of the distortion function (Yaari, 1987; Chateauneuf et al., 2004) will be shwon to correspond to a known specific property of the capacity. Accordingly, we now focus on some properties of “distortion” functions which are relevant for our purpose. Let f : [0, 1] → [0, 1] with f (0) = 0 and f (1) = 1. The function f is said to be f (x) star-shaped at t ∈ [0, 1], if x 7→ f (t)− is non-decreasing on [0, 1] \ {t}. In the following t−x Remark 3.1 and Lemma 3.1, we present some characterizations of star-shaped functions which will be useful thereafter. The proofs of most of these facts follow in a straight-forward manner from the definitions. We give a proof of Lemma 3.1 which appears to be more relevant. Remark 3.1. (i) The function f is star-shaped at 0 if and only if f |[0, p] ≤ [ f (pp) ] id 1 for each p ∈ (0, 1]. f (y)− f (x) (ii) The function f is star-shaped at 0 if and only if f (x) , ∀ 0 < x < y ≤ 1. x ≤ y−x ¯( p) f (iii) The function f is star-shaped at 1 if and only if f¯|[0, p] ≥ [ p ] id for each p ∈ (0, 1]. Where f¯(·) = 1 − f (1 − ·) is the conjugate function of f . f (y) f (x) (iv) The function f is star-shaped at 1 if and only if 1−1−y ≥ f (y)− , ∀ 0 ≤ x < y < 1. y−x Facts (ii) and (iv) of Remark 3.1 make it possible to give a geometrical interpretation of starshapedness at 0 and 1 (See Fig. 1). Indeed a function f is star-shaped at 0 and 1 if and only if for any a ∈ [0, 1], the cord connecting (a, f (a)) to the origin and the cord connecting it to (1, f (1) = 1) are above the curve of f on the interval [0, a] (respectively [a, 1]). While the minimum of concave functions is concave, the minimum of convex functions is star-shaped at 0 and 1. More precisely: Lemma 3.1. The function f is star-shaped at 0 and 1 if and only if f = mini∈I f i where I is a non-empty set and f i : [0, 1] → [0, 1] is a convex function satisfying f i (0) = 0 and f i (1) = 1, ∀i ∈ I . Proof. Necessity. Let G = {g : [0, 1] → [0, 1] s.t. g(0) = 0, g(1) = 1, g is convex and g ≥ f }. Then G 6= ∅ because id ∈ G. We prove f = ming∈G g. Let x ∈ (0, 1) and let gx : [0, 1] → [0, 1] be defined by f (x) t if t ∈ [0, x], x gx (t) = 1 − f (x) 1 − f (x) t +1− if t ∈ [x, 1]. 1−x 1−x 1 The symbol id represents the identity mapping of the interval [0, 1], that is the mapping [0, 1] → [0, 1], x 7→ x.
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Fig. 1. An example of a function that is star-shaped at 0 and 1.
Then gx is convex ( f (x) x ≤
1− f (x) 1−x
by star-shapedness at 0 and 1), gx ≥ f and gx (x) = f (x).
Sufficiency. Let 0 < x < y ≤ 1. Then, there exists j ∈ I such that f (y) = f j (y). Hence f j (x) f j (y) − f j (x) f j (y) − f (x) f (x) f (y) − f (x) ≤ ≤ ≤ = · x x y−x y−x y−x Then f is star-shaped at 0 (by Remark 3.1(ii)). Let 0 ≤ x < y < 1. Then there exists i ∈ I such that f (x) = f i (x). Hence 1 − f (y) 1 − f i (y) f i (y) − f i (x) f (y) − f i (x) f (y) − f (x) ≥ ≥ ≥ = . 1−y 1−y y−x y−x y−x Then f is star-shaped at 1 (by Remark 3.1(iv)).
Remark 3.2. (i) If f is convex then f is star-shaped at every point of [0, 1]. (ii) If f is star-shaped at 0 and 1 then f is non-decreasing on [0, 1] and continuous on [0, 1). Proof. (ii) The non-decreasing property of f follows directly from star-shapedness at 0. f (t) Continuity arises from star-shapedness at 1. The functions f and t 7→ 1−1−t are monotone and thus have finite limits on the right (and respectively left) at each point of [0, 1) (and respectively f (x) f (x + ) (0, 1]). Let x ∈ [0, 1). We have 1−1−x ≤ 1−1−x . Then f (x + ) ≤ f (x). Since f is nonf (x ) f (x) decreasing, f (x) = f (x + ) follows. Similarly, for x ∈ (0, 1) we have 1−1−x ≤ 1−1−x − − implying f (x ) ≥ f (x) and f (x) = f (x ) by the same argument as above. Hence f is continuous at each x ∈ (0, 1) and f is continuous on the right at 0. −
Let us now recall how a distorted probability is defined. A probability P on (Ω , A) is said to be non-atomic if ∀A ∈ A, ∀α ∈ [0, P(A)], ∃B ∈ A
such that B ⊂ A and P(B) = α.
In the sequel of this section, A is a σ -algebra of subsets of Ω , P is a non-atomic probability on (Ω , A) and f : [0, 1] → [0, 1] is a non-decreasing function satisfying f (0) = 0 and f (1) = 1. The capacity v defined by v = f ◦ P will be called a distorted probability.
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In the following proposition, we give a necessary and sufficient condition on the transformation f for balancedness of a restriction of v.2 Proposition 3.1. Let A ∈ A be such that P(A) 6= 0. Then C(v|A ) 6= ∅ if and only if f |[0,P(A)] ≤ [ f (P(A)) P(A) ]id. Proof. Only if part. Fix A ∈ A with P(A) 6= 0 and C(v|A ) 6= ∅. First, we show f (t P(A)) ≤ t v(A), for all t ∈ [0, 1]. Denote N∗ the set of positive integers. Since P is non-atomic, there exist pairwise disjoint sets A1 , . . . , An ∈ A ∩ A such that P(Ai ) = P(A) for all i ∈ {1, . . . , n}. n Let m ∈ {1, . . . , n}. Denote C = {I ⊂ {1, . . . , n} : |I | = m}. The cardinality of C is n! |C| = Cnm = m!(n−m)! . For I ∈ C, let A I = ∪i∈I Ai . A I ∈ A ∩ A and P(A I ) = mn P(A). P m−1 ∗ Furthermore Cn−1 A = I ∈C A∗I because for i fixed in {1, . . . , n}, the number of subsets I ∈ C P m−1 m−1 containing i is Cn−1 . Since C(v|A ) 6= ∅, Corollary 2.1 yields Cn−1 v(A) ≥ I ∈C v(A I ) = m−1 P C m m m m n−1 m I ∈C f ( n P(A)) = C n f ( n P(A)), that is, f ( n P(A)) ≤ Cnm v(A) = n v(A). Therefore f (r P(A)) ≤ r v(A), for all r ∈ Q ∩ [0, 1]. Now, let t ∈ [0, 1) and let (rn )n∈N be a sequence of rational numbers which decreases towards t. Then f (rn P(A)) ≤ rn v(A), for all n. Taking the limit when n goes to infinity, we get f (t P(A)) ≤ f (t + P(A)) ≤ t v(A). Hence f (t P(A)) ≤ t v(A), for all t ∈ [0, 1) and the inequality is clearly satisfied at 1. Finally, applying x the previous inequality to t = P(A) when x ∈ [0, P(A)], we obtain the desired result. If part. Let λ =
v(A) P(A)
P. Then λ ∈ C(v|A ).
Proposition 3.1 provides interesting characterizations of balancedness and total balancedness of v in terms of geometrical properties of the transformation function f . Taking A = Ω in Proposition 3.1 we get Corollary 3.1. The capacity v is balanced if and only if f ≤ id. Combining Proposition 3.1 and Remark 3.1(i) yields Corollary 3.2. The capacity v is totally balanced if and only if f is star-shaped at 0. Star-shapedness of f at 1 characterizes the total anti-balancedness of v. ¯ As noticed in the previous section, total balancedness and total anti-balancedness are not dual properties of each other. This is easier to see in the case of distorted probabilities as shown by the following example. Example 3.1. Define f on [0, 1] by f (x) = x if x ∈ [0, 21 ], f (x) = 21 if x ∈ [ 12 , 1) and f (1) = 1. Let P be the Lebesgue measure on Ω = [0, 1]. Then, the corresponding capacity v = f ◦ P is not totally balanced but v¯ is totally antibalanced. Notice that f is star-shaped at 1 and not star-shaped at 0. Proposition 3.2. The capacity v¯ is totally antibalanced if and only if f is star-shaped at 1. Proof. The proof is similar to the proof of Proposition 3.1. Assume v¯ is totally antibalanced. To show that f is star-shaped at 1, it suffices to prove that for all n ∈ N∗ , for all m ∈ {0, . . . , n − 1} 2 We are indebted to a referee for suggesting to adapt our initial proofs in deriving first Proposition 3.1.
Z. Aouani, A. Chateauneuf / Mathematical Social Sciences 56 (2008) 185–194
one has
1− f ( mn ) 1− m m
≤
1− f ( m+1 n ) . 1− m+1 n
191
Let n ∈ N∗ and let (Ai )1≤i≤n be a partition of Ω composed by
elements of A such that P(Ai ) = n1 for each i ∈ {1..., n}. Consider A = ∪n−m j=1 A j where Pn−m n−m ∗ ∗ 0 ≤ m < n − 1. Then (n − m − 1)A = k=1 (∪ j=1 j6=k A j ) . Thus, by Corollary 2.3, Pn−m n−m (n − m − 1)v(A) ¯ ≤ ¯ j=1 A j ). Hence (n − m − 1)[1 − f (1 − n−m k=1 v(∪ n )] ≤ (n − j6=k
m)[1 − f (1 − 0 0
n−m−1 )], n
that is,
1− f ( mn ) 1− mn
≤
1− f ( m+1 n ) . 1− m+1 n
Conversely, assume f is star-shaped at 1 and let A ∈ A. Then, under the usual convention ¯ 1−v( A) = 1, λ(·) = 1−P( ¯ P(·) is in AC(v¯ |A ). A)
Remark 3.3. It can readily be verified that total balancedness of a game v and total antibalancedness of its dual v¯ are necessary conditions for exactness of v. As shown by the following Proposition 3.3, it turns out that when v is a distorted probability, these conditions nicely prove to be sufficient for exactness. This justified our interest in exactness in the framework of distorted probabilities, furthermore it seems (to the best of our knowledge) to be an open problem whether this characterization remains valid for general games or even capacities. The present authors fail to prove the result beyond a finite Ω of more than four elements. Proposition 3.3. The capacity v is exact if and only if f is star-shaped at 0 and 1. Proof. If v is exact then v is totally balanced and v¯ is totally anti-balanced. Hence f is starshaped at 0 and 1. If f is star-shaped at 0 and 1 then we have f = mini∈I f i where I 6= ∅ and the f i ’s are convex satisfying f i (0) = 0 and f i (1) = 1. Thus, v = f ◦ P = mini∈I vi , where vi = f i ◦ P are convex capacities (since f i ’s are) therefore exact. Let A ∈ A. There exists i ∈ I such that v(A) = vi (A). Since vi is exact, there exists λi ∈ C(vi ) such that λi (A) = vi (A) = v(A). Notice that C(vi ) ⊂ C(v) because vi ≥ v. Hence λi ∈ C(v) and λi (A) = v(A). In order to further illustrate the flexibility of distorted probabilities, we provide the explicit expression, when v = f ◦ P is convex,3 of a probability in the core of v which coincides with v on a given finite chain of elements of the σ -algebra A. On an infinite σ -algebra, such a constructive proof cannot be provided and only an existence proof which usually leans on the use of Hahn–Banach Theorem (see e.g. Proposition 10.1 in Denneberg (1994)) can be offered. The proof of the related Proposition 3.4 is inspired by Wallner (2003). Proposition 3.4. If f is convex and ∅ = A0 ⊆ A1 ⊆ · · · ⊆ An = Ω is a finite chain in A, then the probability defined (with the convention 00 = 1) by λ(·) = =
n X j=1 n X j=1
[v(A j ) − v(A j−1 )]P(·|A j \ A j−1 ) v(A j ) − v(A j−1 ) P[· ∩ (A j \ A j−1 )] P(A j ) − P(A j−1 )
belongs to C(v) and satisfies λ(A j ) = v(A j ), for all j ∈ {0, . . . , n}. 3 Here we assume that f is continuous and we skip the proof of the well known fact that for a continuous distorted probability, v convex is equivalent to f convex.
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Proof. It is clear that λ is a probability and that it coincides with v on the chain. Let B ∈ A. We have ¯ =1− λ(B) = 1 − λ( B)
n X v(A j ) − v(A j−1 ) ¯ ¯ − P(A j−1 ∩ B)]. [P(A j ∩ B) P(A ) − P(A ) j j−1 j=1
¯ = P(A ∪ B) − P(B) to get Notice that P(A ∩ B) λ(B) = 1 −
n X v(A j ) − v(A j−1 ) [P(A j ∪ B) − P(A j−1 ∪ B)]. P(A j ) − P(A j−1 ) j=1
Since f is convex, v(A j ) − v(A j−1 ) [P(A j ∪ B) − P(A j−1 ∪ B)] ≤ [v(A j ∪ B) − v(A j−1 ∪ B)]. P(A j ) − P(A j−1 ) P Then λ(B) ≥ 1 − nj=1 [v(A j ∪ B) − v(A j−1 ∪ B)] = v(B). That is λ ∈ C(v). 4. Yaari’s model, capacities and risk aversion Let us first recall Yaari’s model (Yaari, 1987). Let (Ω , A, P) be a probability space, with P a non-atomic σ -additive probability. Let be the preference relation of the decision maker (DM, for short) defined on the set B(Ω , A) of measurable and bounded random variables X on Ω . In Yaari’s model, preferences are represented by the Choquet integral with respect to a capacity v. Furthermore, v is a transformation of P. Namely, there exists a unique, continuous, and non-decreasing function f : [0, 1] → [0, 1] satisfying f (0) = 0 and f (1) = 1, such that is represented by the functional Z Z 0 Z +∞ I (X ) = X dv = (v(X ≥ t) − 1)dt + v(X ≥ t)dt, −∞
0
where v = f ◦ P. It is well known (see e.g. Yaari (1987)) that for such preferences the two classical “extreme” risk aversion attitudes, i.e. weak risk aversion (E(X ) X for all X ∈ B(Ω , A)) and strong risk aversion (X Y if X second order stochastically dominates Y ) are characterized respectively by f ( p) ≤ p for all p ∈ [0, 1] and f convex. Indeed in view of Corollary 3.1 and of footnote 3 page 7, we obtain that in fact weak risk aversion and strong risk aversion are characterized respectively by v balanced and v convex. A legitimate question is then to ask whether in our framework totally balanced, totally antibalanced, or else exact capacities would inherit from some clearcut behavior towards risk. Note that strong risk aversion interprets merely by the fact that if E(X ) = E(Y ) and if for any p ∈ [0, 1] the expected gains for the p% smallest values are greater for X than for Y then the DM will weakly prefer X to Y . Left monotone risk aversion, introduced by Jewitt (1989) under the denomination of locationindependent risk, is weaker than strong risk aversion since it requires, for X and Y such that E(X ) = E(Y ), to weakly prefer X to Y not only if the expected gains for the p% smallest values are greater for X than for Y , but also if the additional expected gains of X upon Y for the p% smallest gains are non-increasing in p. The following table gives an example of a pair X and Y
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193
of random variables such that X is weakly preferred to Y in the left monotone sense. Probability X Y
1 5 −1250 −2000
1 5 −750 −1000
1 5 0 0
1 5 0 1000
1 5 2000 2000.
This is typically the kind of risk aversion that fits the actual behavior of insurees who usually prove to be particularly sensitive to high losses. As a matter of fact, it turns out that the famous result of Arrow (1965) – proving that in the framework of the expected utility model, the optimal insurance contract for a EU strongly risk averse DM is a contract with deductible – remains valid (see Vergnaud (1997)) for any left monotone risk averse DM (not necessarily strongly risk averse) regardless of the decision model under risk. It turns out (see e.g. Chateauneuf et al. (2004)) that left monotone risk aversion is characterized by f star-shaped at 1, consequently we obtain that left monotone risk aversion is characterized by v¯ totally antibalanced (see Proposition 3.2). Symmetrically, right monotone risk aversion requires, for X and Y such that E(X ) = E(Y ), to weakly prefer X to Y not only if the expected gains for the p% smallest values are greater for X than for Y , but also if the additional expected gains of Y upon X for the p% highest values are non-decreasing in p, this last property strengthening the riskier feature of Y when compared to X . The table below gives an example of a pair X and Y of random variables such that X is right monotone less risky than Y . Probability X Y
1 5 −2000 −2000
1 5 0 −1000
1 5 0 0
1 5 750 1000
1 5 1250 2000.
As proved in Chateauneuf et al. (2004), right monotone risk aversion is characterized by f starshaped at 0, accordingly right monotone risk aversion is characterized by v totally balanced (see Corollary 3.2). So by taking also into account Proposition 3.3, we may summarize increasing aversion attitudes towards risk4 in the following Proposition 4.1. Proposition 4.1. In Yaari’s model (i) (ii) (iii) (iv) (v)
The DM is weakly risk averse if and only if v is balanced. The DM is left monotone risk averse if and only if v¯ is totally antibalanced. The DM is right monotone risk averse if and only if v is totally balanced. The DM is both left and right monotone risk averse if and only if v is exact. The DM is strongly risk averse if and only if v is convex.
5. Concluding remarks In this paper we aimed at showing that for distorted probabilities, classical properties of capacities as balancedness, total-balancedness, and exactness prove to be related to simple properties of the distortion function, and moreover that these classical properties of the capacity are intimately related to meaningful risk aversion attitudes for decision makers a` la Yaari. This 4 It should be noted that more precisely (v) ⇒ (iv) ⇒ (iii), (iv) ⇒ (ii), (ii) or (iii) ⇒ (i), but no implication exists between (ii) and (iii).
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suggests that under uncertainty it might be fruitful to investigate for the Choquet expected utility model Schmeidler (1989), Gilboa (1987) the kind of uncertainty aversion that is linked, for instance, with the use of totally balanced or exact capacities in the same way as it has already been done for convex or balanced capacities (see e.g. Schmeidler (1989) and Chateauneuf and Tallon (2002)). Acknowledgments The helpful comments and suggestions of two anonymous referees and of an associate editor are gratefully acknowledged. References Arrow, K., 1965. Aspects of the Theory of Risk-Bearing. Yrjo Jahnsson Saatio, Helsinki, ch. The theory of risk aversion. Chateauneuf, A., Cohen, M., Meilijson, I., 2004. Four notions of mean-preserving increase in risk, risk attitudes and applications to the rank-dependent expected utility model. Journal of Mathematical Economics 547–571. Chateauneuf, A., Tallon, J.-M., 2002. Diversification, convex preferences and non-empty core in the choquet expected utility model. Economic Theory 19, 509–523. Delbaen, F., 1974. Convex games and extreme points. Journal of Mathematical Analysis and Applications 45, 210–233. Denneberg, D., 1994. Non-Additive Measure and Integral. Kluwer Academic Publishers. Fan, K., 1956. On systems of linear inequalities. Annals of Mathematical Studies 38, 99–156. Gilboa, I., 1987. Expexted utility theory with purely subjective non-additive probabilities. Journal of Mathematical Economics 16, 65–88. Gilboa, I., Lehrer, E., 1991. The value of information — an axiomatic approach. Journal of Mathematical Economics 20, 443–459. Jewitt, I., 1989. Choosing between risky prospects: The characterization of comparative statics results, and location independent risk. Management Science 35, 60–70. Kannai, Y., 1969. Countably additive measures in cores of games. Journal of Mathematical Analysis and Applications 27, 227–240. Schmeidler, D., 1972. Cores of exact games 1. Journal of Mathematical Analysis and Applications 40, 214–225. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57, 571–587. Vergnaud, J.C., 1997. Analysis of risk in a non-expected utility framework and applications to the optimality of the deductible. Finance 18, 155–167. Wallner, A., 2003. Bi-elastic neighborhood models. In: Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications. pp. 590–605. Yaari, M.E., 1987. The dual theory of choice under risk. Econometrica 55, 95–115.
Exact capacities and star-shaped distorted probabilities Zaier Aouani a,∗ , Alain Chateauneuf b a University of Kansas, Economics Department, 415 Snow Hall, Lawrence, KS 66045, USA b Universit´e Paris I, PSE-CES, 106-112 boulevard de l’Hopital, 75647 Paris Cedex 13, France
Received 22 February 2006; received in revised form 18 January 2008; accepted 18 January 2008 Available online 13 February 2008
Abstract We are interested in capacities which are deformations of probability, i.e. v = f ◦ P. We characterize balanced, totally balanced, and exact capacities by properties concerning the probability transformation function, f . These results allow us to obtain simple new characterizations of a large pattern of risk aversions relevant to Yaari’s dual theory of choice under risk. c 2008 Elsevier B.V. All rights reserved.
JEL classification: C71; D80 Keywords: Capacity; Exact; Balanced; Star-shaped function; Risk aversion
1. Introduction Set functions, which are not necessarily additive, are frequently used in decision theory. Indeed, they have different interpretations: on the one hand, they represent transferable utility or the characteristic function of a cooperative game; while on the other hand, they represent non-additive probabilities. In this work, we are interested in capacities which are distorted probabilities, i.e. deformations of a given probability, v = f ◦ P where P is a non-atomic additive probability on a measurable space (Ω , A) where Ω is a non-empty set, A a σ -algebra of subsets of Ω , and f : [0, 1] → [0, 1] is a non-decreasing function, satisfying f (0) = 0 and f (1) = 1. Because of their special form, ∗ Corresponding author.
E-mail addresses: [email protected], [email protected] (Z. Aouani), [email protected] (A. Chateauneuf). c 2008 Elsevier B.V. All rights reserved. 0165-4896/$ - see front matter doi:10.1016/j.mathsocsci.2008.01.006
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distorted probabilities play an important role in mathematical economics, specifically in risk theory and game theory. The main purpose of this work is to characterize a wide classical range of monotone normalized games, i.e. capacities, by properties concerning the probability transformation f . An important motivation for such characterizations is that, building upon (Chateauneuf et al., 2004), this will allow us to directly obtain simple new characterizations of a large pattern of risk attitudes relevant to Yaari’s dual theory of choice under risk (Yaari, 1987), in terms of simple specific properties of the capacity arising in Yaari’s model. The paper is organized as follows: in Section 2, we introduce our notation, definitions and preliminary results. Section 3 is devoted to the characterization of balanced, totally balanced, and exact capacities by properties related to f . The most salient result – related to Wallner (2003) who dealt with robust statistics – concerns exact capacities which are proved to be obtained through star-shaped distortions. In Section 4, results of Section 3 are straightforwardly applied to derive new characterizations of several kinds of risk aversion for decision makers a` la Yaari. Section 5 concludes the paper. 2. Definitions, notation and preliminary results A game is a triple (Ω , A, v) where Ω is a non-empty set, A is an algebra of subsets of Ω and v is a function v : A → R+ such that v(∅) = 0. The game v is called a (normalized) capacity if (i) v is normalized i.e. v(Ω ) = 1, and (ii) v is monotone, i.e. A, B ∈ A and A ⊂ B implies v(A) ≤ ¯ for all A ∈ A. Elements v(B). Each capacity v has a dual capacity v¯ defined by v(A) ¯ = 1 − v( A) of Ω are usually called players or states of nature and the elements of A are called coalitions or events, depending on whether the context is game theory or decision under uncertainty. For each A ∈ A, v(A) is interpreted as a measurement of the power of coalition A or as the subjective likelihood of the event A. If A is an algebra of subsets of Ω and A ∈ A, the family A ∩ A := {B ∩ A, B ∈ A} is an algebra of subsets of A. If A ∈ A, then the game (A, A ∩ A, v|A∩A ) is called the restriction of v to the coalition A. The game v|A∩A will be simply denoted v|A . The core of a game (Ω , A, v) is the set C(v) = {λ : A → R+ , λ is a finitely additive measure, λ(Ω ) = v(Ω ) and λ ≥ v}. A game v is said to be: (a) balanced if C(v) 6= ∅. (b) totally balanced if for all A ∈ A, its restriction to A is balanced. (c) exact if it is balanced and for all A ∈ A, v(A) = min{µ(A) : µ ∈ C(v)}. The minimum in the preceding expression is reached since C(v) is compact under the weak? topology in the space of additive and bounded measures. (d) convex if A1 and A2 ∈ A implies v(A1 ) + v(A2 ) ≤ v(A1 ∪ A2 ) + v(A1 ∩ A2 ). The following implications are well-known (see for example Schmeidler (1972)): v is convex ⇒ v is exact ⇒ v is totally balanced ⇒ v is balanced. Next, we overview some earlier classical characterizations of exact, balanced, totally balanced and convex games. For all A ⊆ Ω , let A∗ denote the characteristic function of A, i.e. the function defined on Ω , being equal to 1 on A and 0 elsewhere.
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Theorem 2.1. Let A be a σ -algebra of subsets of Ω and v a game. The game v is exact if and only if for all n ∈ N∗ , a ∈ R, a1 , . . . , an ∈ R+ , A, A1 , . . . , An ∈ A, A∗ + aΩ ∗ ≥
n X
ai Ai∗ ⇒ v(A) + av(Ω ) ≥
i=1
n X
ai v(Ai ).
i=1
To prove Theorem 2.1, Kannai (1969) used a fundamental result, due to Fan (1956), on linear systems of inequalities in a real normed vector space. Schmeidler (1972) proved the same result using a separation theorem. ∗ Corollary 2.1. The game P v is balanced if and Pnonly if ∀n ∈ N , a1 , . . . , an ≥ 0 and n ∗ ∗ A1 , . . . , An ∈ A we have: i=1 ai Ai ≤ Ω ⇒ i=1 ai v(Ai ) ≤ v(Ω ). ∗ Corollary 2.2. The game v isPtotally balanced if and Pn only if ∀n ∈ N , ∀a1 , . . . , an ≥ 0 and n ∗ ∗ A, A1 , . . . , An ∈ A we have: i=1 ai Ai ≤ A ⇒ i=1 ai v(Ai ) ≤ v(A).
The following theorem characterizes convex games (see e.g. Delbaen (1974)). Theorem 2.2. The game v is convex if and only if for any finite sequence ∅ = A0 ⊆ A1 ⊆ · · · ⊆ An = Ω in A, there exists λ ∈ C(v) such that λ(A0 ) = v(A0 ), . . . , λ(An ) = v(An ). Our Proposition 3.4 provides, when v is a convex distorted probability, an explicit expression for a probability λ satisfying the above condition in Theorem 2.2. The anticore of a game v, introduced by Gilboa and Lehrer (1991), is the set of all finitely additive measures that are setwise dominated by v, that is, AC(v) = {λ : A → R+ , λ is a finitely additive measure with λ(Ω ) = v(Ω ) and λ ≤ v}. The game v is said to be antibalanced if AC(v) 6= ∅. The game v is said to be totally antibalanced if for all A ∈ A, v|A is antibalanced. Because v and v¯ play dual roles, it is obvious that C(v) = AC(v) ¯ and therefore a game is balanced if and only if its dual is antibalanced. As the following example shows, this duality relationship is no longer valid in the case of total balancedness since the restriction of a dual is not the dual of the restriction i.e. (v|A ) 6= v¯|A . Example 2.1. Let Ω = {1, 2, 3, 4} and for A a subset of Ω ; 1 if 1 ∈ A and |A| ≥ 3, v(A) = 0 otherwise. The game v is totally balanced. From monotonicity of v, Corollary 2.2 is trivially satisfied if v(A) = 0. If v(A) = 1, either |A| = 3 and without loss of generality we may assume A = {1, 2, hence if there exists i 0 ∈ {1, 2, . . . , n} such that Ai0 = A, clearly ai0 ≤ 1 and P3}, n ai v(Ai ) ≤ ai0 ≤ v(A), or A = Ω and it suffices to check that the probability therefore i=1 P defined by P({1}) = 1 belongs to C(v). However, v¯ is not totally antibalanced because AC(v¯|{2,3} ) = ∅ since v({2}) ¯ = v({3}) ¯ = 0 and v({2, ¯ 3}) = 1. The same techniques used to prove Theorem 2.1 lead to the following characterization of antibalanced games. Proposition 2.1. The game v is antibalanced if and ∀a1 , . . . , an ≥ 0 and Pn only if∗ ∀n ∗∈ N,P n A1 , . . . , An ∈ A, the following implication holds: i=1 ai Ai ≥ Ω ⇒ i=1 ai v(Ai ) ≥ v(Ω ). From Proposition 2.1 we may state:
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Corollary 2.3. The game v is totally antibalanced if and only if ∀n ∈ N, P∀a1 , .∗. . , an ∗≥ 0, ∀A ∈ A and ∀A , . . . , A ∈ A ∩ A, the following implication holds: n 1 i ai Ai ≥ A ⇒ P a v(A ) ≥ v(A). i i i Notice however that now we require the Ai ’s to be subsets of A. This condition was omitted in Corollary 2.2 because it would be superfluous. 3. Distorted pobabilities As pointed out in the introduction, an important motivation for this paper is to detect in the framework of Yaari’s dual theory of choice under risk, simple specific properties of the underlying capacity, characterizing a large known pattern of risk attitudes. This will be achieved as soon as known characterizations of risk aversion in terms of the distortion function (Yaari, 1987; Chateauneuf et al., 2004) will be shwon to correspond to a known specific property of the capacity. Accordingly, we now focus on some properties of “distortion” functions which are relevant for our purpose. Let f : [0, 1] → [0, 1] with f (0) = 0 and f (1) = 1. The function f is said to be f (x) star-shaped at t ∈ [0, 1], if x 7→ f (t)− is non-decreasing on [0, 1] \ {t}. In the following t−x Remark 3.1 and Lemma 3.1, we present some characterizations of star-shaped functions which will be useful thereafter. The proofs of most of these facts follow in a straight-forward manner from the definitions. We give a proof of Lemma 3.1 which appears to be more relevant. Remark 3.1. (i) The function f is star-shaped at 0 if and only if f |[0, p] ≤ [ f (pp) ] id 1 for each p ∈ (0, 1]. f (y)− f (x) (ii) The function f is star-shaped at 0 if and only if f (x) , ∀ 0 < x < y ≤ 1. x ≤ y−x ¯( p) f (iii) The function f is star-shaped at 1 if and only if f¯|[0, p] ≥ [ p ] id for each p ∈ (0, 1]. Where f¯(·) = 1 − f (1 − ·) is the conjugate function of f . f (y) f (x) (iv) The function f is star-shaped at 1 if and only if 1−1−y ≥ f (y)− , ∀ 0 ≤ x < y < 1. y−x Facts (ii) and (iv) of Remark 3.1 make it possible to give a geometrical interpretation of starshapedness at 0 and 1 (See Fig. 1). Indeed a function f is star-shaped at 0 and 1 if and only if for any a ∈ [0, 1], the cord connecting (a, f (a)) to the origin and the cord connecting it to (1, f (1) = 1) are above the curve of f on the interval [0, a] (respectively [a, 1]). While the minimum of concave functions is concave, the minimum of convex functions is star-shaped at 0 and 1. More precisely: Lemma 3.1. The function f is star-shaped at 0 and 1 if and only if f = mini∈I f i where I is a non-empty set and f i : [0, 1] → [0, 1] is a convex function satisfying f i (0) = 0 and f i (1) = 1, ∀i ∈ I . Proof. Necessity. Let G = {g : [0, 1] → [0, 1] s.t. g(0) = 0, g(1) = 1, g is convex and g ≥ f }. Then G 6= ∅ because id ∈ G. We prove f = ming∈G g. Let x ∈ (0, 1) and let gx : [0, 1] → [0, 1] be defined by f (x) t if t ∈ [0, x], x gx (t) = 1 − f (x) 1 − f (x) t +1− if t ∈ [x, 1]. 1−x 1−x 1 The symbol id represents the identity mapping of the interval [0, 1], that is the mapping [0, 1] → [0, 1], x 7→ x.
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Fig. 1. An example of a function that is star-shaped at 0 and 1.
Then gx is convex ( f (x) x ≤
1− f (x) 1−x
by star-shapedness at 0 and 1), gx ≥ f and gx (x) = f (x).
Sufficiency. Let 0 < x < y ≤ 1. Then, there exists j ∈ I such that f (y) = f j (y). Hence f j (x) f j (y) − f j (x) f j (y) − f (x) f (x) f (y) − f (x) ≤ ≤ ≤ = · x x y−x y−x y−x Then f is star-shaped at 0 (by Remark 3.1(ii)). Let 0 ≤ x < y < 1. Then there exists i ∈ I such that f (x) = f i (x). Hence 1 − f (y) 1 − f i (y) f i (y) − f i (x) f (y) − f i (x) f (y) − f (x) ≥ ≥ ≥ = . 1−y 1−y y−x y−x y−x Then f is star-shaped at 1 (by Remark 3.1(iv)).
Remark 3.2. (i) If f is convex then f is star-shaped at every point of [0, 1]. (ii) If f is star-shaped at 0 and 1 then f is non-decreasing on [0, 1] and continuous on [0, 1). Proof. (ii) The non-decreasing property of f follows directly from star-shapedness at 0. f (t) Continuity arises from star-shapedness at 1. The functions f and t 7→ 1−1−t are monotone and thus have finite limits on the right (and respectively left) at each point of [0, 1) (and respectively f (x) f (x + ) (0, 1]). Let x ∈ [0, 1). We have 1−1−x ≤ 1−1−x . Then f (x + ) ≤ f (x). Since f is nonf (x ) f (x) decreasing, f (x) = f (x + ) follows. Similarly, for x ∈ (0, 1) we have 1−1−x ≤ 1−1−x − − implying f (x ) ≥ f (x) and f (x) = f (x ) by the same argument as above. Hence f is continuous at each x ∈ (0, 1) and f is continuous on the right at 0. −
Let us now recall how a distorted probability is defined. A probability P on (Ω , A) is said to be non-atomic if ∀A ∈ A, ∀α ∈ [0, P(A)], ∃B ∈ A
such that B ⊂ A and P(B) = α.
In the sequel of this section, A is a σ -algebra of subsets of Ω , P is a non-atomic probability on (Ω , A) and f : [0, 1] → [0, 1] is a non-decreasing function satisfying f (0) = 0 and f (1) = 1. The capacity v defined by v = f ◦ P will be called a distorted probability.
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In the following proposition, we give a necessary and sufficient condition on the transformation f for balancedness of a restriction of v.2 Proposition 3.1. Let A ∈ A be such that P(A) 6= 0. Then C(v|A ) 6= ∅ if and only if f |[0,P(A)] ≤ [ f (P(A)) P(A) ]id. Proof. Only if part. Fix A ∈ A with P(A) 6= 0 and C(v|A ) 6= ∅. First, we show f (t P(A)) ≤ t v(A), for all t ∈ [0, 1]. Denote N∗ the set of positive integers. Since P is non-atomic, there exist pairwise disjoint sets A1 , . . . , An ∈ A ∩ A such that P(Ai ) = P(A) for all i ∈ {1, . . . , n}. n Let m ∈ {1, . . . , n}. Denote C = {I ⊂ {1, . . . , n} : |I | = m}. The cardinality of C is n! |C| = Cnm = m!(n−m)! . For I ∈ C, let A I = ∪i∈I Ai . A I ∈ A ∩ A and P(A I ) = mn P(A). P m−1 ∗ Furthermore Cn−1 A = I ∈C A∗I because for i fixed in {1, . . . , n}, the number of subsets I ∈ C P m−1 m−1 containing i is Cn−1 . Since C(v|A ) 6= ∅, Corollary 2.1 yields Cn−1 v(A) ≥ I ∈C v(A I ) = m−1 P C m m m m n−1 m I ∈C f ( n P(A)) = C n f ( n P(A)), that is, f ( n P(A)) ≤ Cnm v(A) = n v(A). Therefore f (r P(A)) ≤ r v(A), for all r ∈ Q ∩ [0, 1]. Now, let t ∈ [0, 1) and let (rn )n∈N be a sequence of rational numbers which decreases towards t. Then f (rn P(A)) ≤ rn v(A), for all n. Taking the limit when n goes to infinity, we get f (t P(A)) ≤ f (t + P(A)) ≤ t v(A). Hence f (t P(A)) ≤ t v(A), for all t ∈ [0, 1) and the inequality is clearly satisfied at 1. Finally, applying x the previous inequality to t = P(A) when x ∈ [0, P(A)], we obtain the desired result. If part. Let λ =
v(A) P(A)
P. Then λ ∈ C(v|A ).
Proposition 3.1 provides interesting characterizations of balancedness and total balancedness of v in terms of geometrical properties of the transformation function f . Taking A = Ω in Proposition 3.1 we get Corollary 3.1. The capacity v is balanced if and only if f ≤ id. Combining Proposition 3.1 and Remark 3.1(i) yields Corollary 3.2. The capacity v is totally balanced if and only if f is star-shaped at 0. Star-shapedness of f at 1 characterizes the total anti-balancedness of v. ¯ As noticed in the previous section, total balancedness and total anti-balancedness are not dual properties of each other. This is easier to see in the case of distorted probabilities as shown by the following example. Example 3.1. Define f on [0, 1] by f (x) = x if x ∈ [0, 21 ], f (x) = 21 if x ∈ [ 12 , 1) and f (1) = 1. Let P be the Lebesgue measure on Ω = [0, 1]. Then, the corresponding capacity v = f ◦ P is not totally balanced but v¯ is totally antibalanced. Notice that f is star-shaped at 1 and not star-shaped at 0. Proposition 3.2. The capacity v¯ is totally antibalanced if and only if f is star-shaped at 1. Proof. The proof is similar to the proof of Proposition 3.1. Assume v¯ is totally antibalanced. To show that f is star-shaped at 1, it suffices to prove that for all n ∈ N∗ , for all m ∈ {0, . . . , n − 1} 2 We are indebted to a referee for suggesting to adapt our initial proofs in deriving first Proposition 3.1.
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one has
1− f ( mn ) 1− m m
≤
1− f ( m+1 n ) . 1− m+1 n
191
Let n ∈ N∗ and let (Ai )1≤i≤n be a partition of Ω composed by
elements of A such that P(Ai ) = n1 for each i ∈ {1..., n}. Consider A = ∪n−m j=1 A j where Pn−m n−m ∗ ∗ 0 ≤ m < n − 1. Then (n − m − 1)A = k=1 (∪ j=1 j6=k A j ) . Thus, by Corollary 2.3, Pn−m n−m (n − m − 1)v(A) ¯ ≤ ¯ j=1 A j ). Hence (n − m − 1)[1 − f (1 − n−m k=1 v(∪ n )] ≤ (n − j6=k
m)[1 − f (1 − 0 0
n−m−1 )], n
that is,
1− f ( mn ) 1− mn
≤
1− f ( m+1 n ) . 1− m+1 n
Conversely, assume f is star-shaped at 1 and let A ∈ A. Then, under the usual convention ¯ 1−v( A) = 1, λ(·) = 1−P( ¯ P(·) is in AC(v¯ |A ). A)
Remark 3.3. It can readily be verified that total balancedness of a game v and total antibalancedness of its dual v¯ are necessary conditions for exactness of v. As shown by the following Proposition 3.3, it turns out that when v is a distorted probability, these conditions nicely prove to be sufficient for exactness. This justified our interest in exactness in the framework of distorted probabilities, furthermore it seems (to the best of our knowledge) to be an open problem whether this characterization remains valid for general games or even capacities. The present authors fail to prove the result beyond a finite Ω of more than four elements. Proposition 3.3. The capacity v is exact if and only if f is star-shaped at 0 and 1. Proof. If v is exact then v is totally balanced and v¯ is totally anti-balanced. Hence f is starshaped at 0 and 1. If f is star-shaped at 0 and 1 then we have f = mini∈I f i where I 6= ∅ and the f i ’s are convex satisfying f i (0) = 0 and f i (1) = 1. Thus, v = f ◦ P = mini∈I vi , where vi = f i ◦ P are convex capacities (since f i ’s are) therefore exact. Let A ∈ A. There exists i ∈ I such that v(A) = vi (A). Since vi is exact, there exists λi ∈ C(vi ) such that λi (A) = vi (A) = v(A). Notice that C(vi ) ⊂ C(v) because vi ≥ v. Hence λi ∈ C(v) and λi (A) = v(A). In order to further illustrate the flexibility of distorted probabilities, we provide the explicit expression, when v = f ◦ P is convex,3 of a probability in the core of v which coincides with v on a given finite chain of elements of the σ -algebra A. On an infinite σ -algebra, such a constructive proof cannot be provided and only an existence proof which usually leans on the use of Hahn–Banach Theorem (see e.g. Proposition 10.1 in Denneberg (1994)) can be offered. The proof of the related Proposition 3.4 is inspired by Wallner (2003). Proposition 3.4. If f is convex and ∅ = A0 ⊆ A1 ⊆ · · · ⊆ An = Ω is a finite chain in A, then the probability defined (with the convention 00 = 1) by λ(·) = =
n X j=1 n X j=1
[v(A j ) − v(A j−1 )]P(·|A j \ A j−1 ) v(A j ) − v(A j−1 ) P[· ∩ (A j \ A j−1 )] P(A j ) − P(A j−1 )
belongs to C(v) and satisfies λ(A j ) = v(A j ), for all j ∈ {0, . . . , n}. 3 Here we assume that f is continuous and we skip the proof of the well known fact that for a continuous distorted probability, v convex is equivalent to f convex.
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Proof. It is clear that λ is a probability and that it coincides with v on the chain. Let B ∈ A. We have ¯ =1− λ(B) = 1 − λ( B)
n X v(A j ) − v(A j−1 ) ¯ ¯ − P(A j−1 ∩ B)]. [P(A j ∩ B) P(A ) − P(A ) j j−1 j=1
¯ = P(A ∪ B) − P(B) to get Notice that P(A ∩ B) λ(B) = 1 −
n X v(A j ) − v(A j−1 ) [P(A j ∪ B) − P(A j−1 ∪ B)]. P(A j ) − P(A j−1 ) j=1
Since f is convex, v(A j ) − v(A j−1 ) [P(A j ∪ B) − P(A j−1 ∪ B)] ≤ [v(A j ∪ B) − v(A j−1 ∪ B)]. P(A j ) − P(A j−1 ) P Then λ(B) ≥ 1 − nj=1 [v(A j ∪ B) − v(A j−1 ∪ B)] = v(B). That is λ ∈ C(v). 4. Yaari’s model, capacities and risk aversion Let us first recall Yaari’s model (Yaari, 1987). Let (Ω , A, P) be a probability space, with P a non-atomic σ -additive probability. Let be the preference relation of the decision maker (DM, for short) defined on the set B(Ω , A) of measurable and bounded random variables X on Ω . In Yaari’s model, preferences are represented by the Choquet integral with respect to a capacity v. Furthermore, v is a transformation of P. Namely, there exists a unique, continuous, and non-decreasing function f : [0, 1] → [0, 1] satisfying f (0) = 0 and f (1) = 1, such that is represented by the functional Z Z 0 Z +∞ I (X ) = X dv = (v(X ≥ t) − 1)dt + v(X ≥ t)dt, −∞
0
where v = f ◦ P. It is well known (see e.g. Yaari (1987)) that for such preferences the two classical “extreme” risk aversion attitudes, i.e. weak risk aversion (E(X ) X for all X ∈ B(Ω , A)) and strong risk aversion (X Y if X second order stochastically dominates Y ) are characterized respectively by f ( p) ≤ p for all p ∈ [0, 1] and f convex. Indeed in view of Corollary 3.1 and of footnote 3 page 7, we obtain that in fact weak risk aversion and strong risk aversion are characterized respectively by v balanced and v convex. A legitimate question is then to ask whether in our framework totally balanced, totally antibalanced, or else exact capacities would inherit from some clearcut behavior towards risk. Note that strong risk aversion interprets merely by the fact that if E(X ) = E(Y ) and if for any p ∈ [0, 1] the expected gains for the p% smallest values are greater for X than for Y then the DM will weakly prefer X to Y . Left monotone risk aversion, introduced by Jewitt (1989) under the denomination of locationindependent risk, is weaker than strong risk aversion since it requires, for X and Y such that E(X ) = E(Y ), to weakly prefer X to Y not only if the expected gains for the p% smallest values are greater for X than for Y , but also if the additional expected gains of X upon Y for the p% smallest gains are non-increasing in p. The following table gives an example of a pair X and Y
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193
of random variables such that X is weakly preferred to Y in the left monotone sense. Probability X Y
1 5 −1250 −2000
1 5 −750 −1000
1 5 0 0
1 5 0 1000
1 5 2000 2000.
This is typically the kind of risk aversion that fits the actual behavior of insurees who usually prove to be particularly sensitive to high losses. As a matter of fact, it turns out that the famous result of Arrow (1965) – proving that in the framework of the expected utility model, the optimal insurance contract for a EU strongly risk averse DM is a contract with deductible – remains valid (see Vergnaud (1997)) for any left monotone risk averse DM (not necessarily strongly risk averse) regardless of the decision model under risk. It turns out (see e.g. Chateauneuf et al. (2004)) that left monotone risk aversion is characterized by f star-shaped at 1, consequently we obtain that left monotone risk aversion is characterized by v¯ totally antibalanced (see Proposition 3.2). Symmetrically, right monotone risk aversion requires, for X and Y such that E(X ) = E(Y ), to weakly prefer X to Y not only if the expected gains for the p% smallest values are greater for X than for Y , but also if the additional expected gains of Y upon X for the p% highest values are non-decreasing in p, this last property strengthening the riskier feature of Y when compared to X . The table below gives an example of a pair X and Y of random variables such that X is right monotone less risky than Y . Probability X Y
1 5 −2000 −2000
1 5 0 −1000
1 5 0 0
1 5 750 1000
1 5 1250 2000.
As proved in Chateauneuf et al. (2004), right monotone risk aversion is characterized by f starshaped at 0, accordingly right monotone risk aversion is characterized by v totally balanced (see Corollary 3.2). So by taking also into account Proposition 3.3, we may summarize increasing aversion attitudes towards risk4 in the following Proposition 4.1. Proposition 4.1. In Yaari’s model (i) (ii) (iii) (iv) (v)
The DM is weakly risk averse if and only if v is balanced. The DM is left monotone risk averse if and only if v¯ is totally antibalanced. The DM is right monotone risk averse if and only if v is totally balanced. The DM is both left and right monotone risk averse if and only if v is exact. The DM is strongly risk averse if and only if v is convex.
5. Concluding remarks In this paper we aimed at showing that for distorted probabilities, classical properties of capacities as balancedness, total-balancedness, and exactness prove to be related to simple properties of the distortion function, and moreover that these classical properties of the capacity are intimately related to meaningful risk aversion attitudes for decision makers a` la Yaari. This 4 It should be noted that more precisely (v) ⇒ (iv) ⇒ (iii), (iv) ⇒ (ii), (ii) or (iii) ⇒ (i), but no implication exists between (ii) and (iii).
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suggests that under uncertainty it might be fruitful to investigate for the Choquet expected utility model Schmeidler (1989), Gilboa (1987) the kind of uncertainty aversion that is linked, for instance, with the use of totally balanced or exact capacities in the same way as it has already been done for convex or balanced capacities (see e.g. Schmeidler (1989) and Chateauneuf and Tallon (2002)). Acknowledgments The helpful comments and suggestions of two anonymous referees and of an associate editor are gratefully acknowledged. References Arrow, K., 1965. Aspects of the Theory of Risk-Bearing. Yrjo Jahnsson Saatio, Helsinki, ch. The theory of risk aversion. Chateauneuf, A., Cohen, M., Meilijson, I., 2004. Four notions of mean-preserving increase in risk, risk attitudes and applications to the rank-dependent expected utility model. Journal of Mathematical Economics 547–571. Chateauneuf, A., Tallon, J.-M., 2002. Diversification, convex preferences and non-empty core in the choquet expected utility model. Economic Theory 19, 509–523. Delbaen, F., 1974. Convex games and extreme points. Journal of Mathematical Analysis and Applications 45, 210–233. Denneberg, D., 1994. Non-Additive Measure and Integral. Kluwer Academic Publishers. Fan, K., 1956. On systems of linear inequalities. Annals of Mathematical Studies 38, 99–156. Gilboa, I., 1987. Expexted utility theory with purely subjective non-additive probabilities. Journal of Mathematical Economics 16, 65–88. Gilboa, I., Lehrer, E., 1991. The value of information — an axiomatic approach. Journal of Mathematical Economics 20, 443–459. Jewitt, I., 1989. Choosing between risky prospects: The characterization of comparative statics results, and location independent risk. Management Science 35, 60–70. Kannai, Y., 1969. Countably additive measures in cores of games. Journal of Mathematical Analysis and Applications 27, 227–240. Schmeidler, D., 1972. Cores of exact games 1. Journal of Mathematical Analysis and Applications 40, 214–225. Schmeidler, D., 1989. Subjective probability and expected utility without additivity. Econometrica 57, 571–587. Vergnaud, J.C., 1997. Analysis of risk in a non-expected utility framework and applications to the optimality of the deductible. Finance 18, 155–167. Wallner, A., 2003. Bi-elastic neighborhood models. In: Proceedings of the Third International Symposium on Imprecise Probabilities and Their Applications. pp. 590–605. Yaari, M.E., 1987. The dual theory of choice under risk. Econometrica 55, 95–115.