choquet expected utility model : a new approach to ...

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observed behaviors, the most famous evidence being Allais paradox under risk ... In the previous presentation of Ellsberg paradox, uncertainty is modelled.
CHOQUET EXPECTED UTILITY MODEL : A NEW APPROACH TO INDIVIDUAL BEHAVIOR UNDER UNCERTAINTY AND TO SOCIAL WELFARE Alain CHATEAUNEUF∗(CERMSEM) and Michèle COHEN†(EUREQua), Université de Paris I‡,§

Abstract The Choquet Expected Utility (CEU) model is a model of decision making under uncertainty generalizing the Expected Utility (EU) model. Under the comonotone independence axiom, an appealing and intuitive axiom requiring that the usual independence axiom holds only when hedging effects are absent, preferences turn out to be represented through a Choquet integral. This model allows for taking into account a fuller array of behaviors under risk and under uncertainty. As a result, the model offers a simple theoretical foundation for explaining (i) actual economic phenomena in finance, in insurance, phenomena that cannot be accounted for in the framework of EU theory, (ii) new indexes for inequality measurement.

Contents 1 Introduction

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2 Decision under uncertainty 2.1 Ellsberg’s paradox . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Schmeidler’s model. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Comonotonic independence . . . . . . . . . . . . . . . . .

3 3 5 5

∗ e-mail

: [email protected] : [email protected] ‡ Maison de l’Economie, 106-112, Bld de l’Hopital, 75647, Paris, Cedex 13, France. § We would like to thank Jean-Marc Tallon and two anonymous referee for their helpful comments. † e-mail

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2.3

2.4 2.5

2.2.2 Schmeidler’s representation theorem . . . . . . Choquet Expected Utility (C.E.U.) models . . . . . . 2.3.1 The comonotonic sure thing principle . . . . . 2.3.2 Uncertainty aversion . . . . . . . . . . . . . . . CEU models and the upper-lower probability approach Some economic applications . . . . . . . . . . . . . . . 2.5.1 Optimal portfolio choice . . . . . . . . . . . . . 2.5.2 Asset pricing . . . . . . . . . . . . . . . . . . .

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7 8 9 9 10 11 11 12

3 Decision under risk 13 3.1 EU model and Allais’ paradox . . . . . . . . . . . . . . . . . . . . 14 3.2 Rank Dependent Expected Utility (RDEU) model . . . . . . . . 14 3.2.1 Definition of RDEU model . . . . . . . . . . . . . . . . . 14 3.2.2 RDEU is a Choquet integral . . . . . . . . . . . . . . . . 15 3.2.3 Key axiom of RDEU’s axiomatization : comonotonic surething principle . . . . . . . . . . . . . . . . . . . . . . . . 15 3.3 From CEU model to RDEU model using first order stochastic dominance (Wakker) . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3.1 From simplified Schmeidler’s model to Yaari’s model . . . 16 3.3.2 From CEU to RDEU . . . . . . . . . . . . . . . . . . . . . 17 3.4 Risk aversion in RDEU model . . . . . . . . . . . . . . . . . . . . 17 3.4.1 Strong risk aversion . . . . . . . . . . . . . . . . . . . . . 17 3.4.2 Monotone risk aversion . . . . . . . . . . . . . . . . . . . 18 3.4.3 Left monotone risk aversion . . . . . . . . . . . . . . . . . 18 3.4.4 Characterization of different notions of risk aversion in the RDEU model . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.5 Economic applications . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5.1 Porfolio choice . . . . . . . . . . . . . . . . . . . . . . . . 20 3.5.2 Complete insurance . . . . . . . . . . . . . . . . . . . . . 20 3.5.3 Optimality of deductible insurance . . . . . . . . . . . . . 20 4 Inequality measurement

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5 Concluding remarks

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1

Introduction

The classical model of decision under risk, the Expected Utility model (von Neumann and Morgenstern (1947)), and under uncertainty, the Subjective Expected Utility model (Savage, (1954)) have been proved to be often violated by observed behaviors, the most famous evidence being Allais paradox under risk and the Ellsberg paradox under uncertainty. Among others, these two paradoxes have called into question these classical models.

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To take into account these behaviors, Schmeidler (1982,1989) under uncertainty, and Quiggin (1982) and Yaari(1987) under risk have built new axiomatizations of behavior . Under a key axiom, namely the comonotone independence axiom, an appealing and intuitive axiom requiring that the usual independence axiom holds only when hedging effects are absent, Schmeidler, Quiggin and Yaari have, independently, succeeded to characterize the preferences by means of a functional that turned out to be a Choquet integral, under uncertainty as well as under risk. Choquet integral thus proved to be an important tool for decision making under risk and uncertainty. Moreover, not only these models - with the generic term of Choquet Expected Utility (CEU) - explain the paradoxes but they also offer simple but flexible representations, allow for more diversified patterns of behavior under uncertainty as well as under risk, and especially allow to separate perception of uncertainty or of risk from the valuation of outcomes. The aim of this paper is essentially to emphasize the role of the models of behavior based on Choquet integral. Moreover, some economic examples are provided in order to illustrate the ability of these models to explain individual and social choices.

2

Decision under uncertainty

By decision under uncertainty, we mean, in contrast with decision under risk, situations when there does not exist a given objective probability distribution on the set S of states of the world, available to the decision maker.

2.1

Ellsberg’s paradox

Let us consider the following version of Ellsberg paradox (Ellsberg, (1961)). Subjects are informed that a ball will be drawn at random from an urn that contains 90 balls : 30 are red (R), and each other ball is either black (B), or yellow (Y ). Subjects are requested to express their preferences between betting on R (act g1 ) or betting on B (act g2 ) and then, independently, between betting on R ∪ Y (act g3 ) or on B ∪ Y (act g4 ). Table 1 below summarizes the corresponding outcomes :

g1 g2 g3 g4

Red

Black

Yellow

$100 $0 $100 $0

$0 $100 $0 $100

$0 $0 $100 $100

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Typical preferences are g1 ≻ g2 and g4 ≻ g3 1 , hence the Sure-Thing Principle is violated, since changing the common outcome $0 under Y , of g1 and g2 , into the common outcome $100 under Y , should have left preference unchanged i.e. should have led to g3 ≻ g4 . Therefore many subjects behave in a ”paradoxical way”, in the sense that they are not subjective expected utility maximizers. Moreover, as noticed by Machina and Schmeidler (1992), nor are such subjects probabilistically sophisticated : this means that they do not ascribe subjective probabilities pR , pB , pY to states of nature (i.e. elementary events R, B, Y ) and then use first order stochastic dominance axiom2 - a widely accepted rule for partially ordered random variables. Otherwise, g1 ≻ g2 would imply pR > pB and g4 ≻ g3 would imply pB + pY > pR + pY : a contradiction. In the previous presentation of Ellsberg paradox, uncertainty is modelled through the set of the states of the world S = {R, B, Y } and bets are interpreted as acts (i.e. mappings) g : S → R. Let us see now that Ellsberg’s paradox is robust, when, in the spirit of Schmeidler (1989), we consider Ellsberg’s experiment in the Anscombe and Aumann’s framework (1963). Uncertainty now concerns the composition of the urn. The set Ω of states of nature is composed of sixty one possible states of nature : Ω = {0, 1, ..., k, ..., 60} where k stands for the elementary event : ”the number of black balls is k”. Bets (i.e. proposals) g are now considered as horse lotteries Xg : Ω → Y i.e. mapping from Ω to Y where Y is the set of all lotteries on C = {0, 100} (i.e. of all probabilities distributions over C). k 60−k Thus Xg (k) = 30 90 δ g(R) + 90 δ g(B) + 90 δ g(Y ) is the lottery faced by the decision maker (DM) if his bet is g and if the number of black balls is k ; namely, in such a case, the DM will earn $g(R) with probability 30 90 , $g(B) k with probability 90 , and $g(Y ) with probability 60−k . Suppose as implicitely 90 assumed by Schmeidler as by Anscombe and Aumann that, under risk, our DM is an expected utility maximizer with von Neumann Morgenstern’s utility function u (assumed without loss of generality to satisfy u(0) = 0, u(100) = 1). Again in the Anscombe-Aumann framework, the expected utility model under uncertainty cannot explain the typical preferences described above. Actually, suppose the DM assigns probabilities to elementary events k, and acts according to the expected  utility model of Anscombe-Aumann, i.e. weakly prefers h to g if and only if pk u(Xh (k)) ≥  pk u(Xg (k)). Hence simple computations  show that g1 ≻ g2 would give 30 > kpk while g4 ≻ g3 would give 30 < kpk , a contradiction. Remark 1 Notice that, in all section 2, in accordance with Schmeidler’s model, L0 will denote the set of acts X from a finite set of states of nature S to Y, the set of all lotteries on a given consequence set C, and then will refer to AnscombeAumann’s framework ; whereas V will denote the set of acts X from a set of 1 g ≻ g means g is strictly preferred to g , the 1 2 1 2 2 Let us recall that, if X and Y are real random

same applies to g4 ≻ g3 . variables, the first order stochastic dominance rule stipulates that if ∀t ∈ R, P {X ≥ t} ≥ P {Y ≥ t} , then X should be weakly preferred to Y , the preference becoming strict if P {X ≥ t} > P {Y ≥ t} , for some t0 ∈ R.

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states of nature S to R hence referring to Savage’s framework (see also footnote 8 in section 2.2.2.).

2.2

Schmeidler’s model.

In order to explain such parad.oxes and to separate perception of uncertainty from valuation of outcomes, Schmeidler (1989) has proposed a model which relaxes the usual independence condition while offering a flexible but simple formula. 2.2.1

Comonotonic independence

As was previously pointed out, Schmeidler (1982, 1989) has developed his model in the Anscombe-Aumann’s framework. Here, for sake of simplicity, we will assume the set S of states of nature to be finite, and the events to be the elements of A = 2S . Define the acts as the set L0 of horses lotteries i.e. of mappings from S to Y where Y is the set of lotteries on a given consequences set C (i.e. Y is the set of probabilities distributions over C with finite support).

denotes the preference relation of a DM over the set of acts L0 where, as usually, X Y means X is -weakly- preferred to Y , X ≻ Y means X is strictly preferred to Y and X ∼ Y means the DM is indifferent between X and Y . Let us recall that SEU (subjective expected utility)3 obtains mainly through the following axiom. Axiom 1 Independence condition (Anscombe and Aumann)4 For all X, Y, Z in L0 , and for all α in (0, 1) : X ≻ Y implies αX +(1−α)Z ≻ αY + (1 − α)Z. In order to weaken this axiom, Schmeidler introduced the following definition of comonotonic acts5 : Definition 1 Acts X and Y are said to be comonotonic6 , if for no s and t in S, X(s) ≻ X(t) and Y (t) ≻ Y (s). Let us notice that contrary to definition 4 , where the outcome set is the ordered set R, the order on L0 is implied by the preference relation for constant acts. Hence the following weakening of the independence axiom : 3 The preferences on L are represented by SEU model if there exists a probability distri0 bution P on S and a utility function u: Y → R, which is continuous, strictly increasing such   that : X Y iff S u(X)dP ≥ S u(Y )dP 4 Let us recall that, for all X, Z of L , and all α of [0, 1], the act αX + (1 − α)Z is such 0 that, for every s of S, αX(s) + (1 − α)Z(s) is the usual convex mixture of the two lotteries X(s) and Z(s). 5 Comonotonicity stands for common monotonicity. 6 In Denneberg, section 3, this volume, definition 1 corresponds to a characterization of comonotonic acts.

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Axiom 2 Comonotonic independence of Schmeidler For all pairwise7 comonotonic acts X, Y and Z in L0 and for all α in (0, 1) : X ≻ Y implies αX + (1 − α)Z ≻ αY + (1 − α)Z. Roughly speaking, comonotonic independence requires the direction of preferences to be retained, provided hedging is not involved. This intuition which is crucial in Schmeidler’s model will appear more transparent when we will examine Schmeidler’s representation theorem (1986) in section 2.2.28 . With the help of comonotonic independence and a few usual axioms, Schmeidler (1982, 1989) characterized a model in which the typical preferences of section 2.1 become admissible. Schmeidler proved that the preference relation on L0 is represented through a Choquet integral with respect to a unique capacity v (instead of a unique probability P ), that   is for all X and Y of L0 : X Y if and only if u(X(.))dv ≥ u(Y (.))dv where u is a von Neumann utility function on the set Y of lotteries. More precisely :

Definition 2 A (normalized) capacity v on (S, A = 2S ) is a monotone set function from A to [0, 1] (∀A, B ∈ A, A ⊂ B ⇒ v(A) ≤ v(B)) satisfying v(φ) = 0, v(S) = 1.  Definition 3 The Choquet integral Ch u(X)dv is defined by : 

Ch

u(X)dv =



0

[v(u(X) > t) − 1] dt +

−∞



∞

v(u(X) > t)dt

(1)

0

 Notice that, if v proves to be a probability measure P, Ch u(X)dv reduces to EP (u(X)), the mathematical expectation of u(X) with respect to P , hence the Choquet expected utility -CEU- model of Schmeidler generalizes the classical subjective expected utility -SEU-model of Anscombe-Aumann.

y1  ...  yi  ...  yn , Ai ∈ A, (Ai ) partition of Ai (i.e. A∗i (s) = 1 if s ∈ Ai , 0 otherwise),

i=n 

yi A∗i , where yi ∈ Y, i=1 of S, A∗i characteristic function n  one obtains : u(X) = ai A∗i i=1

Remark 2 Let X belongs to L0 , we can write : X =

with a1 = u(y1 ) ≤ ... ≤ ai = u(yi ) ≤ ... ≤ an = u(yn ) and the Choquet integral is then  given by: Ch u(X)dv = a1 + (a2 − a1 )v [u(X) ≥ a2 ] + ... + (an − an−1 )v [u(X) ≥ an ] .  Remark 3 Ch u(X)dv has a meaningful interpretation : the DM computes the value of X by taking for sure the minimum expected payoff a1 and adds to this payoff the successive possible additional payoffs ai+1 −ai , 1 ≤ i ≤ n−1, weighted by his personal estimation v [u(X) ≥ ai+1 ] of their occurence. 7 Actually the weaker assumption ”X,Z and Y,Z comonotonic” can be substituted, in order to get Schmeidler’s model. 8 The reader can find further information on comonotonicity in Chateauneuf, Cohen and Kast (1997).

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2.2.2

Schmeidler’s representation theorem

From now on, we consider a DM faced with choices among acts X, the set V of such acts consisting of all bounded real-valued, A-measurable functions on S, X : (S, A)→ R, where S is a set of states of nature and A a σ-algebra of events (i.e. subsets of S). This DM is supposed to face uncertainty, and will denote the preference relation on V of the agent. In this framework, referred to as Savage’s framework9 in the sequel for sake of simplification, which fits the first and simple presentation of Ellsberg’s paradox, a natural translation of the comonotonic independence axiom of Schmeidler is a follows : Axiom 3 Comonotonic independence (Chateauneuf, (1994)) : F orX, Y, Z X

∈ V, Xand Y comonotone, Y and Z comonotone, then, ∼ Y ⇒ X +Z ∼Y +Z

where in accordance with definition 1 : Definition 4 X, Y ∈ V are said to be comonotonic if ”they vary in the same direction”. More precisely if (X(s) − X(t))(Y (s) − Y (t)) ≥ 0, ∀s, t ∈ S. Thus, in case of asymmetric reduction of uncertainty (through hedging effects), attitudes towards uncertainty can be taken into account as in example below where acts offer payments depending on the realization of event A or of _ the complementary one A. Aր Assume:

and let :

X

Z

25,000

_

Aց

15,000

Aր

15,000

_

Aր ∼

Aց

Y

_

Aց

12,000 30,000

25,000

Z is comonotonic with Y but not with X, Z is a hedge against X but not against Y , hence an uncertainty-averse DM might exhibit, after addition of Z, the strict preference : Aր X +Z

_

Aց

40,000

Aր ≻

Y +Z

40,000

_

Aց

27,000 55,000

For typical preferences of Ellsberg’s paradox, a similar interpretation in terms of asymmetric reduction of uncertainty paired with aversion to uncertainty, can be provided (see e.g. Schmeidler (1989), Chateauneuf (1994)). 9 Although Savage framework allows for general sets C of outcomes ; taking C = R as here, permits a simple exposition of the main features of CEU models.

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With the help of the comonotonic independence axiom and further simple requirements, it is then possible to derive a simplified version of Schmeidler’s model where preferences can be represented by a Choquet integral with respect to a capacity v, namely :  X, Y ∈ V, X Y iff Ch Xdv ≥ Ch Y dv (see Chateauneuf(1994)). Such a result is derived from the fundamental theorem of Schmeidler (1986)10 , which characterizes the Choquet integral (Choquet (1953)), and appeared as a crucial tool for Schmeidler’s model and more generally for Choquet Expected Utility models. Theorem 1 Schmeidler’s representation theorem (1986) : Let I : V → R satisfying I(S ∗ ) = 1 be given. Suppose also that the funtional I satisfies : (i)Comonotonic additivity : X and Y comonotonic implies I(X + Y ) = I(X) + I(Y ) (hence I(0) = 0) (ii) Monotonicity : X ≥ Y on S implies I(X) ≥ I(Y ) Then, defining v(A) = I(A∗ ) on A we have for all X in V :  0 ∞ I(X) = Ch Xdv i.e.I(X) = −∞ [v(X > t) − 1] dt + 0 [v(X > t)] dt

2.3

Choquet Expected Utility (C.E.U.) models

By Choquet expected utility models, we mean those non-additive models directly connected with the Choquet integral which, following pionner’s work of Schmeidler (1982, 1989) in the Anscombe-Aumann framework, have been derived in Savage framework as for example by Gilboa (1987) or Wakker(1990). Referring to the Savage framework introduced in Subsection 2.2.2, one obtains: Definition 5 The Choquet Expected Utility (C.E.U.) model stipulates that the DM ranks acts X with the help of a utility-on-wealth function u: R → R, which is continuous, strictly increasing and cardinal (i.e., defined up to a positive affine transformation) and with the help of his personal evaluation set function  v that is a capacity. The ranking of acts X is performed through I(u(X)) = Ch u(X)dv, the Choquet integral of u(X) with respect to the capacity v, defined for X ∈ V by    0

u(X)dv =

Ch

∞

[v(u(X) > t) − 1] dt +

−∞

[v(u(X) > t)] dt

(2)

0

Note that, in this general CEU model, u is not necessarily linear. Let us now give the predominant axiom to axiomatize CEU theory : 1 0 It is worth noticing that Schmeidler became aware that the integral he discovered in his representation theorem has been previously discovered by Choquet in 1953 only through private discussions with Mertens, when Mertens drew his attention on the paper of Dellacherie(1970) who states the opposite direction namely : The Choquet integral is comonotonic additive and monotone.

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2.3.1

The comonotonic sure thing principle

A central feature of the CEU model is that it aims to take into account possible hedging effects. A crucial axiom for axiomatization of CEU (see e.g. Gilboa (1987), Chew and Wakker (1996)) is the comonotonic sure-thing principle, a weakening of the sure thing principle, which can be stated as : Axiom 4 Comonotonic sure thing principle n n   Let X = xi A∗i and Y = yi A∗i , where {Ai } is a partition of S and i=1

i=1

x1 ≤ ... ≤ xi ≤ ... ≤ xn ; y1 ≤ ... ≤ yi ≤ ... ≤ yn are such that xi0 = yi0 for some 1 ≤ i0 ≤ n. Then X Y implies X ′ Y ′ for acts X ′ and Y ′ obtained from the acts X and Y by merely replacing the i0 ’th common outcome by any other common outcome that preserves the rank i0 in each of the two sequences x and y. This axiom expresses that as long as acts remain comonotone (i.e., no hedging effect happens), there is no reason to change the direction of preferences when a common outcome is modified. Note, however, that even jointly with standard axioms of weak order, continuity and monotonicity, the comonotonic sure-thing principle fails to fully characterize CEU. We now turn to the ability of Schmeidler’s model to handle uncertainty aversion (and symmetricaly uncertainty appeal). 2.3.2

Uncertainty aversion

In his seminal papers, Schmeidler (1982, 1989) has shown the great ability of his model to capture the concept of uncertainty aversion. He defined uncertainty aversion through convexity of preferences i.e. : ∀X, Y ∈ L0 , ∀α ∈ [0, 1] , X ∼ Y ⇒ αX + (1 − α)Y X, interpreting this axiom as ”smoothing” or averaging potential outcomes makes the DM better off. This definition revealed as particularily meaningful since as proved by Schmeidler (1986, 1989), uncertainty aversion is equivalent to the capacity v being convex i.e. ∀A, B ∈ A, v(A ∪ B) + v(A ∩ B) ≥ v(A) + v(B) and since furthermore one has : Proposition 1 Schmeidler(1986) Let I : V → R a Choquet integral with respect to a capacity v, i.e.∀X ∈ V, I(X) = Ch Xdv, then the following two conditions are equivalent : (i) v is convex    (ii) For all X in V : Ch Xdv = M in XdP, P ∈ core(v) , where core(v) = {additive probability measures P on A s.t. P (A) ≥ v(A), ∀A ∈ A} This proposition offers an attractive interpretation of uncertainty aversion in terms of pessimism : in Schmeidler’s model, an uncertainty averse DM behaves as conceiving as possible any probability distribution in the core of v, and by pessimism, feels that the worst of these distributions  will happen, namely : ∀X, Y ∈ L0 , X Y iff M in u(X)dP ≥ M in u(Y )dP. P ∈core(v) S

P ∈core(v) S

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Such an interpretation would remain true for the CEU model (i.e. in Savage’s framework ) since for such a model, convexity of preferences is equivalent to v convex and u concave (see Chateauneuf, Dana and Tallon (1997)). A direct interpretation in terms of hedging effects can moreover be obtained in the simple case of a CEU DM with constant marginal utility of wealth (u(x) = x, ∀x ∈ R), since there, convexity of preferences is equivalent to the following uncertainty aversion axiom (Chateauneuf (1994)): Axiom 5 Uncertainty aversion

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F orX, Y, Z ∈ V, Y and Z comonotone, then, X ∼ Y ⇒ X + Z Y + Z This axiom allows for taking into account hedging effects : since Z is not a hedge against Y but may be a hedge against X, hence X + Z may display a reduction of uncertainty with respect to Y + Z, and therefore X + Z may be preferred to Y + Z by an uncertainty averse DM. Such an interpretation fits particularly well with Ellsberg example. Actually, describe uncertainty in Ellsberg’s paradox in the first simple way of subsection 2.1: S = {R, B, Y } , A = 2S . Let P be the set of possible probability distributions on (S, 2S ) and let v be defined by v(A) =Inf P (A), ∀A ∈ A, one obtains: P

v

φ

R

B

Y

R∪B

R∪Y

B∪Y

S

0

1/3

0

0

1/3

1/3

2/3

1

That v is convex and that P = core(v) is straightforward to check.  (P = Pr obabilitydistributions on (S, 2S ) such that P (R) = 1/3 ) Compute I(X) = Ch Xdv for any act X : I(g1 ) = 1/3 × 100 > I(g2 ) = 0 × 100, hence g1 ≻ g2 I(g4 ) = 2/3 × 100 > I(g3 ) = 1/3 × 100, hence g4 ≻ g3

2.4

CEU models and the upper-lower probability approach

As previously shown, the uncertainty environment P in Ellsberg’s paradox can be summarized through the lower probability v(.) =Inf P (.), which moreover P ∈P

proves to exhibit the nice property of being convex, hence allowing for easy computations as shown by the re-statement of Proposition 7: Corollary 1 Let v be a capacity on (S, A), then the following two assertions are equivalent: 1 1 Note that uncertainty aversion axiom implies comonotonic independence, hence characterizes the simplified version of Schmeidler’s model (see section 2.2.2), where furthermore v is convex.

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(i) v is convex    (ii) ∀X ∈ V, min XdP, P ∈ core(v) = Ch Xdv

Such situations, summarized by a convex lower probability (i.e. a convex capacity) have been named regular uncertainty by Jaffray (1989a, 1989b). It happens that regular uncertainty arises in natural circumstances as was shown by Dempster (1967). Thus, let us assume as Dempster that (Ω, 2Ω , π) is a finite probability space endowed with a multivalued mapping Γ from Ω to A∗ = A − {φ}, where A = 2S and S the set of states of the world is finite. Interpret Γ as informing us that if ̟ ∈ Ω occurs, then Γ(̟)obtains (such a space (Ω, 2Ω , π) is called a message space), then one can assert that each event A ∈ A  is at least as likely to obtain as any event with probability v(A) where v(A) = m(B), B⊂A  Γ({̟})and v proves to be a belief function (i.e. a where m(B) = {̟∈Ω,Γ(̟)=B}

particular convex function (see e.g. Shafer (1976)). In such situations of regular uncertainty i.e. when the set P of possible  probability distributions P on (S, 2S ) is defined by P= P on (S, 2S ), P ≥ f where f(A) =Inf P (A), and f is convex (note that the lower probability is P ∈P

now noted f , and that F the upper probability is defined by F (A) = 1 − _ f(A), ∀A ∈ A), one might suspect that a CEU DM would not necessarily be uncertainty averse i.e. would not necessarily exhibit a subjective estimation of events through v = f , but for instance would merely exhibit a subjective estimation of events through a capacity v = αf + (1 − α)F with α ∈ [0, 1] . Such a behavior with α interpreted as a Hurwicz coefficient of pessimism, has been thoroughly studied by Jaffray and Philippe (1997) and shown to be both consistent with the C.E.U. model and Jaffray’s model (1989a,1989b).

2.5

Some economic applications

Here we focus on two examples of economic applications. As other examples of economic applications (such a list being far from exhaustive), the readers can consult Chateauneuf, Dana and Tallon (1997) for a study of optimal risk-sharing rules and equilibria, Dow and Werlang (1992b) for an explanation of the excess volatility of stock prices, Epstein and Wang (1994, 1995) for a formal model of asset pricing under uncertainty leading to indeterminations of equilibria and possible security price booms and crashes. 2.5.1

Optimal portfolio choice

A major economic application of the Choquet Expected Utility model was obtained by Dow and Werlang (1992a), when revisiting the basic portfolio problem. Let w > 0 be the initial certain wealth of a DM facing the question of buying or selling short an uncertain asset X ∈ V, whose price is p > 0. Under EU theory, the agent is assumed to assign probabilities P (A) to events A and to rank assets X ∈ V by comparing expected utilities of X, with a concave utility function u concavity of u conveying, in the framework of EU model, uncertainty aversion. 11

According to a famous result of Arrow (1965), an EU agent will invest in an asset X if and only if the expected value EP (X) of this asset exceeds the price, and will wish to sell the asset short if and only if the expected value is lower than the price of the asset. Consequently, the investor will have no position in the asset if and only if the price is exactly EP (X). Dow and Werlang (1992a) used Choquet Expected Utility theory to show a result intuitively plausible and compatible with observed investment behavior: there is a range of prices at which the investor has no position in the asset. At prices below these, the investor holds a positive amount of the asset and at higher prices he holds a short position. More precisely, the agent is assumed to be uncertainty averse hence (see subsection 2.2.2) to rank assets X by performing the Choquet Expected Utility I(u(X)) of X, with concave u and convex v. Then, Dow and Werlang proved that, for any asset X, the reservation prices only depend of the DM’s attitudes towards uncertainty. They are respectively I(X) = Min {EP (X), P ∈ core(v)} and −I(−X) = M ax {EP (X), P ∈ core(v)} Therefore, the length of the no-trade interval (I(X), −I(−X)) is intimately connected with the size of core(v). Obviously, if core(v) reduces to a single probability law, one obtains the result of Arrow (Arrow, 1965). 2.5.2

Asset pricing

In Chateauneuf, Kast and Lapied (1996), for any asset X, I(X) (resp. −I(−X)) denotes the selling (resp. buying) price of asset X by a broker12 . Three axioms on the functional I : V → R are proposed in order to model the dealer’s behavior. The first two ones are standard: requiring on one hand that if an asset Y pays more in all states than an asset X, its price must be higher, and on the other hand assuming no transaction costs on riskless assets, i.e., I(αS ∗ ) = α, ∀α ∈ R. The third axiom, named “comonotonicity premium” stipulates that ∀X, Y ∈ V, I(X + Y ) ≤ I(X) + I(Y ), with equality holding if X and Y are comonotone. This axiom aims at taking into account a potential reduction of uncertainty (when the portfolio X + Y is sold instead of X or Y alone), which would induce the dealer to sell X + Y at a discount compared to X and Y . However, there are situations where no hedging effects occur when adding two assets X and Y : this happens when X and Y are comonotone.  These axioms are equivalent to I(X) = Ch Xdv with v concave13 . Such a pricing scheme provides an explanation for possible positive profits for the broker: since I(X) ≥ −I(−X), all the more so the dealer behaves as if he conceived as possible all the probabilities P s.t. P ≤ v and evaluates the selling price I(X) by I(X) = Max {EP (X), P ≤ v} and the buying price by −I(−X) = Min {EP (X), P ≤ v}. Furthermore, as usual in finance, no 1 2 Notice that this notation is consistent with the fact that selling asset X is the same as buying the short position −X. 1 3 A capacity v is concave if : ∀A, B ∈ A, v(A ∪ B) + v(A ∩ B) ≤ v(A) + v(B)

12

arbritrage opportunity is left to the consumer (i.e., investor) in the sense that he cannot form a portfolio yielding positive payments at a strictly negative cost; this can be easily checked from subadditivity, positive homogeneity and positivity of the price function I. Lastly, such a price functional is consistent with the observed violation of the put-call parity (see e.g. Gould and Galai, (1974), Galai, (1983)), and Lefoll, (1994), namely, with I(P ) ≤ I(C) + I(−S) + K, i.e., it is consistent with the observed fact that for a call C and a put P with exercise price K and underlying security S, the price of the put P is smaller than the price of the replicating portfolio C − S + K.

3

Decision under risk

We assume now that the DM is informed of the value of an objective probability measure P on (S, A) and thus, the DM is now faced to decision under risk. Moreover, to simplify the exposition, we also assume that P is σ−additive and non-atomic (i.e.∀A ∈ A, such that P (A) > 0, ∀α ∈ (0, 1] , ∃B ∈ A, B ⊂ A , such that P (B) = αP (A)) ; thus, V generates any bounded random variable. Any X of V is then a random variable and has a probability distribution PX . Let FX denote the cumulative distribution function of X (FX (x) = P {X ≤ x}) and E (X) its expected value. Finally, let L be the set of probability distribution functions of random variables in V. Since each X of V induces a probability distribution L(X) on R, the preference relation on V induces a preference relation on L denoted also provided the assumption that, if two random variables have the same probability distribution, they are equivalent. Thus, under risk, and with this important assumption, called neutrality axiom by Yaari (1987), but implicit for many authors, the axiomatization on (V, ) can be replaced by an axiomatization on (L, ). Remark 4 Any discrete random variable Z of V can be written as Z = (x1 , A1 ; . . . ; xk , Ak ; . . . ; xn , An ), where (Ai ) (i = 1, ..., n) is a partition of S and xi the outcomes on each Ai . Under risk, the distribution of this discrete random variable Z will be denoted by : L(Z) =  (x1 , p1 ; . . . ; xk , pk ; . . . ; xn , pn ) with14 x1 ≤ x2 ≤ . . . ≤ xn , pi = P (Ai ) ≥ 0, and pi = 1. For convenience of exposition, we will also write distribution L(Z) in the equivalent form : L(Z) = (x1 , 1 − q1 ; x2 , q1 − q2 ; . . . ; xn−1 , qn−2 − qn−1 ; xn , qn−1 ) where qi =

j=n 

(3)

pj .

j=i+1

In all this section, we will identify any outcome c with the constant random variable δ {c} . 1 4 Outcomes

will always be ranked in the non-decreasing order.

13

3.1

EU model and Allais’ paradox

Let us first recall briefly the Allais paradox (Allais (1953)) that gave rise to the new models of behavior under risk ; in fact we take an example from Kahneman and Tversky (1979) similar to Allais paradox : Subjects are requested to express their preferences first between these two prospects : A : winning $3, 000 with probability 1 or B : winning $4, 000 with probability 0.8 and then independently between C : winning $3, 000 with probability 0.25 or D : winning $4, 000 with probability 0.20. Typical preferences are A ≻ B and D ≻ C. Since C = 0, 25A + 0, 75δ 0 and D = 0, 25B + 0, 75δ 0 (where δ 0 is the prospect : winning 0 with probability 1), such preferences violate then the independence axiom, the key axiom of Expected Utility (EU) theory15 . Not only observed behaviors are in contradiction with EU theory, but also the EU model raised a theoretical difficulty, namely, the interpretation of the funtion u (called von Neumann’s utility) characterizing the DM’s behavior : as pointed out by Allais himself, the function u has, in fact a double role of expressing the DM’s attitude with respect to risk (concavity of u implying risk aversion) and the DM’s valuation of differences of preferences under certainty (concavity of u implying then diminishing marginal utility of wealth). The RDEU model that we will present in the next section, not only will disantangle attitude under risk and satisfaction of wealth, but also will be compatible with observed behaviors in Allais paradox.

3.2 3.2.1

Rank Dependent Expected Utility (RDEU) model Definition of RDEU model

The RDEU (Rank Dependent Expected Utility) model is due to Quiggin (1982) under the denomination of Anticipated Utility. Variants of this model are due to Yaari (1987), Segal (1987, 1993) and Allais (1988). More general axiomatizations can be found in Wakker (1994), Chateauneuf (1998). Definition 6 A DM behaves in accordance with the rank-dependent expected utility (RDEU) model if the DM’s preferences on (L, ) are characterized by two functions u and f : a continuous, increasing, cardinal utility function u : R → R (that plays the role of utility on certainty) and an increasing probability -perception function f : [0, 1] → [0, 1] that satisfies f (0) = 0, f (1) = 1. Such a DM prefers the random variable X to the random variable Y if and only if V (X) ≥ V (Y ), where the functional V is given by : V (Z) = Vu,f (Z) =



0

[f(P (u(Z) > t)) − 1] dt +

−∞



∞

f (P (u(Z) > t))dt . (4)

0

1 5 Let us recall that prefences on (L, ) satisfy Expected Utility model if they are represented  by the functional U (X) = u(x)dF (x) where u is an increasing function from R to R, defined up to a strictly positive affine transformation (i.e. X Y ⇔ U(X) ≥ U(Y )).

14

• If the perception function f is the identity function f (p) ≡ p, then V (Z) = Vu,I (Z) is the expected utility E[u(Z)] of the random variable. • The Yaari functional (see Yaari (1987)) is the special case V (Z) = VI,f (Z), in which the utility u is the identity function u(x) ≡ x. In fact, Yaari axiomatized his model independently. • If both perception and utility are identity functions, then V (Z) = VI,I (Z) is simply the expected value E[Z] of the random variable. When Z is discrete, V (Z) can be written as  ∞ u(x)df(1 − F (x)) = u(x1 ) + f(q1 )[u(x2 ) − u(x1 )] V (Z) = − −∞

+f(q2 )[u(x3 ) − u(x2 )] + · · · + f (qn−1 )[u(xn ) − u(xn−1 ]

(5)

We can then interpretate the behavior of a RDEU decision maker: he takes for sure the utility of the worst outcome u(x1 ) and weights the additional possible increases of utility u(xi ) − u(xi−1 ) by his personal transformation f (qi ) of the probability vi of having at least xi . According to this interpretation, if the decision maker behaves in such a way that f(p) ≤ p, it means that he underestimates all the additional utilities of gains. In this sense, we will call him pessimistic under risk. In the same way, u reflecting now his satisfaction for wealth,concavity of u reveals diminishing marginal utility. Allais paradox is then explainable by RDEU theory for any f satisfying : f (0, 8) = f (0, 20/0, 25) < f (0, 2)/f(0, 25). 3.2.2

RDEU is a Choquet integral

In RDEU model, f has the properties of a distorted function (see Denneberg, section 8, this volume). The corresponding ”distorted” probability f oP is then a capacity and the RDEU integral is a Choquet integral with respect to this capacity v = f oP . More precisely, V (Z) = Ch u(Z)d(foP ) ∞ ∞ = − −∞ u(x)df(P (Z > x)) = − −∞ u(x)df (1 − F (x))

Remark 5 Let us notice by now, that if f is a convex function, then v = f oP is a convex16 capacity (see e.g. Chateauneuf, (1991) or Denneberg, (1994)). Moreover, if f is below the diagonal (i.e. satisfies f (p) ≤ p, ∀p ∈ [0, 1]), then it can be easily seen that core(v) = φ. 3.2.3

Key axiom of RDEU’s axiomatization : comonotonic surething principle

The key axiom of the RDEU model is the following : 1 6 Supermodular

or 2-monotone in the terminology of Denneberg( this volume).

15

Axiom 6 Comonotonic Sure-Thing Principle under risk17 : Let P , and Q be two lotteries of L. P = (x1 , p1 ; . . . ; xk , pk ; . . . ; xn , pn ) and Q = (y1 , p1 ; . . . ; yk , pk ; . . . ; yn , pn ) be such that xi0 = yi0 ; then P Q implies P ′ Q′ for lotteries obtained from lotteries P and Q by merely replacing the ith common outcome xi0 , by a common outcome x′i0 again in ith rank both in 0 0 ′ P and Q′ . This axiom from Chateauneuf (1998), is very similar to Green and Julien’s ordinal independence axiom (1988), to Segal’s irrelevance axiom (1987), and comonotonic independence in Chew and Wakker (1996)( see also Chew-Epstein (1989), Quiggin (1989), Wakker, (1994) ; it is clearly much weaker than Savage’s sure-thing principle that requires no restriction on x’i0 . This attractive axiom is a central necessary axiom in the characterization of RDEU, but, as in the case of CEU, even jointly with the standard axioms of weak order, continuity, monotony, this axiom fails to fully characterize RDEU. A complete characterization of RDEU model can be, for instance, obtained with the help of noncontradictory comonotonic tradeoffs (see Wakker (1994)), or else with a comonotonic mixture independence axiom, an adaptation of mixture independence, which underlines the role played not only by comonotonicity, but also by the extrema outcomes (see Chateauneuf, 1998).

3.3

From CEU model to RDEU model using first order stochastic dominance (Wakker)

It has been recognized by several authors including Wakker (1990), Chateauneuf (1991), that RDEU models under risk can be derived from models under uncertainty by merely postulating the respect of first order stochastic dominance. We will use this approach, first to get Yaari’s model from the simplified version of Schmeidler’s model (section 2.2), then to get RDEU model from Choquet Expected Utility model. Being under risk, we also suppose that the objective probability P is compatible with the preference relation on (V, ). More precisely, we suppose : Axiom 7 First order stochastic dominance : [A, B ∈ A, P (A) ≥ P (B)] ⇒ A B. Let us notice that this axiom implies the neutrality axiom stated at the beginning of the section. 3.3.1

From simplified Schmeidler’s model to Yaari’s model

Let us suppose that the preference relation on (V, ) satisfies comonotonic independence axiom 2, moreover the usual axioms of non-trivial weak-order, continuity, monotonicity. The preference relation is then represented by a Choquet 1 7 According to remark 3, the justification of the denomination of this axiom results from a natural interpretation of P, Q, P’, Q’ as probability distributions of pairwise comonotonic random variables.

16

integral with respect to a capacity v such that A B implies v(A) ≥ v(B). The axioms imply then that P (A) ≥ P (B) imply v(A) ≥ v(B). This gives us an intuition of the result : There exists a unique distorted function f ( increasing function f : [0, 1] → [0, 1] satisfying f (0) = 0, f(1) = 1) such that v = foP . It can then be readily seen that the simplified Schmeidler’s model reduces to Yaari’s model under the assumption of first order stochastic dominance (Chateauneuf, 1994, Wakker, 1990). 3.3.2

From CEU to RDEU

Let us suppose that the preference relation on (V, ) satisfies all the axioms to get general CEU model characterized by v and u (see definition 6), then, A B implies v(A) ≥ v(B). If moreover, the objective measure P on S satisfies first order stochastic dominance, again, since P (A) ≥ P (B) imply v(A) ≥ v(B), there exists then a unique distorted function f such that v = foP . We get the following result due to Wakker (1990) : Let the preference relation on (V, ) satisfies all the axioms to get general CEU, and let P be a probability distribution on S satisfying first order stotastic dominance, then the preference relation on (V, ) can be represented by the RDEU model.

3.4

Risk aversion in RDEU model

The most natural way to define risk aversion is the following : a weak risk averse DM is the one who always prefers to any random variable X the certainty of its expected value E(X). An other possible way to define some type of risk aversion is to define it as aversion to some type of (mean preserving) increasing risk. All kinds of stochastic orders can then generate as many different kinds of risk aversion. There exists, then, many different definitions of risk aversion but their different meanings have been hidden by the fact that, under expected utility theory, all are equivalent : they all reduce to the concavity of the utility function. The result is that, for an EU decision maker, weak risk aversion implies automatically aversion to any mean preserving increase in risk (MPIR). Let us give some usual definitions of (mean preserving) increasing risk and their corresponding definitions of risk aversion: 3.4.1

Strong risk aversion

t Y is a general mean preserving increase in risk (MPIR) of X if −∞ FY (x)dx ≥ t  +∞  +∞ F (x)dx for all t ∈ R and −∞ FY (x)dx = −∞ FX (x)dx. This usual −∞ X concept of (mean preserving) increasing risk is classically used since Rothschild and Stiglitz (1970) and we define the corresponding notion of strong risk aversion : A DM is then strongly risk averse if for any X and Y in V such that Y is a MPIR of X, the DM prefers X to Y . 17

3.4.2

Monotone risk aversion

Quiggin (1992) brought to light that strong risk aversion may be a too strong concept, i.e., that second degree dominance is perhaps too weak an order of risk, and introduced a new notion, monotone (mean-preserving) increase in risk, defined in terms of comonotonic random variables instead of a general mean-preserving increasing risk : Y is a monotone increase in risk (MIR) of X if and only if18 Y = X + Z, d

where Z is such that E(Z) = 0 and X and Z are comonotonic19 . A DM is thus monotone risk averse if he is averse to any monotone increase in risk (i.e. for every pair (X,Y) where Y is MIR of X, he always prefers X to Y ). The notion of monotone risk aversion20 is particularly fitted to RDEU theory where comonotonicity plays a fundamental part at the axiomatic level. 3.4.3

Left monotone risk aversion

The order induced by monotone increasing risk is a very partial order since it can order very few pairs of random variables. The following notion compares more pairs and this notion of increasing risk is asymmetric in the sense that it treats differently downside and upside risks. This notion will prove to be particularily fitted with deductible insurance (see section 3.5.3.). The following definition is due to Jewitt21 (1989) under the name of Locationindependent Risk (see also Lansberger and Meilijson (1994a). The motivation of Jewitt was to find a notion of increase in risk that models coherent behavior in a context of partial insurance22 . Let us first recall that F −1 (p) = inf {z ∈ R|F (z) ≥ p} , and then we can interpret F −1 (p) as the highest gain among the least favourable p% of the outcomes. Y is said to be a left monotone mean preserving increase in risk (LIR)  FY−1 (p)  FX−1 (p) of X if −∞ FY (x)dx ≥ −∞ FX (x)dx for all p ∈ (0, 1)23 . Again, we define the corresponding notion of left monotone risk aversion : A DM is left monotone risk averse (respectively, left monotone risk seeking) if for any X and Y in V such that Y is a left monotone MPIR of X, the DM prefers X to Y (respectively, Y to X). 1 8 = means d 1 9 It follows

equality of probability distributions.

from the results of Landsberger & Meilijson (1994b) that the notion of monotone increase in risk coincides with the statistical notion of Bickel & Lehmann dispersion (1976, 1979) for random variables with equal means : Y is said more dispersed than X in the sense −1 −1 of Bickel-Lehmann if FY−1 (q) − FY−1 (p) ≥ FX (q) − FX (p), where F −1 from (0, 1] into R by F −1 (p) = inf {z ∈ R|F (z) ≥ p}, for all 0 < p < q < 1. We can notice that all the interquantile intervals are shorter for X than for Y . 2 0 Applications of this notion of risk aversion can be found e.g. in Cohen (1995). 2 1 In Jewitt, the notion is given for X and Y with possibly unequal means. 2 2 See also Lansberger and Meilijson (1994a) on this subject. 2 3 It is instructive to notice that in the integral definition above, the upper limits of integration are arbitrary quantiles corresponding to equal cumulative probabilities p. If the upper limits of integration are taken to be arbitrary but equal to each other, the corresponding integral condition becomes the usual definition of MPIR.

18

Remark 6 It can be readily seen that Strong risk aversion ⇒Left motone risk aversion⇒ Monotone risk aversion ⇒ Weak risk aversion, but the contrary is only true in the EU model. 3.4.4

Characterization of different notions of risk aversion in the RDEU model

Contrary to EU model, in RDEU model, each of the different notions of aversion to risk has a specific characterization24 . Gathering several results in different papers, we get the following results: Let a RDEU decision maker be characterized by two differentiable functions u and f : He is then strongly risk averse if and only if the perception function f is convex25 and the utility function u is concave (see Chew, Karni and Safra, 1987). He is left monotone risk averse if and only if the perception function f is star-shaped26 at 1 and the utility function u is concave (see Chateauneuf, Cohen and Meilijson, 1998). The characterization of monotone risk aversion is based on two following (v) 1−v indices : Pf = inf [ 1−f f (v) / v ], called index of pessimism, which is ≥ 1 as 0