constant risk aversion - boston college

journal of economic theory 83, 19�42 (1998)

Constant Risk Aversion*

Zvi Safra

Faculty of Management, Tel Aviv University, Tel Aviv, 69978, Israelsafraz�post.tau.ac.il

and

Uzi Segal

Department of Economics, University of Western Ontario, London, Ontario, N6A 5C2, Canadasegal�sscl.uwo.ca

Received April 6, 1997; revised June 10, 1998

Constant risk aversion means that adding a constant to all outcomes of twodistributions, or multiplying all their outcomes by the same positive number, willnot change the preference relation between them. We prove several representationtheorems, where constant risk aversion is combined with other axioms to implyspecific functional forms. Among other things, we obtain a form of disappointmentaversion theory without using the concept of reference point in the axioms, and aform of the rank dependent model without making references to the ranking of theoutcomes. This axiomatization leads to a natural generalization of the Gini index.Journal of Economic Literature Classification Number: D81. � 1998 Academic Press

1. INTRODUCTION

Constant risk aversion means that adding the same constant to all out-comes of two distributions, or multiplying all their outcomes by the samepositive constant, will not change the preference relation between them.Within the expected utility framework, this assumption implies expectedvalue maximization. But there are many (non-expected utility) functionalsthat satisfy this requirement, for example, the dual theory of Yaari [30],or functions offered by Roberts [23] and by Smorodinsky [27] (seeExample 1 in Section 2 below).

article no. ET972457

190022-0531�98 �25.00

Copyright � 1998 by Academic PressAll rights of reproduction in any form reserved.

* We thank Chew Soo Hong, Jim Davies, Larry Epstein, Joel Sobel, Shlomo Yitzhaki, andPeter Wakker for their useful comments and suggestions. Zvi Safra thanks the Johns HopkinsUniversity and the Australian National University for their hospitality and the Israel Institutefor Business Research for financial support. Uzi Segal thanks the Social Science andHumanities Research Council of Canada for financial support.

In this paper we prove several representation theorems, where constantrisk aversion is combined with some other known axioms to imply specificfunctional forms. We first show that non-trivial (that is, non-expectedvalue) functionals that satisfy constant risk aversion cannot be Fre� chetdifferentiable. This differentiability is the key assumption in Machina'sanalysis [19], and is widely used in the decision theoretic literature. Sincethey are not Fre� chet differentiable, constant risk aversion functionalscannot be approximated (in the L1 norm) by expected utility preferences.These results are presented in Section 3.

A possible relaxation of Fre� chet differentiability is the requirement thatrepresentation functionals are only Gateaux differentiable. This requires thederivative with respect to : of V((1&:) F+:G ) to exist, and to becontinuous and linear in G&F. Many constant risk aversion functionalssatisfy this assumption, but as we show in Section 4, adding betweenness tothe list of axioms (FtG implies :F+(1&:) GtF for all : # [0, 1]) permitsonly one functional form, which is Gul's disappointment aversion theory[16] with a linear utility function u. According to this theory, the decisionmaker evaluates outcomes that are better than the certainty equivalent ofa lottery by using an expected utility functional with a utility function u. Hesimilarly evaluates outcomes that are worse than the certainty equivalent(with the same function u). Finally, the value of a lottery is a weighted sumof these two evaluations. In this theory, the certainty equivalent serves asa natural reference point, to which Gul's axioms make an explicit reference.Our axioms do not require any such explicit dependency, and the referencepoint is obtained as part of the results of the model, rather than as part ofits assumptions.

One of the most popular alternatives to expected utility is the rankdependent model, given by V(F )=� u(x) dg(F(x)). (This model has severaldifferent versions, see Weymark [29], Quiggin [21], and other citationsin Section 5 below.) This functional form is consistent with constant riskaversion whenever u is a linear function, which is Yaari's dual theory [30].Although many axiomatizations of the rank dependent model exist,1 theyall depend on one crucial assumption, namely, that the value of anoutcome depends on its relative position. This is of course a key feature ofthe rank dependent model, from which its name is derived. But it would benice to be able to obtain this property as a result, rather than as anassumption of the model. In Section 5 we offer an axiomatization of a non-trivial special case of Yaari's functional, where none of the axioms makesan explicit appeal to the relative position of any of the outcomes. The keyadded axiom is mixture symmetry, which states that if FtG, then for all: # [0, 1], :F+(1&:) Gt(1&:) F+:G (see Chew, Epstein, and Segal

20 SAFRA AND SEGAL

1 We have counted more than ten axiomatizations of this model.

[7]). The functional form we obtain is a weighted average of the expectedvalue functional, and the Gini inequality index.

Two recent works on bargaining with non-expected utility preferencesmake use of homogeneity of preferences with respect to probabilities (seeRubinstein, Safra, and Thomson [24] and Grant and Kajii [14]). Weoffer a slightly stronger assumption, where FtG iff for every : # [0, 1],:F+(1&:) $0 t:G+(1&:) $0 ($0 is the distribution of the degeneratelottery that yields the outcome zero with probability one). Together withconstant risk aversion, this axiom implies Yaari's representation with aprobability transformation function of the form g( p)=1&(1& p)t. Weprove this result in the last section of the paper.

2. DEFINITIONS

Let 0=(S, 7, P) be a measure space and let X be the set of real boundedrandom variables with non-negative outcomes on it. For X # X, let FX bethe distribution function of X. Denote by F the set of distribution functionsobtained from elements of X. With a slight abuse of notations, we denoteby a the constant random variable with the value a, and its distributionfunction by $a . For X # X, let X

�be the lowest possible value of X (that is,

X�

is the supremum of the values of x for which FX (x)=0). Observe thatfor X # X and a>&X

�, X+a # X, and the distribution FX+a is given by

FX+a(x)=FX (x&a). Throughout the paper, when we use the notationX+a or F+a we assume that a>&X

�. Also, for X # X and *>0, *X # X,

and we define the distribution *_FX :=F*X by (*_FX)(x)=FX (x�*).On X we assume the existence of a complete and transitive preference

relation p. We assume throughout the paper that if FX=FY , then XtY.Therefore, p induces an order on F, which we also denote p. Assumefurther that p is continuous (with respect to the weak topology), andmonotonic (with respect to first order stochastic dominance). It thenfollows that every F # F has a unique certainty equivalent x # [0, �),satisfying Ft$x (recall that for every F # F there exists x such thatF(x)=1). We restrict attention to preference relations satisfying the followingassumption.

Constant Risk Aversion. XpY iff for every a>max[&X�, &Y

�] and for

every *>0, *(X+a)p*(Y+a). Or equivalently, FpG iff for every sucha and *, *_(F+a)p*_(G+a).

Note that p satisfies constant risk aversion if it satisfies both constantabsolute risk aversion and constant relative risk aversion. Also, itsrepresentation functional V: F � R which is defined implicitly by Ft$V(F )

21CONSTANT RISK AVERSION

(that is, V(F ) is the certainty equivalent of F ), satisfies V(*_(F+a))=*[V(F )+a]. In such a case we say that V satisfies constant risk aversion.The following are examples for such functionals.

Example 1. (1) V(F )=� x dg(F(x)) (Yaari's dual theory [30]).

(2) V(F )=+F+_F W((1�_F)_[F&+F]) for some functional W,where +F is the expected value of F and _F is its standard deviation(Roberts [23]).

(3) V(F )=arg mint � |x&t| c+1 dF(x) for some c>0 (Smorodinsky[27]).

The next lemma shows that the set of functionals satisfying constant riskaversion is much larger than the above list. Moreover, from two such func-tionals more functionals can be created.

Lemma 1. Let V be the set of all functionals that satisfy constant riskaversion. Then for every V$�V, the functionals V1 and V2 are in V, whereV1(F )=inf[V(F ): V # V$] and V 2(F )=sup[V(F ): V # V$].

Proof. For a given F, there is a sequence Vi in V$ such that V1(F )=lim Vi (F ). Hence V1(*_(F+a))�lim Vi (*_(F+a))=lim *(Vi (F )+a)=*(V1(F )+a). On the other hand, there is a sequence V$i in V$ suchthat for G=*_(F+a), V1(G )=lim V$i (G ), hence V1(F )�lim V$i (F )=lim V$i (*&1_(G&*a)) = lim *&1(V$i (G ) & *a) = *&1(V1(G ) & *a). Thesetwo inequalities imply V1(F)�*&1(V1(*_(F+a))&*a)�V1(F ), henceV1(*_(F+a))=*(V 1(F )+a).

The proof of the sup case is similar. K

3. SMOOTH PREFERENCES

Machina [19] introduced the concept of smooth representations, that is,representations that are Fre� chet differentiable.

Definition 1. The function V: F � R is Fre� chet differentiable if forevery F # F there exists a ``local utility'' function u( } ; F ): R � R such thatfor every G # F,

V(G )&V(F )=| u(x; F ) d[G(x)&F(x))+o(&G&F&), (1)

where & }& is the L1 -norm.

22 SAFRA AND SEGAL

In other words, V is Fre� chet differentiable if for every F, the functionalV behaves around F like an expected utility representation with thevon Neumann and Morgenstern utility function u( } ; F ). Machina [19]demonstrated how under this assumption, many results in decision theorycan be extended to non-expected utility models. Obviously, if V(F ) alwaysequals E[F ], the expected value of F, then V is smooth and satisfiesconstant risk aversion. But we have seen above that many other functionalssatisfy constant risk aversion. It is therefore natural to ask whether any ofthem is Fre� chet differentiable.

Theorem 1. The following two conditions are equivalent.

(1) V is an expected value functional, that is, V(F )=E[F]=� x dF(x).

(2) V satisfies constant risk aversion and is Fre� chet differentiable.

Proof. Obviously, (1) O (2). To see why (2) O (1), consider first thefamily of local utilities u( } ; $x). Since local utility functions are unique upto scalar addition,2 we assume that for every x, u(x; $x)=x.

Consider now the lottery ( y, p; 1, 1& p) for arbitrary y and p. Using thelocal utility u( } ; $1) we obtain (recall that u(1; $1)=1)

V( y, p; 1, 1& p)&V($1)=[ pu( y; $1)+1& p]&1+o( p( y&1)). (2)

Similarly, for ( y+:, p; 1+:, 1& p) we obtain

V( y+:, p; 1+:, 1& p)&V($1+:)

=[ pu( y+:; $1+:)+(1& p)(1+:)]&(1+:)+o( p( y&1)). (3)

By constant risk aversion, the left hand sides of Eqs. (2) and (3) are thesame. Subtract Eq. (2) from Eq. (3) to obtain

0= p _u( y+:; $1+:)&u( y; $1)&:+o( p( y&1))

p & .

Divide by p, and then take the limit as p � 0 to obtain

u( y+:; $1+:)=u( y; $1)+:.

Set x= y+: and k=1+:, and obtain for x�k&1,

u(x; $k)=u(x&k+1; $1)+k&1. (4)


2 For this and other statements concerning local utilities, see Machina [19].

Similarly to Eq. (2) we obtain for *>0

V(*y, p; *, 1& p)&V($*)=[ pu(*y; $*)+*(1& p)]&*+o( p*( y&1)). (5)

By constant risk aversion, the left hand side of Eq. (5) equals * times theleft hand side of Eq. (2). Hence

pu(*y; $*)&*p+o( p*( y&1))=*[ pu( y; $1)& p+o( p( y&1))].

Divide both sides of the last equation by p, and then take the limit as p � 0to obtain

u(*y; $*)=*u( y; $1).

Set x=*y and k=* to obtain for x�k&1,

u(x; $k)=ku \xk

; $1+ (6)

Let h( } )=u( } ; $1) and obtain from Eqs. (4) and (6)

h(x&k+1)+k&1=kh \xk+ . (7)

Since h is increasing, it is almost everywhere differentiable. Pick a pointx*>1 at which h$ exists. Differentiate both sides of Eq. (7) and obtain thatfor x=kx*,

h$(k(x*&1)+1)=h$(x*).

Since x*>1, it follows that h$(z) is constant for z>1.By similar arguments, there is x*<1 at which h is differentiable. We now

obtain that h is differentiable on (0, 1), and that its derivative there isconstant. In other words, there are two numbers, s and t such that

�u(x; $1)�x

={ s, x<1t, x>1.

From Eq. (6) it now follows that

�u(x; $k)�x

={ s, x<kt, x>k.

(8)

Claim 1. Let V be a Fre� chet differentiable functional, and let u( } ; $x*)be its local utility at $x* . Then for almost all x*, u(x; $x*) is differentiablewith respect to its first argument at x=x*.

24 SAFRA AND SEGAL

Proof of Claim 1. By monotonicity, the functional V satisfiesV($x)>V($y) � x>y, hence the set of points where �V($x)��x does notexist is of measure zero. The claim now follows from the equivalence of thefollowing two conditions.

(1) The derivative �V($x)��x exists at x=x*.

(2) The local utility u(x; $x*) is differentiable with respect to its firstargument at x=x*.

To see why (1) and (2) are equivalent, note that

V($x*+=)&V($x*)=u(x*+=; $x*)&u(x*; $x*)+o(=).

Divide both sides by = and let = � 0 to obtain that V($x) is differentiablewith respect to x at x=x* iff u(x; $x*) is differentiable with respect to itsfirst argument at x=x*. K

Since V is Fre� chet differentiable,3 it follows from Claim 1 that for almostall k, u( } ; $k) is differentiable with respect to the first argument at k. Hencein Eq. (8), s=t, and since V($k)=k, it follows that u(x; $k)=sx+(1&s) k.

Fix the probabilities p1 , ..., pn , and consider the space of lotteries(x1 , p1 ; ...; xn , pn). These lotteries can be represented as vectors of the form(x1 , ..., xn) # Rn. For y�0, let dy=( y, ..., y) be the point on the maindiagonal corresponding to the lottery $y . Pick a point x* not on the maindiagonal of Rn and y such that x*t$y , and let H* be the two dimensionalplane containing x* and the main diagonal. It follows from Roberts [23,p. 430] that indifference curves on H* below the main diagonal are linearand parallel to each other, and so are indifference curves above the maindiagonal. In other words, x*tdy iff for all : # [0, 1], :x*+(1&:) dy tdy .Note that this mixture is with respect to outcomes, not with respect todistributions.

The local utility function at $y is given by u(x; $y)=sx+(1&s) y, hence

0=V(x*)&V(dy)=V(:x*+(1&:) dy)&V(dy)

= :n

i=1

pi[s(:xi*+(1&:) y)+(1&s) y]&[sy+(1&s) y]+o(:)

=:s _:n

i

pi xi*& y&+o(:).


3 Up to this point, we only used Gateaux, rather than Fre� chet, differentiability (seeSection 4).

Divide by :s, and then let : � 0 to obtain �ni=1 p ixi*= y, which is

the expected value functional. By continuity, V(F ) is the expected valuefunctional for all F. K

Following this result, Chambers and Quiggin [3] proved that, assumingdifferentiability, decreasing absolute risk aversion implies increasing relativerisk aversion.

4. BETWEENNESS AND GA� TEAUX DIFFERENTIABILITY

The last section suggests that in the presence of constant risk aversion,the assumption of Fre� chet differentiability is too strong. Weaker notions ofdifferentiability exist, and at least one of them got special attention in theliterature.

Definition 2. A functional V is Gateaux differentiable at F (Zeidler[32, p. 191]) if for every G,

$V(F, G&F ) :=��t

V((1&t) F+tG ) } t=0

exists and if $V(F, G&F ) is a continuous linear function of G&F. V isGateaux differentiable if it is Gateaux differentiable at F for every F.

If V is Fre� chet differentiable, then it is also Gateaux differentiable, butthe opposite is not true. For example, the rank dependent model isGateaux, but not Fre� chet differentiable (see Chew, Karni, and Safra [8]).4

In this section we assume that preferences can be represented by Gateauxdifferentiable functionals, and that they satisfy the following betweennessassumption.

Betweenness. FpG implies that for every : # [0, 1], Fp:F+(1&:) GpG (see Chew [4, 5] and Dekel [9]).

We say that the functional V satisfies betweenness if V(F )�V(G )implies that for every : # [0, 1], V(F )�V(:F+(1&:) G)�V(G ). In thissection we characterize functionals that satisfy constant risk aversion,Gateaux differentiability, and betweenness. It turns out that the onlyfunctional to satisfy these three axioms is a special case of Gul [16]disappointment aversion theory (which by itself is a special case of Chew'ssemi-weighted utility theory [5]).

26 SAFRA AND SEGAL

4 The minimum of two Gateaux differentiable functionals is not necessarily Gateauxdifferentiable. Suppose that for : # [0, :*), V 1(:F+(1&:) G )>V 2(:F+(1&:) G ), but for: # (:*, 1], V1(:F+(1&:) G )<V 2(:F+(1&:) G ). Let H=:*F+(1&:*) G and let V*=min[V 1, V2] to obtain that $V*(H, F&H ){&$V*(H, G&H ).

Definition 3. V is a Disappointment Aversion functional (see Gul[16]) if it is given by

V(F )=#(:)

: |x>C(F )

u(x) dF(x)+1&#(:)

1&: |x<C(F )

u(x) dF(x), (9)

where : is the probability that F yields an outcome above its certaintyequivalent C(F ), and #(:)=:�[1+(1&:) ;] for some number ;.

According to disappointment aversion theory, the decision makerevaluates outcomes that are better than the certainty equivalent of a lotteryby using an expected utility functional with a utility function u. Hesimilarly evaluates outcomes that are worse than the certainty equivalent.Finally, the value of a lottery is a weighted sum of these two evaluations.

Theorem 2. The following two conditions are equivalent.

(1) V is Gateaux differentiable, and satisfies constant risk aversion andbetweenness.

(2) V is a disappointment aversion functional with the linear utilityu(x)=x.

Proof. If V satisfies betweenness, then each of its indifference curves canbe obtained from an expected utility functional. Assuming as before that forevery k, V($k)=k, it follows that there are utility functions uk : [0, �) � Rsuch that Ft$k iff

| uk (x) dF(x)=uk (k)=k. (10)

Choose k>m>0, and let (x, p; 0, 1& p)t(k, 1). Then (mx�k, p; 0, 1& p)t(m, 1). By Eq. (10), puk (x)=k and pum(mx�k)=m, hence

uk (x)=km

um \mxk + . (11)

Let Fk, m=[Ft$k : F(k&m)=0]. That is, Fk, m consists of those distribu-tions whose certainty equivalent is k, and whose lowest outcome is not lessthan k&m. On Fk, m the preference relation p satisfies � uk (x) dF=k and� um(x&k+m) dF=m, or equivalently, � um(x&k+m) dF+k&m=k.The reason is that Ft$k iff F&k+mt$m . Consider the two expectedutility preferences on [F: F(k&m)=0] that are represented by � uk (x) dF(x)and � v(x) dF(x), where v(x)=um(x&k+m)+k&m. Since they share anindifference curve (the one that goes through $k), they are the same, hence


v is a linear transformation of uk . Also, since uk (k)=v(k)=k, there exists%>0 such that

v(x)=um(x&k+m)+k&m=%uk (x)+(1&%) k.

Together with Eq. (11), this implies

um(x&k+m)+k&m=%km

um \mxk ++(1&%) k.

Let y=x&k+m and obtain

um( y)=%km

um \my+mk&m2

k ++k&%k. (12)

We want to show that %=1.Since V is Gateaux differentiable, it follows from the proof of Theorem 1

(see Footnote 3) that

�um(x)�x

={ s, x<mt, x>m.

For y{m we obtain

u$m( y)=%u$m \my+mk&m2

k + .

But m+(m�k)( y&m) # ( y, m) (or # (m, y)), hence %=1. This is the case ofdisappointment aversion theory, where u(x)=x and ;=(s�t)&1.

In the Appendix we show that disappointment aversion function withlinear utility function u is Gateaux differentiable. K

Theorem 2 strongly depends on the assumption that the functional isGateaux differentiable. For a functional that satisfies betweenness and isnot disappointment aversion (and therefore, by Theorem 2, is not Gateauxdifferentiable), see Example 1(3), which is taken from Smorodinsky [27].

In disappointment aversion theory, the certainty equivalent of a lotteryserves as a natural reference point, which Gul's axioms explicitly use. Ouraxioms do not refer to any special point, and the reference point isobtained as part of the results of the model, rather than as part of itsassumptions, even if only for a special case of this theory.

28 SAFRA AND SEGAL

5. MIXTURE SYMMETRY

As mentioned in the Introduction, Yaari's dual theory [30], given byV(F )=� x dg(F(x)), satisfies constant risk aversion. This functional is aspecial case of the rank dependent model, V(F )=� u(x) dg(F(x)) where theutility function u: R+ � R and the probability transformation functiong: [0, 1] � [0, 1] are strictly monotonic, g(0)=0 and g(1)=1 (seeWeymark [29] and Quiggin [21]). This family of functionals receivedmany different axiomatizations (e.g., Chew and Epstein [6], Segal [26],Quiggin and Wakker [22], Wakker [28] and Fishburn and Luce [12]),but all these axiomatizations make use of the order of the outcomes. Forexample, Quiggin's axiom 4 of [21] implies expected utility if non-orderedoutcomes are allowed.5

Since rank dependent functionals evaluate outcomes not only by theirvalue, but also by their relative rank as compared to other possible out-comes, axioms that presuppose attitudes that are based on outcomes'relative rank are arguably less convincing than axioms that do not makean explicit appeal to such ranks. The aim of this section is to offer what webelieve to be the first axiomatization of a non-trivial set of rank dependentfunctionals where none of the axioms refers to the order of the outcomes,or treats an outcome differently based on its rank. We will use the followingterms.

Mixture Symmetry. FtG implies for all : # [0, 1], :F+(1&:) Gt

(1&:) F+:G (see Chew, Epstein, and Segal [7]).

Non-betweenness. There exist FtG and : # [0, 1] such that Ft% :F+(1&:) G.

Quasi Concavity�Quasi convexity. FpG implies that for every : # [0, 1],:F+(1&a) GpG�Fp:F+(1&:) G.

Similarly to betweenness, these definitions can be extended to the func-tional V. The next lemma shows that quasi concavity�betweenness�quasiconvexity along one indifference curve implies global quasi concavity�betweenness�quasi convexity. Formally,

Lemma 2. Suppose that p satisfies constant risk aversion. If there isa*>0 such that FtGt$a* implies for all : # (0, 1), (1) :F+(1&:) GoF ;(2) Ft:F+(1&:) G; (3) Fo:F+(1&:) G, then the preference relation(1) is quasi concave; (2) satisfies the betweenness assumption; (3) is quasiconvex, respectively.


5 Miyamoto and Wakker [20] combine the rank dependent model with absolute and,separately, relative risk aversion. They too assume ordering of outcomes.

Proof. We prove the case of betweenness; the other two cases aresimilar. Let FtGt$a for a{0. Then

a*a

_Fta*a

_Gta*a

_$a=$a* .

Hence, by the betweenness assumption,

\:,a*a

_Ft: _a*a

_F&+(1&:) _a*a

_G&O \:,

aa*

_\a*a

_F+ta

a*_\: _a*

a_F&+(1&:) _a*

a_G&+

O \:, Ft:F+(1&:) G.

Since all outcomes are non-negative, Ft$0 implies F=$0 , hence thelemma. K

Theorem 3. The following two conditions on p are equivalent.

(1) It satisfies constant risk aversion, non-betweenness, and mixturesymmetry.

(2) It can be represented by a rank dependent functional with linearutility (that is, Yaari 's dual theory functional [30]) and quadratic probabilitytransformation function of the form g( p)= p+cp&cp2 for some c #[&1, 0) _ (0, 1].

Proof. Obviously, (2) implies (1), so we show that (1) implies (2). Asis proved in [7], mixture symmetry implies that the domain of p can bedivided into three regions A, B, and C, AoBoC, such that on B, psatisfies betweenness, on A and on C, p can be represented by (notnecessarily the same) quadratic functional of the form

V(x1 , p1 ; ...; xn , pn)= :n

i=1

:n

j=1

pi p j.l(xi , x j), l=A, C, (13)

where .l is symmetric, l=A, C. Moreover, V is quasi concave on A andquasi convex on C.

By Lemma 2, only one of the three regions is not empty, and by the non-betweenness assumption, V is quadratic throughout. In other words, p canbe represented by a strictly quadratic function � i � j pi pj.(x i , xj). Definea function v: R+ _[0, 1] � R+ implicitly by

(x, p; 0, 1& p)t(v(x, p), 1). (14)

30 SAFRA AND SEGAL

For *>0, v(*x, p)=*v(x, p). So for a fixed p the function v is homo-geneous of degree 1, hence

v(x, p)=\( p) x. (15)

Assume without loss of generality that .(0, 0)=0. Let q(x) :=.(x, x)and r(x) :=.(x, 0), hence q(0)=0. Then from Eqs. (13) and (14) it followsthat

p2q(x)+2p(1& p) r(x)=q(x\( p)). (16)

In Lemma 3 below we prove that the two functions q and \ are tricedifferentiable. Differentiate both sides three times with respect to p toobtain

0#\$$$( p) xq$(x\( p))+3\"( p) \$( p) x2q"(x\( p))

+[\$( p)]3x3q$$$(x\( p)). (17)

Since this equation holds for every x, it follows that all the coefficientsof x on the right-hand side of Eq. (17) are zero. In particular,\$$$( p) q$(x\( p))#0. By monotonicity, q$>0, hence \ is quadratic. Since byEq. (14), \(0)=0 and \(1)=1, it follows that

\( p)=cp2+(1&c) p. (18)

Also, [\$]3q$$$#0. Since \ is not constant, it follows that q$$$#0, hence qis quadratic. In other words, together with the assumption that q(0)=0, weobtain

.(x, x)=ax2+bx. (19)

From Eqs. (14) and (15) it follows that (x, p; 0, 1& p)t(\( p) x, 1). Byconstant absolute risk aversion we obtain for every x> y

(x& y, p; 0, 1& p)t(\( p)[x& y], 1)

O (x, p; y, 1& p)t(\( p) x+(1&\( p)) y, 1)

O p2.(x, x)+(1& p)2.( y, y)+2p(1&p) .(x, y)

=.(\( p) x+(1&\( p)) y, \( p) x+(1&\( p)) y)

O 2p(1& p) .(x, y)

=a[\( p) x+(1&\( p)) y]2+b[\( p) x+(1& p( p)) y]

& p2[ax2+bx]&(1& p)2 [ay2+by].


Substitute Eq. (18) into this last equality to obtain

(2p&2p2) .(x, y)=a([cp2+(1&c) p] x+[1&cp2&(1&c) p] y)2

+b([cp2+(1&c) p] x+[1&cp2&(1&c) p] y)

& p2[ax2+bx]&(1& p)2 [ay2+by]. (20)

Comparing the coefficients of powers of p on both sides of this last equationwe get for p4

0=ac2(x& y)2.

Hence either a=0 or c=0. Suppose first that c=0, then \( p)= p.Comparing the coefficients of p2 in Eq. (20) we obtain

.(x, y)=axy+b2

(x+ y).

It follows from Eq. (13) that for X(x1 , p1 ; ...; xn , pn),

V(FX)=a :i

:j

pi pjxixj+b2

:i

:j

pi pj (xi+xj)

=a(E[FX])2+bE[FX]

hence V is an expected value functional, but this contradicts the non-betweenness assumption.

On the other hand, if c{0 and a=0, then by Eq. (19), b{0. Bycomparing the coefficients of p2 in Eq. (20) we get for x> y

.(x, y)=b2

[(1&c) x+(1+c) y].

Similarly, for y>x,

.(x, y)=b2

[(1&c) y+(1+c) x].

By the monotonicity of . we may assume, without loss of generality, thatb=1. We thus obtain that

.(x, y)= 12 (1&c)(x+ y)+c min[x, y].

Following Chew, Epstein, and Segal [7, Sect. 3, especially footnote 5], weobtain

V(FX)=(1&c) E[FX]+c | x dg*(FX (x)), (21)

32 SAFRA AND SEGAL

where g*( p)=1&(1& p)2. Alternatively, V(FX)=� x dg(FX (x)), whereg( p)= p+cp&cp2. K

Lemma 3. The functions q and p of the proof of Theorem 3 are tricedifferentiable.

Proof. By monotonicity, both q and p are increasing functions, hencealmost everywhere differentiable. The left hand side of Eq. (16) is alwaysdifferentiable with respect to p, hence so is the right hand side of thisequation. Suppose \ is not differentiable at p*. Since q is almost every-where differentiable, there is x such that q is differentiable at x\( p*), a con-tradiction. Since \ is differentiable, it follows by the differentiability of theleft hand side of Eq. (16) that so is q. Differentiating both sides of Eq. (16)with respect to p we thus obtain

2pq(x)+(2&4p) r(x)=\$( p) xq$(x\( p)). (22)

The right-hand side of Eq. (16) is differentiable with respect to x, and sinceq is differentiable, so is r. It follows that the left-hand side of Eq. (22) isdifferentiable with respect to x, hence so is the right-hand side of thisequation, in other words, q" exists. Since \ is differentiable, �q$(\( p))��pexists, and since the left-hand side of Eq. (22) is differentiable with respectto p, so must be \$( p). In other words, \ is twice differentiable. Differentiatingboth sides of Eq. (22) with respect to p we obtain

2q(x)&4r(x)=[\$( p)]2 x2q"(x\( p))+\"( p) xq$(x\( p)). (23)

Similarly to the above analysis, since both sides of Eq. (23) are differen-tiable with respect to x, q$$$ exists, and since both sides are differentiablewith respect to p, \$$$ exists. K

Since betweenness implies mixture symmetry, and since preferencessatisfy either betweenness or non-betweenness, Theorems 2 and 3 imply thefollowing corollary.

Corollary 1. Let V represent the preferences p. Then the followingtwo conditions are equivalent.

(1) The preferences p satisfy constant risk aversion and mixturesymmetry, and V is a Gateaux differentiable functional.

(2) The preferences p can be represented either by Gul 's disappoint-ment aversion functional with the linear utility u(x)=x or by a dual theoryfunctional with a quadratic probability transformation function of the formg( p)= p+cp&cp2 for some c # [&1, 0) _ (0, 1].


Consider again Eq. (21). When c= &1, V(FX)=� x dg� (FX (x)), whereg� ( p)= p2, and when c=1, V(FX)=� x dg*(FX (x)), which is the Ginimeasure of income inequality. Since \( p) is monotonic (see Eq. (15)), itfollows by Eq. (18) that c # [&1, 1]. Since g* is concave it represents riskaversion (see Yaari [30] and Chew, Karni, and Safra [8]), hence� x dg*(F(x))<E[FX]. In other words, the Gini measure is the lowerbound of all the monotonic functionals that satisfy the assumptions ofTheorem 3.

6. ZERO INDEPENDENCE

Suppose FtG. What will be the relation between qF+(1&q) $0 andqG+(1&q) $0? Some existing empirical evidence suggests that if x> y>0and (x, p; 0, 1& p)t( y, 1), then (x, qp; 0, 1&qp)o ( y, q; 0, 1&q) (this iscalled the common ratio effect��see Allais [2], MacCrimmon and Larsson[18], or Kahneman and Tversky [17]). Assuming the rank dependentmodel, Segal [25, 26] connected this effect to the elasticity of the probabilitytransformation function g, and showed that if qFtG iff F+(1&q) $0 t

qG+(1&q) $0 for all F and G with two outcomes at most, then g( p)=1&(1& p)t for some t>0. Stronger results were achieved by Grant and Kajii[15], who proved the same for a wider set of initially possible functionals.They assume that preferences can be represented by a measure of the epi-graphs of cumulative distribution functions,6 and that if F(z)=G(z)=1,then FpG implies that for all q # (0, 1), qF+(1&q) $z pqG+(1&q) $z .In this section we prove that to a certain extent, it is enough to assumeconstant risk aversion (although in that case the utility function will haveto be linear). The formal assumption we use is the following.

Zero Independence. FtG iff for all q # [0, 1], qF+(1&q) $0 tqG+(1&q) $0 .

Slightly weaker assumptions were introduced by Rubinstein, Safra, andThomson [24], and later by Grant and Kajii [14], for the derivation ofa preference-based Nash bargaining solution that applies to generalizedexpected utility preferences. The outcome of zero is considered there to bethe disagreement outcome (the outcome that the bargainers receive if theyfail to reach an agreement). In [24] the similar assumption is calledhomogeneity and it requires that zero independence should hold forG=$x . In [14] it is called weak homogeneity and it requires that zeroindependence should hold for G=$x and for F= p $y+(1& p) $0 . Theseassumptions play a crucial role in establishing the existence and theuniqueness of the ordinal Nash solution.

34 SAFRA AND SEGAL

6 The rank dependent model is a special case of this family.

To prove Theorem 4 we will modify a result from Gilboa and Schmeidler[13] and assume that preferences satisfy diversification (see below). As inSection 2, let 0=(S, 7, P ) be a measure space and let X be the set of allmeasurable bounded random variables on it.

Diversification. Let X, Y # X. If XtY, then :X+(1&:) YpX for all0<:<1. Equivalently, for G, H # F, if GtH, then for all X and Y suchthat G=FX and H=FY , and for all 0<:<1, F:X+(1&:) Y pG.

Lemma 4. The following two conditions are equivalent.

(1) V satisfies constant risk aversion and diversification.

(2) There exists a unique set T of increasing, concave, and onto functionsover [0, 1] such that p can be represented by

V(FX)=ming # T {| x dg(FX (x))= .

Proof. Clearly (2) implies (1) (see Lemma 1 in Section 2 above), so weprove here that (1) implies (2). First note that for each X # X with certaintyequivalent x, the set [Y: YpX] is a convex cone with a vertex at x. Thereason is that for Xtx, constant risk aversion implies that for all :�0,:X+(1&:) xtx, and diversification implies convexity of upper sets.Constant risk aversion also implies that all these cones are parallel shifts ofeach other.

The conditions in (1) imply the axioms of Gilboa and Schmeidler [13].These authors discuss preference relations over a set of ``horse-lotteries''that is, acts with subjective probabilities whose consequences are (possiblydegenerate) roulette lotteries. They show, in their Theorem 1 and Proposi-tion 4.1, that a preference relation satisfies their axioms if, and only if, itcan be represented as a minimum over a family of subjective expectedutility functionals. Translated into our framework, their results imply theexistence of a compact set C of finitely additive measures on 0 such thatXpY iff minQ # C [� X dQ]�minQ # C [� Y dQ]. For a given X, let F Q

X

denote the distribution function of X with respect to the measure Q (F PX

is denoted by FX , as before) and let V: F � R be defined by V(FX)=minQ # C [� z dF Q

X (z)]. Then FX pFY iff V(FX)�V(FY). The followingclaim explains the relationship between the measures in C and the givenprobability measure P.

Claim 2. Every Q # C is absolutely continuous with respect to P. That is,for all Q and E/S, P(E )=0 O Q(E )=0.


Proof of Claim 2. Let E be such that P(E )=0 and let [z; w] stand forthe random variable [z on E; w on E c] (E c is the complement of E ). Bymonotonicity with respect to first-order stochastic dominance, w1>w2

implies [z1 ; w1]o[z2 ; w2]. Let q=maxQ # C [Q(E )]. If there exists Q # C

such that Q(E )>0, then q>0 and [4; 5]o[5&q; 5&(q�2)].On the other hand, for z�w,

minQ # C

[zQ(E )+w(1&Q(E ))]=minQ # C

[w+(z&w) Q(E )]=w+(z&w) q.

Therefore

minQ # C {(5&q) Q(E )+\5&

q2+ (1&Q(E ))==5&

q2

&q2

2.

On the other hand,

minQ # C

[4Q(E )+5(1&Q(E ))]=5&q.

Hence [5&q; 5&(q�2)]p[4; 5], a contradiction. K

Claim 2 implies that, for all X, Q, z1 , and z2 ,

FX (z1)=FX (z2) O F QX (z1)=F Q

X (z2).

Consider now a given X # X and a measure Q # C, and define a functiongQ, X : [0, 1] � [0, 1] by

F QX (F&1

X ( p)), p # Image(FX)

gQ, X ( p)={0, p=0

l( p), otherwise,

where l is the piece-wise linear function, defined on the complement of theimage of FX , that makes gQ, X continuous. By the claim, gQ, X is welldefined. Clearly, it is onto and non-decreasing, and it satisfies � z dF Q

X (z)=� z dgQ, X (FX (z)).

For each X # X there exists Q(X) # C such that

V(FX)=minQ # C {| z dF Q

X (z)==| z dF Q(X )X (z)=| z dgX (FX (z)),

where gX= gQ(X ), X .Let Com(X )=[Y: \s1 , s2 # S, (X(s1)&X(s2))(Y(s1)&Y(s2))�0] (that

is, the set of all random variables that are comonotone with X ). Restricted

36 SAFRA AND SEGAL

to Com(X ), indifference sets of � z dgZ(FY (z)) are hyperplanes in X. There-fore, for Y # Com(X ),

V(FY)= minZ # Com(X ), ZtY {| z dF Q(Z)

Y (z)== min

Z # Com(X ), ZtY {| z dgZ(FY (z))= .

Next, we discuss the case of non-comonotone random variables. DefineW*: F � R by

W*(FX)= minZ # X, ZtX {| z dgZ(FX (z))= .

By definition, W*(FX)�V(FX). Suppose there exists X such that W*(FX)=x~ <x=V(FX). Then there exists X� � Com(X ) such that W*(FX)=� z dgX� (FX (z)).

Let X9 n be the set of all random variables Y # X that have at most ndifferent outcomes y1� } } } � yn and satisfy Pr(Y= yi)=1�n. Assume firstthat there exist n and X� such that X, X� # X9 n and Fx=FX� . Clearly,Q(X� )=Q(X� ). Therefore, gX� = gX� , which implies W*(FX)=V(FX).

If there is no such n then, by continuity, there exists n large enough forwhich there exist Xn, X� n # X9 n that satisfy W*(FX n)=� z dgX� n(FX n(z))<x~ +1�2(x&x~ ) and V(FX n)>x~ +1�2(x&x~ ), contradiction. Hence, forT=[gZ : Z # X],

V(FX)=ming # T {| z dg(FX (z))= .

It remains to show that the functions gZ are concave. Assume, withoutloss of generality, that there exist n and X # int(X9 n) such that, for somei0 # [2, ..., n&1],

2gX \i0

n +< gX \i0&1)n ++ gX \(i0+1)

n + .

Let Q=Q(X ) and denote qi=Q(X=xi). By definition, gX (i�n)=� ij=1 qj .

Therefore, qi0<qi0+1 . Take =>0 small enough and consider X(=) # int(X9 n)

with the values x1< } } } <x i0&=<xi0+1+=< } } } <xn . By the construction

of Q, X(=)oX. This, however, contradicts risk aversion (note that riskaversion is implied by diversification, see Dekel [10]). K


Theorem 4. The following two conditions on p are equivalent.

(1) It satisfies constant risk aversion, diversification, and zero inde-pendence.

(2) It can be represented by V(F)=� x dg(F(x), where g( p)=1&(1& p)t for some t�1.

Proof. Obviously (2) O (1). We prove that (1) O (2) for finite lotteries(that is, for lotteries with a finite number of different outcomes) by inductionon the number of the nonzero outcomes. Continuity is then used to get thedesired representation for all F # F.

By Lemma 4 there is a family of probability transformation functions[g: : : # A] such that for every F, V(F )=min: � x dg:(F(x)). For lotteriesof the form (x, p; 0, 1& p) we obtain

V(x, p; 0, 1& p)=min:

x[1& g:(1& p)].

Define f:( p)=1& g:(1& p) and h( p)=min: f:( p) and obtain thatV(x, p; 0, 1& p)=xh( p).

By zero independence, (x, p; 0, 1& p)t$y implies for all q # [0, 1],(x, pq; 0, 1& pq)t( y, q; 0, 1&q), hence xh( p)= y and xh( pq)= yh(q).Combining the two we obtain

h( pq)=h( p) h(q).

The solution of this functional equation is h( p)= pt (see Acze� l [1, p. 41]).Suppose we have already proved that for lotteries with at most n prizes

V(F )=� x dg(F(x))=� x d(1&(1&F(x))t)=� x d(&(1&F(x))t). That is,for x1� } } } �xn ,

V(x1 , p1 ; ...; xn , pn)=xnp tn+ :

n&1

u=1

xi _\:n

j=i

p j+t

&\ :n

j=i+1

pj+t

& .

We will now prove it for n+1. Let x1� } } } �xn+1 , and consider thelottery X=(x1 , p1 ; ...; xn+1 , pn+1). By constant risk aversion,

V(FX)=x1+V(FX&x1)=x1+V(0, p1 ; ...; xn+1&x1 , pn+1).

By continuity there is y such that FX&x1 t( y, 1& pi ; 0, p1). By zero inde-pendence FX* t$y , where

X*=\x2&x1 ,p2

1& p1

; ...; xn+1&x1 ,pn+1

1& p1+ .

38 SAFRA AND SEGAL

By the induction hypothesis, V(FX*)=V($y) implies

(xn+1&x1) \ pn+1

1& p1+t

+ :n

i=2

(xi&x1) _\ :n+1

j=i

p j

1& p1+t

&= y. (24)

Also, since FX&x1 t( y, 1& p1 ; 0, p1), it follows by the constant riskaversion properties of V that V(FX)=x1+ y(1& p1)t. Substitute intoEq. (24) to obtain

V(FX)=x1+(1& p1)t {(xn+1&x1) \ pn+1

1& p1+t

+ :n

i=2

(x i&x1) _\ :n+1

j=i

pj

1& p1+t

&\ :n+1

j=i+1

pj

1& p1 +t

&==xn+1p t

n+1+ :n

i=1

x i _\ :n+1

j=i

pj+t

&\ :n+1

j=i+1

pj+t

&which proves the induction hypothesis. K

The functional form of Theorem 4 has been previously appeared in theliterature on income distribution under the name of ``the S-Gini family''(see Donaldson and Weymark [11] and Yitzhaki [31]).

APPENDIX: DISAPPOINTMENT AVERSION THEORY ANDGA� TEAUX DIFFERENTIABILITY

In this appendix we prove that the disappointment aversion with linearutility is Gateaux differentiable. In our case, we assume that V($k)=k.Therefore, for fixed s and t, the value of V at F is the number k that solvesthe implicit equation

|k

0[sx+(1&s) k] dF(x)+|

�

k[tx+(1&t) k] dF(x)&k=0.

For given F and G, let H:=:G+(1&:) F, and obtain that the value ofV(H:) is the number k(:) that solves

|k(:)

0.(x, k(:)) dH:(x)+|

�

k(:)�(x, k(:)) dH:(x)&k(:)=0,


where .(x, k(:))=sx+(1&s) k(:) and �(x, k(:))=tx+(1&t) k(:).Define

J(:, k)=|k

0.(x, k) dH:(x)+|

�

k�(x, k) dH:(x)&k

=|�

0�(x, k) dH:(x)+(s&t) |

k

0(x&k) dH:(x)&k.

By the implicit function theorem, �k��:=&J:�Jk . Trivially,

�J�:

=|k

0.(x, k) d[G(x)&F(x)]+|

�

k�(x, k) d[G(x)&F(x)]. (25)

We compute next the derivative of J with respect to k. We need only thederivative at :=0, in which case H:=F. Therefore, the derivative of J withrespect to k at :=0 is given by

(1&t) |�

0dF(x)+(s&t) lim

= � 0

1= _|

k+=

0(x&k&=) dF(x)&|

k

0(x&k) dF(x)&&1

=&t+(s&t) lim= � 0

1= _|

k+=

0(x&k&=) dF(x)&|

k+=

0(x&k) dF(x)&

+(s&t) lim= � 0

1= |

k+=

k(x&k) dF(x)

=&t&(s&t) lim= � 0

F(k+=)+(s&t) lim= � 0

1= |

k+=

k(x&k) dF(x).

The expression

|k+=

k(x&k) dF(x) (26)

is bounded from above by the maximal change in x&k multiplied by themaximal change in the cumulative probability, that is, by =[F(k+=)&F(k)].The integral at Eq. (26) is bounded from below by = multiplied by the minimalpossible change in the value of F, namely by lim= � 0 F(k+=)&F(k).Hence, =&1 times the integral at Eq. (26) is between lim= � 0 F(k+=)&F(k)and F(k+=)&F(k). It thus follows that

(s&t) lim= � 0

1= |

k+=

k(x&k) dF(x)=(s&t)[ lim

= � 0F(k+=)&F(k)]

40 SAFRA AND SEGAL

and

�J�k } :=0

=&t&(s&t) F(k)=t(1&F(k))&sF(k)<0.

From Eq. (25) it now follows that

�k�: } :=0

=�k

0 .(x, k) d[G(x)&F(x)]+��k .(x, k) d[G(x)&F(x)]

t+(s&t) F(k).

Obviously, this expression is continuous in G&F and also linear inG&F. K

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42 SAFRA AND SEGAL

constant risk aversion - boston college

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