Abstract
Various notions of risk aversion can be distinguished for the class
of rank-dependent expected utility (RDEU) preferences. We provide
the …rst complete characterization of the RDEU orderings that are
risk-averse in the sense of Jewitt (1989). We also extend Chew, Karni
and Safra’s (1987) important characterization of strong risk aversion
(Rothschild and Stiglitz, 1970) by relaxing strict monotonicity and
di¤erentiability assumptions, and allowing for discontinuities in the
probability transformation function. The important special case of
maximin choice falls within this relaxed RDEU class. It is shown that
any strongly risk-averse RDEU order is a convex combination of maximin and another RDEU order with concave utility and continuous,
concave probability transformation. Our proof of the result on strong
risk aversion is also simpler (as well as more general) than that of
Chew, Karni and Safra (1987).
1
Risk Aversion in RDEU
Matthew J. Ryan
February 2006
1
Introduction
The rank-dependent expected utility (RDEU) model is the most widely used,
and arguably the most empirically successful, generalization of expected utility (EU).1 Given a cumulative distribution function F on [0; 1], the RDEU
model evaluates F according to the functional:
Z 1
U (F ) =
u d (h F )
(1)
0
In addition to the utility function u, this expression involves a non-decreasing
transformation function h : [0; 1] ! [0; 1], satisfying h (0) = 0 and h (1) = 1.
The EU special case is obtained if h is linear.
Risk attitude in the RDEU model resides jointly in u and h. In particular,
aversion to risk can be separated from diminishing marginal utility of wealth
(Chateauneuf and Cohen, 1994; Wakker, 1994). Moreover, unlike EU, RDEU
preferences allow us to discriminate amongst di¤erent notions of risk aversion
(Cohen, 1995).
Chew, Karni and Safra (1987, Corollary 2) were the …rst to demonstrate
that risk aversion in the sense of Rothschild and Stiglitz (1970) –also known
Special thanks to Suren Basov, Simon Grant, Warren Moors, John Quiggin and
Arkadii Slinko for valuable comments on earlier drafts. My thanks also to seminar audiences at the Universities of Auckland and Melbourne. The usual disclaimer applies.
1
RDEU began life as anticipated utility (Quiggin, 1982). Yaari (1987) introduced an
important special case which he called the dual theory. References to the Quiggin-Yaari
functional under the rank-dependent nomenclature seem to have begun with Chew, Karni
and Safra (1987). The RDEU (sometimes, RDU) terminology is now …rmly established.
2
as strong risk aversion – imposes straightforward and independent restrictions on u and h: each function must be concave. Chew, Karni and Safra’s
result is an important benchmark in RDEU theory. Unfortunately, their
proof requires strong assumptions on u and h, including strict monotonicity
and di¤erentiability, and employs an argument based on Gâteaux derivatives
–not a standard part of most economists’toolkit.
In Section 3, we extend Chew, Karni and Safra’s result by showing that
it su¢ ces to assume weak monotonicity of u and h, and continuity of u.2 We
also employ a more straightforward method of proof.
Transformation functions that are dis-continuous or non-strictly increasing are more than just an esoteric curiosity. They include all-but-one members of the NEO-additive class of RDEU preferences (Chateauneuf, Eichberger and Grant, 2004). This is a tractable, two-parameter generalization
of EU that o¤ers a more realistic description of human decision-making. The
added realism comes from the fact that all NEO-additive transformations
exhibit the inverse-S shape consistently observed in experimental studies of
risky choice (Wakker, 2001).
The NEO-additive transformation functions are those that are linear on
(0; 1). Functions within this class may be discontinuous at zero or one, and
may be constant on (0; 1). The indicator function for (0; 1], for example, is
a NEO-additive transformation and corresponds to maximin choice.
Under our weaker assumptions on u and h, we show that any strongly
risk-averse RDEU functional is a convex combination of maximin and another
RDEU functional with concave and continuous u and h. In other words,
maximin is the canonical example of a strongly risk-averse RDEU ordering
with a discontinuous transformation function.
More recently, Chateauneuf, Cohen and Meilijson (2005)3 have provided a
characterization of monotone risk aversion (Quiggin, 1991) for RDEU preferences. This is a less demanding notion than strong risk aversion, and imposes
a joint restriction on u and h: non-concavity of the utility function may be
compensated through su¢ ciently pronounced “pessimism”embodied in h.
We adapt the ideas of Chateauneuf, Cohen and Meilijson (2005) in Section
4 to provide a characterization of another important risk aversion concept
–Jewitt’s (1989) notion of aversion to location-independent risk. A random
2
Note that U cannot be continuous with respect to the weak convergence topology
unless h is continuous.
3
This paper …rst appeared in the Cahiers d’Ecomath in 1997.
3
variable X2 is said to exhibit greater location-independent risk than X1 if
the di¤erence
E [X1 j X1
E [X2 j X2
x1 (p)]
x2 (p)]
is non-increasing in p 2 (0; 1), where xi (p) denotes the (100p)th percentile of
Xi . This notion was originally motivated by expected utility considerations,
but has proved useful beyond the EU context. For example, Jewitt’s is
the weakest notion of risk aversion consistent with Arrow’s famous theorem
on the optimality of deductible insurance: see Landsberger and Meilijson,
(1994b, p.664), Vergnaud (1997) and Chateauneuf, Cohen and Vergnaud
(2001). Ours is the …rst characterization of the RDEU preferences that are
risk averse in this sense.
The following section presents some preliminary ideas. The RDEU model
is described …rst. We then recall four notions of risk aversion, and show that
no two are equivalent within the RDEU class. For each notion of risk aversion, we also describe a class of “elementary transformations”of distribution
functions that add risk of the appropriate sort. These elementary transformations are essential to the proofs that follow. Section 3 proves our result
on the characterization of strong risk aversion. Results for Jewitt’s locationindependent notion are in Section 4. Section 5 o¤ers concluding remarks.
Less informative proofs are relegated to an Appendix.
2
2.1
Preliminaries
Rank-Dependent Expected Utility (RDEU)
Let D denote the set of all cumulative distribution functions (CDF’s) on the
unit interval [0; 1]. Capital letters, such as F and G, will be used to denote
elements of D, and the same letters with an overbar –F , G, and so forth –
to denote the corresponding decumulative distribution functions (DDF’s). If
c 2 [0; 1] then c 2 D is the indicator function for [c; 1]. For any F 2 D we
use (F ) to denote the mean of F .
We also de…ne, for any F 2 D, the following “inverses”:
F
F
1
1
(p) =
(p) = F
1
(1
inf fx j F (x) > pg if p < 1
sup fx j F (x) < pg if p = 1
p) =
inf x j F (x) < p
sup x j F (x) > p
4
if p > 0
if p = 0
It is useful to observe that if F 1 is interpreted as a random variable on
[0; 1], endowed with the usual Borel -algebra and Lebesgue measure, then
its CDF is F .
An RDEU preference ordering % on D is such that:
Z 1
Z 1
F %G ,
u d (h F )
u d (h G)
(2)
0
0
for some utility function u : [0; 1] ! [0; 1] and CDF transformation function
h : [0; 1] ! [0; 1]. Throughout the paper, we shall maintain the assumptions
that u and h are non-decreasing and normalized: u (0) = h (0) = 0 and
u (1) = h (1) = 1. We also assume that u is continuous (and hence surjective)
to ensure that the Riemann-Stieltjes integrals in (2) exist (Wheeden and
Zygmund, 1977, Theorem 2.24).
Let us pause to consider an important special case of (2) in which h is
the indicator function for (0; 1]. Then:
F %G
,
u F
1
(0)
u G
1
(0)
(3)
In other words, each distribution is assigned a value equal to the in…mum of
utility over outcomes in its support. If u is strictly increasing, this is maximin
choice.
For our purposes, it is convenient to re-express the functional (1) in an
equivalent form. To do so, we …rst de…ne the following dual to h:4
h (z) = 1
h (1
z) .
Thus, h : [0; 1] ! [0; 1] is also non-decreasing and normalized. We observe,
using integration by parts (ibid., Theorem 2.21), that
Z 1
Z 1
u d (h F ) =
h F du.
0
0
For example, if F is the CDF of a discrete random variable with values
x1 < x 2 <
then
Z
0
where x0 = 0.
4
1
u d (h F ) =
n
X
h
< xn
F (xi 1 ) [u (xi )
u (xi 1 )]
i=1
Chateauneuf, Cohen and Meilijson (2005) call h the probability perception function.
5
2.2
Notions of risk aversion
Recall the notion of risk aversion proposed by Rothschild and Stiglitz (1970),
also known as strong risk aversion (Cohen, 1995). We say that F second
order stochastically dominates (SOSD) G if
Z x
F G dz
0
0
for every x 2 [0; 1].
De…nition 1 Preferences % on D exhibit strong risk aversion if F % G
whenever F and G have the same means and F SOSD G.
This is the strongest of the extant risk aversion concepts. It is also immediate from (3) and the de…nition of SOSD that:
Lemma 2 If h is the indicator function for (0; 1], then the RDEU ordering
(2) exhibits strong risk aversion.
We next recall four alternative notions of risk aversion. Say that F is less
location-independent risky than G (F LLIR G) if the di¤erence
E F
1
1
jF
F
1
(p)
E G
1
jG
1
G
1
(p)
is non-increasing in p 2 (0; 1). This is equivalent to
F
1
(p)
Z
F
1
(p)
F (z) dz
G
0
1
(p)
Z
G
1
(p)
G (z) dz
(4)
0
for all p 2 (0; 1).5 Chateauneuf, Cohen and Meilijson (2004, §2.3.4) show
that the LLIR order on equal-mean distributions is important in optimal
search problems, such as a buyer who pays a …xed search cost per seller
sampled while searching for the best price. Assuming that the buyer follows
5
I have expressed all de…nitions in terms of the DDF’s for comparison with Figure 1.
But Jewitt’s notion is easier to parse in its equivalent CDF version (Jewitt, 1989, Theorem
3):
Z F 1 (p)
Z G 1 (p)
F (z) dz
G (z) dz.
0
0
6
an optimal stopping rule, the expected total cost (expected price plus search
cost) and the expected search cost are both monotone with respect to LLIR: if
sellers’prices are independent random draws from G then the expected total
cost (respectively, search cost) is lower (respectively, higher) than if prices
are drawn from F , where F LLIR G.
Jewitt (1989) introduced the following:
De…nition 3 Preferences % on D exhibit Jewitt risk aversion if F % G
whenever F and G have the same means and F LLIR G.
We de…ne F to be less Bickel-Lehmann dispersed than G (F LBLD G) if
1
G
F is non-increasing on (0; 1). The following notion is due to Quiggin
(1991) and Landsberger and Meilijson (1994a):
1
De…nition 4 Preferences % on D are monotone risk averse if F % G
whenever F and G have the same means and F LBLD G.
The …nal two concepts are based on the idea that one may increase risk by
splitting an atom while preserving the mean. They do not de…ne increases in
risk starting from any atomless distribution. First is the well-known concept
of weak risk aversion, which requires that any distribution G is weakly dispreferred to the distribution that places unit mass on the mean of G.
De…nition 5 Preferences % on D exhibit weak risk aversion if
(G)
% G.
Finally, we have the less familiar conditional certainty equivalent risk aversion.
De…nition 6 Preferences % on D exhibit conditional certainty equivalent risk aversion if pF + (1 p) (G) % pF + (1 p) G for any F; G 2 D
and any p 2 [0; 1].
To understand the di¤erences between these concepts, it is useful to consider the discrete CDF’s F and G whose associated DDF’s are depicted in
Figure 1. The bold lines describe G and the lighter lines F . The di¤erence
in means
Z
Z
x dF
x dG
is equal to the area of region A less that of region B. As indicated in Figure
1, we assume
(a a) (
) = b b
(5)
so that both distributions share the same mean.
7
1
a
A
a
b
B
b
α
α
β
β
1
(α −α )(a − a )= (β − β )(b − b )
Figure 1: DDF’s F and G (bold) with equal means
Notice that F has more weight concentrated near the “middle” of the
distribution than G: moving from the random variable F 1 to the random
variable G 1 , weight is shifted from to and from to . When should we
regard F 1 as “less risky” than G 1 , and hence require that F % G for any
risk-averse decision-maker? This depends on which notion of risk aversion
we adopt.
For example, it is clear that F SOSD G in Figure 1. We say that G is
an elementary mean-preserving spread (MPS) of F .6 More precisely, let us
de…ne
M
a; a; b; b; ; ; ;
=
0
b<b
0
<
<
1;
1 and (5) .
a<a
Then G is an elementary MPS of F if F; G 2 D are …nite step functions and
there is some a; a; b; b; ; ; ;
2 M with F , G constant on [ ; ) and
6
Compare Rothschild and Stiglitz (1970, p.229) and Dasgupta, Sen and Starrett (1973,
Lemma 2(iii)). Hong and Hui (1995, p.413) de…ne this term slightly di¤erently, requiring
that all probabilities are rational and
=
.
8
;
, and
F (x)
8
< a
G (x) =
b
:
a if x 2 [ ; )
b if x 2 ;
0
otherwise
(6)
It is straightforward to observe that F SOSD G whenever G is an elementary
MPS of F . Thus, a necessary condition for strong risk aversion is aversion
to elementary MPS’s: that is, F % G whenever G is an elementary MPS of
F . In fact, this condition is also su¢ cient –see Theorem 10.
Returning to Figure 1, F LLIR G i¤ a = 1, since otherwise condition (4)
is violated for p 2 (a; a). Say that G is an elementary Jewitt mean-preserving
spread (JMPS) of F if G is an elementary MPS of F and (6) holds for some
2 M with a = 1. The concept of an elementary JMPS
a; a; b; b; ; ; ;
strengthens that of an elementary MPS by requiring that the worst possible
outcome become even worse. A necessary condition for Jewitt risk aversion
is aversion to elementary JMPS’s: F % G whenever G is an elementary
JMPS of F . Once again, this condition turns out to be su¢ cient as well –
see Theorem 20.
For F LBLD G in Figure 1 we need both a = 1 and b = 0. Say that G is an
elementary mean-preserving out-stretch (MPOS) of F if G is an elementary
2 M with a = 1 and
JMPS of F and (6) holds for some a; a; b; b; ; ; ;
b = 0.7 In this case, not only must the worst outcome become worse, but the
best outcome must also become better. Therefore, a necessary condition for
monotone risk aversion is aversion to elementary MPOS’s: F % G whenever
G is an elementary MPOS of F .
For F in Figure 1 is the indicator for [0; (G)) i¤ G is an elementary
MPOS of F and = . It follows that a necessary condition for weak risk
aversion is that F % G whenever G is an elementary MPOS of F and (6)
holds for some a; a; b; b; ; ; ;
2 M with a = 1, b = 0 and = .
Finally, a necessary condition for conditional certainty equivalent risk
aversion is that F % G whenever G is an elementary MPS of F and (6) holds
for some a; a; b; b; ; ; ;
2 M with = .
The discussion of Figure 1 suggests a clear ordering amongst the …rst
four risk aversion concepts. This impression is con…rmed by the following
Proposition, whose proof may be found in the Appendix:8
7
Compare Chateauneuf, Cohen and Meilijson (2005, p.662).
The content of Proposition 7 is well known. However, proofs of its various parts are
scattered through the literature, so a proof is included here for completeness and ease of
8
9
Proposition 7 The following implications hold:
% exhibit strong risk aversion
+
% exhibit Jewitt risk aversion
+
% exhibit monotone risk aversion
+
% exhibit weak risk aversion
However, none of the converse implications holds within the RDEU preference
class.
Only by admitting non-linear transformation functions can we distinguish
these various notions of risk aversion. Within the class of expected utility
preferences, all four notions are equivalent. This follows, for example, from
Proposition 7 and Rothschild and Stiglitz (1970, Theorem 2).
Since weak risk aversion is clearly implied by conditional certainty equivalent risk aversion, weak risk aversion is the weakest of the …ve notions. We
shall later show that conditional certainty equivalent risk aversion is identical
to strong risk aversion for RDEU preferences (Corollary 13).9
Finally, let us observe that F % G in Figure 1 if and only if
Z 1
Z 1
h G du,
h F du
0
0
which is equivalent to
u( )
u( )
h (a)
a
u
h (a)
a
u
!
h
b
b
h (b)
b
!
(7)
given the equal-mean condition (5). We therefore obtain:
Proposition 8 The RDEU preferences (2) exhibit
(i) aversion to elementary MPS’s i¤ (7) for all a; a; b; b; ; ; ;
2 M;
reference.
9
See also Theorem 2 in Hong and Hui (1995), which implies this result for the case of
u and h continuous and strictly increasing.
10
(ii) aversion to elementary JMPS’s i¤ (7) for all a; a; b; b; ; ; ;
with a = 1;
2M
(iii) aversion to elementary MPOS’s i¤ (7) for all a; a; b; b; ; ; ;
with a = 1 and b = 0; and
2M
(iv) weak risk aversion only if (7) for all a; a; b; b; ; ; ;
a = 1, b = 0 and = .
2 M with
(v) conditional certainty equivalent risk aversion only if inequality (7) holds
2 M with = .
for all a; a; b; b; ; ; ;
Propositions 7 and 8 are fundamental to the arguments that follow. We
…rst use Proposition 8(iv) to establish the following useful fact:
Lemma 9 Unless h is the indicator on (0; 1], a necessary condition for the
preferences % in (2) to exhibit weak risk aversion is that there exists some
x^ 2 (0; 1] with u strictly increasing on [0; x^] and u (^
x) = 1.
Proof. See the Appendix.
Lemma 9 and Proposition 7 show that risk aversion of any sort imposes
strict monotonicity of utility up to its maximum value, unless preferences are
of the maximin variety (3).
3
Strongly risk-averse RDEU preferences
The main result of this section is that the strongly risk averse RDEU orderings are precisely those with u and h concave, or h the indicator for (0; 1]
(Corollary 12). To clarify the underlying logic, we argue in two steps. First,
we prove the result under a slight strengthening of our assumptions on u,
which supposes the existence of some x 2 (0; 1) at which u is di¤erentiable
with u0 (x) > 0. In general, continuity, monotonicity and surjectivity do
not su¢ ce for this – recall the Cantor-Lebesgue function. Next, we show
that strong risk aversion implies a strengthening Lemma 9: the derivative of
u must be strictly positive at every point of di¤erentiability in (0; x^). The
main result then follows as a corollary.
11
Theorem 10 Suppose there exists an x 2 (0; 1) at which u is di¤erentiable
with u0 (x) > 0. Then the preferences % in (2) exhibit strong risk aversion i¤
(i) h is concave, and (ii) either u is concave or h is the indicator for (0; 1].
This result shows that strong risk aversion imposes a good deal of continuity and monotonicity on the transformation function. Concavity of h
excludes discontinuities at any x > 0, and further implies that h is strictly
increasing up to its maximum value.
Theorem 10 also reveals that maximin choice is fundamental to strong
risk aversion. Since h can be discontinuous only at zero, we may write:
Z 1
u d (h F ) =
0
(
1
u (F
h (0+) u (F
1
(0)) + [1
(0))
h (0+)]
where
R1
0
if h (0+) = 1
~ F
ud h
if h (0+) < 1
(8)
h (0+) = lim h (z)
z#0
and
~ (z) =
h
(
h(z) h(0+)
1 h(0+)
0
if z 2 (0; 1]
.
if z = 0
~ is continuous. Thus, any strongly risk-averse RDEU ordering is
Note that h
a convex combination of maximin and a strongly risk-averse RDEU ordering
with a continuous (and concave) probability transformation function.
Proof of Theorem 10. We prove Theorem 10 in two steps: (I) we …rst
show that % exhibit aversion to elementary MPS’s i¤ (i) and (ii) hold; and
then (II) we show that aversion to elementary MPS’s su¢ ces for strong risk
aversion.
Aversion to elementary MPS’s i¤ (i) and (ii).
Su¢ ciency is obvious from Proposition 8. For the necessity, let x 2 (0; 1)
be such
that u0 (x) exists and u0 (x) > 0. Consider sequences f n g1
n=1 and
n
o
n 1
n
n
" x,
in [0; 1] such that
# x and
n=1
(a
a) (x
n
) = b
12
b
n
x
for each n.10 Then
u (x)
x
u(
n
n
)
h (a)
a
0
h (a)
a
@
for each n. Taking limits:
h (a)
a
h
h (a)
a
n
u
u (x)
n
b
x
1
A
h
b
b
h (b)
b
h (b)
b
b
for any a < a b < b. But this just says that h is convex, and hence h is
concave. Therefore, aversion to elementary MPS’s implies (i).
We may apply a similar argument to show u concave if there is some
z 2 (0; 1) at which h0 (z) exists and h0 (z) > 0. But concavity of h ensures that
this is so, unless h is the indicator for (0; 1]. See this as follows. Concavity
entails that h is strictly increasing up to its maximum value, and is Lipschitz
continuous on any closed subinterval of (0; 1) –Wheeden and Zygmund (1977,
Theorem 7.43). Therefore, either h (x) = 1 for all x > 0, or there exists a
subinterval [x; y] (0; 1) with h (x) < h (y) and some z 2 (x; y) at which h
is di¤erentiable and h0 (z) > 0 (ibid., Theorem 7.43).
Aversion to elementary MPS’s implies strong risk aversion.
If h is the indicator for (0; 1], then preferences exhibit strong risk aversion
by Lemma 2. Suppose, therefore, that h is not the indicator for (0; 1].
Rothschild and Stiglitz (1970, Lemmas 1 and 2) show that F and G are
n 1
the L1 -limits of sequences fF n g1
n=1 and fG gn=1 of CDF’s with …nite ranges,
such that Gn is obtained from F n by a (…nite) sequence of elementary MPS’s.
However, potential discontinuity of h at zero dictates the need for extra care
when inferring F % G, since standard arguments based on the continuity of
integrals no longer apply.11
10
That is:
n
n
+ (1
)
(a
(a a)
a) + b
= x
for each n, where
=
b
.
11
Since maximin preferences are not continuous with repect to the weak convergence
topology, we cannot apply the technique in Hong and Hui (1995, §4) either, as their Lemma
1 requires continuous preferences.
13
!
~ is continuous:12
Recall the decomposition (8). Since h
Z 1
Z 1
n
~ F
~ F du.
h
du !
h
0
and
Z
0
1
~
h
G
n
0
du !
Z
1
~
h
G du.
0
It is also obvious from Rothschild and Stiglitz’s construction of fF n g1
n=1 and
n 1
n 1
1
n 1
1
fG gn=1 that (F ) (0) ! F (0) and (G ) (0) ! G (0). Thence,
using (8) and the continuity of u:
Z 1
Z 1
n
h F du
h F du !
0
0
and
Z
1
h
G
n
0
Since
Z
1
h
F
n
du !
Z
du
Z
1
h
G du.
h
G
0
1
n
du
0
0
for each n, the desired result follows.
Lemma 11 Let x^ be as de…ned in Lemma 9. Unless h is the indicator for
(0; 1], a necessary condition for the preferences % in (2) to exhibit strong risk
aversion is that there not exist any z 2 (0; x^) at which u is di¤erentiable and
u0 (z) = 0.
Proof. See the Appendix.
Since u is monotone, it is di¤erentiable almost everywhere (Wheeden and
Zygmund, 1977, Corollary 7.23). Lemma 11 implies that its derivative is
strictly positive at every point of di¤erentiability in (0; x^). We may now
immediately deduce the main result of this section:
12
Recall that u is strictly increasing on [0; x
^] with u (^
x) = 1. We may therefore treat du
as a measure on the Borel subsets of [0; 1] that is absolutely continuous with respect to
the uniform measure on the Borel subsets of [0; x
^].
14
Corollary 12 The preferences % in (2) exhibit strong risk aversion i¤ (i) h
is concave, and (ii) either u is concave or h is the indicator for (0; 1].
We also obtain:
Corollary 13 The preferences % in (2) exhibit strong risk aversion i¤ they
exhibit conditional certainty equivalent risk aversion.
Proof. The “only if” part is obvious. For the “if” part, note that, in
the proving the equivalence of aversion to elementary MPS’s and (i) and
(ii) in Theorem 10, we used = = x. Therefore, the same proof shows
that the necessary condition for conditional certainty equivalent risk aversion
(Proposition 8(v)) implies (i) and (ii). Hence conditional certainty equivalent
risk aversion implies strong risk aversion.
So far as we are aware, Chew, Karni and Safra (1987, Corollary 2) is
the …rst characterization of strong risk aversion in the RDEU context. Their
result assumes u and h are strictly increasing and di¤erentiable. This ensures
the Gâteaux di¤erentiability of the RDEU functional; a fact which Chew,
Karni and Safra exploit in their proof. Our proof makes no appeal to such
exotica. Indeed, if we had assumed di¤erentiability of u and h, our proof
would become very straightforward. The only technical complexities arise
from the need to accommodate potential discontinuities in h and potential
lack of absolute continuity of u. But apart from an appeal to results on
Lipschitz continuity at one point, even these di¢ culties are overcome entirely
by consideration of the fundamental inequality (7).
After completing a …rst draft of our proof, we learned of three related results. Grant and Kajii (1994, Proposition 2) characterize strong risk aversion
for their AUSI-EU model, in which h (z) = z with > 0. The published
version of their paper – Grant and Kajii (1998) – makes mention of this
result, but excludes the proof.
Schmidt and Zank (2002, Theorem 1) characterize strong risk aversion
for cumulative prospect theory (a generalization of RDEU), assuming strictly
increasing utility and probability transformation functions. Also, Schmidt
and Zank restrict attention to random variables with …nitely many outcomes.
Nevertheless, their arguments share many common features with ours.
Finally, Hong and Hui (1995) state a result similar to Theorem 10, but under the stronger assumptions that both u and h are continuous and strictly
15
increasing. Their proof, too, has many similarities to ours. However, the
proof of their result is actually incomplete (ibid., Appendix 2). They implicitly assume that u and h (denoted v and g respectively in their paper)
each have a strictly positive derivative somewhere on the interior of their domains. However, as the Cantor-Lebesgue function illustrates, this need not
be so, even for continuous, strictly increasing and surjective functions.13 Our
Lemma 11 and the third paragraph in the proof of Theorem 10 provide the
necessary auxiliary arguments.
We may use Corollary 12 to obtain the following characterization of the
SOSD relation between equal-mean distributions.
Corollary 14 Suppose that F and G have equal means. Then F SOSD G
i¤
Z 1
Z 1
u d (h F )
u d (h G)
0
0
for any concave, non-decreasing and continuous u : [0; 1] ! R and any
concave, non-decreasing h : [0; 1] ! [0; 1] with h (0) = 0 and h (1) = 1.
Proof. The “only if” part follows directly from Corollary 12. In particular, any non-constant utility function can be normalized by a positive a¢ ne
transformation; and the result is trivial for constant utility functions.
Conversely, suppose
Z x
Z x
G dz
F dz >
0
0
for some x 2 (0; 1). Taking h to be the identity function and
u (z) =
we …nd that G F since
Z 1
Z
u dF =
0
z if z < x
x if z x
x
F dz <
Z
0
0
x
G dz =
Z
1
u dG.
0
The following analogues of Corollary 14 are well-known.
13
An early draft of the present paper fell into the same error. We are most grateful to
Suren Basov for having pointed it out.
16
Proposition 15 If F and G have equal means, then F SOSD G i¤
Z 1
Z 1
u dG
u dF
0
0
for any concave, non-decreasing and continuous function u : [0; 1] ! R.
Proof. The “only if” direction is provided by Corollary 14. The converse
follows by the same argument as in the proof of Corollary 14.
Proposition 16 If F and G have equal means, then F SOSD G i¤
Z 1
Z 1
z d (h G)
z d (h F )
0
0
for any concave, non-decreasing function h : [0; 1] ! [0; 1] that satis…es
h (0) = 0 and h (1) = 1.
Proof. The “only if” direction again follows from Corollary 14. For the
converse, if
Z x
Z x
F dz >
G dz
0
for some x 2 (0; 1), set
h (z) =
and observe that G
0
8
< 0
:
z
x
if z
(1 x)
x
1
x
if z > 1
x
F (with u the identity function).
Proposition 15 is familiar to economists from the work of Rothschild and
Stiglitz (1970). Proposition 16 is a less well-known result of Hardy, Littlewood and Polya (1929, Theorem 10).14 It is used by Yaari (1987, Theorem 2)
to show that RDEU preferences exhibit strong risk aversion when u is linear
and h is concave, non-decreasing.15
14
In fact, Hardy, Littlewood and Polya’s result makes no reference to h being normalized
–which is clearly redundant –but does make the further assumption that h is continuous.
The latter may also be dropped.
15
Yaari also makes the assumption that h is continuous, since his axioms ensure that
such exists.
17
4
RDEU and Jewitt risk aversion
Chateauneuf, Cohen and Meilijson (2005) characterize monotone risk aversion for u and h strictly increasing. Adapting aspects of their argument, we
here provide an analogous characterization of the RDEU orders that are risk
averse in the sense of Jewitt (1989). However, our proof does not require
strict monotonicity of u or h.
We begin with some de…nitions. Let
P^h =
1
inf
0
0<z <z<1
and
Qxu^ =
u
sup
<
<
h (z)
z
h (z)
1 z
u
x
^
!
h (z 0 )
z0
u( )
u( )
.
When x^ = 1, Chateauneuf, Cohen and Meilijson (2005) refer to Qxu^ as an
index of “greediness”for u.16 It measures the extent of non-concavity of the
utility function. Note that Qxu^
1 when u is strictly increasing on [0; x^],
with equality precisely when u is concave.
The quantity P^h is a variant on
Ph =
1
inf
0<z<1
h (z)
1 z
h (z)
z
which Chateauneuf, Cohen and Meilijson (2005) describe as an index of “pessimism” for h . Consider, for example, a two-outcome lottery whose better
outcome occurs with probability z. Then
1
h (z)
1 z
>
h (z)
z
implies a pessimistic transformation of the odds ratio: the relative weighting
of the bad outcome is increased. For example, if h is the indicator for f1g,
corresponding to maximin preferences, then Ph = 1 so h is maximally
“pessimistic”. If we call
Ph (z) =
1
h (z)
1 z
16
h (z)
z
(9)
This index is denoted Gu in their paper. We have avoided this notation because of
potential confusion with a CDF.
18
the degree of pessimism at z, then monotone risk aversion requires that
the degree of pessimism over all z 2 (0; 1) is bounded below by Qxu^
1
(Chateauneuf, Cohen and Meilijson, 2005, Theorem 1).
Suppose, for example, that Ph < 1, which implies Ph (z) < 1 for some
z 2 (0; 1). Then the RDEU preferences (2) cannot be monotone risk averse
(or even weakly risk averse),17 and it is easy to see why by considering Figure
1. Let a = 1, b = a = z, b = 0 and = = x. In this case, the random
variable G 1 shifts weight from x and distributes it in a mean-preserving
fashion over and . If u were linear, then Ph (z) < 1 would imply a strict
preference for G over F . But even if u is not linear, provided we can …nd
some x 2 (0; 1) at which it is di¤erentiable with u0 (x) > 0, then choosing
and close enough to x delivers the same conclusion.
Now consider P^h . It is obvious that P^h
Ph . In particular, we may
18
^
have Ph < 1 < Ph . We show below that aversion to location-independent
risk requires P^h
Qxu^ , which is a more demanding requirement than that
for monotone risk aversion (as we should expect). For example, if h is
linear in a neighbourhood of 1, then P^h
1 so it is necessary for u to be
concave in order for preferences to be Jewitt risk averse. This is not the case
for monotone risk aversion – see Chateauneuf, Cohen and Meilijson (2005,
Example 2).
More generally, we can see that P^h < 1 is inconsistent with Jewitt risk
aversion as follows. Suppose
1
1 h (z)
<
0
h (z) h (z )
z
z
z0
for some z; z 0 with 0 < z 0 < z < 1. Setting a = 1, b = a = z, b = z 0 and
= = x in Figure 1, we infer a strict preference for G over F when u is
linear, and hence a violation of Jewitt risk aversion. Since Lemma 19 ensures
the existence of x 2 (0; 1) at which u is di¤erentiable with u0 (x) > 0, then we
may again choose and close enough to x to arrive at the same conclusion.
The following lemmata will be useful for proving the main results of this
section (Theorems 20 and 21).
17
18
This fact is also demonstrated by Chateauneuf and Cohen (1994).
For example, if
8
0
if z < 12
<
h (z) =
: 3
1
if z 12
2z
2
then Ph =
3
2
while P^h = 0.
19
Lemma 17 [Chateauneuf, Cohen and Meilijson (2005, Lemma 1)] If u is
strictly increasing on [0; x^], then
!
u
u
u( ) u( )
Qxu^ =
sup
<
(
for any
<
)
(
x
^
)=
> 0.
Lemma 18 If h (x) =x is non-increasing on (0; 1], then
P^h =
1
inf
0
b<b
a<1
h (a)
1 a
h
b
b
h (b)
b
!
.
Lemma 19 Unless h is the indicator on (0; 1], a necessary condition for the
preferences % in (2) to exhibit Jewitt risk aversion is that there not exist any
z 2 (0; x^) at which u is di¤erentiable and u0 (z) = 0.
We these preliminaries in hand, we may proceed to the characterization
of Jewitt risk aversion. As an intermediate step, we …rst demonstrate the
following theorem, which is of independent interest.
Theorem 20 The preferences % in (2) exhibit Jewitt risk aversion if and
only if they exhibit aversion to elementary JMPS’s.
Proof. The “only if”part is Proposition 8(ii), so we consider the “if”direction here.
The argument is more complex than the corresponding part of the proof
of Theorem 10 for two reasons. First, suppose F n and Gn are …nite step
functions with equal means, such that F n LLIR Gn . Because of the need to
anchor a = 1, it will not in general be possible to transform F n into Gn by
a sequence of elementary JMPS’s. Second, even if we can express F and G
as the limits of …nite step functions, because of potential discontinuities in
~ in (8) may
h we need to modify the convergence argument. In particular, h
no longer be continuous.
Suppose that F LLIR G. Landsberger and Meilijson (1994, Theorem
1) guarantees the existence of a sequence fFm g1
m=0 such that: F0 = F ,
1
fFm (x)gm=0 converges to G (x) at every point x at which the latter is continuous, and Fm is a mean-preserving left stretch of Fm 1 (ibid., De…nition 1)
20
for every m > 0. For our purposes, the essential content of the last statement
is that, for each m > 0, Fm has the same mean as Fm 1 and Fm 1 crosses
Fm 1 1 at most once in (0; 1) (ibid., Lemma 1).
We …rst show that aversion to elementary JMPS’s implies Fm 1 % Fm for
each m > 0. To do so, we adapt arguments from Chateauneuf, Cohen and
Meilijson (2005).
Fix some m > 0. It is easy to see that we may obtain Fm 1 and Fm
1
(respectively) as the L1 -limits of sequences Fmn 1 n=1 and fFmn g1
n=1 of CDF’s
n
with …nite ranges such that, for each n, Fm 1 has the same mean as Fmn and
1
Fmn 1
crosses (Fmn ) 1 at most once in (0; 1). We now show that aversion
to elementary JMPS’s implies Fmn 1 % Fmn for each n.
Fix n 1. We may assume, without loss of generality, that Fmn 1 and Fmn
are non-identical. Because Fmn 1 and Fmn have …nite ranges, we may write
Z
1
h
n
Fm 1
Z
du
1
n
F m du =
h
0
0
s
X
h (pi 1 )] u xm
i
[h (pi )
1
u (xm
i )
(10)
i=1
with s > 1,
0 = p0 < p 1 <
< ps
1
n
1
1
n
< ps = 1
1
(pi ) and xm
(pi ). Furthermore, since Fmn 1
and xm
= Fm 1
i = Fm
i
and Fmn have equal means and cross only once in (0; 1), there also exists
k 2 f1; 2; :::; s 1g such that
xm
i
1
xm
i
0
xm
j
1
xm
j
and
u xm
i
whenever i
1
u (xm
k )
u xm
j
k < j. We may therefore choose
u( )
u( )
u xm
i
1
u (xm
i )
u xm
j
1
u xm
j
<
1
<
such that
for all j > k
and
u
u
21
for all i
k.
Hence, from (10) we have
Z 1
n
h Fm
Z
du
1
0
[1
n
h
F m du
0
h (pk )] [u ( )
Let
1
u ( )]
(
h (pk ) u
)
=
(11)
u
.
In virtue of Lemma 17 it is without loss of generality to suppose that
=
pk
.
(1 pk )
Therefore, aversion to elementary JMPS’s implies
[1
h (pk )] [u ( )
u ( )]
h (pk ) u
u
n
which, together with (11), gives the desired result: Fmn n
1 % Fm .
Fmn 1
Next, consider the sequences of random variables19
1
1
d
d
0,
1
o1
and
n=1
d
(Fmn ) 1 n=1 . Since Fmn 1
! Fm 1 1 and (Fmn ) 1 ! Fm 1 (where “!”
denotes convergence in distribution) and u is continuous, it follows that
u
Fjn
1 d
! u Fj
1
for each j 2 fm 1; mg (Rao, 1973, p.124). Therefore (Denneberg, 1994,
Proposition 8.9):
Z
Z
Z
Z
1
1
n
n
u d h Fj =
u Fj
dh !
u Fj
dh =
u d (h Fj )
for j 2 fm 1; mg, and hence Fm 1 % Fm .
To recap: we have shown that aversion to elementary JMPS’s implies
F % Fm for each m > 0. It remains only to show that
Z
Z
u d (h Fm ) !
u d (h G) .
19
These should all be considered as random variables on [0; 1] endowed with the Borel
-algebra and Lebesgue measure.
22
But this follows by the same sort of argument as we deployed in the preceding
paragraph. This completes the proof.
We can now prove:
Theorem 21 The preferences % in (2) exhibit Jewitt risk aversion only if
h (x)
x
is non-increasing for x 2 (0; 1]. A necessary and su¢ cient condition is that
EITHER (i) h is the indicator on (0; 1]; OR OTHERWISE (ii) there exists
x^ 2 (0; 1] with u strictly increasing on [0; x^] and u (^
x) = 1, and P^h
Qxu^ .
Corollary 22 When u is concave, the preferences % in (2) exhibit Jewitt
risk aversion if and only if
h (x)
x
is non-increasing on (0; 1]. If h is the identity, then u concave is both necessary and su¢ cient.
Proof of Theorem 21. Given Theorem 20, it su¢ ces to show that the
preferences exhibit aversion to elementary JMPS’s i¤ EITHER (i) OR OTHERWISE (ii).
Given Lemma 19, we may argue as in the proof of Theorem 10 to deduce
the necessity of
h (b)
h b
1 h (a)
1 a
b b
whenever 0
b<b
a < 1. This is equivalent to
1
h (x)
1 x
(12)
non-decreasing in x on [0; 1), which is in turn equivalent to h (x) x nonincreasing on (0; 1]. Using Lemmas 2 and 9 we further deduce the necessary
condition that u be strictly increasing up to its maximum value, unless h is
the indicator for (0; 1].
Let us therefore exclude the trivial case (i), and assume that u is strictly
increasing up to its maximum value, which is reached at x^. It remains to
demonstrate the necessity and su¢ ciency of P^h
Qxu^ .
23
Observe that P^h is well-de…ned (since (12) is non-increasing on [0; 1), so
h (x) < 1 when x < 1) and …nite (since h is not the indicator for f1g). Note
also that P^h
Qxu^ entails P^h
1, and hence that h (x) x is non-increasing
on (0; 1].
Preferences therefore exhibit aversion to elementary JMPS’s i¤ for any
a; b; b with 0 b < b a < 1,
u
sup
<
0
(
)
(
<
!
u
1
) = (b b)
u( )
1 h (a)
1 a
u( )
h
(b)
.
h (b)
b b
(1 a)
Since u is constant on [^
x; 1] this is equivalent to
u
sup
<
0
(
)
(
<
!
u
x
^
) = (b b)
u( )
1 h (a)
1 a
u( )
h
(b)
h (b)
b b
(1 a)
By Lemma 17 it follows that aversion to JMPS obtains i¤
Qxu^
1
inf
0
b<b
which is equivalent to P^h
a<1
h (a)
1 a
h
b
b
h (b)
b
!
,
Qxu^ by Lemma 18.
Corollary 22 follows straightforwardly, so we omit its proof. Indeed,
Chateauneuf, Cohen and Meilijson (2004, Theorem 4) already implies most
of Corollary 22.
We now easily obtain the following characterization of the LLIR order on
equal-mean distributions:
Corollary 23 Given F and G with equal means, F LLIR G i¤
Z 1
Z 1
u d (h F )
u d (h G)
0
0
for any concave, non-decreasing and continuous u : [0; 1] ! R and any
non-decreasing h : [0; 1] ! [0; 1] that satis…es h (0) = 0 and h (x) =x nonincreasing on (0; 1].
24
.
Proof. Theorem 21 gives the “only if”part. For the converse, suppose
Z
F
1 (p)
F (z) dz >
0
Z
1 (p)
G
G (z) dz
0
for some p 2 (0; 1). We shall separate the analysis into two cases.
Case I: F
1
(p)
1
G
1
p
Z
(p). In this case:
p
F
1
dz
=
F
1
1
p
(p)
0
<
=
G
1
p
1
Z
1
p
(p)
p
1
G
Z
F
1 (p)
F dx
0
Z
G
1 (p)
G dx
0
dz
0
If we let u be the identity function and de…ne
z=p if z 2 [0; p]
1
if z > p
h (z) =
then h (z) =z is non-increasing on (0; 1],
Z 1
Z
1 p
1
F dh =
F
p 0
0
and
Z
1
G
0
so G
1
1
dh =
p
F as required.
Z
1
dz
1
dz,
p
G
0
Case II: F 1 (p) > G 1 (p). By right-continuity it follows that there is some
" 2 (0; 1 p) such that
F
1
(z) > G
1
(z)
for all z 2 [p; p + "]. If G 1 (0) > F 1 (0) then G
F according to
maximin preferences (which are Jewitt risk-averse), so let us further
25
assume G
and
1
(0)
1
(0). We may now choose u to be the identity
8
0
if z = 0
>
>
<
"+z
if z 2 (0; p)
h (z) =
p + " if z 2 [p; p + "]
>
>
:
z
if z > p + "
F
Observe that h (z) =z is non-increasing on (0; 1]. Furthermore, since F
and G have equal means:
Z 1
Z 1
Z p+"
1
1
F dh =
F dz
F 1 dz + "F 1 (0)
0
p
0
<
Z
1
G
1
dz
=
p+"
G
1
dz + "G
1
(0)
p
0
Z
Z
1
G
1
dh
0
so G
F.
Chateauneuf, Cohen and Meilijson (2004, Theorem 4(iii)) demonstrate
that, for equal-mean distributions F and G, F LLIR G if all RDEU maximizers with linear u and h (z) =z non-increasing on (0; 1] weakly prefer F to
G. In this connection, note that the “if” part of our Corollary 23 is proved
using linear utility functions.
5
Concluding remarks
Figure 1 and the associated inequality (7) prove to be simple but powerful tools for understanding various risk aversion concepts within the RDEU
framework. As we have seen, a simple argument based on (7) can be used
to characterize the strongly risk averse RDEU orders, even under very weak
maintained assumptions about u and h. This characterization reveals the
prominent role of maximin behavior in strong aversion to risk.
It is also possible to fully characterize risk aversion in the sense of Jewitt
(1989) as the requirement that (7) hold whenever a = 1. This simple test
implies the necessary –and, given u increasing up to its maximum, su¢ cient
26
–condition that P^h
Qxu^ . Corollary 22 describes important special cases in
which this condition may be signi…cantly simpli…ed.
As corollaries to our main results, one also obtains characterizations of
the underlying “more risky than”orders SOSD and LLIR on D. These may
be of independent interest.
Appendix: Proofs
Proof of Proposition 7. For this proof, it is convenient to work with the
various de…nitions re-expressed in terms of CDF’s, rather than DDF’s.
If F = (G) , then F 1 is constant on (0; 1) and hence G 1 F 1 is
non-decreasing on (0; 1). It follows immediately that monotone risk aversion
implies weak risk aversion.
Jewitt (1989, pp.68-69) shows that
Z
Z
1 (p)
F
F (z) dz
0
G
1 (p)
G (z) dz
0
is equivalent to
G
1
(p)
F
1
1
p
(p)
Z
p
G
1
(!)
F
1
(!) d!
(13)
0
If G 1 F 1 is non-decreasing on (0; 1), then (13) holds for all p 2 (0; 1).
Therefore, Jewitt risk aversion implies monotone risk aversion.
Finally, suppose (13) holds for all p 2 (0; 1). If F and G have the same
mean, then
Z
1
G
1
(!)
F
1
(!) d! = 0.
G
1
(!)
F
1
(!) d!
0
This fact and (13) gives
Z
p
0
0
for all p 2 [0; 1] with equality for p = 1. It follows that F SOSD G (Roell,
1987, p.151; Vergnaud, 1997, Proposition 1). Therefore, strong risk aversion
implies Jewitt risk aversion.
27
Next, consider the converses. To see that Jewitt need not imply strong
risk aversion, choose u linear and any non-concave h with
h (x)
x
non-increasing on [0; 1] (compare Theorem 10 in Section 3 and Corollary 22
in Section 4).
Likewise, to see that monotone risk aversion does not imply Jewitt risk
aversion, choose u linear and h satisfying h (x) x but not such that
h (x)
x
is non-increasing on [0; 1] (compare Chateauneuf, Cohen and Meilijson (2004,
Theorem 3(i)) and Corollary 22 in Section 4 below).20
Finally, Chateauneuf and Cohen (1994, Example 3 with h = f1 ) reveals
that weak risk aversion does not imply monotone risk aversion. In particular,
recalling Chateauneuf, Cohen and Meilijson (2005, Theorem 1), we observe
that Q1u = 1 while Pf is …nite for this example.
Proof of Lemma 9. Suppose that h is not the indicator on (0; 1] and
there exist ; 2 [0; 1) with < and u ( ) = u ( ) < 1. We deduce a
contradiction follows. Since u is continuous and monotone, it is without loss
of generality to assume
= sup fz 2 [0; 1] j u (z) = u ( )g
(14)
Set = , a = 1 and b = 0 in Figure 1. Then weak risk aversion requires
that F % G (Proposition 8(iv)). If we can …nd 2 ( ; 1] and a = b 2 (0; 1)
such that h b > 0 and
1
b (
,
) = b
=
1
b
b
(15)
we are done, since (7) then implies u
= u ( ), which contradicts (14).
But h b > 0 for any b su¢ ciently close to 1 – recall that h is not the
indicator on (0; 1] –so we can always satisfy (15) with 2 ( ; 1].
20
A suitable example may be found in Quiggin (1991, pp.243-4).
28
Proof of Lemma 11. Suppose h is not the indicator on (0; 1] and there
exists z 2 (0; x^) with u0 (z) = 0. We shall deduce a contradiction.
Let p 2 (0; 1). We show that h has a left-hand derivative equal to zero
at p. To do so, consider Figure 1 and …x a = 1, a = b = p, = = z and
= 1. Now choose a sequence fbn g1
bn < p for each n and
n=1 such that 0
lim bn = p.
n!1
We need to show that
lim
n!1
Let f
n 1
gn=1
n
be a sequence with 0
(1
p) (z
h (bn )
= 0
bn
h (p)
p
n
(16)
< z and
bn ) (1
) = (p
z)
for each n,21 and
lim
n
n!1
= z.
Since u0 (z) = 0 and u is strictly increasing on (0; x^) we have
lim
n!1
u (z)
z
u(
n
)
n
= 0
and
u (1) u (z)
> 0.
1 z
Therefore, strong risk aversion and (7) give (16) as required.
Now let p 2 [0; 1). Using a similar argument, we may show that h has
a right-hand derivative equal to zero at p. In this case, we start by
n n…xing
o1
n 1
a = 1, b = p, = = z, = 1; and choose sequences fa gn=1 and b
with p < an = b
n
n=1
1 for each n and
lim an = p.
n!1
The rest of the argument follows, mutatis mutandis, as before.
21
This is always possible, if necessary by …rst eliminating …nitely many initial members
1
of the sequence fbn gn=1 .
29
We have therefore established that h has a derivative equal to zero at
every p 2 (0; 1), and a right-hand derivative equal to zero at p = 0. Given
the normalization of h , it follows that h is the indicator function for f1g.
But this implies that h is the indicator for (0; 1] –a contradiction.
Proof of Lemma 18. The arguments used in the proof of Chateauneuf, Cohen and Meilijson (2005, Proposition 2(vi)) are easily adpated to the present
case.
Proof of Lemma 19. Observe that throughout the proof of Lemma 11 we
…x a = 1. Therefore, we can use the same argument to prove the present
Lemma, substituting “Jewitt risk aversion”for “strong risk aversion”where
the latter appears.
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