Ambiguity Aversion, Robustness, and the Variational Representation of Preferences
Author(s): Fabio Maccheroni, Massimo Marinacci, Aldo Rustichini
Source: Econometrica, Vol. 74, No. 6 (Nov., 2006), pp. 1447-1498
Published by: The Econometric Society
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Econometrica,Vol. 74, No. 6 (November, 2006), 1447-1498
AMBIGUITY AVERSION, ROBUSTNESS, AND THE VARIATIONAL
REPRESENTATION OF PREFERENCES
BY FABIO MACCHERONI, MASSIMO MARINACCI, AND ALDO RUSTICHINI1
We characterize,in the Anscombe-Aumann framework,the preferences for which
there are a utilityfunction u on outcomes and an ambiguityindex c on the set of probabilities on the states of the world such that, for all acts f and g,
ftg
+c(p)
min(Ju(f)dp+
c(p) >min u(g)dp
The function u representsthe decision maker'sriskattitudes,while the index c captures
his ambiguityattitudes. These preferences include the multiple priors preferences of
Gilboa and Schmeidler and the multiplier preferences of Hansen and Sargent. This
provides a rigorousdecision-theoreticfoundation for the latter model, which has been
widely used in macroeconomicsand finance.
KEYWORDS:
Ambiguityaversion,model uncertainty,robustness.
1. INTRODUCTION
AMBIGUITY HAS BEENA CLASSICISSUEin decision theory since the seminal
work of Ellsberg (1961). The fundamental feature of ambiguity pointed out
by Ellsberg is that there may be no belief on the states of the world that the
decision maker holds and that rationalizes his choices.
A widely used class of preferences that model ambiguity are the multiple
priorspreferencesaxiomatized by Gilboa and Schmeidler (1989) (also known
as maxmin expected utility preferences). Agents with these preferences rank
payoff profiles f accordingto the criterion
(1)
u(f)dp,
V(f) =min
peCf
where C is a given convex subset of the set A of all probabilitieson states. The
set C is interpreted as a set of priors held by agents, and ambiguityis reflected
by the multiplicityof the priors.
1An extended version of this paper was previouslycirculatedwith the title "VariationalRepresentation of Preferencesunder Ambiguity"ICER WorkingPaper5/2004, March 2004. We thank
Erio Castagnoli,Rose-Anne Dana, LarryEpstein, Peter Klibanoff,Bart Lipman,MarkMachina,
JianjunMiao, SujoyMukerji,Emre Ozdenoren,Ben Polak, and, especially,Andy Postlewaiteand
four anonymousreferees for helpful discussions and suggestions.We also thank several seminar
audiences. Part of this research was done while the first two authors were visiting the Department of Economics of Boston University,which is thanked for its hospitality. They also gratefully acknowledgethe financial support of the Ministero dell'Istruzione,dell'Universithe della
Ricerca. Rustichinigratefullyacknowledgesthe financialsupport of the National Science Foundation (Grant SES-04-52477).
1447
1448
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
In the past few years the possibility that agents may not hold a single belief on the states of the world has widely informed the research in macroeconomics and finance. In particular,there has been a growing dissatisfactionin
macroeconomicstowardthe stronguniformityon the agents' beliefs that is imposed by the rationalexpectationshypothesis.Under this hypothesis,all agents
share the same probabilitydistributionon some relevant economic phenomenon and each agent has to be firmlyconvinced that the model he has adopted
is the correct one. This is a strong requirement because agents can have different models, each of them being only an approximationof the underlying
true model, and they may be aware of the possibilitythat their model is misspecified. A weakening of this requirementallows agents to entertain different
priors on the economy.
A large part of the research has modeled ambiguitywith multiple priors
models (see, for instance, Epstein and Wang (1994) and Chen and Epstein
(2002)). More recently,a different alternativehas been explored, startingwith
the work of Hansen and Sargent (2000, 2001) which in turn builds on earlier
work in the field of robustcontrol in the engineeringand optimal controlliterature. In this robustpreferences approach,the objectivefunctions of the agents
take into account the possibilitythat their model q may not be the correctone,
but only an approximation.Specifically,agents rankpayoff profiles f according
to the choice criterion
(2)
V(f)=-min
fu(f)dp
+ OR(p
Iq)},
where R(. IIq):A -* [0, oo] is the relative entropy with respect to q (see Section 4.2 for the definition). Preferences represented by criterion (2) are called
multiplierpreferences.2
Agents who behave according to this choice criterion are considering the
possibility that q may not be the appropriate law that governs the phenomenon in which they are interested and for this reason they take into account
other possible models p. The relative likelihood of these alternative models
is measured by their relative entropy, while the positive parameter 0 reflects
the weight that agents are giving to the possibilitythat q might not be the correct model. As the parameter 0 becomes larger,agents focus more on q as the
correct model, giving less importanceto possible alternativemodels p (Proposition 22).
Hansen and Sargent (2000) have pointed out that this model uncertainty
can be viewed as the outcome of ambiguity,possibly resulting from the poor
quality of the informationon which agents base the choice of their model. For
this reason, the motivation behind multiplier preferences is closely connected
2Hansen and Sargent (2001) also considered a class of multiple priors preferenceswith C =
{pe A: R(p IIq) < r7},which they called constraintpreferences.
VARIATIONALREPRESENTATIONOF PREFERENCES
1449
to the motivation that underlies multiple priors preferences. So far, however,
this connection has been stated simply as intuitively appealing rather than established on formal grounds. In particular,no behavioral foundation of the
preferences in (2) has been provided.
Here we establish this connection by presenting a general class of preferences with common behavioralfeatures that includes both multiplierand multiple prior preferences as special cases. The nature of the connection between
the two main models that we have been discussingso far can be firstestablished
formally.The multiple priors criterion (1) can be written as
(3)
V(f) = min
u(f)dp+8
c(p)},
where Sc: A -- [0, oc] is the indicator function of C (in the sense of convex
analysis;see Rockafellar (1970)) given by
c(P)
pEj
O,
oc,
if
C,
otherwise.
Like the relative entropy, the indicator function also is a convex function defined on the simplex A.
This reformulationclarifies the formal connection with the multiplier preferences in (2) and suggests the general representation
(4)
V(f) = min
u(f) dp + c(p) ,
where c: A -- [0, oo] is a convex function on the simplex. In this paper, we
show that the connection is substantiveand not formal by establishingthat the
two models have a common behavioralfoundation.
We first axiomatize (Theorem 3) the representation (4) by showing how it
rests on a simple set of axioms that generalizes the multiple priors axiomatization of Gilboa and Schmeidler (1989). We then show (Proposition 8) how
to interpret in a rigorous way the function c as an index of ambiguity aversion: the lower is c, the higher is the ambiguityaversion exhibitedby the agent.
The relative entropy OR(p 11q) and the indicator function 8c(p) can thus be
viewed as special instances of ambiguityindices. The assumptionson behavior
for this general representation are surprisinglyclose to those given by Gilboa
and Schmeidler (1989), and result from a simple weakening of their certainty
independence axiom, so the similaritybetween the different representations
has a sound behavioralfoundation.
Once we have established this common structure, we can analyze the relationship between ambiguity aversion and probabilistic sophistication. For
example, the multiplier preferences used by Hansen and Sargent are probabilisticallysophisticated (and the same is true for their constraintpreferences).
1450
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
Is this a fundamental property of robust preferences that makes them profoundly different from multiple priors? In the common structure provided
by the representation (4), the answer is simple and clear. The preferences
in (4) are probabilisticallysophisticated if and only if the ambiguityindex c is
symmetric.For multiple priorspreferences, this propertytranslatesinto a symmetry property of the set C. For multiplier preferences (and for constraint
preferences), the condition is reflected by the symmetryof the relative entropy.
The propertyof being probabilisticallysophisticated is therefore a property
of the specific ambiguityindex, not of the model. Being a symmetryproperty,
this property is fragile: small perturbationsdestroy it and produce Ellsbergtype behavior. Even if one adopts the view that such behavior is necessaryfor
ambiguityaversion,the preferences in (4) are typicallyambiguityaverse (in the
precise sense of Theorem 14).
Although our original motivation came from multiple priors and multiplier
preferences, the class of preferenceswe axiomatizegoes well beyond these two
classes of preferences. In particular,multiplier preferences are a very special
example of a new class of preferences, called divergencepreferences,that we
introduce and study in this paper. These preferences are able to accommodate
Ellsberg-typebehavior and, unlike multiple priorspreferences,they are in general smooth (Proposition 23), an importantfeature for applications.This new
class of preferences can provide a tractablealternativeto multiple priorspreferences in economic applications that deal with ambiguity,and the works of
Hansen and Sargent can be viewed as an instance of this (see Theorem 16).
This claim is further substantiated by the observation that divergence
preferences include as a special case a third classic class of preferences-the
mean-variancepreferencesof Markowitz(1952) and Tobin (1958). Recall that
mean-variance preferences are represented by the preference functional
(5)
V(f)
=
f dq -
Var(f).
In Theorem 24, we show that, on the domain of monotonicityof V, the equality
f dq-
= min
Var(f)
20
~
~p(-AI
f dp+ OG(pIIq)
holds, where G(- I q): A -- [0, oo] is the relative Gini concentrationindex (see
Section 4.2.2 for the definition). As a result, the mean-variance preference
functional (5) is a special case of our representation(4). Interestingly,the associated index of ambiguityaversion is the relative version of the classic Gini index. After Shannon'sentropy,a second classic concentrationindex thus comes
up in our analysis.
Summingup, in this paper we generalize a popular class of preferences that
deal with ambiguity-the multiple priors preferences-and in this way we are
VARIATIONALREPRESENTATIONOF PREFERENCES
1451
able to introduce divergence preferences, a large new class of preferences under ambiguitythat are in general smooth and that include, as special cases,
two widely used classes of preferences, the multiplier preferences of Hansen
and Sargent and the mean-variance preferences of Markowitzand Tobin.We
are thus able to provide both a rigorous decision-theoretic foundation on two
widely used classes of preferences and a setting in which the two most classic concentration indices, Shannon's entropy and Gini's index, have a natural
decision-theoretic interpretation. We finally characterize the preferences in
this class that are probabilisticallysophisticated and show that this property
coincides with a symmetrypropertyof the cost function. This property is fragile and so the preferences we characterize are typically not probabilistically
sophisticated.
1.1. AmbiguityAversion
In addition to Schmeidler's (1989) original notion based on preference for
randomization (see Axiom A.5 in Section 3), there are two main notions of
ambiguityaversion in the literature:those proposed by Epstein (1999) and by
Ghirardatoand Marinacci (2002). The key difference in the two approaches
lies in the different notion of ambiguityneutralitythey use: while Ghirardato
and Marinacci (2002) identify ambiguity neutrality with subjective expected
utility,Epstein (1999) more generallyidentifies ambiguityneutralitywith probabilistic sophistication. In a nutshell, Ghirardato and Marinacci (2002) claim
that, unless the setting is rich enough, probabilisticallysophisticated preferences may be compatible with behavior that intuitivelycan be viewed as generated by ambiguity.For this reason, they consider only subjective expected
utility preferences as ambiguityneutral preferences.3
For the general class of preferences we axiomatize, the relative merits of
these two notions of ambiguityaversion are the same as for the special case
represented by multiple priors preferences. Although in the paper we adopt
the view and terminology of Ghirardato and Marinacci (2002), the appeal of
our analysis does not depend on this choice; in particular,we expect that the
ambiguityfeatures of our preferences can be studied along the lines of Epstein
(1999), in the same way as it has been done in Epstein (1999) for multiple
priorspreferences.
To clarifythis issue further,in Section 3.5 we study the form that probabilistic sophisticationtakes in our setting. As we alreadymentioned, we show that
our preferences are probabilisticallysophisticatedonce their ambiguityindices
satisfy a symmetryproperty.As a result, probabilisticsophistication is not peculiar to some particularspecificationof our preferences, but, to the contrary,
andMarinacci(2002)for a detailedpresentation
3Wereferto Epstein(1999)andGhirardato
and motivationof their approaches.Notice that the notion of ambiguityaversion in Ghirardato
staticsexercises
andMarinacci
(2002)iswhatprovidesa foundationforthe standardcomparative
in ambiguity
formultiplepriorspreferencesthatarebasedon the sizeof the set of priors.
1452
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
all examples of our preferences will include special cases of probabilistically
sophisticated preferences. For instance, both multiplier and mean-variance
preferences are easily seen to be examples of probabilisticallysophisticateddivergence preferences, although Example 17 shows that this is not the case for
general divergencepreferences, even for those that are very close to multiplier
and mean-variance preferences.
We close by briefly mentioning a possible alternative interpretationof our
preferences. Consider an agent who has to make choices with only limited
information and without a full understandingof what is going on. Some recent psychologicalliterature (e.g., Keren and Gerritsen (1999), Kiihbergerand
Perner (2003)) suggests that in this case the agent may behave as if he were
playingagainst an informed opponent who might take advantageof this uncertainty and turn it against him.
This is a psychological attitude that can be relevant in many choice situations, including Ellsberg-type choice situations, and our preference functional (4) can be viewed as modeling such psychological treat. In fact, agents
who rank payoff profiles accordingto (4) can be viewed as believing they are
playing a zero-sum game against (a malevolent) Nature. This view, however,
has not been firmly established in the psychological and neuroscience literatures, and it is the object of currentresearch (see Hsu, Bhatt, Adolphs, Tranel,
and Camerer (2005) and Rustichini(2005)); for this reason, we do not expatiate further on this interpretation.
The paper is organized as follows. After introducingthe setup in Section 2,
we present the main representationresult in Section 3. In the same section, we
discuss the ambiguityattitudes featured by the preferences we axiomatizeand
we give conditions that make them probabilisticallysophisticated.In Section 4,
we studytwo importantexamplesof our preferences, that is, the multiplepriors
preferences of Gilboa and Schmeidler(1989) and divergencepreference, a new
class of preferences that includes, as special cases, multiplierpreferences and
mean-variance preferences. Proofs and related material are collected in the
Appendixes.
2. SETUP
Consider a set S of states of the world, an algebra X of subsets of S called
events,and a set X of consequences.We denote by .T the set of all the (simple)
acts: finite-valuedfunctions f :S -- X, which are X-measurable.Moreover,we
denote by Bo(2) the set of all real-valued X-measurablesimple functions, so
that u(f) E Bo(V) whenever u: X -- R, and we denote by B(s) the supnorm
closure of BO(X).
Given any x E X, define x E T to be the constant act such that x(s) = x for
all s E S. With the usual slight abuse of notation, we thus identify X with the
subset of the constant acts in F. If f E F, x E X, and A E I, we denote by
xAf E FTthe act that yields x if s E A and f(s) if s A.
VARIATIONALREPRESENTATIONOF PREFERENCES
1453
We assume additionally that X is a convex subset of a vector space. For
instance, this is the case if X is the set of all the lotteries on a set of prizes,
as happens in the classic setting of Anscombe and Aumann (1963). Using the
linear structureof X, we can define as usual for every f, g E T and a E [0, 1]
the act af + (1 - a)g E F, whichyields oaf(s)+ (1 - a)g(s) E X for every s E S.
We model the decision maker'spreferenceson Tf by a binary relation >. As
usual, >-and - denote, respectively,the asymmetricand symmetricparts of >-.
If f E F, an element xf E X is a certaintyequivalentfor f if f - xf.
3. REPRESENTATION
3.1. Axioms
In the sequel we make use of the following properties of >:
AXIOMA.1-Weak Order: If f, g, h E ?, (a) either f > g or g >- f, and
(b) f > g and g> h implyf > h.
A.2-Weak CertaintyIndependence:If f, g E ?, x, y EX, and a E
AXIOM
(0, 1),
af + (1 - a)x > Lag+ (1 - a)x
+ (1 - aa)y ag + (1 - a)y.
c
= of
AXIOMA.3 -Continuity: If f, g, h E Y, the sets {a E [0, 1]: af + (1 a)gg> h} and {a e [0, 1]: h > af + (1 - a)g} are closed.
AXIOMA.4-Monotonicity: If f, g E ? and f(s)
f rg.
-
g(s) for all s E S, then
AXIOMA.5-Uncertainty Aversion: If f, g E ? and a E (0, 1),
f~g
=
af +(1-a)g
rjf.
AXIOMA.6-Nondegeneracy: f >-g for some f, g E F.
Axioms A.1, A.3, A.4, and A.6 are standardassumptions.Axioms A.3 and A.6
are technical assumptions,while Axioms A.1 and A.4 require preferences to
be complete, transitive, and monotone. The latter requirement is basically a
state-independencecondition,which saysthat decision makers always(weakly)
prefer acts that deliver statewise (weakly)better payoffs, regardlessof the state
where the better payoffs occur. If a preference relation >- satisfiesAxioms A.1,
A.3, and A.4, then each act f E F admits a certainty equivalent xf E X. (See
the proof of Lemma 28 in Appendix B.)
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
1454
Axiom A.5, due to Schmeidler (1989), is a smoothing axiom that can be interpreted as an ambiguityaversion axiom, as discussed at length in Gilboa and
Schmeidler (1989), Schmeidler (1989), Epstein (1999), and Ghirardato and
Marinacci(2002).
Axiom A.2 is a weak independence axiom-a variationof an axiomof Gilboa
and Schmeidler (1989). It requires independence only with respect to mixing
with constant acts, providedthe mixingweights are kept constant.Axiom A.2 is
weaker than the original axiom of Gilboa and Schmeidler (1989), and it is this
weakening that makes it possible to go beyond the multiple priors model. Because of its importancefor our derivation,we devote the rest of this subsection
to Axiom A.2.
Consider the following strongerversion of Axiom A.2:
AXIOMA.2'-Certainty Independence: If f, g e T, x e X, and a E (0, 1),
then
f
g
'
af + (1 - a)x
ag + (1 - a)x.
Axiom A.2' is the original axiom of Gilboa and Schmeidler (1989). The next
lemma shows how it strengthensAxiom A.2.
LEMMA1: A binaryrelation - on F satisfiesAxiom A.2' if and only if,for all
, x, y E X, anda, E (0, 1],
f,g
af + (1 - a)x > ag + (1 - a)x
=
pf + (1 - P)y
3pg+ (1 3)y.
Axiom A.2 is therefore the special case of Axiom A.2' in which the mixing
coefficients a and /3 are required to be equal. At a conceptual level, Lemma 1
shows that Gilboa and Schmeidler's(1989) certaintyindependence axiomactually involves two types of independence: independence relative to mixingwith
constants and independence relative to the weights used in such mixing. Our
Axiom A.2 retains the firstform of independence, but not the second. In other
words, we allow for preference reversals in mixing with constants unless the
weights themselves are kept constant.
This is a significant weakening of the certainty independence axiom, and
its motivation is best seen when the weights a and /3 are very different, say
a is close to 1 and p is close to 0. Intuitively, acts af + (1 - a)x and ag +
(1 - a)x can then involve far more uncertaintythan acts Pf + (1 - P)y and
p/g + (1 - P/3)y,which are almost constant acts. As a result, we expect that,
at least in some situations, the rankingbetween the genuinely uncertain acts
af + (1 - a)x and ag + (1 - a)x can well differ from that between the almost
constant acts Pf + (1 - P)y and pg + (1 - P)y.
VARIATIONALREPRESENTATIONOF PREFERENCES
1455
Needless to say, even though we believe that such reversals can well occur
(from both a positive and normativestandpoint), the only way to test them, and
so test the plausibilityof Axioms A.2 and A.2', is by runningexperiments.This
is possible because both Axioms A.2 and A.2' have clear behavioral implications. For instance, the following (thought) experiment gives a simple testable
way to compare Axioms A.2 and A.2', and runningthis type of experimentswill
be the subject of future research.
2: Consider an urn that contains 90 black and white balls in unEXAMPLE
known proportion, and the bets (payoffs are in dollars)
t> 0
Black
White
ft
t
t
gt
3t
0.01t
that is, f, pays t dollars whatever happens, while g, pays 3t dollars if a black
ball is drawn and t cents otherwise. For example,
10
fio
g10
Black
10
30
104
White
10
and f10o4
0.1
g104
Black
White
10,000
30,000
10,000
100
Assume the decision maker'spreferences satisfyAxioms A.1 and A.3-A.6, and
she displaysconstant relative risk aversion y E (0, 1).4 If her preferences satisfy
Axiom A.2', then
either ft~, g, for all t or gt - f, for all t.
If, in contrast, her preferences only satisfy Axiom A.2 and not Axiom A.2',
there might exist a thresholdt such that
ft g, forallt>t
and gt f,
for all t < t.
This reversal is compatible with Axiom A.2, but it would reveal a violation of
Axiom A.2'.
We close by observingthat, in terms of preference functionals,by Theorem 3
and Proposition 19 all preference functionals (4) satisfyAxiom A.2 and violate
Axiom A.2' unless they reduce to the multiple priors form (1).
4Noticethat,forall t > - > 0 andy, thereexistf, g, x, y, a, p as discussedin the textsuchthat
- g,.
af + (1 - a)x - ft, ag + (1 - a)x - gt, Pf + (1 - P)y - f,, and pg + (1 - P)y
1456
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
3.2. Main Result
We can now state our main result, which characterizespreferences that satisfy Axioms A.1-A.6. Here A = A(2) denotes the set of all finitely additive
probabilities on I endowed with the weak* topology5 and c: A -- [0, 00] is
said to be groundedif its infimumvalue is zero.
3: Let >. be a binaryrelationon F. Thefollowing conditionsare
THEOREM
equivalent:
(i) Therelation>- satisfiesAxioms A.1-A.6.
(ii) Thereexistsa nonconstantaffinefunction u :X - IRand a grounded,convex, and lower semicontinuousfunction c: A -+ [0, oo] such that,for all
f, g e F,
(6)
min(f u(f)dp + c(p) > min(f u(g)dp+ c(p)).
f jg
g
For each u thereis a (unique) minimal c*:A - [0, oo] that satisfies (6) and is
given by
(7)
c*(p)= sup(u(xf)
fGT(
u(f)dp).
The representation(6) involves the minimizationof a convex lower semicontinuous function, which is the most classic variationalproblem. This motivates
the following definition:
4: A preference - on F is called variationalif it satisfies AxDEFINITION
ioms A.1-A.6.
By Theorem 3, variationalpreferences can be represented by a pair (u, c*).
From now on, when we consider a variationalpreference, we will write u and c*
to denote the elements of such a pair. Next we give the uniqueness properties
of this representation.
5: Twopairs (uo, c*) and (u, c*) representthe same variational
COROLLARY
- as in Theorem
3 if and only if thereexist a > 0 and P/ IRsuch that
preference
u = auo +
p and c* =
aco.
In Theorem 3 we saw that c* is the minimal nonnegative function on A for
which the representation (6) holds. More is true when u(X) = {u(x): x e X}
is unbounded (either below or above):
5That is, the
pd(A)
- p(A) for
o(A(X,),
all A
Bo(2)) topology where a net
•
.
{PdideD
converges to p if and only if
VARIATIONALREPRESENTATIONOF PREFERENCES
1457
6: Let > be a variationalpreferencewith u(X) unbounded.
PROPOSITION
Then thefunction c* definedin (7) is the unique nonnegative,grounded,convex,
and lowersemicontinuousfunction on Afor which (6) holds.
As shown in Lemma 29 in Appendix B, the assumption that u(X) is unbounded (above or below) is equivalent to the following axiom (see Kopylov
(2001)).
AXIOMA.7 -Unboundedness: Thereexist x >-y in X such that,for all a E
(0, 1), there exists z E X that satisfies either y >- az + (1 - a)x or az + (1 a)y >-x.
We call the variationalpreferences that satisfyAxiom A.7 unbounded.
3.3. AmbiguityAttitudes
We now study the ambiguityattitudes featured by variational preferences.
We follow the approach proposed in Ghirardato and Marinacci (2002), to
which we refer for a detailed discussion of the notions we use.
Begin with a comparativenotion: given two preferences >1 and >2, say that
(1 is more ambiguityaversethan (2 if, for all f E F and x E X,
(8)
f >- x
=
f2
f x.
To introduce an absolute notion of ambiguity aversion, as in Ghirardato and
Marinacci(2002) we consider subjectiveexpected utility (SEU) preferences as
benchmarks for ambiguityneutrality.We then say that a preference relation
b is ambiguityaverseif it is more ambiguityaverse than some SEU preference.
We now apply these notions to our setting. The first thing to observe is that
variationalpreferences are alwaysambiguityaverse.
7: Each variationalpreferenceis ambiguityaverse.
PROPOSITION
Since variationalpreferences satisfy Axiom A.5 and the choice rule that results from (6) is a maxmin rule, intuitively it is not surprisingthat variational
preferences alwaysdisplaya negative attitude toward ambiguity.Proposition 7
makes this intuition precise.
Next we show that comparative ambiguity attitudes for variational preferences are determined by the function c*. Here u1 m u2 means that there exist
a > 0 and /3 E R such that ul = au2 + / .
8: Given two variationalpreferences>- and -2, the following
PROPOSITION
conditionsare equivalent:
(i) Therelation -1 is moreambiguityaversethan j2.
1458
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
u2and c
(ii) ul
c* (providedul
<_
= u2).
Given that u1a U2, the assumption ul = u2 is just a common normalization
of the two utility indices. Therefore, Proposition 8 says that more ambiguity
aversepreference relations are characterized,up to a normalization,by smaller
functions c*. Therefore, the function c* can be interpretedas an indexof ambiguityaversion.
We now give few simple examples that illustrate this interpretation of the
function c*.
EXAMPLE
9: By Proposition 8, maximalambiguityaversion is characterized
by c*(p) = 0 for each p e A. In this case, (6) becomes
f
min u(f) dp > minf u(g)dp,
g
that is,
f -g
min
u(fS(s))
sES
> min
u(g(s)),
seS
a form that clearly reflects extreme ambiguityaversion.
EXAMPLE
10: Minimal ambiguity aversion corresponds here to ambiguity
neutralitybecause, by Proposition 7, all variationalpreferences are ambiguity
averse. Therefore, the least ambiguityaverse functions c* are those associated
with SEU preferences. As will be shown in Corollary20, when a SEU preference is unbounded, c* takes the stark form
c(p =,
=c0,
if p = q,
otherwise,
where q is the subjectiveprobabilityassociated with the preference.
11: Denote by c* the ambiguityneutral index of Example 10 and
EXAMPLE
denote by
the maximal ambiguity index of Example 9. Given a e (0, 1),
suppose thecm
right-handside of (6) is
(1- a)u(f) dq+amins
u(f(s))
> (1 -a)
sSE
u(g)dq + aminu(g(s)),
f
which is the well known e-contaminated model. In this case,
c*(p) = min (1 P1,P2P2EA
a)cq(P2)+
= 5(1-a)q+aA(P),
Cm(P):(1
- a)p2 + ap = p}
VARIATIONALREPRESENTATIONOF PREFERENCES
1459
and this is a simple example of an index c* that does not display extreme ambiguity attitudes.
According to Proposition 8, variationalpreferences become more and more
(less and less, resp.) ambiguity averse as their ambiguity indices become
smaller and smaller (larger and larger, resp.). It is then natural to wonder
what happens at the limit, when they go to 0 or to oc. The next result answers
this question. In reading it, recall that
< c, implies dom c, dom c,+1 and
c1+•
argmin c, C argmin cn+1.6
12: (i) If c, 4 and limncn~(p)= 0 wheneverthis limit is finite,
PROPOSITION
then
(9)
limminf u(f)dp + cn(p)
=
n-+0o pEAj
min
JpEUn dom-cCJ
u(f) dp
for all f E 'F.
(ii) If c, and limn cn (p) = 0c wheneverthis limit is not 0, then
(10)
lim
n-+ c minf
pEAI
u(f)dp+cn(p)
=
min
PEn argmincnJ
u(f ) dp
for all f E F.
Proposition 12 shows that the limit behavior of variational preferences is
describedby multiple priors preferences, but the size of the sets of priors they
feature is very different. In fact, in (9) the relevant set of priors is given by
U, dom c,, whereas in (10) it is given by the much smaller set n,, argminc,.
For example, Proposition22 will show that for an importantclass of variational
preferences, the set ", argmin c, is just a singleton, so that the limit preference
in (10) is actuallya SEU preference, while the set U, dom c, is very large.
We close with few remarks. First, observe that Lemma 32 in Appendix B
shows that the set that Ghirardato and Marinacci (2002) call benchmark
measures-those probabilities that correspond to SEU preferences less ambiguity averse than --is given here by argmin c* = {p E A: c*(p) = 0}.
Second, notice that by standard convex analysis results (see Rockafellar
(1970)), Example 11 can be immediately generalized as follows: the ambiguity index of a convex combinationof preference functionals that represent unbounded variational preferences is given by the infimal convolution of their
ambiguityindices.
6Thetermdomc denotesthe effectivedomain{c < oc} of c, whereasargminc,= {p E A:
c,(p) = 0} becausec, is grounded.Observethat domcnrepresentsthe set of all probabilities
that the decision maker considersto be relevantwhen rankingacts using the ambiguityindex cn,
whereas the smaller set argmincn contains only the probabilities that are getting the highest
weight by this decision maker.
1460
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
Finally, Proposition 12 is an example of a result about the limit behaviorof
sequences of variationalpreferences
V.(f)
min
u,(f)
dp + c,(p)),
a type of result that can be importantin some applications to enable dealing
with stability issues. These limit results involve the convergence of minima,
a classic problem in variational analysis. A noteworthy feature of variational
preferences is that the lower semicontinuityof the functions f u,(f) dp+c,(p)
on A makes it possible to use the powerfulde Giorgi-Wijsmantheory of F convergence (often called epiconvergence; see, e.g., Dal Maso (1993)) to study
the behavior of the sequence {V,(f)}, and, for example, to determine the
conditions under which it converges to some preference functional V(f) =
minpEa(f u(f) dp + c(p)) for suitable limit functions u and c.
3.4. An Extension:CountableAdditivity
In Theorem 3 we considered the set A of all finitely additive probabilities.
In applications, however, it is often important to consider countably additive
probabilities,which have very convenient analytical properties. For example,
we will see momentarilythat this is the case for divergence preferences, and
therefore for the multiplierpreferences of Hansen and Sargent (2001) and for
the mean-variancepreferencesof Markowitz(1952) and Tobin(1958). For this
reason, here we consider this technical extension.
Fortunately,in our setting we can still use the monotone continuity axiom
introduced by Arrow (1970) to derive a SEU representationwith a countably
additive subjectiveprobability(see Chateauneuf, Maccheroni,Marinacci,and
Tallon (2005)).
AXIOMA.8-Monotone Continuity: If f, g E FT,x E X, {En,1> E with
E D2E2 D ... and n,,>E, = 0, then f >- g implies that thereexists no > 1 such
that xE,,f >- g.
Next we state the countablyadditiveversion of Theorem 3. Here Ad = AU (X)
denotes the set of all countablyadditiveprobabilitiesdefined on a ur-algebraZ,
while A((q) denotes the subset of Ad that consists of all probabilities that are
absolutelycontinuous with respect to q; i.e., A'((q) = {p E d: p <<q)}.
THEOREM
13: Let > be an unboundedvariationalpreference.Thefollowing
conditionsare equivalent:
(i) Therelation satisfiesAxiom A.8.
(ii) For each t > 0, {p E A : c*(p) < t} is a (weakly)compactsubsetof AU.
VARIATIONALREPRESENTATIONOF PREFERENCES
1461
In this case, thereexistsq E Az such that,for all f, g E F,
(11)
f g
u(f
(f~~~(S(R
Amin
u(f)dp+c*(p)>
u(g)dp+c*(p).
qmin
Lemma 30 in Appendix B shows that even when the preference is not unbounded, Axiom A.8 still implies the countable additivityof the probabilities
involved in the representation.
In view of these results, we call the variational preferences that satisfy Axiom A.8 continuous.
pEA'Or'(q)
PEAO"(q)
3.5. ProbabilisticSophistication
In this section we characterizevariational preferences that are probabilistically sophisticated, an important property of preferences introduced by
Machina and Schmeidler (1992) that some authors (notably Epstein (1999))
identify with ambiguityneutrality(or absence of ambiguityaltogether).
Our main finding is that what makes a variational preference probabilistically sophisticated is a symmetry property of the ambiguity index. As a result, all classes of variational preferences (for example, multiple priors and
divergence preferences) contain a subclass of probabilisticallysophisticated
preferences characterizedby a suitable symmetryproperty of the associated
ambiguityindices.
Specifically, given a countably additive probability q on the o--algebraZ,
a preference relation > is probabilisticallysophisticated(with respect to q) if,
given f and g in F,
q(s E S: f(s) = x) = q(s E S: g(s) = x) for all x E X = f ~ g.
For example, if S is finite and q is uniform, then probabilistic sophistication
amounts to permutation invariance (i.e., each act is indifferent to all its permutations).
A preference satisfies (firstorder)stochasticdominance (with respect to q)
if, given f and g in F,
x) for all x E X =
x) < q(s c S:g(s)
f g.
q(s E S: f(s)
Notice that a preference that satisfies stochastic dominance is probabilistically
sophisticated.
To characterizeprobabilisticsophistication,we need to introduce a few well
known notions from the theory of stochastic orders (see, e.g., Chong and Rice
(1971) and Schmeidler (1979)). Define a partial order >-,, on AZ(q) by
p
p' iff fJf(
'dq
>cx
dq>
f'
J'
(dpf)dq
dq j~
1462
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
for every convex function 0 on R. This is the so-called convexorderon probability distributions,and by classic results of Rothschild and Stiglitz (1970) and
Marshalland Olkin (1979) we have p >-~x p' when the "masses"dp(s) are more
scatteredwith respect to dq(s) than the masses dp'(s).
An important property of the convex order >-c is that its symmetric part ~cx
coincides with the identical distributionof the densities with respect to q. That
is, given p and p' in A "(q),
(12)
p ex, '
iff
dp
q seS:
(s)<t
dq
=q( sS:
dp' (s)<t
VteR.
dq
For p in AU(q),the set O(p) =- {p' e A (q): p' -,, p} is called the orbitof p.
A function c: A -+ [0, oo] is rearrangementinvariant(with respect to q) if
dom cC AO(q)and, given p and p' in Ad(q),
P
c(p) = c(p'),
cx pi
whereas it is Shurconvex(with respect to q) if dom c C A'(q) and, given p and
p' in A (q),
(13)
p
pc'p'
=
c(p) > c(p').
A Shur convex function is clearly rearrangementinvariant,whereas the converse is, in general, false. A subset C of Ao(q) is Shur convex (with respect
to q) if its indicatorfunction 8c is Shur convex.7Finally,we say that q in Ad is
adequateif either q is nonatomic or S is finite and q is uniform.
We can now state our characterizationresult, which shows that rearrangement invarianceis the symmetrypropertyof ambiguityindices that characterizes probabilisticallysophisticatedpreferences.
THEOREM14: Let > be a continuous unbounded variationalpreference.If
q E Ao is adequate, then the following conditions are equivalent (with respect
to q):
(i) Therelation satisfiesstochasticdominance.
(ii) Therelation> is probabilisticallysophisticated.
invariant.
(iii) Thefunction c* is rearrangement
c*
Shur
The
is
convex.
function
(iv)
Moreover,for any variationalpreference>, (iv) implies (i) even if q is not adequate.
7Thatis, {p' e A•(q): p' -<cxp}
C for every p E C.
VARIATIONALREPRESENTATIONOF PREFERENCES
1463
The proof of Theorem 14 builds on the results of Luxemburg (1967) and
Chong and Rice (1971), as well as on some recent elaborations of these results
provided by Dana (2005).
As a first straightforwardapplication of Theorem 14, observe that if a set
of priors is Shur convex, then the corresponding (continuous) multiple priors
preferences are probabilisticallysophisticated and the converse is true in the
adequate case. More is true in this case: multiple priors preferences are probabilisticallysophisticated if and only if the indicator functions Sc of their sets
of priors C are rearrangementinvariant,that is, if and only if the sets C are
orbit-closed (O(p) c C for every p e C).8
To furtherillustrateTheorem 14, we now introduce a new class of variational
preferences that plays an importantrole in the rest of the paper. As before, assume there is an underlyingprobabilitymeasure q E Ad. Given a X-measurable
function w:S --+ R with infs,s w(s) > 0 and f w dq = 1, and a convex contin-
uous function &P
:R+ *R+
such that 0 (1) = 0 and limt,,,
(t)/t = 00, the
w-weighted 0 divergence of p E A with respect to q is given by
(14)
w(s)
Dw(pIIq)=
00,
A(q),
(s))
dqr(s), otherwise.
ifspe
Here w is the (normalized) weighting function, and in the special case of uniform weighting w(s) = 1 for all s E S, we just write DO(p IIq) and we get
back to standard divergences, which are a widely used concept of "distance
between distributions"in statistics and informationtheory (see, e.g., Liese and
Vajda(1987)). The two most important divergences are the relativeentropy(or
Kullback-Leibler divergence) given by 4 (t) = t In t - t + 1 and the relative Gini
concentration index (or X2 divergence) given by 0 (t) = 2-' (t - 1)2
The next lemma collects the most important properties of weighted divergences.
LEMMA15: A weighted divergence Dw(. 1q) : A -- [0, oo] is a grounded, con-
vex,and lowersemicontinuousfunction, and the sets
(15)
{p EA:D (p IIq) < t}
are (weakly)compactsubsetsof Ad(q) for all t E R. Moreover,Do(. IIq) :A --+ R
is Shurconvexwheneverw is uniform.
8Thisfact can be used to provide an alternativederivationof some of the results of Marinacci
(2002), which showed that multiple priorspreferences that are probabilisticallysophisticatedreduce to subjectiveexpected utilitywhen there exists even a single nontrivialunambiguousevent.
The direct proofs in Marinacci(2002) are, however, shorter and more insightfulfor the problem
with which that articlewas dealing.
1464
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
Thanks to the foregoing properties, preferences represented by the functional
(16)
V(f) =
u(f)dp +
minf
OD(p
I
q)}
where 0 > 0 and u: X -+ IRis an affine function, belong to the class of variational preferences. In view of their importance,we call them divergencepreferences; that is, > on Y is a divergence preference if
ft>g
min
>
u(f)dp+ OD(p IIq)
minEm
u(g)dp + OD(p
| q) }
THEOREM
16: Suppose u(X) is unbounded.Then divergencepreferencesare
continuousvariationalpreferenceswithindexof ambiguityaversiongivenby
=
c*(p) OD=(p
|| q) Vp EA.
In particular,thesepreferencesare probabilisticallysophisticatedwheneverw is
uniform.
Divergence preferences are a very importantclass of variationalpreferences
and we further study them in the next section. Unlike multiple priors preferences, they are in general smooth (see Proposition 23), a noteworthyfeature
for applications.
For a finite state space S, X = R, u(t) = t, and w uniform, some classes of
divergence preferences have been considered by Ben-Tal(1985) and Ben-Tal,
Ben-Israel, and Teboulle (1991). In the next section, we will study two significant examplesof divergencepreferences,correspondingto the relativeentropy
and to the relative Gini concentration index. Here it is important to observe
that, by Theorem 16, all divergence preferences represented by
V(f)
in
u(f)dp+OD(p(lq)p
,
are examples=Iqmm
of probabilisticallysophisticatedvariationalpreferences.'
the
next example shows that even under minimal nonuniformiHowever,
ties of the weighting function, divergencepreferences are in general not probabilisticallysophisticated and they exhibit Ellsberg-typebehavior. Therefore,
'Analogously, Theorem 14 and Lemma 15 show that all multiplepriorspreferenceswith sets of
priors {p e A'I(q):D4(p IIq) < r}j are probabilisticallysophisticated.These preferencesinclude
the constraintpreferences of Hansen and Sargent(2001), mentioned in footnote 2.
VARIATIONALREPRESENTATIONOF PREFERENCES
1465
the probabilisticsophisticationof divergence preferences cruciallydepends on
the uniformityof the weight w.
17: Considera standardEllsberg three color urn,with 30 red balls
EXAMPLE
and 60 balls either green or blue. As usual, consider the bets:
Red
fR
1
Green
0
fG
0
1
0
1
0
1
fRuB
fGUB
Blue
0
0
1
1
where fR pays one dollar if a red ball is drawn and nothing otherwise, fG pays
one dollar if a green ball is drawnand nothing otherwise, and so on. As is well
known, Ellsberg (1961) arguedthat most subjects rankthese acts as
(17)
fR >f
and
fRUB-<fGUB.
Consider a decision maker who has divergence preferences represented by
the preference functional V given by (16). Here it is natural to consider a
uniform q on the three states. Without loss of generality, set u(0) = 0 and
u(1) = 1. By Theorem 16, when w is uniform, it cannot be the case that
V(fR) > V(fG) and V(fRUB) < V(fGUB). However, consider the weighting function w:{R, G, B} -+ IRgiven by
w(R) = 1.01,
w(G) = 0.99,
and
w(B) = 1,
which is only slightlynonuniform.If we set 0 = 1 and take either the weighted
relative entropy 0 (t) = t ln t - t + 1 or the weighted Gini relative index P(t) =
2-' (t - 1)2, then some simple computations (availableupon request) show that
V(fR) > V(fG)
and V(fRUB) < V(fGUB),
thus deliveringthe Ellsberg pattern (17).
3.6. Smoothness
Most economic models are based on the optimization of an objective function. When this function is differentiable, solving the optimization problem
is easier and the solution has appealing properties. There is a well established
set of techniques (first-ordernecessaryconditions, envelope theorems, implicit
function theorems, and so on) that are extremely useful in both finding a solution of an optimizationproblem and characterizingits properties.For example,
they make it possible to carryout comparative statics exercises, a key feature
in most economic models.
1466
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
In view of all this, it is importantto studywhether our variationalpreference
functionals are differentiable. In this section (see Theorem 18), we fully characterize the differentiabilityproperties of variational preference functionals
and show that they are adequate for economic applications.
Throughout this section, we assume that X is the set of all monetary lotteries, that is, the set of all finitely supported probabilitymeasures on R. An
act f of F is monetaryif f(s) is a degenerate lottery for every s e S, that is,
if f(s) e IR(with the usual identificationof z IRwith the degenerate lottery
dz E X). The set of all monetary acts can thus be identified with Bo(2).
We consider a variationalpreference functional
V(f) = min u(f) dp+ c*(p)}
restrictedto Bo(I), that is, restrictedto monetaryacts. We also make the standard assumptionthat the associated utility function u is concave (thus reflecting risk aversion), strictlyincreasing,and differentiableon R.
To state our results, we need some standard notions of calculus in vector
spaces (see Rockafellar (1970) and Phelps (1992)). Given f E Bo(I), the diR
rectionalderivativeof V : Bo(E) -+ R at f is the functional V'(f; .): Bo()
-defined by
+ th) - V(f)
V(f
V'(f; h) = lim V(f
hV(f th)t
t4o
Bo().
The functional V is (Gateaux) differentiableat f if V'(f; -) is linear and supnorm continuous on Bo(2). In this case, V'(f; .) is the (Gateaux) differential
of V at f.
The superdifferentialof V at f is the set 8V(f) of all linear and supnorm
continuous functionals L: Bo(1) -*+ R such that
V'(f; h) < L(h)
VheBo(X).
In particular,dV(f) is a singleton if and only if V is differentiableat f. In this
case, dV(f) consists only of the differential V'(f; -) ; i.e., dV(f) = {V'(f; -)}.
We are now ready to state our result. It provides an explicit formula for
the superdifferentialdV(f) at every f E Bo(E) and a full characterizationof
differentiability,along with an explicit formula for the differential.
18: For all f e Bo(e),
THEOREM
(18)
dV(f)-=
u'(f)dr:reargmin
u(f) dp+c*(p)).
VARIATIONALREPRESENTATIONOF PREFERENCES
1467
In particular,V is everywheredifferentiableon Bo(X) if and only if c*is essentially
strictlyconvex.'0In this case,
(19)
V'(f; h) = f hu'(f) dr,
where{r} = argminpEa(fu(f) dp + c*(p)).
The strict convexity of the ambiguity index thus characterizes everywhere
differentiablevariationalpreference functionals. For example, Proposition 23
will show that divergence preferences have a strictly convex ambiguityindex,
provided 0 is strictly convex. By Theorem 18, they are everywhere differentiable.
Variationalpreferences that feature an index c that is not strictlyconvex are,
by Theorem 18, not everywhere differentiable in general. This is a large class
of preferences, which includes, but is much larger than, that of multiple priors
preferences.
However, although there are plenty of examples of variational preferences
that are not everywheredifferentiable,since the convex combination of a convex cost function and a strictlyconvex function is strictly convex, one can approximate arbitrarilywell any variational preference with another one that is
everywheredifferentiable.
In view of all this, the interest of Theorem 18 is both theoretical and practical. In applications, the explicit formulas (18) and (19) are very important because they make possible the explicit resolution of optimal problems
based on the variationalpreference functional V. It is worth observing that in
Maccheroni, Marinacci,and Rustichini (2006) we show that a version of these
formulasholds also for dynamicvariationalpreferences, the intertemporalversion of the variationalpreferences we are introducingin this paper.
On the theoretical side, Theorem 18 is interestingbecause in some economic
applications of ambiguity aversion, the lack of smoothness (and in particular the existence of "kinks"in the indifference curves) has played a key role.
For example, it has been used to justify nonparticipationin asset markets (see
Epstein and Wang (1994)). By fully characterizingthe differentiabilityof variational preferences, Theorem 18 clarifiesthe scope of these results. In particular, it shows that although kinks are featured by some important classes of
ambiguityaverse preferences, they are far from being a property of ambiguity
aversionper se. Indeed, we just observed in the preceding text that it is always
possible to approximateany variationalpreference arbitrarilyclosely with another one that is everywheredifferentiable.
10Here u'(f) dr denotes the functional on Bo(E) that associates f hu'(f) dr to every
h E Bo(2), and c is essentially strictly convex if it is strictly convex on line segments in
argminpeA(fu(f) dp + c*(p)). Clearly,if c is strictlyconvex, then a fortiori it is essenUfeB0(V)
tially strictlyconvex.
1468
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
4. SPECIAL CASES
In this section we study in some more detail two important classes of variational preferences: the multiple priors preferences of Gilboa and Schmeidler
(1989) and the divergence preferences we just introduced. In particular,we
show that two importantclasses of preferences-the multiplierpreferences of
Hansen and Sargent (2001) and the mean-variance preferences of Markowitz
(1952) and Tobin (1958), are special cases of divergence, and therefore variational, preferences.
4.1. MultiplePriorsPreferences
Begin with the multiple priors choice model axiomatized by Gilboa and
Schmeidler (1989). As we mentioned in Section 3.1, the multiple priorsmodel
is characterizedby Axiom A.2', a stronger version of our independence Axiom A.2. Next we show in greater detail the relationshipbetween Axiom A.2'
and the variationalformula (6).
In particular,when Axiom A.2' replaces Axiom A.2, the only probabilities
in A that "matter"in the representation (6) are those to which the decision
maker attributes "maximumweight," that is, those in argminc*. The set of
priors C used in the multiple priors model is then given by {p e A: c*(p) = 0}.
19: Let > be a variationalpreference.Thefollowing conditions
PROPOSITION
are equivalent:
(i) Therelation satisfiesAxiom A.2'.
(ii) For all f E T,
(20)
min
u(f)dp+c*(p)p)
=
mm
u(f)dp.
is also equivalentto the following
If, in addition, is unbounded,then (ii)inf=)
condition:
(iii) Thefunction c* takeson only values 0 and oc.
>
The characterizationof the multiple priors model via Axioms A.1, A.2', and
A.3-A.6 is due to Gilboa and Schmeidler(1989). Proposition 19 shows how the
multiple priors model fits in the representationwe established in Theorem 3.
As is well known, the standardSEU model is the special case of the multiple
priors model characterizedby the following strongerversion of Axiom A.5.
AXIOMA.5'-Uncertainty Neutrality: If f, g e F and a e (0, 1),
af + (1 - a)g f.
f~g g
In terms of our representation, by Theorem 3 and Proposition 19 we can
make the following statement:
VARIATIONALREPRESENTATIONOF PREFERENCES
1469
20: Let > be a variationalpreference.Thefollowing conditions
COROLLARY
are equivalent:
(i) The relation> satisfiesAxiom A.5'.
(ii) The relation - is SEU.
satisfiesAxiom A.2' and {p e A: c*(p) = 0} is a singleton.
(iii) The relation
If, in addition, is unbounded,then (iii) is also equivalentto the statement:
(iv) Thereexistsq e A such that c*(q) = 0 and c*(p) = oc for everyp 0 q.
4.2. DivergencePreferences
In the previous section we introduced divergence preferences so as to illustrate our results on probabilisticsophistication.Here we discuss their ambiguity attitudes.
Recall that a preference > on T is a divergence preference if
ftg
min
d
u(f)dp+
O6D'(p 1|q)}
> minf u(g)dp+ OD(p q) ,
where 0 > 0, q is a countably additive probability on the o--algebra Z,
u:X -* R is an affine function, and D q(.IIq): A -> [0, oo] is the w-weighted
4 divergence given by (14).11
By Theorem 16, all divergence preferences are examples of continuous variational preferences. As a result, to determine their ambiguityattitudes,we can
invoke Propositions 7 and 8. By the former result, divergence preferences are
ambiguity averse. As to comparative attitudes, the next simple consequence
of Proposition 8 shows that they depend only on the parameter 0, which can
therefore be interpreted as a coefficient of ambiguityaversion.
21: Given two (w, 4) divergencepreferences>- and >2, thefolCOROLLARY
are equivalent:
conditions
lowing
relation
The
(i)
>1 is moreambiguityaversethan
2*.
0 02 (provided = u2).
and
u,
u2
(ii)
ul
<_
According to Corollary21, divergence preferences become more and more
(less and less, resp.) ambiguityaverse as the parameter 0 becomes closer and
closer to 0 (closer and closer to oc, resp.).
The limit cases where 0 goes either to 0 or to oo are described by Proposition 12, which takes an especially stark form for divergence preferences under
"1Whenwe want to be specific about w and p, we speak of (w, 4) divergence preferences.
1470
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
some very mild assumptions.To see this, we need a piece of notation: given a
simple measurablefunction ?p:S -- R, set
essmin
seS
= max{t e R:q({s e S: (s) > t}) = 1}.
•p(s)
For example, when q has a finite support supp(q), we have
(21)
essmin~p(s)=
min ?p(s).
seS
sEsupp(q)
PROPOSITION
22: (i) If w e Lo(q), then for all f
lim min fu(f) dp + ODW(pq)
OtO pe"W(q)Ij
=
F,
essminu(f(s)).
seS
(ii) If 0 is strictlyconvex,thenfor all f e F,
lim min{
u(f)
dp+ODp(p
Otoo pEMO(q)
fU0J
jq)
=u(f)dq.
When w E L"(q), divergence preferences therefore tend more and more,
as 0 - 0, to rank acts according to the very cautious criterion given by
essmin,,s u(f(s)). In contrast, when ( is strictly convex, divergence preferences tend more and more, as 0 -+ o, to rank acts accordingto the SEU criterion given by f u(f) dq.
The next result, a simple consequence of Theorem 18, shows that divergence
preferences are smooth under the assumption that 0 is strictlyconvex (most
examples of divergence Dw satisfy this condition; see, e.g., Liese and Vajda
(1987)). This is an important feature of divergence preferences that makes
them especially suited for optimization problems and that also differentiates
them from multiple priors preferences, which are not everywhere differentiable (see, e.g., Epstein and Wang(1994, p. 295)).
As we did for Theorem 18, we assume that X is the set of all monetary
lotteries and we regard Bo(.) as the collection of all monetary acts. We also
assume that the utility function u is concave, strictlyincreasing, and differentiable on R.
23: If 4 is strictlyconvex, then Dw(.1I q):A(X) - [0, oc] is
PROPOSITION
its effectivedomain and the variationalpreferencefunctional
on
convex
strictly
V : Bo(E) --+R, given by
V(f=min
OD(p
0 II q)f
PEA(X)
f u(f)dp+
1~piJYfBZ)
Vfe
B0(B),
VARIATIONALREPRESENTATIONOF PREFERENCES
1471
is everywheredifferentiablefor all 0 > 0. In this case,
(22) V'(f;h)= f hu'(f)dr,
where{r} = argminpEA(fu(f) dp + ODw(pIIq)).
Consider, for example, multiplier preferences, a special class of divergence
preferences in which c*(p) = OR(p IIq). By some well known properties of the
relative entropy (see Dupuis and Ellis (1997, p. 34)), formula (22) takes, in this
case, the neat form
(f h)
exp(- (f) dq
dq
dq
h)exp(-)
( f;hu'(f)
f exp(-
0)
for all f, h E Bo(V).
Unlike the multiple priors case, for divergence preferences, we do not have
an additional axiom that, on top of Axioms A.1-A.6, would deliver them (for
multiplepriorspreferences, the needed extra axiomwas Axiom A.2'). We hope
that this will be achieved in later work and in this regard it is worth observing
that the ambiguityindex D'(p IIq) is additively separable, a strong structural
property.
After having established the main properties of divergence preferences, we
now move on to discuss two fundamental examples of this class of variational
preferences.
4.2.1. Entropicand multiplierpreferences
We say that a preference >- on F. is an entropicpreferenceif
f
g
(min
min
u(f) dp +ORW(p| q))
(f
u(g)dp+ORW(p| q)),
where 0 > 0, q e A , u: X -- R is an affine function, and RW"(I q) : A -+ [0, oo]
is the weighted relative entropy given by
RW(p 11q) =
s
w(s)
00,
5
dq
-q(s)log
if pE
dAc (q),
otherwise.
dq
q(s)
- -(s)
dq
+ 1 dq(s),
1472
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
Since the entropy RW(p IIq) is a special case of divergence Dv(p IIq) defined
in (14), where 4)(t) = t log t - t + 1, entropic preferences are an example of divergence preferences. Hence, by Theorem 16, they are continuous variational
preferences,with index of ambiguityaversion given by
c*(p) = ORW(pIIq) Vp E A.
When w is uniform, is probabilisticallysophisticatedand it features (a positive multiple of) the standardrelative entropy R(. 1Iq) as the index of ambiguity aversion. This is the case considered by Hansen and Sargent (see, e.g.,
Hansen and Sargent (2000, 2001)), who called this class of entropic preferences in which acts are ranked accordingto
V(f)
=min
(f u(f)dp+ OR(p| q))
multiplierpreferences.Although multiplierpreferences are probabilisticallysophisticated, Example 17 shows that this not the case for general entropic
preferences that have nonuniformweightingfunctions w. These "nonuniform"
entropic preferences thus provide a specification of preferences that can, in
general, produce Ellsberg-typebehavior-and so are ambiguityaverse according to all notions of ambiguityavailable in literature-but that also retain the
good analyticaltractabilityof multiplierpreferences.
As to the ambiguity attitudes featured by entropic preferences, by Corollary 21 they are characterizedby the parameter 0 as follows: the lower 0 is,
the more ambiguity averse is the entropic preference. The parameter 0 can
therefore be interpreted as a coefficient of ambiguityaversion.
We have shown how entropic,and hence multiplier,preferences are a special
case of divergence preferences. As we already observed, for divergencepreferences we do not have an additional axiom that, on top of Axioms A.1-A.6,
would deliver them (even though we have been able to point out some strong
structuralproperties that their ambiguityindices satisfy). On the other hand,
we view entropic preferences as essentially an analyticallyconvenient specification of variationalpreferences, much in the same way as, for example,CobbDouglas preferences are an analyticallyconvenient specificationof homothetic
preferences. As a result, in our setting there might not exist behaviorallysignificant axioms that would characterize entropic preferences (as we are not
aware of any behaviorallysignificantaxiom that characterizesCobb-Douglas
preferences). Similarconsiderationsapply to the Gini preferences that we will
introduce momentarily.
For their macroeconomic applications, Hansen and Sargent are mostly interested in dynamic choice problems. Although our model is static, in the
follow-up paper (Maccheroni, Marinacci,and Rustichini (2006)), we provide
a dynamicversion of it and, inter alia, we are able to provide some dynamic
VARIATIONALREPRESENTATIONOF PREFERENCES
1473
specificationsof multiplierpreferences that are time consistent. As is often the
case in choice theory, also here the analysisof the static model is key to paving
the way for its dynamicextension.
We close by discussing the related work of Wang (2003), who (in a quite
different setting) recentlyproposed an axiomatizationof a class of preferences
that include multiplierpreferences as special cases. He considered preferences
over triplets (f, C, q), where f is a payoff profile, q is a reference probability,
and C C A is a confidence region. For such preferences, he axiomatized the
representation
(23)
V(f, C, q)=min{u(f)
peCIf
dp +OR(p li q)J.
His modeling is very differentfrom ours: in our setup, preferences are defined
only on acts, and we derive simultaneouslyboth the utility index u and the ambiguityindex c; that is, uncertaintyis subjective.In Wang(2003), both C and q
are exogenous, and so uncertaintyis objective; moreover, agents' preferences
are defined on the significantlylarger set of all possible triplets that consist of
payoff profile, confidence region, and reference model.
In any case, observe that when (23) is viewed as a preference functional
on F, then it actually represents variational preferences that have as ambiguity index the sum of Sc and OR(-IIq). As a result, Wang'spreferences are a
special case of variationalpreferences once they are interpretedin our setting.
4.2.2. Gini and mean-variancepreferences
We say that a preference ? on F is a Ginipreferenceif
fg
min(f u(f) dp+?OGW(p11q))
min
u(g)+POGw(p||q)),
dp
•
where 0 > 0, q e An, u: X -- R
is an affine function, and G"(. IIq): A -- [0, co]
is the weighted relative Gini index given by
G"(p IIq)
=
w(s)dp
00c,
(s) - 1
dq(s),
if p
EA•(q),
otherwise.
Like the weighted relative entropy R"(p IIq), the Gini index GW(p 11q)
also is a special case of divergence Dw(p IIq) defined in (14), with 4(t) =
1474
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
2-1(t - 1)2.12 As a result, by Theorem 16, Gini preferences are continuousvari-
ational preferences, with ambiguityindex OG"(. IIq).
In particular, - is probabilisticallysophisticatedwhen w is uniform. In this
case, when X is the set of all monetary lotteries and u(t) = t for all t E R,
we call the restrictionof these preferences to the collection Bo(X) of all monetary acts monotone mean-variancepreferences,written >mmv; that is, for all
f, ge Bo(1),
mmV
f
g
i
Q
(ffdp+
OG(p I q))
gdp+OG(p
iq))
>amin(
Since the Gini index is, along with Shannon's entropy, a classic concentration
index, monotone mean-variance preferences are a natural example of divergence preferences. However, we are not considering them just for this: their
main interest lies in the close connection they have with mean-variance preferences.
In fact, consider the classic mean-variance preferences of Markowitz(1952)
and Tobin(1958) that are defined on Bo(X) by
(24)
f
-mv g
f dq
-
g dq
Var(f)
--
Var(g),
where Var is the variance with respect to q. These preferences are not
monotone unless their domain is suitably restricted to the set M on which
the (Gateaux) differential of the mean-variance functional f ? f fdq (1/20) Var(f) is positive (as a linear functional). The convex set M, called the
domain of monotonicity of >mv, is where these preferences do not violate the
monotonicityAxiom A.4; that is, where they are economically meaningful.
24: Thedomain of monotonicityM of >-mvis the set
THEOREM
jf
e Bo() :f -
f
f dq 0,
<
q-a.s.}.
Moreover,
ftmg
f d (p+
(min
'min(
G(q)
I
gdp + OG(p q))
12Theclassic Gini concentration index can be obtained by normalizationfrom the relative
index (withuniformw) in the same way Shannon'sentropycan be obtainedfrom relativeentropy.
VARIATIONALREPRESENTATIONOF PREFERENCES
1475
for all f, g E M.
By Theorem 24, mean-variance preferences coincide with monotone meanvariance preferences once they are restricted on their domain of monotonicity M, which is where they are meaningful.
Inter alia, this result suggests that monotone mean-variance preferences
are the natural adjusted version of mean-variance preferences that satisfy
monotonicity. This insight is developed at length in Maccheroni, Marinacci,
Rustichini, and Taboga (2004), and we refer the interested reader to that paper for a detailed analysisof this class of variationalpreferences.
Istitutodi Metodi Quantitativiand IGIER, UniversitaBocconi, Milano, 20100
Italy,
Dipartimentodi Statisticae MatematicaApplicataand CollegioCarloAlberto,
Universitui
di Torino,Torino,10122 Italy;[email protected],
and
Dept. of Economics, Universityof Minnesota,Minneapolis,MN 55455, U S.A.
ManuscriptreceivedAugust,2004;final revisionreceivedApril,2006.
APPENDIX A: NIVELOIDS
The set of all functions in Bo(2) (resp. B(2)) that take values in the interval
K C R is denoted by B0(2, K) (resp. B(X, K)).
When endowed with the supnorm,Bo(Y) is a normed vector space and B(2?)
is a Banach space. The norm dual of Bo0() (resp. B(.)) is the space ba(.)
R endowed with the
of all bounded and finitely additive set functions
:'X-for all ?cE B0(E) (resp.
total variationnorm, the dualitybeing (?P,/t) = f cpdlt
B(,Y)) and all / E ba(2) (see, e.g., Dundorf and Schwartz(1958, p. 258)).
As is well known,the weak* topologies o-(ba(.X),Bo(N)) and o-(ba(2), B(.))
coincide on A(Y). Moreover, a subset of A7(X) is weakly* compact if and only
if it is weakly compact (i.e., compact in the weak topology of the Banach space
ba (X)). For pI, I E B(s), we write ?p> tp if ?p(s) > f&(s)for all s E S.
if
A functionalI: 0 -- R, defined on a nonemptysubset P of B(2), is a niveloid
I((p) - I($i) < sup((p - 0)
for all ?p, f E 0; see Dolecki and Greco (1995). Clearly a niveloid is Lipschitzcontinuous in the supnorm. A niveloid I is normalizedif I(kls) = k for all
k e R such that k1s ec0. With a little abuse, we sometimes write k instead
of k1s.
LEMMA25: Let 0 E int(K).A functional I: BO0(.,K) -+ R is a niveloid if and
only if,for all cp, f E Bo(Z, K), k E K, and a E (0, 1),
(i) c > q impliesI(pc) > I(ti), and
(ii) I(atp + (1 - a)k) = I(acp) + (1 - a)k.
1476
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
In this case, I is concave if and only if
(iii) I(tf) = I(Q?p)implies I(aqf + (1 - a) p) > I((p).
Properties (i) and (ii) are called monotonicityand verticalinvariance,respectively. The proof of Lemma 25 can be found in Maccheroni, Marinacci,and
Rustichini (2005). If I:(P
I(I) = sup [I()
-*
R is a niveloid, then
+ inf(qi(s) - p(s))]
iV e B(I)
is the least niveloid on B(s) that extends I (see Dolecki and Greco (1995)).
Moreover:
* If ( is convex and I is concave, then I is concave.
* If (P+ R Bo(),13 then I is the unique niveloid on B(X) that extends I (see
Maccheroni,
_ Marinacci,and Rustichini (2005)).
If 0 is convex and I: (P -- R is a concave niveloid, direct application of
the Fenchel-Moreau theorem (see, e.g., Phelps (1992, p. 42)) to I guarantees
that
I(p)
=
minLEba(5)((p, /-) - I*(Aj)), where I*(1i) = infqB(Y)((q, p ) - I(0)) is the
Fenchel conjugate of I. If At is not positive, there exists qp> 0 such that
a((p, /t) - I(0) for all a >0, whence
(?, /t) < 0. Then (ap, ) - I(a?p)
=
=
If
+ b) =
,t(S)
1, choose q _ B(2). Then (41 + b, F) I*(QL) -00.
I(0q
and so I*(1i) = -oc. That is,
(q, /-) - I(jq) + b(CL(S) - 1) for all b
•R
(cp) = min ((?, p) - I*(p));
(25)
pEA(?)
see also Fllmer and Schied (2002, Theorem 4.12). Set I*(p) = I*(p) for each
pe
p E A(s) and set d,I(4o) = {p
A():I(qj)
- I(?p) < (q -
n,p) for each
The next two results are proved in Maccheroni, Marinacci, and Rustichini
(2005).
LEMMA26: Let 0 be a convexsubsetof B(s) that contains at least one constantfunction, and let I : -- R be a concaveand normalizedniveloid.Then:
(i) Foreach p e A(E), I*(p) = infEP((I, p) -I(-q)).
(ii) I* :A(X) -+ [-oo, 0] is concave and weakly*uppersemicontinuous.
p) - I*(p)) and it is not
(iii) For each Ee 0, ,I(p?) =
argminpEA(z•(((p,
empty.
= {I* = 0} = argmaxpeAJ()I*(p) for all k E R such that
(iv) dI(kls)
kls E (P.
13p?+ R is the set {~c+ b: 0E, b E R}. An importantspecial case in which P + R = Bo(E) is
unbounded.
when 0 = Bo(2, K) and K is•p
VARIATIONALREPRESENTATIONOF PREFERENCES
1477
(v) I* is the maximalfunctional R" A(X) - [-oc, 0] such that
(26)
inf ((p, p)- R(p))
I(po)= pEA(l)
Vo E P.
Moreover,if P + R 2 Bo(2), 1*is the uniqueconcave and weakly*upper
semicontinuous function R :A()) - [-oo, 0] such that (26) holds.
(vi) If (26) holds and T 0 is such that sups,,s (s) - infses ii(s) < b for all
j
, then
_
iqe
(27)
I(f) =
((p , p) - R(p))
inf
{peAd():R(p)>-b}
V i E T.
PROPOSITION27: Let I: Bo(2, K) --+ R be a normalized concave niveloid,
Thenthefollowing conditionsare equivawith K unboundedand L a ur-algebra.
lent:
E . withE1 D E2 D . and n,• En =
(i) If c, q E Bo(X, K), k E K,
{En•}>i
that thereexistsno > 1 such.that
0, then I (p) > I ( /) implies
I(klEn0
+ (lE)>
1).
(ii) The set {p E A(X) : I*(p) > b} is a weaklycompact subset of A((Y) for
each b< 0.
(iii) Thereexists q E d"(X) such that {p E A() :I*(p) > b} is a weaklycompact subsetof Ao(X, q) for each b < 0, and,for each 4 E Bo(I, K),
(28)
I(G) =
mi
pEA0(.,q)
APPENDIX
((, p) -
*(p)).
B: PROOFS OF THE RESULTS IN THE MAIN TEXT
The main result proved in this appendix is Theorem 3. Its proof proceeds
as follows: using Lemma 28, we first show that if - satisfies Axioms A.1-A.6,
then there exists a nonconstantaffine function u: X -- R and a normalized and
concave niveloid I: Bo(0, u(X))
R such that f >- g if and only if I(u(f))
>
-A delivers the desired variational
I(u(g)). Then (25) obtained in Appendix
representation I(u(f))
= minp,,(n)(f u(f) dp + c(p)).
We now move to the proofs. The standardproof of Lemma 1 is omitted.
LEMMA28: A binaryrelation > on F satisfiesAxioms A.1-A.4 and A.6 if
and only if thereexista nonconstantaffinefunction u :X -- IRand a normalized
niveloid I :Bo(., u(X)) --+ R such that
ft g
I(u(f))
> I(u(g)).
PROOF:Assume that > on . satisfiesAxioms A.1-A.4 and A.6. Let x, y E X
be such that x - y. If there exists z E X such that x + z
y + z, with-
1478
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
out loss of generality ix ?+ z >- y + 1z; by Axiom A.2 (we can replace z
with x to obtain),lx + x > y + x and (we can replace z with y to obtain) ix ?+ y >- y ?+ y, and we conclude that x >- y, which is absurd. Then
the hypotheses of the mixture space theorem (Hernstein and Milnor (1953))
are satisfied and there exists an affine function u:X -+ R such that x 'e y if
and only if u(x) > u(y). By Axiom A.6 there exist f, g e F such that f >-g. Let
x, y E X be such that x >- f(s) and g(s) > y for all s e S. Then x >
f
>- g > y
implies x >-y and u cannot be constant. Moreover, u is unique up to positive
affine transformationsand we can assume 0 E int(u(X)).
For all f e F, let x,y E X be such that x - f(s) >- y for all s E S. Then
x > f > y. By Axiom A.3, the sets {aE [0, 1]:ax + (1 - a)y >- f} and {a E
[0, 1]: ff ax + (1 - a)y) are closed; they are nonempty because 1 belongs to
the first and 0 belongs to the second; their union is the whole [0, 1]. Because
[0, 1] is connected, their intersection is not empty;hence, there exists P3 [0, 1]
such that fx + (1 - 3)y - f. In particular,any act f admits a certaintyequivalent xf E X.
f
and f
If
~
xf, set U(f) = u(xf).
The function U is well defined because
f
- xf
~ y, with xf, yf EX implies xf - yf and u(xf) = u(yf). Clearly,f > g
if and only if xf xg if and only if u(xf) > u(xg) if and only if U(f) > U(g).
Therefore, U represents>-.
If f F, then u(f) e Bo(1, u(X)). Conversely, if p E B0o(, u(X)), then
?p(s) = u(xi) if s e Ai for suitable xl, ..., XN EX and a partition {(A, A2,
S.., AN) of S in X. Therefore, setting f(s) = x, if s e Ai, we have ?q= u(f).
We can conclude that Bo(1, u(X)) =
E F). Moreover, u(f) = u(g) if
S if and only if f(s) - g(s) for all s e S,
and only if u(f(s)) = u(g(s)) for all s E{u(f):'f
and, by Axiom A.4, f -- g or, equivalently,U(f) = U(g).
Define I('p) = U(f) if cp= u(f). By what we have just observed, I:Bo(,
u(X)) -+ R is well defined. If ?c= u(f), 0f= u(g) e Bo(Z, u(X)), and p > q ,
then u(f(s)) > u(g(s)) for all s E S and f(s) - g(s) for all s e S, SOf g,
U(f) > U(g), and I(?p) = I(u(f))
= U(f) > U(g) = I(u(g)) = I(iq). There-
fore, I is monotonic. Take k e u(X), say k = u(x). Then I(k1s) = I(u(x)) =
U(x) = u(x) = k. Therefore, I is normalized.
Take a E (0, 1), ?P= u(f) e Bo(., u(X)), and k = u(xk) e u(X); denote
by x0 an element in X such that u(xo) = 0. Choose x, y E X such that x>
f (s) >-y for all s e S. Then ax +(1- a)xo > af (s) + (1- a)xo >- ay + (1- a)xo
for all s e S. The technique used in the second paragraphof this proof yields
the existence of p E [0, 1] such that P(ax + (1 - a)xo) + (1 - P)(ay + (1 a)xo) - af + (1 - a)xo, i.e., az + (1 - a)xo - af + (1 - a)xo, where z =
pix + (1 - P)y E X. Then, by Axiom A.2, az + (1 - a)xk - af + (1 - a)xk
and
I(ap? +(1 - a)k)
= I(u(af + (1 - a)xk))
=
u(az + (1 - a)Xk)
VARIATIONALREPRESENTATIONOF PREFERENCES
1479
= au(z) + (1 - a)k = au(z) + (1 - a)O + (1 - a)k
= u(az + (1 - a)xo) + (1 - a)k
= I(u(af + (1 - a)xo)) + (1 - a)k = I(ap?) + (1 - a)k.
By Lemma 25, I is a niveloid, and we alreadyproved that it is normalized.
Conversely, assume there exist a nonconstant affine function u: X -- R
and a normalized niveloid I: Bo(Z, u(X)) -WR such that f - g if and only
if I(u(f)) > I(u(g)).
Choose c E R such that 0 e int(u(X) + c) and set v = u + c. Define
J:
v(X)) --+ R by J((p) = I('p - c) + c. Notice that J is a normalized
Bo(,, and
niveloidl4
f >g
> I(u(g))
I(u(f))
SI(u(f)+
SI(v(f)
c > I(u(g) + c-c)
c-c)+
- c) + c > I(v(g) - c) + c
SJ(v(f))
+ c
> J(v(g)).
Clearly,> satisfies Axiom A.1.
If f, g E
x, y c X, and a E (0, 1), then av(h), (1 - a)v(z), av(h) + (1 •,
e
a)v(z)
Bo(Y, v(X)) for h = f, g and z = x, y. Moreover,
af + (1 - a)x > ag + (1 - a)x
=
=
=
=
=
+ (1 - a)v(x)) > J(av(g) + (1 - a)v(x))
J(av(f)) + (1 - a)v(x) > J(av(g)) + (1 - a)v(x)
J(av(f)) + (1 - a)v(y) > J(av(g)) + (1 - a)v(y)
J(av(f)
J(av(f)
+ (1 - a)v(y)) > J(av(g) + (1 - a)v(y))
af + (1 - a)y > ag + (1 - a)y
and Axiom A.2 holds.
If f, g, h E , a E [0, 1], and there exists a, E [0, 1] such that ac, - a and
af + (1 - an)g9 h for all n > 1, then v(anf + (1 - an)g) = av(f) + (1 an)v(g) converges uniformly to av(f) + (1 - a)v(g) = v(af + (1 - a)g). The
inequality J(v(anf + (1 - an)g)) > J(v(h)) for all n > 1 and the continuity
of J guarantee J(v(af + (1 - a)g)) > J(v(h)). Therefore, {a E [0, 1]: af +
14Infact, for all ?p,4 E Bo(2, v(X)),
J(p) - J(f)
= I(p( - c) + c - I((
- c) - c < sup((p - c) - (qf - c)) = sup(?p - qf).
Moreover,for all t E v(X), J(t) = I(t - c) + c = t - c + c = t.
1480
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
(1 - a)g - h} is closed. A similar argument shows that {a E [0, 1]: h > af +
(1 - a)g} too is closed, and Axiom A.3 holds.
Given f, g E T, f(s) > g(s) for all s E S if and only if J(v(f(s))) > J(v(g(s)))
for all s if and only if v(f(s)) > v(g(s)) for all s, then monotonicityof J yields
J(v(f)) > J(v(g)). This shows Axiom A.4.
Finally,because v is not constant and it represents on X, there exist x >- y
and Axiom A.6 holds too.
Q.E.D.
PROOFSOF THEOREM3 AND OF PROPOSITION6: Assume > satisfies Ax-
ioms A.1-A.6. By Lemma 28, there is a nonconstant affine function u:X -R
and a normalized niveloid I: Bo(X, u(X)) --+ R such that f - g if and only if
I(u(f))
>_I(u(g)).
Next we show that Axiom A.5
implies that I: B0o(, u(X)) -* R is concave.
Let ?q,1'fe B0o(, u(X)) be such that I(qp) = I((qi) and a E (0, 1). If f, g e .F
are such that op= u(f) and
ii
= u(g), then f - g and, by Axiom A.5, af + (1 -
a)g > f, that is,
I(a••
+ (1 - a)q/) = I(au(f) + (1 - a)u(g))
= I(u(af + (1 - a)g))
> I(u(f)) = I(p).
Lemma 25 guarantees the concavityof I.
The functional I: Bo(X, u(X)) --+R is, therefore, a concave and normalized
niveloid. For all p e A(I), set c*(p) = -I*(p). Lemma 26 guarantees that
c*(p) =
inf
qEBoB( ', u(X ))
((i, p) -I(q))=sup(u(xf)
fET
-
u(f)dp /
for all p E A(.) (where xf is a certaintyequivalent for f), that c* is nonnegative, grounded, convex, and weakly* lower semicontinuous,and that
I((p)
= min
+pc*(p)
(pdp
Y
Vp
Bo(,,u(X));
see also (25).
Let c: A(s) -+ [0, 00] be a grounded, convex, and weakly* lower semicontinuous function such that
fig
p(min
u(f)dp+ c(p)
> min
u(g)dp+ c(p))
VARIATIONALREPRESENTATIONOF PREFERENCES
1481
For all o = u(f) E Bo(X, u(X)),
I(w)=
I(u(f))
=
u(x) = mi
(f
= mm
PEA(-Y)
(/
p(fu(f)dp+c(p))
u(x)
dp +c(p)
= minm
peA(sY)(fdp
(J
+c(p).
Lemma 26 guarantees c* < c (this concludes the proof that (i) implies (ii)).
Moreover, if u(X) is unbounded, again Lemma 26 guarantees c = c* (this
proves Proposition 6).
For the converse, notice that u and I(cp) = minpeA(X)(f~pdp + c(p)) are,
respectively,a nonconstant affine function and a normalized niveloid that represents >. By Lemma 28, > satisfies Axioms A.1-A.4 and A.6. Concavityof I
Q.E.D.
guaranteesAxiom A.5.15
PROOF OF COROLLARY5: Let (uo, c;) represent > as in Theorem 3. If
(u, c*) is another representation of > (as in Theorem 3), by (6), u and uo are
affine representationsof the restrictionof - to X. Hence, by standarduniqueness results, there exist a > 0 and / E IRsuch that u = auo + P. By (7),
c*(p)= sup(u(xf)-
u(f) dp)
=sup
fE-F(
•auo(xf)+1-f(auo(f)+1)dp)
as desired. The converse is trivial.
=ac•*(p),
Q.E.D.
LEMMA29: Let > be a binaryrelationon X representedby an affinefunction
u :X -- R. The set u(X) is unbounded(eitherbelow or above) if and only if j
satisfiesAxiom A.7.
The standardproof is omitted. Proposition 7 is a consequence of Lemma 32.
PROOF OF PROPOSITION8: (ii) trivially implies (i). As to the converse, let
(ui, c7) represent i as in Theorem 3, i = 1, 2, and set Ii(cp)= minpac()(f pdpc7(p)) for all coE Bo(0, ui(X)).
15Iff - g and a E (0, 1), then
I(u(af + (1 - a)g)) = I(au(f) + (1 - a)u(g))
> aI(u(f)) + (1 - a)I(u(g)) = I(u(f)).
1482
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
By (8) and the fact that u1 and u2 are not constant, we can choose
= = For all f e ?, if f ~1x, then f >2 X;therefore, Ii(u(f)) = u(x) <
ul u2 u.
I2(u(f)). This implies I < I12and
=
-
sup
? EBo(1,u(X))
(Ip)
cl(P)
<
fJ/
( dp)
sup
?PEBo(X,u(X))
\f)
I2(P)-fqDdp)=c2(p)
for all p e A(?).
Q.E.D.
PROOFOF PROPOSITION
12: Observe that the functions c, are weakly*
lower semicontinuous on A and so is f u(f) dp + Cn(p) for each n. Using this
observation,we now prove (i) and (ii).
(i) The decreasing sequence {f u(f) dp + c,(p)}n pointwise convergesto
(29)
f
U(f) dp
(P)
•Undomcn
Hence, by Dal Maso (1993, Proposition5.7), this sequence F-converges to
/
u(f) dp +
c, (P)
Un,dom
the weakly* lower semicontinuousenvelope of (29). By Dal Maso (1993, Theorem 7.4), this implies
limmin
u(f)dp+cn(p) =minf
u(f)dp +
cn
5U,,dom
(ii) Since argmin c, = (p E A: cn(p) = 0), the increasingsequence If u(f) dp+
Dal Maso (1993, p. 47), F-converges)
c,(p))}, pointwise converges (and so, by
to
J
u(f) dp + Sn
argmincn
(p)
Because A is weak-* compact, Dal Maso (1993, Theorem 7.8) implies
limmin
n pA I
u(f ) dp + cn(P)
)d
=min
+
u(f) dp +
S
8FnnargminCn(P)
Q.E.D.
VARIATIONALREPRESENTATIONOF PREFERENCES
1483
PROOFOF THEOREM13: Let (u, c*) represent >- as in Theorem 3 and set,
for all cpe B0o(, u(X)), I(Qp)= minpeA( )(f dp + c*(p)). It is easy to check
if I•psatisfiescondition of
that - satisfiesAxiom A.8 on .F if and
only
(i) Proposition 27 on Bo(E, u(X)). Unboundedness of u(X) and the relation c* = -I* allow us to apply Proposition 27 and to obtain the desired equivalence. Q.E.D.
LEMMA
30: Let >- be a variationalpreferencethat satisfiesAxiom A.8. Then
inf
Sg
(f u(f) dp + c*(p)
>
u(f) dp+c*(p))
Vf, gEc.
Xinf(
PROOF:Let (u, c*) represent >- as in Theorem 3 and set I(p) =
minpEA(,)(f dp + c*(p)) for all cpE Bo(E, u(X)). Let p e int(Bo( , u(X))),
E, { 0, and e > 0 such that cp- se Bo(E, u(X)). Then
f
I(lEcp + (minp En-E)E,)
-
I(p) < I(
-
lE) - I( P)
< -ep(E,) < O
for all pee dI(q(). Consider a sequence {kj}jl in u(X) such kj < I(?O)and
kj f I(qo).By Axiom A.8, I satisfies (i) of Proposition27 on Bo(Y, u(X)). Then,
for all j > 1, there is no > 1 such that kj < I(cplE o + (min q E
Hence,
8)lEn0).
m
>
because
the
limn-o I(~plEc + (min - e)lEn)
sequence I(?plEc + (min c kj
limit
to
the
for
j -- oc, we obtain
e) 1En) is increasing. Passing
(30)
lim I(GplE + (minp - 8)1E,)
IE =
n---> oc
)
and p(E,) -- 0 (uniformlywith respect to pe dI(?p)); that is, OdI(p) is a
(weaklycompact) subset of A0(Y). By Lemma 26, minPEA(X)(f?pdp + c*(p)) is
attained for all 'p e int(Bo(Z, u(X))) in (dI(c) and hence in) Az(E). Hence,
I(cp)= (minf~pdp+c*(p)) and
J('p)=
)in(f
'pdpc+ c*(p)
coincide on int(Bo(Z, u(X))) and, being continuous, they coincide on Bo( ,
Q.E.D.
u(X)).
The results of Section 3.5 require some notation and preliminaries.For all
cp e L'(, q), set G,(t) = q({Qp> t}) for each t e R and set T~(b) = inf{t e
R: G,(t) < b} for each b e [0, 1].
1484
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
The function G, is the survivalfunction of ?c, and 7, is the decreasingrearrangementof cp.Two functions c', / e L'(Z, q) are equimeasurableif G,=
GQ(if and only if 7, = F,). We refer to Chong and Rice (1971) for a comprehensive study of equimeasurability.The preorder -<cx defined in Section 3.5
can be naturally regarded as a relation on L'(Z, q) by putting p -<,, ji if and
only if f P(cp)dq < f P(qj) dq for every convex 0 on R.
Analogously, the definition of Shur convexity (resp. rearrangementinvariance) can be spelled for functions T: L(Z, q) -- (-oo, 0o] by requiringthat
p -<cx
implies T(?9) < T(Iq) (resp. p
cx
implies T(o) = T(ql)).
Put ~p-<-<,cxq if and only if fo'1,(b) db < fo'F(b) db for all t E [0, 1]. Then
-,,<c
if and only if qp-<-<c
,
and
f
p dq =
f qjdq
(see, e.g., Chong (1976)).
Moreover, the equimeasurabilityrelation is the common symmetricpart -,cx
of <-<cx and -<cx. Simple manipulation of the results of Luxemburg (1967,
pp. 125-126) yields the following lemma:
LEMMA31-Luxemburg:
Let q be adequate. If T: Bo(Z, K) -+ (-00, 00] is
then
the
invariant,
rearrangement
function definedfor all p e L' (Z, q) by H (S) =
is
5'
Shur convex. If H is monotonic, then p
(p/
dq T(
-<-<cx
supEBO(Z,K) (f
q,))
H
H
implies (?) < (p').
PROOF:Let c' E L1'(, q). By Luxemburg(1967, Theorem 9.1) for all qJE
Bo(Z1,K),
(31)
maxJpfi'dq:
~'~
f' e
=
Bo(,, K)
fF,(t)Fq(t)
dt.
Therefore, H(p) < sup,B,,,(,K)( f0 ,(t)q,(t)dt - T(q,)). Again by (31),
e Bo(,, K) there exists fi' e Bo(V, K) with q,'
such that
q,
-- q'
=
then
dq. Since T(q')
f; F,(t)FI(t) dt =
f~,c'
T(q,),
for all
J
T(q) =
IF,(t)F,(t)dt-
p'dq-
T(qj) =
Jp'dq-
T(')
and
=
H(•o)
sup
qEBo(1,K)
0
F(t)Fq(t)dt-
T(q)
.
If ? -<,x 9?',an inequalityof Hardy(see, e.g., Chong and Rice (1971, pp. 57-58))
delivers H(Sp) < H(?o'). Let cp-<-<,cx '. By Chong (1976, Theorem 1.1), there
is a nonnegative Sc"e L' (2, q) such that Sc+ Sc"-<,, S'. If H is monotonic, then
H((p) < H(Sc + So")< H(cp').
Q.E.D.
1485
VARIATIONALREPRESENTATIONOF PREFERENCES
REMARK1: Analogously, if q is adequate and T: L'(2, q) -+ (-oo, oo] is
rearrangementinvariant (resp., q e A'z(2) and T is Shur convex), then the
function defined for all cpe Bo(X) by H(cp) - sup,eL1(2,q)(f cp dq - T(f)) is
Shur convex. If H is monotonic, then ?o
9' implies H(?p) H(p'). (If T is
Shur convex and q is not adequate, use -<-<•
Chong and Rice (1971,_Theorem 13.8)
rather than Luxemburg(1967, Theorem 9.1).) See also Dana (2005).
PROOFOF THEOREM14: For f e F, qf denotes the finite support probability on X defined by qf(x) = q(f-'(x)) for all x E X. We prove (i) =4 (ii) =
(iii)
(iv) = (i).
=:== (ii) This step can be proved by a routine argument.
(i)
(ii)
(iii) Let >-be a continuous unboundedvariationalpreference, which
=
is also probabilistically
sophisticated with respect to an adequate q E A?(E).
Let (u, c*) represent > as in Theorem 3 and set, for all p? Bo(2, u(X)),
I(p?) = minp A)s,(f
(p
dp + c*(p)). For all p E A,
*(p)=sup u(xf)fET
\
u(f) dp)=
Jf
/
sup (I(j)- f
EBo(2,u(X))
\
Jf
jdp).
Theorem 13 guarantees that {c* < oo} C Ao(2). If p e A'( )\A'(, q), there
exists A E Y such that p(A) > 0 and q(A) = 0. If u(X) is unbounded below, without loss of generality assume u(X) 2 (-oo, 0]. Let x, e u-l(-n) and
y E u-1(0). Consider the act f, = x,Ay and the constant act y. Since qf, = 6y =
qy, by probabilisticsophistication, f - y and I(-nlA) = I(u(fn)) = 0 for all
n> 1; therefore,
c*(p)
-
sup(I(-nlA)
nal
\(Jf
lA dp) =
00.
If u(X) is unbounded above, without loss of generality assume u(X) D [0, oo).
Since qf, Let x,, E u-l'(n) and y e u-'(0). Consider the act f, =
yAx,.
~
=
=
all
n
then
and
therefore,
n for
> 1;
I(nlAc)
I(u(fn))
fn x,
qx,,
c*(p) > sup (I(nlAc)
n>l
-
8x,
-
- n(1 - p(A))) = oo.
nlA dp)/ = sup(n
n>l
We conclude that {c* < oo}Al(
, q). In particular, 40 = ?o'q-a.s. implies
C
=
I(p) I(4').
Assume cp, E Bo(X, u(X)) are equimeasurable. Therefore (see, e.g.,
Chong and Rice (1971, p. 12)), there exist x1, ... , xe X, with u(x1) > ... >
u(x,), and two partitions {A1, A2, ..., A,} and {B1, B2, ..., Bj} of S in 2, with
q(Ai) = q(Bi) for all i = 1, 2, ..., n, such that q-a.s.
n
n
=
LU(Xi)1A,
i=1
and
i=1
=•u(xYi)lBi.
1486
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
If f(s) = xi for all s E Ai and g(s) = xi for all s E Bi, then f,g e IF and
qf = qg. By probabilisticsophisticationof -, we obtain f - g, whence I(?p) =
I(u(f)) = I(u(g)) = I(if) and I is rearrangementinvariant.Setting T(f) =
-I(-q#), T:Bo(2, -u(X)) -- R is rearrangementinvarianttoo. For all p E
Ao(E, q),
c*(p)
sup
(J
sup
(
4EBo(X,u(X))
dp)
dq
dq- T()
T(
-
,EBo(,,-u(X))
=
By Lemma 31, H(p?)
-ux(f (p dq - T(tI)) for all ?pc L'(V, q)
supo(EBO,(x
is Shur convex; therefore,
is
c*(p) Shur convex and a fortiori rearrangement
invariant.
(iii) =? (iv) Consider T: L'(C, q) -+ [0, co] defined as
(32)
T(f) =
c*(p,),
if J > 0 q-a.s., f/
00,
otherwise,
dq= 1,
where p, is the element of AO(I, q) such that dp,/dq =
set
S4
L'(X,
q):
i>
0
q-a.s.
and
/idq=
. Denote by P the
1.
Let q/'~cx i. Then q E P if and only if q' e P (see, e.g., Chong and Rice (1971,
pp. 15-16)). If 4,, 4' P, then T(4) = oo = T(q'). If 4, 4' e P, then q and 4'
are the Radon-Nikodym derivativesof p, and p,,. By the rearrangementinvariance of c*, T(4') = c*(po,) = c*(pq,)= T(qI) and so T is rearrangement
invariant.By Theorem 13, (p A: c*(p) < t} is a weakly compact subset of A7
for each t > 0; therefore, {1 E L'(Z, q): T(q ) < t} is a weakly compact subset
of L'(E, q) and a fortiori T is weakly lower semicontinuous. Since obviously
T is convex, Luxemburg(1967, Theorem 13.3) guarantees that T (hence c*) is
Shur convex.'6
(iv) =* (i) Let be a variational preference. Let (u, c) represent > as in
Theorem 3 and assume c is Shur convex (with respect to q).17 Set, for all p e
definitionscaveat:Luxemburg(1967, pp. 123-124) defined Shur convexityof T by
16Different
His Theorem13.3shows
invariance,
rearrangement
convexity,andweaklowersemicontinuity.
that if T has these features, then o -<,, so'implies T((o) < T(?p').
'7Noticethatwe arenot assumingthat> is unboundedor continuous,or thatc = c*,orthatq
is adequate.
VARIATIONALREPRESENTATIONOF PREFERENCES
1487
B0(o), I(p) = minpeA(d)(f?odp + c(p)). Then
-I() =-
-p pdq
inf
sup (
.N,
q/ELIq)
(fJ;,q}
c(p)
pdql-T(q),
where T is defined as in (32), replacing c* with c. Let I ' -<c, . If /, V P, then
T(q) = oc > T(f'). Else if e P, f q' dq = f if dq = 1 and ip' > 0 q-a.s. (see,
/,
e.g., Chong and Rice (1971, p. 62)); that is, q' E P. The Shur convexity of c
ensures that T(i') = c(py,) < c(pq,) = T(if). Thus, T is Shur convex. By Remark 1, H(cp) = -I(-?p) is Shurconvex. Next we show that >- satisfies stochastic dominance. Assume that q({s E S: f(s)
x}) q({s E S:g(s)
x}) for all
x E X. For all t E u(X), q({s E S: u(f(s)) > t}) >_<
q({s E S: u(g(s)) > tj), and
this is a fortiori true if t V u(X). We conclude that Gu(f), Gu(g),F >F ,
<
F Uf) Fu(g (see, e.g., Chong and Rice (1971, pp. 30-31)), and -u(f) -<-<x
-u(g). Since H is monotonic, Remark 1 yields -H(-u(f))
> -H(-u(g)), so
that I(u(f)) > I(u(g)) and f ?- g.
Q.E.D.
PROOFOF LEMMA15: Groundedness (in particular, D (q IIq) = 0) and
convexityare trivial,so is Shurconvexityif w is uniform.Weak*lower semicontinuity descends from the fact that the sets {p E A: Dw(p IIq) < t} are weakly
compact in AU(q) for all t E R. The remaining part of the proof is devoted to
show this compactness feature.
We first consider a simple Y-measurable function w with min,,s w(s) > 0,
without requiringf w dq = 1. Set w = minSEsw(s) and without loss of generality suppose t > 0. By definition, {p E A: D(p IIq) < t} A'(q), hence
_
0
=
A
[
)<t
t}.
t}
q)
q)
:Dw(p
{p
11
{p
II < 0-D4(E AU:D•(p
Denote the set on the right-handside by D. We show that (a) limq(B>+op(B) =
0 uniformly with respect to p E D and (b) if {ppnl}l
D and p,(B) -- p(B) for
all B E Z, then p E D. Then a classical result of Bartle,
_ Dunford, and Schwartz
D
Dunford
and Schwartz(1958,
that
is
guarantees
weakly compact (see, e.g.,
Chapter IV)).s
(a) Notice that D'(p IIq) < D'(p IIq) for all p E A", so that
S= {p e d":D'(p
IIq) <
t}
{p E A: D(p
IIq)
< t}= C.
1"Part(a) guaranteesthat D is relativelysequentiallyweakly compact;that is, every sequence
{p,n in D admits a weaklyconvergentsubsequence {p,j }. Part (b) guaranteesthat the limit of p,j
belongs to D; that is, D is sequentiallyweakly compact. The Eberlein-Smuliantheorem guaran-
tees that D is weakly compact.
1488
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
Next we show that
p(B) = 0 uniformlywith respect to p E C and a
limq(B0>o
fortioriwith respect to p E D. Let p E C. Clearly,
JA
U'w
dq<
_ wo?-dq )dq<t+1
Y
f
w \9/U•d q)
= 0+, there exists 6 > 0 such that
WO(T)
For all e > 0, because lim,,,,
_
b
<b
(b)
0<
forallAeX.
if
b>
,
2(t + 1)
w4
28'
in particular,04(b) > 0. For all B such that q(B) < 8 and all p E C,
p(B) =
dq +
-2nld
Sdq
W
qdq-=
18
< --q(B)
252 8nt
21
<-8+
++w
>
dq nU
.
dp
dq
( LP)
fL)
S1 dq
w
dq
dq
WC•Pdo
p
8Ed
2
dq
wI
n•>
2(t+1)J
-
(ddq<e.
dq
This concludes the proof of (a).
Let u (B) = fBw dq for all B E I. Notice that
(33)
[ fdp.d,
D'(p IIq)= I
dq
if p E zAU(q),
(q),
otherwise.
00,
Denote by H,, the set of all finite partitions 7r of S in X finer than rw,=
{w- (b) :b e w(S)}. For all r E Hw,,define on Ad the function
Dw(pl,
0
IIq,)=A
E,7 q(A)
q(A)
(A),
with the conventions p -A) = 0 if p(A) = 0 and p(
)t(A) =- o if p(A) > 0,
when q(A) = 0. Notice that because rTis finer than r,,,, then for all A e:r,
)
w(A) is a singleton (improperlydenoted by w(A)) and
(A)=
wdq = w(A)q(A).
CLAIM1: Wehave Dl(p IIq) = sup,,,
D'(p, IIq,) for all p E A'.
VARIATIONALREPRESENTATIONOF PREFERENCES
1489
PROOF:Let p <<q. Take an increasinglyfiner sequence 7rn in IH such that
dp/dq is (U, l irr,)-measurable.Then
q-a.e.
)1A
dp
q(A) IA -- dq
Continuityof 4 and the Fa(see, e.g., Billingsley (1995, p. 470)), hence ctL-a.e.
tou lemma imply
J
< liminf
()dp\
O(p(A)N)
AEi
q(A) (A
n
,Aq)
-
: (p
< sup
AtE!!w
(A).
The converse inequality is guaranteed if f 4)(L)d-=
coc.Else, if dq(L)is
Jensen
the
conditional
inequalityguarantees that, for all iTre ,,
/t-integrable,
EM?
where m =
?dq)
Ix/It(S)
Em(
Em7dqI
m-a.e.,
)
\dq
and E is the expectation. Hence,
=-Em(Em
((o
\)
V
(dp7
dq)
)
1
dp
>Em
d
-( E kEm (dp
for all r E H,, but
dqdp
(dq
E:m(A)O
AE~r:m(A)ZO
A(I(S)
Ar:
dq
m(A)JA
1
L(A)
(w(A)q(A)
AE7T:
q(A)0
1A
-
AE-i: q(A) wO
w(A)q(A)w(A)
= 1A
AET
dpd)
(A) A(S)Adq
q *(A)
dq
(A))
1490
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
Therefore,
f
dp
()=d.
t(S)Em( (dp)
-
(
dq
dq)(SE
>/-t(S)Em(C(
t
(A),
A1A(P(A)))
(q(A)
(p(A)A)
AE7T
so that
fS
dp
) L supAE.
riH
p()j )
q(A)
( A).
If p E Ao\Ar(q), there exists B E . such that p(B) > 0 and q(B) = 0, and we
can assume it belongs to some r 1Hw,. Therefore,
supp
reHU'
Ar,
(A)
PS(p(A)p(B)
p(B
Kq(A) i(A)> 4 q(B))
B)=
oc-=
D(p
w
11q).
This concludes the proof of the claim.
Q.E.D.
(b) For all Ir E H,, set D, = {p E A':DI(p,1 IIq,) < t}. We show that
- p(B) for all B E I imply p E D,. First notice that
D), and p,(B)
{Pn}n
p E AO•l(for the Vitali-Hahn-Saks theorem). For all A e 7Tsuch that q(A) = 0,
then p,(A) = 0 for all n > 1 (else Dw((p,), 1Iq,) = cx)and hence
Pnq(A)
(0) = (kp (A))
q(A))
q(A)
For all A E Trsuch that q(A) > 0, clearly pn(A)
AE7T
h(p(A)
q(A) j(A)
p(A).
We conclude that
(Pn(A)YA)<
= lim
A
q(A)
t,
as wanted. Now (b) descends from Claim 1 and the observationthat
p={pe dA:Dw(p IIq) < t}
= pe6":
sup D(p,
"rrEHw
II
q,)5
t =
N
rEn-w
,.
VARIATIONALREPRESENTATIONOF PREFERENCES
1491
This completes the proof when w is simple. Suppose now that w is any
2-measurable function with w = infSEsw(s) > 0. Then there exists a sequence
of simple 2-measurable functions w, such that w, T w and minses w,n(s) > w
for all n > 1. By Levi's monotone convergence theorem, D "(p | q) T DW(p |
q) for all p E A-(q), therefore
D= {pe A'(q):Dw(p
=p
IIq) < t}
A'(q):supDn(pII
nl
q)<
tJ
=
n
n>l
We conclude that D as well is weakly compact.
Q.E.D.
PROOFOF THEOREM16: Lemma 15 and Theorem 3 guarantee that diver-
gence preferences are variational.Unboundedness of u together with Proposition 6 guarantees that c*(-) = OD'(- 11q). Finally, Theorem 13 and Lemma 15
Q.E.D.
imply that > satisfiesAxiom A.8.
PROOF OF THEOREM 18: Set I(cp)= minpe,,A)(fpdp
+ c*(p)) for all
p B0(Z). Because -c* coincides with the Fenchel conjugate of I (see
Lemma 26) on A(2) and I*(/t) = - 00 for all/t e ba(Y)\A(2), then
d(?p)= argmin (
U
qpeBo(2)
U
II((p)=
dp + c*(p)) V
Bo(),
argminfu(f)dp+c*(p)
fEBo(2)
and I is Gateaux differentiableon Bo(2) if and only if P*is strictlyconcave on
line segments in UEBO(l) I(cp), that is, if and only if c* is essentially strictly
convex.19
Let f E Bo(X). Set A = {u'(f) drIr e argminpEa(fu(f) dp + c*(p))} and notice that A is the image of argminpEa(fu(f) dp + c*(p)) = 8I(u(f)) through
the map from A(2) to ba(2) that associates to p(.) in A(X) the bounded and
finitely additive set function
u'(f) dp in ba(2). Because this map is linear
4.)
and a(ba(2), Bo(2))-o-(ba(2), Bo(2))-continuous, and because dI(u(f)) is
weakly* compact and convex, then A is or(ba(X),Bo(,))-compact and convex.
19Thisis proved in Bauschke, Borwein, and Combettes (2001). Notice that our definition of
essentialstrictconvexityis weakerthantheirsandit guaranteesthatstrictconvexityimpliesessentialstrictconvexity.
1492
E MACCHERONI,M. MARINACCI,AND A. RUSTICHINI
By standardresults,
(34)
I'(u(f); u'(f)h)=
mm(f
u'(f)hdp
= minf h dp,
I-A
EA
'Vhe Bo().
J
Let h E Bo(2) and s E S. If h(s) 0 O,then
u(f + th)(s) = u(f (s) + th(s))
= u(f(s)) + u'(f(s))th(s) + o(th(s))
for th(s) -+ 0, that is,
lim
u(f (s) + th(s)) - u(f (s)) - u'(f (s))th(s)
-
th(s)
th(s)-0O
and so
(35)
lim
u(f (s) + th(s)) - u(f (s)) - u'(f (s))h(s)t
t
= 0.
t3,O
Clearly (35) holds also if h(s) = 0. Because f and h are simple, the preceding
limit is uniformwith respect to s e S. Therefore, for t 4 0,
+
<
I(u(f th)) - I(u(f))
-t
< (I(u(f + th)) - I(u(f)
+ I(u(f) + tu'(f)h) I(u(f + th)) - I(u(f)
t
n
/la
minhd
+ tu'(f)h)
minf
I(u(f))))/t-
h
di
+ tu'(f)h)
+ tu'(f)h) - I(u(f))
? I(u(f)
t
Ilu(f + th) - u(f) - tu'(f)hll
t
minf hd
XEA
+ o(1),
where the last inequalitydescends from the Lipschitzcontinuityof I and (34).
Uniformity of limit (35) then delivers
im I(u(f + th)) I(u(f)) -minm hd
tq0
t
/eA
d
VARIATIONALREPRESENTATIONOF PREFERENCES
1493
that is, OV(f) = A.
for all h E B0o() or V'(f; -) = min, Ef(.-)
d/u,
if
c*
then
is essentially strictlyconvex,
Now,
dI(u(f)) is a singleton for every
=
in
dr
and
Ir E (u(f))} is a singleton too. Conversely,
f B0o()
dV(f)
{u'(f)
assume that V is Gateaux differentiable on Bo(0) and per contra that c* is
not essentially strictly convex. Then there exists ?0E B0o() such that I(40p)
contains two distinct elements r, and r2. Because u: R - R is concave, then
it is unbounded below and there is b E R such that p?+ b E Bo(2, u(R)). In
particular,there exists f E Bo(V) such that u(f) = ??+ b and
rl,
min(fdp+c*(p)
r2E
I(4)-=-arg
(cf(+b)dp+c*(p))=
=-argmin
d(u(f)).
It follows that u'(f) dr1,u'(f) dr2 e
Since V is Gateaux differentiable
•V(f).
on Bo(X), then u'(f) dr1= u'(f) dr2,
that is,
(36)
f
hu'(f) dr =
f
hu'(f) dr2 VhE Bo().
Because u is strictlymonotonic, concave, and differentiable,then u'(z) # 0 for
= fJi dr2 for all &E Bo(Y), contradicting
all z E R and (36) implies f f
drQ.E.D.
rl r2.
PROOF OF PROPOSITION19: Let (u, c*) represent - as in Theorem 3 and
without loss of generality assume [-1, 1] u(X).
_ Theorem 1), there is a weakly*
(i) =, (ii) By Gilboa and Schmeidler (1989,
compact and convex set C A(2) such that u(xf) - minPEC
f u(f) dp for all
f E F and each xf - f. By Theorem
3,
_
(37)
c*(p) = sup(minf u(f)dq rf.T
?qECff
u(f) dp
d)
Vp
A().
Suppose p E C. Then c*(p) < 0. Because c* is nonnegative, we have c*(p) = 0.
Next, suppose po V C. By the separatinghyperplanetheorem, there is a simple
measurablefunction p: S -+ u(X) such that f cpdp > f1? dpo for each p E C.
Hence, taking f E T such that 40= u(f), minpc f u(f) dp - f u(f) dpo > 0,
which in turn implies c*(po) > 0. We conclude that c*(p) = 0 if and only if
p E C. Therefore, for all f E F,
min(fu(f)
pE(f)
dp +c*(p)) = u()
=
min
= pE{c*=0}
u(f) dp
minf
peCf
u(f) dp.
1494
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
(ii) =4 (i) and (iii) =- (ii) are trivial.Now assume u(X) is unboundedabove
(resp., below).
(ii) ?= (iii) For all pEA(E),
c*(p)= sup
fey
S
mm
in
u(f)dq-
u(f)dp
qEf{c*=O)f}
sup (minf pdq-
oPeBo(X,u(X))
\q
({c*=0}J
Jf
pdp.
Suppose, c*(po) > 0. There exist a nonnegative (resp. nonpositive), simple,
measurable function p: S -- u(X) and e > 0 such that minpeIc=01o
f dp f podpo > e, but nqpE Bo(V, u(X)) for all n E N and
c*(po)>
minf
n dp -
for all n E N. We conclude c*(po) =
>n
n
•pdpo
0
.
Q.E.D.
The proof of Corollary20 is omitted. Just notice that Axiom A.5' can be used
to obtain affinityof the functionalI that appearsin Lemma 28 in the same way
in which Axiom A.5 is used to obtain its concavityat the beginningof the proof
of Theorem 3.
LEMMA
32: Let - be a variationalpreferencerepresentedby (u, c*) as in Theorem 3 and let q e A(2). Thefollowingconditionsare equivalent:
(i) q correspondsto a SEU preferencethat is less ambiguityaversethan >-.
(ii) c*(q) = 0.
(iii) q e 81(k) for some (all) k e u(X), where I(?p) = minpa,,()(f(p dp +
E B0o(, u(X)).
c*(p)) for all Op
In particular,any variationalpreferenceis ambiguityaverse.
PROOF:(i) =? (ii) Suppose to is a SEU preference, with associated subjective probabilityq and nonconstant affine utility index uo such that > is more
ambiguityaverse than >o. By (8), we can assume uo= u. By Proposition8, and
c* < co, and by Corollary20, co(q) = 0; hence, 0 < c*(q) < c(*(q)= 0.
(ii) =? (iii) We have c*(q) = 0 if and only if supEB,,0(z,,,,x(IX(p) -
f ?pdq) =
0
if and only if I((p) Sf p dq for all 'p e Bo(2, u(X)) if and only if I(Gp)- I(k) <
f 'p dq - f k dq for all 'p e B0o(?,u(X)) and all k e u(X) if and only if q e
lI(k) for all k e u(X).
(iii) =4 (i) If q e dl(k) for some k e u(X), then I('p) - I(k) < f p dq k
f dq for all 'p e B0o(, u(X)) and then I('p) <f p dq for all 'p e B0o(, u(X)).
Denote by o0the SEU preference,with associated subjectiveprobabilityq and
utility index u. Notice that for all fe•F and x e X, f x implies I(u(f)) >
u(x), a fortiori f u(f) dq > u(x) and f -o x.
VARIATIONALREPRESENTATIONOF PREFERENCES
1495
Ambiguity aversion of >- now follows from the observation that
Q.E.D.
argminpEA() c*(p) is nonempty and minpEA(.) c*(p) =0.
PROOFOFPROPOSITION
22: (i) Let 0, 4 0. By Proposition 12,
limmin f
+p u(f) dp OnD(p 1q
= minff u(f) dp+
{D(-lIq)<oo}(P)J
If w E LO(q) and w = infs s w(s), then wDp(p I| q) < Dl(p IIq) < IlwllKx
D4(p IIq), which implies domDp(. IIq) = domDw(. IIq). In turn, this implies
limminf fu(f)dp
n
+ OnD(pl q)
peA IfJpedomD(.IlIq)
J
u(f) dp.
mDinf
q)
Next we show that, for all cpE Bo( ),
(38)
inf
f
pedomD4(.I q)J
= inf f dp.
pdp
peA"(C,q)J
with a, > ... > a, and
every p E A(X, q), define p,E Bo(2) by {Ai}il
Let cp=
ajlAj,
a partition of S in . For
•,l
(A>
i: q(Ai)>0
q•
) i:Ai+
i:
1Ai
pq(Ai)=0
It is easy to see that q, is nonnegative and f rp dq = 1. Call p' the element of
is a simple function, then DO(p' II
A(~(, q) such that dp'/dq = fp,. Since
l
p,
q) = f 4(frjp)dq E R, so that p' E domDo(.
I| q). Since f cpdp' = p dp, we
then have I{fp dp:p e A(X, q)I}C {f p dp:p E domD(. IIq)}, which yields
(38) since the converse inclusion is trivial.
It remains to prove that
(39)
inf f
J
pedAa(q)
dp = ess
seS
min•p(s). >
Here essmin,,s p(s) = min{ja
0}. Let i* E {1,...,
:q(Ai)
n} be such that
= min{ai: q(Ai) > 0} and let qA,. be the conditional distribution of q
aoi
Then qAi, E A?(q), and f p dqA, = ess min,,s 4?(s). This proves (39).
on
Ai,.
Let
6,, f oc. By Proposition 23, strict convexity of 4 implies that of
(ii)
D (p 11q) on its effective domain. Hence, argmin
1Iq) - {q} for
each n and so the result follows from Proposition 12. O,,D(p
Q.E.D.
1496
E MACCHERONI, M. MARINACCI,AND A. RUSTICHINI
PROOF OF PROPOSITION23: By Theorem 18, it is enough to show that
Dw(. I q): A(,) -- [0, 00] is strictlyconvex on its effective domain. Let p, p' E
domD'(. IIq), p p', and a e (0, 1). Let pt(B) = fBwdq for all B E .
The assumption infses
w(s) > 0 guarantees that the measure pt is equivalent to the probability q. By (33), p, p' e domD(. IIq) if and only if
f()\
<
(dp'') d
d,, f
00
if and only if
d(L),
ity and nonnegativityof 0 guarantee that
< dp
(1 - a)
( ad
-q
dp
<a(OP( -(s)
dq
) e L1(X,
•(
). Convex-
dp'\ (s)
dq /
N
d
dp'
+ (1- a)o4 -(s) )VseS;
/(dq
strict convexity,p : p', and a E (0, 1) imply that the second inequalityis strict
on a set E E X with q(E) > 0, and hence g(E) > 0. Therefore, 0(aL
+ (1 -
a )) E (L
L1 , g) and
(s) I (1- a)4 d (s)0
dp
-a
S
S
dp +(1-a)dp'\d>0,
dq dF
q )-J.
that is, Dw(ap + (1 - a)p' IIq) < aD (p IIq) + (1 - a)Dw(p' IIq), as desired.
Q.E.D.
+ R defined by J(p) =
is
concave
and
Gateaux
differentiable.
dq
Concavity is
(20)-'
Varq(Gp)
f
trivial.Moreover, for all p, I e Bo(V) and t E IR,
PROOFOF THEOREM24: The functional J: -B
Bo()
~
J(
+ to) =
(s +
tip)dq - 1Varq(?q + tf)
20
(
+[J(1-i
0
+ tq)
J(
t--+O ~t
J(
+[f
p
Jpdq ))dq]t+J(p),
so that
(40)
J'(Gp;q) =
limt
f
-0
dq
dq.
That is, for all p e B0o(2),the Gateaux differentialof J at (pis representedby a
measure with Radon-Nikodym derivative with respect to q given by 1 - 1 ('p -
VARIATIONALREPRESENTATIONOF PREFERENCES
1497
f ?pdq). Therefore, J'(cp;-) is positive as a linear functional on B0o() if and
only if
q({sS:p(s)
-fJ
dq })1.
This relation characterizesthe elements of the domain of monotonicity M. In
Maccheroni, Marinacci,Rustichini,and Taboga(2004) we show that
(f dp+?OG(p1lq))
mJ()-•=qin
V(
M.
Q.E.D.
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