KIER DISCUSSION PAPER SERIESKYOTO INSTITUTEOFECONOMIC RESEARCH
Discussion Paper No.1097“Arrow-Pratt-Type Measure of Ambiguity Aversion”

Chiaki Hara

September 2023

KYOTO UNIVERSITY
KYOTO, JAPANArrow-Pratt-Type Measure of Ambiguity Aversion
Chiaki Hara∗†
September 8, 2023
Abstract
We deﬁne a measure of ambiguity aversion for ambiguity-averse utility
functions in a way analogous to the Arrow-Pratt measure of risk aversion.
The measure is determined by the second Peano derivative, which exists evenfor non-diﬀerentiable functions, such as maximin and Choquet expected utility functions. Unlike the standard notion of comparative ambiguity aversion,it allows us to compare ambiguity aversion between two utility functions
exhibiting diﬀerent risk attitudes. We introduce a notion of ambiguity premium and show that our measure is related to the second-order, as opposedto the ﬁrst-order, ambiguity premium. We also show that it is related to theﬁrst-order impact on matching probabilities of the size of prizes.
JEL Classiﬁcation Codes: C38, D81, G11.
Keywords: Expected utility functions, risk aversion, ambiguity aversion, ambiguity premium, matching probabilities, Peano derivative.

1

Introduction

A decision maker in the face of uncertainty is said to perceive ambiguity in an
event if he cannot attach any single reliable probability to the event, and, thus,∗

Institute of Economic Research, Kyoto University. Email: hara.chiaki.7x@kyoto-u.ac.jp
Some of the contents of this paper were included in the papers circulated under the title “Comparative Ambiguity Aversion for Smooth Utility Functions,” “Measure of Ambiguity
Aversion,” and so on. The research is funded by the Open Research Area (ORA) for the Social
Sciences “Ambiguity in Dynamic Environments,” the JSPS KAKENHI Grant Number 17K18561,
the Trust Fund Foundation of Japan, Nomura Foundation, and the Joint Usage/Joint Research
Program at the Institute of Economic Research of Kyoto University. I am grateful to Pierpaolo
Battigali, Simone Cerreia-Vioglio, Mira Frick, Yuhki Hosoya, Ryota Iijma, Tomoki Inoue, Peter Klibanoﬀ, Fabio Maccheroni, Massimo Marinacci, Sujoy Mukerji, Jonathan Newton, Yusuke
Osaki, Frank Riedel, Takashi Ui, Koji Shirai, Norio Takeoka, Jan Werner, Michael Zierhut, andPeter Wakker for their insightful comments.
†1needs to take multiple probabilities into consideration for decision making. If thedecision maker tends to prefer state-contingent consequences whose distributions
are independent of the choice of these relevant probabilities to those whose distributions are not, then he is said to be ambiguity-averse. Ambiguity aversion has
been well documented ever since the experiments by Ellsberg (1961). It has been
given a theoretical foundation in the form of axiomatization by Schmeidler (1989),
Gilboa and Schmeidler (1989), and their followers. Its implications on portfolio
choice and asset pricing have been explored by Dow and Werlang (1992), Epsteinand Wang (1994), and many others. More recent studies, such as Ju and Miao(2011), Chen, Ju, and Miao (2014), Jahan-Parvar and Liu (2014), and Gallant,
Jahan-Parvar, and Liu (2019), and Collard, Mukerji, Sheppard, and Tallon (2018)
have shown not only that introducing ambiguity aversion solves otherwise unexplainable puzzles, such as the home bias puzzle, but also how much ambiguity
aversion is needed to solve such puzzles. A typical quantitative exercise there is
to ﬁx a form of the decision maker’s ambiguity-averse utility function and, then,infer values of ambiguity aversion parameters necessary to solve the puzzle basedon experimental evidence or market data.
While these quantitative exercises undoubtedly help us grasp a better understanding of the role of ambiguity aversion in decision making, their approach has
a problem. To see what it is, note that inference of the ambiguity aversion parameter often comes with that of the risk aversion parameter. If two pairs of inferredor calibrated values of risk and ambiguity aversion parameters, say (θ1 , γ1 ) and(θ2 , γ2 ), involve two diﬀerent values of risk aversion parameter, θ1 6= θ2 , then, evenwhen one ambiguity aversion parameter is greater than the other, γ1 > γ2 , wecannot conclude that the decision maker (θ1 , γ1 ) is more ambiguity-averse thanthe decision maker (θ2 , γ2 ). This is because the notion of comparative ambiguity aversion employed by Epstein (1999, Section 2.3), Ghirardato and Marinacci(2002, Deﬁnition 7), and the subsequent literature is restricted the case where thetwo decision makers share the same risk attitudes, with a couple of exceptions towhich we will come back later. Let us illustrate the limitation that the restriction
imposes on the analysis in the setting of the home bias puzzle.
Imagine that there are two Japanese investors of equal wealth, say 10 million
yen, who invest in a Japanese stock, an American stock, and the bond. Supposethat they perceive no ambiguity in the return on the Japanese stock, but they
perceive ambiguity in the return on the American stock. Their portfolios are2presented in Table 1. That is, investor 1 puts just a half of the money into theAmerican stock (1 million yen) that he invests into the Japanese stock (2 millionyen), while investor 2 allocates the same amount of money in the two stocks (4million yen each). Then it seems reasonable to think of investor 1 as being more
ambiguity-averse than investor 2. However, since the total amount that he invests
in the two stock is 3 million yen, which is much less than the total amount (8 million
yen) that investor 2 invests in the two stocks, it seems equally reasonable to thinkof investor 1 as being more risk-averse than investor 2. 1 But, if this is indeed thecase, then we cannot say that investor 1 is more ambiguity-averse than investor
2, because we could do so only when they are equally risk-averse according to
the standard notion of comparative ambiguity aversion. 2 In other words, we haveno theoretical foundation to attribute the diﬀerence in the allocation of wealthbetween the two stocks to the diﬀerence in the degree of the investors’ ambiguity
aversion.
Table 1: Two Japanese investors with equal wealth invest into a Japanese stock,
an American stock, and the bond. The ﬁgures are in million yen.

Japanese stock
American stock
bond
total

Investor 1
2
1
7
10

Investor 2
4
4
2
10

The purpose of this paper is to introduce a measure of ambiguity aversion
along the lines of Arrow-Pratt measure of risk aversion3 under the assumptionthat the risk attitude can be represented by an expected utility function. It allows
us to compare ambiguity attitudes of two decision makers who do not have thesame risk attitudes. In Table 1, for example, the measure oﬀers the scope forconcluding that investor 1 is more risk-averse and ambiguity-averse than investor
2. It also allows us to compare ambiguity attitudes of two decision makers whose1
It was also found in the experiments by Bossaerts et al (2010) that those who invest more instocks (Arrow securities) invest proportionally more in stocks whose returns are ambiguous thanin stocks whose returns are unambiguous.
2
Bossaerts et al (2010) postulated that the investors have α-maximin expected utility functionsof Ghirardato, Maccheroni, and Marinacci (2004) and used the inferred values of α alone tocompare their degrees of ambiguity aversion with no reference to risk aversion.
3
Sujoy Mukerji kindly suggested the title of the paper.

3

ambiguity attitudes are represented by utility functions of diﬀerent forms. To see
this second point, recall that the literature on ambiguity has endeavored to clarify
the relationship between particular forms of utility functions and particular (lessdemanding) variants of the independence axiom in the expected utility theory.
Within a class of ambiguity-averse utility functions of the same form, equivalentconditions have been given, in term of parameters in the common functional form,
for one decision maker to be more ambiguity-averse than another. 4 Since ourmeasure of ambiguity aversion can be deﬁned without relying on any particularfunctional forms, it can also be used to compare two decision makers who areambiguity-averse but satisfy, say, diﬀerent variants of the independence axiom.
A technical contribution of this paper is to use second Peano derivatives 5 toaccommodate some important classes of ambiguity-averse utility functions, mostnotably Choquet expected utilities of Schmeidler (1989) and maximin expectedutilities of Gilboa and Schmeidler (1989), which are not even diﬀerentiable in thestandard sense. The second Peano derivative is deﬁned as twice the coeﬃcientof the second-order term in the Taylor approximation up to the second order. It
does exist under fairly weak conditions and is suﬃcient for our purpose, because
we are interested not in the second derivative of the utility function per se (which
is deﬁned as the derivative of derivatives) but in the second-order approximationthat it provides.
While our measure of ambiguity aversion is deﬁned in terms of utility functions,
it can be characterized in terms of choice behavior, such as matching probabilities
and ambiguity premiums. The matching probability is deﬁned as follows. Let x bea baseline consumption level and ε be the value of the prize of a bet on an event
A. Then, the decision maker consumes x + ε on A and x outside A. The matchingprobability ρ(A) is deﬁned so that the decision maker is indiﬀerent between thisbinary act and the lottery in which x + ε is obtained with probability ρ(A) and x is
obtained with probability 1 − ρ(A). In general, the matching probabilities dependon the choice of x and ε. We will show (in Section 5) that the marginal change
in matching probabilities due to a marginal increase in prizes ε is approximately
proportional to our measure of ambiguity aversion. The ambiguity premium is
4

The α-maximin expected utility functions mentioned in Footnote 2 constitute such an exam-ple.
5

Mukhopadhyay (2012) is a textbook giving some deﬁnitions and facts on Peano derivatives.
Evans and Weil (1981-82) is an accessible survey. They are both concerned only with the univariate case. Massimo Marinacci kindly pointed out to me that the notion of second right-derivatives
that I introduced in an earlier version was, in fact, a multivariate version of the Peano derivative.

4

deﬁned as the maximum consumption level that the decision maker is willing togive up in exchange for the assurance that the true probability coincides with hisbenchmark probability in the sense of Ghirardato and Marinacci (2002). Deﬁnethe risk premium as the maximum consumption level the decision maker is willingto give up in exchange for full insurance under the benchmark probability. We willshow (in Section 6) that our measure of ambiguity aversion is approximately equal
to the ratio of the ambiguity premium to the risk premium.
Let us now turn to some earlier works that are most relevant to this one. Dimmock, Kouwenberg, and Wakker (2016, Section 3) deﬁned a measure of ambiguity
aversion as the shortfall of the matching probabilities ρ(A) deﬁned for various
events A from the subjective probabilities that he would attach to unambiguous
acts.6 They assumed that for preferences in the source method, ρ(A) is uniquelydetermined by A, independently of the choice of the baseline consumption level
x and the prize ε.7 This uniqueness property holds also for all biseparable preferences in the sense of Ghirardato and Marinacci (2002, Deﬁnition 2), including
those represented by Choquet expected utilities and α-maximin expected utilities
of Ghirardato, Maccheroni, and Marinacci (2004).
Wang (2019) used an ingenious idea of restricting preference comparison on aset of “aligned” acts to give a notion of a decision maker as being more ambiguityaverse than another that neither requires nor implies common risk attitudes. His
notion relies on matching probabilities. But, for non-biseparable preferences, suchas the smooth ambiguity model of Klibanoﬀ, Marinacci, and Mukerji (2005, hereafter KMM), the matching probabilities depend on the choice of the baseline consumption level x and the prize ε; and so does Wang’s deﬁnition (Deﬁnition 5 ofhis paper) of the more-ambiguity-averse-than relation. This fact suggests that for
general preferences, the matching probability, as well as the measure of ambiguity
aversion, should be amended by specifying the baseline consumption level x and6
In their setting, the decision maker may use a probability weighting function to calculate
expected utility levels. When he does, the subjective probabilities here should be replaced by thevalues of the probability weighting function taken at the subjective probabilities.
7
If the subjective probability of an event A is denoted by P (A), then the shortfall is equal to
P (A)−ρ(A). In the case of diﬀerentiable utility functions, P is deﬁned as (a scalar multiple of) thegradient of the utility function at a constant act. In some experiments, as in the case of Ellsberg’s
two-color urn, it is deﬁned based on some type of symmetry consideration (exchangeability). For
natural, non-experimental events, Abdellaoui, Baillon, Placido, and Wakker (2011, Section IV)
oﬀered a way to identify it. Denoting the complement of A by Ac , the shortfall of ρ(A) + ρ(Ac )from one can also be taken as a measure of ambiguity aversion; and it can be identiﬁed even when
P is unknown. This method of eliciting ambiguity aversion without identifying P was explored
by Baillon, Huang, Selim, and Wakker (2018) and Baillon, Bleichrodt, Li, and Wakker (2021)

5

the prize ε. This is precisely what we will do in this paper.
Cerreia-Vioglio, Maccheroni, and Marinacci (2022) gave a quadratic approximation of the uncertainty premium for ambiguity-averse utility functions. They
assumed twice continuous Gateaux diﬀerentiability, thereby excluding maximin
and Choquet expected utility functions. They characterized the quadratic approximation of the uncertainty premium with a possibly inﬁnite state space, but did not
discuss its relation to matching probabilities or other characteristics of preferences,such as the curvatures of indiﬀerence curves.
The rest of this paper is organized as follows. Section 2 gives the setup of thepaper. Section 3 deﬁnes a measure of ambiguity aversion. Section 4 characterizes
the measure of ambiguity aversion in terms of utility functions and the curvatures ofindiﬀerence curves. Section 5 relates it to matching probabilities. Section 6 relates
it to ambiguity premiums. Section 7 shows how it can be rewritten for the smoothambiguity model of KMM, and presents a work-out example of the home bias puzzle. All the results so far are obtained under the assumption of twice continuous
diﬀerentiability in the standard sense. Section 8 introduces the notion of Peanodiﬀerentiability, extends the measure of ambiguity aversion to non-diﬀerentiable
but twice Peano diﬀerentiable utility functions, and generalizes the results on ambiguity premiums and matching probabilities in the preceding sections. Section 9
suggests directions of future research. The appendix gathers proofs for the main
results in the twice continuously diﬀerentiable case. The online appendix consistsof three sections. The ﬁrst one contains proofs for the application to the smoothambiguity model. The second one presents an extension of the measure of ambiguity aversion and its relation to matching probabilities and ambiguity premiums
to the case where the decision maker’s risk attitude cannot be represented by any
expected utility function.8 The last one gathers lemmas and and the extension to
the non-diﬀerentiable case.

2

Setup

The state space is a ﬁnite set S. By an abuse of notation, we also write its cardinality as S. As it is ﬁnite, the σ-ﬁeld is the power set and every random variable canbe regarded as a vector in RS . Denote by e the vector in RS whose coordinates are
all equal to one. It represents a proﬁle of state-independent consumption or utility
8Peter Wakker encouraged me to work on this extension.

6

Slevels. Write ∆ = {p ∈ R+
| p · e = 1}. Then, ∆ is the set of all probabilities on
the state space. Each g ∈ RS can be regarded as a random variable on the state
space, and its expectation under p ∈ ∆, E p [g] is equal to the dot product p · g.
Let T be a nonempty open interval of R. It is the set of all possible consumption
levels. We assume that the decision maker has an expected utility function forpure risk (without ambiguity) and denote by v : T → R his Bernoulli utility
function, following the terminology of Mas-Colell, Whinston, and Green (1995).
Let I : v(T )S → R and assume that I is increasing and normalized, that is,I(ye) = y for every y ∈ v(T )S . We refer to I as the aggregator.
Let Π(T ) be the set of all (Borel) probability measures on T . For each f : S →Π(T ), or f ∈ Π(T )S , denote by v ◦ f the vector in v(T )S whose s-th coordinateRis equal to the expected utility level T v(x) d(f (s))(x) given by the lottery f (s) ∈Π(T ). Deﬁne V : Π(T )S → R by letting V (f ) = I(v ◦ f ) for every f ∈ Π(T )S .
This is the form of utility functions that we study in this paper. We then saythat V is deﬁned by (S, T, v, I), or, simply, by (v, I). By Proposition 1 of CerreiaVioglio, Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2011), such an I
exists for each transitive, complete, continuous, and increasing preference relationthat satisfy the independence axiom on the set of lottery acts, to be deﬁned shortly.
An element of Π(T )S , which assigns a lottery to each state, is known as
an Anscombe-Aumann act (named after Anscombe and Aumann (1963)). An
Anscombe-Aumann act f for which f (s) is independent of s is called a lottery act,
and the set of lottery acts can be identiﬁed with Π(T ). Since the aggregator I isnormalized, the preference restricted on Π(T ) is represented by an expected utility
function with a Bernoulli utility function v. The subjective expected utility is thecase where I is the expectation operator under some probability on S. In contrast,
an element of T S assigns a (deterministic) consumption level to each state and istermed a monetary act by Marinacci, Maccheroni, and Rustichini (2006, Section
3.5). We call an act a constant act if it a lottery act and, at the same time, amonetary act. That is, a constant act gives a common deterministic consumption
level over all states. The set of constant acts can thus be identiﬁed with T . The
measure of ambiguity aversion we introduce in this paper is a local measure that
represents ambiguity attitudes around constant acts.
While the use of Anscombe-Aumann acts in the theory of choice may be questionable, as voiced by Kreps (1988, Chapter 7) and Epstein (1999, Section 5), we
have opted for the domain of Anscombe-Aumann acts, rather than that of mone-7tary acts, because, then, V speciﬁes risk attitudes by ranking lottery acts. Once
the Bernoulli utility function v is known, the subsequent analysis is valid (andsimpler) even when the utility function V is restricted to the set of monetary acts.
Bear in mind, though, that the separation between the Bernoulli utility function v
and the aggregator I is incomplete on the set of monetary acts: It is possible that apair (v1 , I1 ) deﬁnes V1 , another, diﬀerent, pair (v2 , I2 ) deﬁnes V2 , and, yet, V1 = V2
on T S . Since our measure of ambiguity aversion depends on how V is decomposed
into v and I, we have chosen the set of Anscombe-Acts as the domain of the utility
function V to circumvent the problems arising from this inseparability.
We assume throughout this paper that v is twice continuously diﬀerentiable
and satisﬁes v ′′ ≤ 0 < v ′ . Then v(T ) is a nonempty open interval of R. We also
assume that v ′′ < 0, except in Section 5, where we explore the relationship of ourmeasure of ambiguity aversion to matching probabilities. In addition, we assume
that I is twice continuously diﬀerentiable until Section 8, where we deal with thecase of non-diﬀerentiable aggregators. The assumption of twice continuous diﬀerentiability is violated by the maximin expected utility of Gilboa and Schmeidler(1989) and Choquet expected utility of Schmeidler (1989), but satisﬁed by thesmooth ambiguity-averse utility functions of KMM and the relative entropy usedby Hansen and Sargent (2001), among others.

3

Measure of ambiguity aversion

In Section 2, we deﬁned a utility function V : Π(T )S → R based on a Bernoulli
utility function v : T → R and a aggregator I : v(T )S → R. In this section,
we deﬁne a measure of ambiguity aversion that is analogous to the Arrow-Prattmeasure of risk aversion for Bernoulli utility functions.
Let x ∈ T and write p = ∇I(v(x)e). Since I is non-decreasing and normalized,p ∈ ∆. For each z ∈ RS , denote the variance of z under p by Varp [z]. It isP
Pequal to s p(s)z(s)2 − ( s p(s)z(s))2 and, thus, to E p [z 2 ] − (E p [z])2 , where z 2 =((z(s))2 )s ∈ RS .
Deﬁnition 1 Let x ∈ T and z ∈ RS . Write p = ∇I(v(x)e) and suppose that8

Varp [z] > 0. Then, we deﬁne−
H x (z) =z ⊤ (v ′ (x)∇2 I(v(x)e)) z
Varp [z].
v ′′ (x)
− ′
v (x)

(1)

The denominator of the right-hand side of (1) is nothing but the Arrow-Prattmeasure of risk aversion of the Bernoulli utility function v at x. The numerator
is determined, in part, by the Hessian of the aggregator I at v(x)e. In the caseof a subjective expected utility function, I is linear, the Hessian is zero, and ourmeasure is also equal to zero. It can, thus, be considered as representing how muchthe utility function V is diﬀerent from the subjective expected utility functions.
The denominator of the right-hand side of (1) is invariant to the aﬃne transformations of v. By Proposition 1 of Cerreia-Vioglio, Ghirardato, Maccheroni,
Marinacci, and Siniscalchi (2011), the numerator is also invariant as long as I istransformed along with an aﬃne transformation of v so that I is normalized and V
represents the same preference. Thus, the value of H x (z) is uniquely determinedby the risk attitudes and the preference relation on T S . The measure H x (z) isinvariant to scalar multiplications and addition of constant acts to z. Hence, toidentify the values of H x (z) for all z, it suﬃces to know the values of H x (z) forthe z with zero mean and unit variance.
A curious aspect of the deﬁnition (1) is that while the ambiguity aversion
seems fully captured by the numerator of the right-hand side, the deﬁnition also
involves the denominator, which is determined solely by the risk aversion. There
are two reasons for employing this deﬁnition. First, as we will see in Sections 5, 6,
and 8, this deﬁnition allows us to obtain tractable asymptotic results on ambiguity
premiums and matching probabilities, even when I is not diﬀerentiable. Second, inmany applications of ambiguity-averse utility functions, most notably in the homebias puzzle, which we will take up in Section 7.2, the ambiguity aversion in excessof the risk aversion is more important than the ambiguity aversion itself. In any
case, we will give equivalent expressions of the numerator in terms of derivatives ofambiguity premiums and matching probabilities in the subsequent analysis. They
may well be helpful for the potential use of the numerator, on its own, as a measureof ambiguity aversion.
The measure of ambiguity aversion can be best interpreted in terms of bench-9mark preferences and benchmark measures in the sense of Ghirardato and Marinacci (2002, Section 2.2). All subjective expected utility functions are ambiguityneutral, and suppose that there is one that is less ambiguity-averse than V according to their Deﬁnition 7. Such a subjective expected utility function is called abenchmark preference of V , and the subjective probability is called a benchmark
measure. As discussed in the introduction, a benchmark preference exhibits thesame risk attitudes as V , which is represented by v. Since the aggregator I is
(twice continuously) diﬀerentiable, the benchmark measure, which we will also refer to as the benchmark probability, is unique and coincides with ∇I(v(x)e) for
any x ∈ T .9 Hence, the measure of ambiguity aversion is equal to the quadratic
form of the Hessian of I divided by the Arrow-Pratt measure of risk aversion andthe variance of deviations from a constant act relative to its benchmark preference. In the rest of this paper, we sometimes refer to ∇I(v(x)e) as the benchmark
probability, even when there is no benchmark preference, as a shorthand for the
subjective probability implicit at the constant act xe.

4

Utility functions and indiﬀerence curves overmonetary acts

In this section, we give an equivalent expression of the measure of ambiguity aversion, (1), in terms of the utility function V . When its domain is restricted to theset of monetary acts, since it is a subset of RS , we can apply the classical demand
theory to ﬁnd its illustrative properties.
Let x ∈ T and p = ∇I(v(x)e). Let I¯ be the expectation operator under p and¯ Deﬁne V ∗ = v −1 ◦V and V̄ ∗ = v −1 ◦ V̄ . Then, V ∗ representsV̄ be deﬁned by (v, I).
the same preference as V , and V̄ ∗ represents the same preference as V̄ . Moreover,
if V ∗ is concave, then it is a least concave utility function in the sense of Debreu(1976),10 and the same can be said for V̄ ∗ . In the following theorem, the utility
9
When there is a benchmark preference, ∇I(v(x)e) is independent of the choice of x ∈ T .
This property is equivalent to Axiom 7, Translation Invariance at Certainty, of Rigotti, Shannon,
and Strzalecki (2008), as their Proposition 8 showed.
10
We need to assume here that V ∗ is concave because it need not be so under the set ofassumptions given so far. Additional assumptions on v necessary to guarantee concavity are
given in Hardy, Littlewood, and Polya (1952, Section 3.16). In fact, then, a property stronger
than least concavity can be obtained: Every concave function that represents the same preference
as V is a concave transformation of V ∗ . This follows from the fact that V ∗ is linear on the set of
constant acts and least concave utility functions are aﬃne transformations of each other.

10

functions V ∗ and V̄ ∗ should be thought as being deﬁned on the set T S of monetaryacts and their Hessians as S × S matrices.
Theorem 1 For all x ∈ T and z ∈ RS ,v ′′ (x)z ⊤ ∇2 V ∗ (xe)z = z ⊤ v ′ (x)∇2 I(v(x)e) z + ′Varp [z].
v (x)

(2)

Since V̄ is a subjective expected utility function under p, a well known resultof Pratt (1964) implies thatz ⊤ ∇2 V̄ ∗ (xe)z =v ′′ (x)Varp [z].
v ′ (x)(3)

Thus, this theorem implies the following equivalent expression of the measure ofambiguity aversion.
Corollary 1 For all x ∈ T and z ∈ RS with Varp [z] > 0,
H x (z) =z ⊤ ∇2 V ∗ (xe)z − z ⊤ ∇2 V̄ ∗ (xe)z.
z ⊤ ∇2 V̄ ∗ (xe)z(4)

That is, H x (z) can be obtained by dividing the diﬀerence between two quadratic
forms z ⊤ ∇2 V ∗ (xe)z and z ⊤ ∇2 V̄ ∗ (xe)z by the latter. Thus, the measure of ambiguity aversion is the ratio of the diﬀerence in the second-order eﬀects on least
concave utility levels between the preferences represented by V and V̄ to that ofthe preference represented by V̄ .
A simple but useful insight can be obtained from Corollary 1 by applying thesecond-order Taylor approximation. Note that ∇V ∗ (xe)z = ∇V̄ ∗ (xe)z = E p [z]and assume that E p [z] = 0. Then,(V ∗ (xe) − V ∗ (xe + εz)) − (V̄ ∗ (xe) − V̄ ∗ (xe + εz))
→ H x (z)∗∗V̄ (xe) − V̄ (xe + εz)
as ε → 0. Since V ∗ (xe + εz) < V ∗ (xe) and V̄ ∗ (xe + εz) < V̄ ∗ (xe), the asymptotic result says that the reduction is larger for V ∗ than for V̄ ∗ , by the factor
approximately equal to H x (z).
We now give yet another equivalent expression of the measure of ambiguity
aversion in terms of indiﬀerence curves, which admits a particularly illuminating
graphical representation. For each z ∈ RS with E p [z] = 0 and Varp [z] > 0, let11

L(z) be the plane spanned by e and z. Then, T S ∩ L(z) is an open subset ofL(z) that contains xe for every x ∈ T . Then the intersection of L(z) with theindiﬀerence hypersurfaces that contain xe, f ∈ T S ∩ L(z) | V (f ) = V (xe) andf ∈ T S ∩ L(z) | V̄ (f ) = V̄ (xe) , are twice continuously diﬀerentiable curves on
L(z). Denote their curvatures by cx (z) and c̄x (z).
Theorem 2 For every x ∈ T and every z ∈ RS with E p [z] = 0 and Varp [z] > 0, ⊤ ′z (v (x)∇2 I(v(x)e)) z v ′′ (x),c (z) = −S
+ ′Varp [z]v (x)v ′′ (x)c̄x (z) = −S 1/2 ′
v (x)
x

1/2

This theorem immediately gives the following equivalent expression for H x (z).
Corollary 2 For every x ∈ T and every z ∈ RS with E p [z] = 0 and Varp [z] > 0,
H x (z) =cx (z) − c̄x (z).
c̄x (z)Since the curvature of a curve at a point measures how much curved it isaround the point, this corollary that the measure of ambiguity aversion represents
how much more curved, in proportion, the indiﬀerence curves of V is than that ofV̄ in the direction z of deviation at the constant act xe.
We can now illustrate the measure of ambiguity aversion in terms of the indifference curves of the utility function V on the set of monetary acts along the lines
of Yaari (1969). Let S = 2. Let x ∈ T and consider the constant act xe. The
corresponding vector of Bernoulli utility proﬁles, v ◦ (xe), coincides with v(x)e.
The indiﬀerence curves of I and I¯ that go through v(x)e are shown on the left
panel of Figure 1. The indiﬀerence curve of I is convex to the origin, which represents ambiguity aversion, while that of I¯ is a straight line with normal vectorp = ∇I(v(x)e). These two indiﬀerence curves correspond to the two indiﬀerence
curves of V and V̄ that go through the constant act xe on the right panel. Since v
is concave, both are convex to the origin, but the indiﬀerence curve of V is moreconvex to the origin than the indiﬀerence curve of V̄ due to the concavity of I.
How much more convex the former is than the latter is represented by their curvatures; and Corollary 1 shows that the measure of ambiguity aversion represents
how much more curved, in proportion, the indiﬀerence curve of V is than that of
12

V̄ .
Figure 1: Measure of ambiguity aversion represented on the indiﬀerence curves of
V and V̄ .

consumption in state 2

Bernoulli utility in state 2
I

V

I

●

v(x)e

V

●

xe

p

p
Bernoulli utility in state 1
I and I

consumption in state 1
V and V

There are only two states (S = 2). On the left panel, the solid curve is theindiﬀerence curve of I that goes through the (Bernoulli) utility proﬁle v(x)e of theconstant act xe, and the dotted line is the indiﬀerence curve of I¯ that goes throughthe same proﬁle. Since v is concave, these indiﬀerence curves are transformed
into more concave ones on the set (plane) of monetary acts on the right panel,
but they are still tangent to the line with normal vector p = ∇I(v(x)e). The
measure of ambiguity aversion quantiﬁes how much more curved, in proportion,
the indiﬀerence curves of V is at xe than that of V̄ .

5

Matching probabilities

In Section 3, we deﬁned our measure of ambiguity aversion, (1), in terms of the
Hessians of aggregators. In this section, we explore its behavioral characterization. Speciﬁcally, we show that it is equal to the ratio of the derivative of thematching probability and its deviation from the derivative of the ﬁctitious matching probability, which is derived from a utility function that coincides with theutility function V on the set T S of monetary acts but exhibits risk neutrality. We
13then give an asymptotic result, establishing that it is approximately equal to theratio of the matching probability and its deviation from the ﬁctitious matchingprobability when acts are almost constant.
Let A ⊂ S and x ∈ T . Denote by eA the indicator function of A, that is,eA (s) = 1 for every s ∈ A and eA (s) = 0 otherwise. For each ε > 0, εeA represents
a bet on the event A with the prize ε, and xe + εeA is a binary act that represents
the consumption plan for a decision maker who holds a benchmark consumption
level x and places the bet. For each r ∈ [0, 1], denote by f x (ε, r) ∈ Π(T ) the
lottery that assigns probability r to x + ε and 1 − r to x. Then, regarding f x (ε, r)
as a lottery act, deﬁne ρx (ε, A) by
V (f x (ε, ρx (ε, A))) = V (xe + εeA ).

(5)

Thus, (5) means that the decision maker is indiﬀerent between the lottery with thewinning probability ρx (ε, A) and the bet on A. The winning probability ρx (ε, A) iscalled the matching probability of the event A at the consumption level x with theprize ε, because the equality means that the probabilistic assessment the decision
maker gives to the winning event A is equal to ρx (ε, A). Since V is an expectedutility function on the set of lottery acts, (5) can be rewritten asρx (ε, A)v(x + ε) + (1 − ρx (ε, A))v(x) = V (xe + εeA ).

(6)

This shows that ρx (ε, A) is a continuously diﬀerentiable function of ε > 0.
Even when ε < 0 (and x + ε ∈ T ), we deﬁne ρx (ε, A) as in (5) and (6). In thiscase, since xe + εeA = (x + ε)e − εeS\A , the bet is on the complementary event
S \ A with the prize −ε, and the decision maker’s initial deterministic consumption
level is x + ε. It can then be shown thatρx (ε, A) = 1 − ρx+ε (−ε, S \ A),(7)

where, since −ε > 0, the right-hand side was deﬁned as in the previous paragraph.
This shows that ρx (ε, A) is a continuously diﬀerentiable function of ε < 0.
For ε = 0, we set ρx (0, A) = p(A), where p(A) = ∇I(v(x)e)eA . Thus p(A) is theprobability of A under the subjective probability implicit at the constant act xe.
We have thus deﬁned the function ε 7→ ρx (ε, A) on an open interval containing 0. If
V represents a subjective expected expected utility function, then ρx (ε, A) = 0 for

14

every ε and ∂ρx (0, A)/∂ε = 0. The next theorem asserts, more generally, that thisfunction is diﬀerentiable at 0 and the derivative is equal to half the denominatorof H x (eA ).
Theorem 3 Let A ⊂ S and x ∈ T . Then the function ε 7→ ρx (ε, A) is diﬀerentiable at 0 and

∂ρx
1
′
2
(0, A) = e⊤
A v (x)∇ I(v(x)e) eA .
∂ε2
To obtain an expression of the numerator of H x (eA ) in terms of matchingprobabilities, write I = v −1 ◦ V . Then, I gives the certainty equivalents in termsof constant acts. It is deﬁned on Π(T )S but think of its domain as restricted on
the set T S of monetary acts for the subsequent analysis. Then, it is continuous,
increasing, and normalized. Let v be the identity function on T and deﬁne a utility
function V by (v, I).11 Since v ◦ f = f and, hence, V (f ) = I(v ◦ f ) = v −1 (V (f ))for every f ∈ T S , V represents the same preference as V on the set of monetaryacts. But, since V is risk-neutral, they diﬀer from each other outside the setof monetary acts. Moreover, any aversion to non-constant acts exhibited by V
is attributed entirely to ambiguity aversion. Thus, any diﬀerence in ambiguity
aversion, and hence in matching probabilities, between V and V is attributed
to the risk aversion represented by the Bernoulli function v. Deﬁne ρx (ε, A) asin (5) for ε 6= 0 using V in place of V , and let ρx (0, A) = ∇I(v(x))eA . Since∇I(v(x))eA = ∇I(v(x))eA = p(A), ρx (0, A) is equal to p(A). Theorem 3 holdsfor (v, I). In particular, ε 7→ ρx (ε, A) is diﬀerentiable at 0. By combining ρx (ε, A)with ρx (ε, A), we obtain the following expression for H x (z).
Corollary 3 Let A ⊂ S and x ∈ T and suppose that 0 < p(A) < 1, then∂ρx(0, A)∂εH x (eA ) =
.
∂ρx
∂ρx
(0, A) −(0, A)∂ε∂ε(8)The ﬁrst-order Taylor approximation shows thatp(A) − ρx (ε, A)→ H x (z)xxp(A) − ρ (ε, A) − (p(A) − ρ (ε, A))
11(9)

The Bernoulli utility function v violates the assumption of negative second derivatives, statedin Section 2. This, however, will not cause any problem for the analysis in this section.

15

as ε → 0. This asymptotic result can be explained as follows. The diﬀerencep(A) − ρx (ε, A) is the reduction in the matching probabilities of the event A causedby an increase in prizes of bets on the event. The diﬀerence p(A) − ρx (ε, A) is also
the reduction in the matching probabilities of the event A but measured for theﬁctitious utility function V . Since V and V represent the same preference for monetary acts but the latter is risk-neutral, the reduction in the matching probabilities
of A for V is due not only to the ambiguity aversion represented by I but also,
erroneously, to the risk aversion represented by v (as it is missed out by v). Thusthe diﬀerence p(A) − ρx (ε, A) − (p(A) − ρx (ε, A)) in the reductions of matchingprobabilities for V and V is due solely to the risk aversion of v. Thus, it can approximate the Arrow-Pratt measure of risk aversion of v. Corollary 3 shows thatthe ratio of the reduction in the matching probabilities due to ambiguity aversion
and the reduction in the ﬁctitious matching probabilities due to risk aversion isapproximately equal to our measure of ambiguity aversion.
As any statement on derivatives is hard to test when there are only ﬁnitely manyobservations (such as investors’ portfolio choices in asset markets and subjects’
responses in laboratory experiments), this asymptotic result may be useful whenidentifying the decision maker’s ambiguity attitude from his choice behavior. The
availability of such an asymptotic result is one of the reasons, stated in Section 3,
why we include the Arrow-Pratt measure in the deﬁnition (1) of the measure ofambiguity aversion.

6

Ambiguity premiums

In this section, we explore another behavioral characterization of the measure ofambiguity aversion. We show that it is equal to the ratio of the second derivatives of
the risk and ambiguity premiums. We then give an asymptotic result, establishingthat it is approximately equal to the ratio of the risk and ambiguity premiums
when acts are almost constant. Finally, we give a graphical illustration of theambiguity premium.
Let x ∈ T and p = ∇I(v(x)e). Let z ∈ T S . For each ε close to 0, denoteby f x (ε, z) ∈ Π(T ) the lottery that coincides with the distribution on T of themonetary act x + εz under the probability p. Then, regarding f x (ε, z) as a lotxtery act, deﬁne κx (ε, z) by V f x−κ (ε,z) (ε, z) = V (xe + εz). That is, κx (ε, z) is
the maximum consumption level that the decision maker is willing to give up in
16

exchange for the assurance that the true probability coincides with his benchmark
probability. It can, thus, be called the ambiguity premium of the act xe + εz withrespect to the utility function V .
As in Section 3, let I¯ be the expectation operator under p and deﬁne V̄ by¯ Since V coincides with V̄ on the set of lottery acts, κx (ε, z) can also be(v, I).
deﬁned byV̄ ((x − κx (ε, z)) e + εz) = V (xe + εz).

(10)

By applying the implicit function theorem to (10), we can see that κx (ε, z) is a twice
continuously diﬀerentiable function of ε. In particular, if V represents a subjective
expected utility function, then κx (ε, z) = 0 for every ε and ∂ 2 κx (0, z)/∂ε2 = 0.
The following theorem, more generally, relates the numerator of the measure ofambiguity aversion (1) to the second derivative of the ambiguity premium.
Theorem 4 For all x ∈ T and z ∈ RS ,
κx (0, z) = 0∂κx(0, z) = 0,∂ε
∂ 2 κx
⊤
′
2
z)
−z
(0,
=
v
(x)∇
I(v(x)ez.
∂ε2
In the terminology analogous to that of Segal and Spivak (1990), the equality∂κ (0, z)/∂ε = 0 means that the utility function V is second-order ambiguityaverse.
The Arrow-Pratt measure of absolute risk aversion, which appears in the denominator of the deﬁnition (1) of the measure of ambiguity aversion, can be written
in terms of the second derivative of the risk premium. This is a well known result due to Pratt (1964, Section 3), but we reproduce the argument here for thesubsequent analysis.
Deﬁne κ̄x (ε, z) by V̄ ((x − κ̄x (ε, z)) e) = V̄ (xe + εz). Then κ̄x (ε, z) is the risk
premium, that is, x − κ̄x (ε, z) is equal to the certainty equivalent of the monetaryx17act xe + εz under the probability p. Pratt (1964) showed that if E p [z] = 0, then∂κ̄x(0, z) = 0,∂ε
∂ 2 κ̄xv ′′ (x)(0,z)=−
Varp [z].
∂ε2v ′ (x)(11)(12)

Hence, Theorem 4 leads to the following characterization of H x (z) in terms ofambiguity and risk premiums.
Corollary 4 For all x ∈ T and z ∈ RS with E p [z] = 0,
∂ 2 κx(0, z)
2
H x (z) = ∂ε.
∂ 2 κ̄x(0, z)∂ε2
By applying the second-order Taylor approximation, we can show that as ε → 0,
κx (ε, z)→ H x (z)κ̄x (ε, z)(13)That is, our measure of ambiguity aversion, H x (z), is approximately equal to theratio of the ambiguity premium κx (ε, z) to the risk premium κ̄x (ε, z). This corollary
will be useful for the quantitative analysis of ambiguity premiums, just as Corollary
3 will be useful for the quantitative analysis of matching probabilities, because any
statement on (second) derivatives, in Theorem 3 or 4, is hard to test when there
are only ﬁnitely many observations.
Before giving a graphical illustration of the ambiguity premium, let us introduce
another type of premium, which can be called the uncertainty premium. Deﬁneκ̌x (ε, z) by V ((x − κ̌x (ε, z)) e) = V (xe + ε). That is, x − κ̌x (ε, z) is the certainty
equivalent of xe + εz with respect to the utility function V , which reﬂects bothrisk and ambiguity aversion. We could deﬁne the part of the uncertainty aversion
that cannot be attributed to risk aversion, κ̌x (ε, z) − κ̄x (ε, z), as the ambiguity
premium. Although this is, in general, not equal to κx (ε, z), they are equal up to
the ﬁrst- and second-order. In particular,∂ 2 κx∂ 2 κ̌x∂ 2 κ̄x
(0,
z)
=
(0,
z)
−
(0, z).
∂ε2∂ε2∂ε2Hence, Theorem 4 can be restated using the right-hand side of the above equality.
18

We can now give a graphical representation of the risk premium and the ambiguity premium. Assume that S = 2. Let x ∈ T and write p = ∇I(v(x)e). Let z
represent a deviation such that E p [z] = 0. Let ε > 0. The constant act xe and themonetary act xe + εz are depicted on Figure 2. The solid curve is the indiﬀerence
curve of the utility function V ; the upper dotted curve is the indiﬀerence curve of
the benchmark utility function V̄ ; and the dashed line is the iso-expectation line,
all going through xe + εz. By the deﬁnition of the uncertainty premium, the solid
indiﬀerence curve intersects the 45-degree line at (x − κ̌x (ε, z)) e; by the deﬁnitionof the risk premium, the upper dotted indiﬀerence curve intersects the 45-degreeline at (x − κ̄x (ε, z)) e; and since E p [z] = 0, the dashed iso-expectation line goesthough xe. Thus, the risk premium κ̄x (ε, z) is the distance between the upper
dotted indiﬀerence cure and the dashed iso-expectation line, measured along the
45-degree line. The lower dotted curve is the indiﬀerence curve of the benchmark
utility function V̄ that goes through (x − κ̌x (ε, z)) e. It is tangent to the solid
indiﬀerence curve, because they are both tangent to the iso-expectation line. It
also goes through (x − κx (ε, z)) e + εz, becauseV̄ ((x − κ̌x (ε, z)) e) = V ((x − κ̌x (ε, z)) e) = V (xe + εz) = V̄ ((x − κx (ε, z)) e + εz) .
Thus, the ambiguity premium κx (ε, z) is the distance between the upper and lower
dotted curves as measured along the 45-degree line going through xe+εz. Corollary
4 shows that our measure of ambiguity aversion is approximately equal to theratio of the distance between the dashed iso-expectation line and the upper dotted
indiﬀerence curve and the distance between the upper and lower dotted indiﬀerence
curves.

7

Smooth ambiguity model

In this section, we deal with the smooth ambiguity model of KMM, as it is theprime example of twice continuously diﬀerentiable aggregators. We show that it
admits an insightful equivalent expression of the measure of ambiguity aversion,
and, then, give a simple numerical example to illustrate how much ambiguity
aversion needs to be introduced to solve the home bias puzzle.

19

Figure 2: Risk premium, ambiguity premium, and the indiﬀerence curves of V andV̄ .

consumption in state 2
V

V

V

xe
●

x
(x - Ǩx(ε,z))e●● (x - κ (ε,z))e

●
●

xe+εz
p

(xe + εz) - κx(ε,z)econsumption in state 1
The solid curve is the indiﬀerence curve of the utility function V ; the upper dottedcurve is the indiﬀerence curve of the benchmark utility function V̄ ; and the dashedline is the iso-expectation line, all going through xe+εz. The measure of ambiguity
aversion is approximately equal to the ratio of the distance between the dashediso-expectation line and the upper dotted indiﬀerence curve, measured along the
45-degree line going through the origin versus the distance between the upper andlower dotted indiﬀerence curves, measured along the 45-degree line going throughxe + εz.

7.1

Equivalent expression of the measure of ambiguity aversion

Let v : T → R be a Bernoulli utility function be as in Section 2, ϕ : v(T ) → R
be a twice continuously diﬀerentiable function that satisfy ϕ′′ ≤ 0 < ϕ′ , and µRS
be a probability measure on ∆ such that ∆ p dµ(p) ∈ R++. This is the reduced
I
S
probability of µ and denoted by p . Deﬁne I : v(T ) → R by letting
I(g) = ϕ

−1

Z


p

ϕ (E [g]) dµ(p)
∆

20

(14)

for every g ∈ v(T )S . As in Section 2, deﬁne V : T S → R by (v, I). Then,Z
ϕ (E p (v ◦ f )) dµ(p).

ϕ(V (f )) =

(15)

∆

This is the smooth ambiguity model axiomatized by KMM. Rigotti, Shannon,
and Strzalecki (2008, Proposition 5) showed that the reduced probability pI is its
benchmark probability.
Let w = ϕ ◦ v, then (15) can be rewritten asZw v −1 (E p (v ◦ f )) dµ(p).

ϕ(V (f )) =∆While ϕ is a Bernoulli utility function over expected utility levels, w is a Bernoulli
utility function over certainty equivalents. Denote by E · [z] the random variable
p 7→ E p [z] deﬁned on the probability space (∆, B(∆), µ). Denote by E µ and Varµbe the expectation operator and the variance operator for the random variancedeﬁned on ∆ under the probability µ. In this class of smooth ambiguity-averse
utility functions, the measure of ambiguity aversion can be written as follows.
Proposition 1 If I is deﬁned by (14), thenw′′ (x) Varµ [E · [z]]
 − w′ (x)x−1
H (z) =  VarpI [z] v ′′ (x)
− ′
v (x)


(16)

I

for all x ∈ T and z ∈ RS with Varp [z] > 0.
This proposition shows that in the case of the smooth ambiguity utility functions, the measure H x (z) of ambiguity aversion consists of two terms. The ﬁrst
term of (16) represents how much in proportion the decision maker is more averseto the variability in certainty equivalents under diﬀerent probabilities on the state
space than to the variability that is independent of the choice of such probabilities. This term is independent of the second-order belief µ and also of the choiceof z. The second term is dependent on the choice of z. Since its numerator isthe variance of the random variable E · [z] : p 7→ E p [z] under the probability µ on∆, it measures how much uncertainty is perceived in the distribution of z. The
denominator is a normalizing factor, as the law of total variance guarantees that21

the fraction lies in [0, 1].12 Since pI is the reduced probability of the second-orderbelief, the second term is determined by the second-order belief, and represents
how much ambiguity is perceived in the direction the proﬁle z of state-dependentutility levels. Thus, the measure of ambiguity aversion can be decomposed into
two parts, the ﬁrst part determined by the decision maker’s inherent aversion to
ambiguity, and the second part determined by the decision maker’s perception of
ambiguity in the environment he is in.13

7.2

Application to the home bias puzzle

We now turn to a simple work-out example in which the measure of ambiguity aversion can be inferred from the observed portfolio choice in the home bias puzzle.
In the following speciﬁcation of utility functions, the robust quadratic approximation Maccheroni, Marinacci, and Ruﬃno (2013) is exact. The general case of anarbitrary number of assets was considered by Hara and Honda (2022).
Three types of assets, the home stock, the foreign stock, and the risk-free bond,can be bought and sold. The state space S coincides with the set of all possible
realizations of a bivariate random variable R = (R1 , R2 ) of home and foreign stocks
returns. The return of the risk-free bond is denoted by Rf .
In the speciﬁcation (v, w, µ) of a KMM utility function, we assume that v has aconstant coeﬃcient θ of absolute risk aversion, and w has a constant coeﬃcient γ
of absolute risk aversion, with θ ≤ γ. Write η = γ/θ − 1. The second-order beliefµ is deﬁned as follows. Let σ > 0 and τ ∈ (0, 3/4], and deﬁne a 2 × 2 matrix ΣR|M
by
ΣR|M =

σ2(1/2)σ 2(1/2)σ 2 (1 − τ )σ 2!
.

Since τ ≤ 3/4, ΣR|M is positive semideﬁnite. Then, for each m ∈ R2 , let thebivariate normal distribution N (m, ΣR|M ) be a ﬁrst-order belief, that is, a proba12
This point can be most clearly seen when z = eA , that is, z is the indictor function of an event
A. Then, the denominator of the second term is equal to pI (A)(1 − pI (A)), while the numerator
is equal to the variance of p 7→ p(A) under µ, which does indeed measure the variability of theprobability of A due to ambiguity perception. Jewitt and Mukerji (2017) gave a comprehensive
analysis on more-ambiguous-than relations between two acts and events.
13
The phrase, “inherent aversion of ambiguity”, has a narrower sense than “ambiguity aversion”
in the “measure of ambiguity aversion”, as the latter includes the perception of ambiguity but
the former does not.

22

bility measure on the state space.14 We let the support of the second-order belief
be the set of all these bivariate normal distributions. As such, it is parameterizedby m ∈ R2 . Let R̄ > Rf and write
ΣM =

0 0
0 τ σ2!
.

Let the second-order belief µ in terms of the parameters m ∈ R2 be the bivariate
normal distribution N (R̄1, ΣM ), where 1 = (1, 1) ∈ R2 . WriteΣR =σ2(1/2)σ 2(1/2)σ 2
σ2!
,then ΣR = ΣR|M + ΣM and, by the law of total variance, the joint distributionof R = (R1 , R2 ) under the reduced probability of µ, which coincides with thebenchmark probability, is N R̄1, ΣR .
The second-order belief µ has the following features. Under the benchmark
probability, the home and foreign stocks share the same expected return R̄ andthe same variance σ 2 , and their correlation coeﬃcient is equal to 1/2. Lewis (1999,
Table 2, Panel A), for example, reported that the mean and standard deviationof the annualized monthly returns of the home (US) stock from January 1970 to
December 1996 are 11.14% and 15.07%, those of the foreign (Europe, Australia, andFar East, measured in dollars) stocks are 12.12% and 16.85%, and the correlationcoeﬃcient between the returns of the two stocks is 0.48. As such, the assumptions
that the two stock returns share the same mean R̄ and the same variance σ 2 andthat their correlation coeﬃcient is 1/2 are not overly unrealistic.
The decision maker perceives no ambiguity in the return of the home stock,
but he is unsure of the expected return of the foreign stock. In particular, he
perceives that the expected return itself is normally distributed and the varianceof the foreign stock return due to this ambiguity has a proportion τ in the (total)
variance under the benchmark probability.
Denoting his total wealth by W , we formulate the decision maker’s portfolio

14

This assumption implies that the state space S is inﬁnite and, thus, violates a maintained
assumption of this paper. This, however, causes no problem because our analysis is concentrated
on the linear subspace spanned by R1 , R2 , and the constant-valued function in the set of all
random variables on S.

23

choice problem as the problem of maximizing


Z

w v
R2

−1



Z
v(a0 Rf + a · r) dN (m, ΣR|M )(r)dN (R̄1, ΣM )(m),

(17)

R2

where r = (r1 , r2 ), a = (a1 , a2 ), and a · r = a1 r1 + a2 r2 , by choosing the money
amounts invested, (a0 , a1 , a2 ) ∈ R3 , subject to the budget constraint a0 + a1 + a2 ≤W . We can then derive the following one-to-one relation between the measure ofambiguity aversion and the fraction of the wealth invested in the foreign stock outof the total wealth invested in the two stocks.
Proposition 2 For every x ∈ R, H x (R1 ) = 0 and H x (R2 ) = ητ . Moreover, if(a0 , a1 , a2 ) ∈ R3 is the solution to (17), thena21=.
a1 + a2
2 (1 + ητ )The expected consumption level under the benchmark probability is equal toa0 Rf + a1 R̄ + a2 R̄. The proposition shows that the equalities H x (R1 ) = 0 andH x (R2 ) = ητ hold not only for x = a0 Rf + a1 R̄ + a2 R̄ but also for any otherx, and that the measure of ambiguity aversion in the direction of R2 and thefraction invested in the foreign stock depend on the CARA coeﬃcient γ of theouter Bernoulli utility function v only through η = γ/θ − 1. This substantiates ourclaim, made right after the deﬁnition (1) of the measure of ambiguity aversion, thatthe ambiguity aversion in excess of the risk aversion is often more important than
the ambiguity aversion itself. The proof shows that of the measure H x (R2 ) = ητ ,
η is equal to the ﬁrst term, and τ is equal to the second term, on the right-handside of (16).
The proposition provide us with simple but interesting quantitative implications. First, the fraction of investment in the foreign stock is a decreasing functionof the measure of ambiguity aversion, starting from 1/2 in the case of ambiguity
neutrality and converging to 0 in the case of unboundedly high ambiguity aversion.
Second, by reverting the equality, we can infer the measure of ambiguity aversion
from the (observed) fraction of the investment into the foreign stock. Lewis (1999)
reported that the fraction was just 8% in the US data, which implies that the
representative American’s measure of ambiguity aversion is equal to 5.25.
The possibility of inferring the measure of ambiguity aversion from observablechoice behavior is of considerable help in any quantitative analysis of ambiguity
24

aversion. Recall, as shown by Corollaries 3 and 4, that the measure of ambiguity
aversion is related to the ambiguity premium and the matching probabilities. Thus,
we can check, by comparing the inferred values with those derived from laboratory
experiments, whether the inferred value is derived solely from ambiguity aversion
or contaminated by institutional factors, such as incomplete asset markets andtransaction costs in the context of the home bias puzzle.

8

Extension to the non-diﬀerentiable case

8.1

Background and motivation

To deﬁne the measure of ambiguity aversion in (1), we assumed that the aggregator
I is twice continuously diﬀerentiable. This assumption excludes, most notably,maximin expected utility of Gilboa and Schmeidler (1989) and Choquet expectedutility of Schmeidler (1989). As they have signiﬁcant behavioral implications,such as the inertia in portfolio choice, we extend the deﬁnition of the measure ofambiguity aversion to such non-diﬀerentiable aggregators and generalize the results
on matching probabilities and ambiguity premiums in Sections 5 and 6. This allows
us to compare the ambiguity aversion exhibited by non-diﬀerentiable aggregatorson the same footing with the ambiguity aversion exhibited by diﬀerentiable ones.
The extension to the non-diﬀerentiable case involves two additional considerations that were unnecessary for the twice continuously diﬀerentiable case. The ﬁrst
one is to give a notion of twice diﬀerentiability that can accommodate maximin
expected utility and Choquet expected utility. Since these utility functions are noteven diﬀerentiable, we need to give one that is particularly suited to our purpose.
Our deﬁnition is based on Peano right-derivatives.
The second consideration we need speciﬁcally for the non-diﬀerentiable case is
the choice of the benchmark probability in the sense of Ghirardato and Marinacci(2002). As is well known, for concave but non-diﬀerentiable aggregators, there may
be multiple benchmark probabilities. When deﬁning the measure of ambiguity
aversion, the choice of the benchmark probability p aﬀects the ambiguity measureH x (z) through the variance Varp [z] of the deviation z from the constant act xe. In
our extended deﬁnition, we will allow p to depend on the choice of z (even after weﬁx the deterministic consumption level x), but require p to minimize its mean E p [z]among all supergradients and, in addition, to maximize its variance Var p [z] among25

all such mean-minimizing supergradients, for a reason that we will elaborate on in
the next subsection.

8.2

Deﬁnitions

We assume that the aggregator I is continuous, non-decreasing, normalized, concave, but not necessarily twice continuously diﬀerentiable. Then, I is locally Lipschitz continuous. It follows from Theorem 3.6(ii) and Theorem 4.8 of Chapter
3 of Delfour (2019) that I is Hadamard right-diﬀerentiable at every point in all
directions in the sense of Deﬁnition 3.1(ii) in Section 3 of Chapter 3 of Delfour(2019). Denote the Gateaux and Hadamard right-derivative of I at g in the
direction z by dG I(g; z) and dH I(g; z). They are positively homogeneous, but
not necessarily linear, in z. They coincide whenever the latter exists. They arelinear in z if I is diﬀerentiable in the standard sense; and we can then writedG I(g; z) = dH I(g; z) = ∇I(g)z using the gradient vector ∇I(g) of I at g.
Below are two versions of twice right-diﬀerentiability that we use in this paper.
It is given in terms of a general function F deﬁned on an open subset A of an
Euclidean space RN of arbitrary dimension, as it will be used not only for theaggregator I also for other functions such as the ambiguity premium.
Deﬁnition 2 Let a ∈ A and b ∈ RN .
1. We say that F is twice Peano-Gateaux right-diﬀerentiable at a in the directionb if F is Gateaux right-diﬀerentiable at a in the direction b and there is an
L ∈ R such that
F (a + εb) − (F (a) + εdG F (a; b))→Lε2
as ε → 0+. We denote 2L by d2G F (a; b) and call it the second Peano-Gateauxright-derivative of F at a in the direction b.
2. We say that F is twice Peano-Hadamard right-diﬀerentiable at a in the direction b if F is Hadamard right-diﬀerentiable at a in all directions near b
and there is an L ∈ R such that
F (a + εz) − (F (a) + εdH F (a; z))
→L
ε2

26

as ε → 0+ and z → b. We denote 2L by d2G F (a; b) and call it the second
Peano-Hadamard right-derivative of F at a in the direction b.
Both of the two deﬁnitions require the function F to satisfy the second-order
Taylor approximation, but they are diﬀerent in that the twice Peano-Gateaux rightdiﬀerentiability is met whenever the direction b of deviation from a is ﬁxed throughout the limiting operation while the twice Peano-Hadamard right-diﬀerentiability
requires the existence of the limit even when the direction of deviation itself varies
around b. The twice Peano-Hadamard right-diﬀerentiability implies twice PeanoGateaux right-diﬀerentiability, and, then, the second Peano-Hadamard and PeanoGateaux right-derivatives coincide. They also coincide whenever N = 1, and wethen refer to them simply as the second Peano right-derivative. They are positively homogeneous of degree two, but not necessarily quadratic, in b. They arequadratic in b if F is twice diﬀerentiable in the standard sense; and we can thenwrite d2G F (a; b) = d2H F (a; b) = b⊤ ∇2 F (a)b using the Hessian ∇2 F (a).
The above deﬁnition of twice right-diﬀerentiability weaker than other known
deﬁnitions of twice right-diﬀerentiability, such as Deﬁnition 3.12 in Section 3 ofChapter 3 of Delfour (2019), Theorem 2.3 of Rockafellar (2000), and Deﬁnition
13.1 of Rockafellar and Wets (1998), mainly in that the diﬀerentiability of the(ﬁrst) right-derivative on a neighborhood of the point a is not a prerequisite. For
example, Deﬁnition 3.12 in Section 3 of Chapter 3 of Delfour (2019) goes as follows:
If F is Gateaux right-diﬀerentiable at every point near a in all directions, and there
is an L ∈ R such thatdG F (a + εc; b) − dG F (a; b)→Lε
as ε → 0+, then F is said to be twice right-diﬀerentiable at a in the direction(b, c). Unfortunately, this condition is not satisﬁed by maximin expected utilities
of Gilboa and Schmeidler (1995).
To extend the measure of ambiguity aversion to the non-diﬀerentiable case,we give the following notation. For each y ∈ v(T ) and z ∈ RS , denote by ∆yI
the set of all supergradients of I at ye, by ∆yI (z) the set of all p ∈ ∆yI such thatp · z ≤ q · z (that is, E p [z] ≤ E q [z]) for every q ∈ ∆yI , and by ΛyI (z) as the set of
all p ∈ ∆yI (z) such that p · z 2 ≥ q · z 2 (that is, E p [z 2 ] ≥ E q [z 2 ], which is equivalent
to Varp [z] ≥ Varq [z]) for every q ∈ ∆yI (z), where z 2 = ((z(s))2 )s ∈ RS . These sets
are nonempty and compact.

27

Deﬁnition 3 Let x ∈ T and z ∈ RS . Suppose that I is twice Peano-Gateauxv(x)right-diﬀerentiable at v(x)e in the direction z and let p ∈ ΛI (z). Then, wedeﬁne−
H x (z) =v ′ (x)d2G I(v(x)e; z)
Varp [z].
v ′′ (x)
− ′
v (x)

(18)

v(x)

If I is twice continuously diﬀerentiable, then ΛI = {∇I(v(x)e)} and d2H I(v(x)e; z) =z ⊤ ∇2 I(v(x)e)z. Thus, the above deﬁnition of H x (z) is, indeed, an extension of (1).
v(x)
The use of a p ∈ ΛI (z) in the deﬁnition of H x (z) can be justiﬁed as follows.
v(x)For each p ∈ ∆I , let I¯p be the expectation operator under p and V̄p be deﬁned byv, I¯p . Then, V̄p represents a benchmark preference of (the preference representedv(x)by) V , and V̄p ≥ V . Thus, the best approximation of V among the V̄p over p ∈ ∆I
is the one that minimizes the values of V̄p . By the Taylor’s theorem,V̄p (xe + εz) ≈ V̄p (xe) +

X
s



1
p(s) v (x)εz(s) + v ′′ (x)(εz(s))2
2



′

1
= v(x) + v ′ (x)E p [z]ε + v ′′ (x)E p [z 2 ]ε2 .
2
To minimize the ﬁrst-order impact on V̄p (xe+εz), we need to minimize the secondv(x)term on the far right-hand side, that is, choose a p ∈ ∆I (z). Once this is done,V̄p (xe + εz) is minimized when the last term is minimized. This is true when E p [z 2 ]v(x)is maximized, that is, when p ∈ ΛI . Note here that the risk aversion, v ′′ < 0,is crucial: When choosing a benchmark preference and probability, we should notignore the second-order impact on V arising from risk aversion. If we did, thenH x (z) would overestimate the ambiguity aversion represented by I.

8.3

Matching probabilities

In this subsection, we generalize the characterization of the measure of ambiguity
aversion in terms of matching probabilities (Theorem 3 and Corollary 3). For asuﬃciently small ε > 0, deﬁne the matching probability ρx (ε, A) just as in (5). Forv(x)ε = 0, we now let ρx (0, A) = p(A), where p ∈ ∆I (eA ) and p(A) = p · eA . Thev(x)value of p(A) is independent of the choice of p ∈ ∆I (eA ).

28

Theorem 5 Let A ⊂ S and x ∈ T . If I is twice Peano-Gateaux right-diﬀerentiableat v(x)e in the direction eA , then ρx ( ·, A) is right-diﬀerentiable at 0 and1dG ρx (0, A) = v ′ (x)d2G I(v(x)e; eA ).
2(19)

In many cases of interest, a symmetry consideration suggests that there is a
v(x)
v(x)
“natural” benchmark probability in ∆I that is outside ∆I (eA ). For example,
in the case of the two-color Ellsberg urn, the natural probability assigns probability
1/2 to the event of drawing a ball of the winning color, while the probability inv(x)∆I (eA ) assigns the lowest probability, among all benchmark probabilities, to thewinning event, which is less than 1/2. Let p∗ be the natural benchmark probabilityv(x)and p be in ∆I (eA ), then the diﬀerence p∗ (A) − p(A) is often taken as a measureof ambiguity aversion. Our measure of ambiguity aversion diﬀers from this one,as it is related to further decrease in matching probabilities ρx (ε, A) as the valueof the prize, ε, increases. When coupled with the often-used measure, however, it
can improve the approximation of the diﬀerence between the natural benchmark
probability and the matching probability viaεp∗ (A) − ρx (ε, A) ≈ (p∗ (A) − p(A)) − v ′ (x)d2G I(v(x)e; eA ).
2v(x)for a small ε > 0, because ρx (0, A) = p(A) for any p ∈ ∆I (eA ).
Deﬁne I, v, V , and ρ as in Section 5, that is, I = v −1 ◦ V , v is the identifyfunction, V is deﬁned by (v, I), and ρ is the matching probability for the utility
function V . Corollary 3 can be extended to the non-diﬀerentiable case as follows.
Corollary 5 Under the assumptions of Theorem 5, I also satisﬁes the assumptionsof Theorem 5 andH x (eA ) =dG ρx (0, A).
dG ρx (0, A) − dG ρx (0, A)(20)

The asymptotic result (9) can also be extended to the non-diﬀerentiable case.
v(x)Corollary 6 Under the assumptions of Theorem 5, for every p ∈ ∆I
p(A) − ρx (ε, A)→ H x (eA )xxp(A) − ρ (ε, A) − (p(A) − ρ (ε, A))

29

(z),

as ε → 0.

8.4

Biseparable preferences

In this subsection, we prove that for the utility functions that represent biseparable preferences, the extended measure of ambiguity aversion is always equal tozero and, then, compare this fact, graphically, with the case of twice continuously
diﬀerentiable utility functions.
The following deﬁnition of biseparability is due to Ghirardato and Marinacci(2001, Section 2.2) and Ghirardato and Marinacci (2002, Section 1).
Deﬁnition 4 We say that V is biseparable if, for every A ⊂ S, there is a ρ(A) ∈[0, 1] such that ρx (ε, A) = ρ(A) for every x ∈ T and every ε > 0.
According to this deﬁnition, a utility function is biseparable if the matchingprobability of any event A does not depend on the consumption levels on A or
outside A, as long as the former is higher than the latter. Examples of biseparable
preferences include Choquet expected utility functions of Schmeidler (1989) andα-maximin expected utility functions of Ghirardato, Maccheroni, and Marinacci(2004).
Proposition 3 Suppose that V is deﬁned by (v, I) and biseparable. Then, for allx ∈ T and A ⊂ S, I is twice Peano-Gateaux right-diﬀerentiable at v(x)e in thev(x)direction eA and d2G I(v(x)e; eA ) = 0. If 0 < p(A) < 1 for any p ∈ ΛI (eA ), thenH x (eA ) = 0.
It can be easily shown that for Choquet expected utility functions and αmaximin expected utility functions, the aggregator I is, in fact, twice PeanoHadamard right-diﬀerentiable, and d2H I(v(x)e; eA ) = 0.
We now turn to the illustration of matching probabilities for biseparable, twice
continuously diﬀerentiable but ambiguity-averse, and subjective expected utility
functions. Let V1 and V2 be two utility functions deﬁned by (v1 , I1 ) and (v2 , I2 ).
Suppose that the preference represented by V2 is a benchmark preference of thepreference represented by V1 , with p the benchmark probability. Then, they sharethe same risk attitudes, by which we can assume that v1 = v2 and write v inplace of v1 and v2 . Moreover, by Theorem 12 of Ghirardato and Marinacci (2002),v(x)p ∈ ∆I1 for every x ∈ T . By the deﬁnition of ρx (0, A), ρx (0, A) ≤ p(A) and30ρx (0, S \ A) ≤ p(S \ A). Thus, ρx (0, A) + ρx (0, S \ A) ≤ 1. The argument so far isapplicable regardless of whether V1 is biseparable or smooth.
Suppose now that V1 is biseparable and A is an ambiguous event for the
preference in the sense of Ghirardato and Marinacci (2002, Section 4). Then,ρx (ε, A) = ρ(A) for every ε > 0. Moreover, by the deﬁnition of ρx (0, A), thisequality holds even when ε = 0. By Proposition 22 of Ghirardato and Marinacci(2002), ρ(A) + ρ(S \ A) < 1. Hence, ρ(A) ≤ p(A) ≤ 1 − ρ(S \ A), and at least oneof these two weak inequalities holds as a strict inequality. As for the case whereε < 0, by (7), ρx (ε, A) = 1 − ρ(S \ A). For a biseparable preference, therefore, thegraph of the matching probability, ε 7→ ρx (ε, A), jumps down at zero, crossing thebenchmark probability p(A), but is constantly equal to ρ(A) on the right of zero,and to 1 − ρ(S \ A) on the left of zero.
Suppose, instead, that V1 is twice continuously diﬀerentiable and A is an ambiguous event for V1 in the sense that H x (eA ) > 0. Then ρx (0, A) = p(A), wherep = ∇I1 (v(x)e), the unique benchmark probability for V1 . Thus, the graph ofthe matching probability ε 7→ ρx (ε, A) goes through the benchmark probabilityp(A) at 0. Moreover, by Theorem 3, it is diﬀerentiable at ε = 0 and, by (8),the derivative is negative. For a twice continuously diﬀerentiable utility function,
therefore, the graph of the matching probability, ε 7→ ρx (ε, A), is diﬀerentiable anddownward-sloping around 0, where it is equal to the benchmark probability.
These observations are gathered in Figure 3, in which we consider a biseparable
utility function VB , a twice continuously diﬀerentiable utility function VS , and an
expected utility function VE . Assume that VE represents a benchmark preference
for VB and also for VS . Let p be the benchmark probability corresponding to VE .
Let A ⊂ S. Assume that A is an ambiguous event for both VB and VS , that is,
ρB (A) + ρB (S \ A) < 1 and HSx (eA ) > 0.
Then, the matching probability ρE (ε, A) for the expected utility function VE isequal to p(A) for every ε. Its graph is horizontal and intercepts the vertical axis atp(A). The matching probability ρS (ε, A) for the twice continuously diﬀerentiableutility function VS is equal to p(A) at ε = 0 and decreasing around ε = 0. Itsgraph intercepts the vertical axis at p(A) and slopes downwards around there.
The matching probability ρB (ε, A) for the biseparable utility function VB is equal
to ρB (A) at every ε ≥ 0 and to 1 − ρB (S \ A) at every ε < 0. The graph ofthe matching probability has a downward jump on the vertical axis, passing p(A),although it is horizontal in either side of the vertical axis.

31

Figure 3: Matching probabilities for a biseparable utility function, a twice continously diﬀerentiable utility function, and an expected utility function
ρn (ε,A) for n = B, S, E

○ 1 - ρB(S \ A)pE (A)
ρB(A) ○
ρS(ε, A)ε0

The dashed lines, with a gap on the vertical axis, constitute the graph of thematching function for a biseparable utility function. The solid downward-sloping
smooth curve is the graph of the matching probability for a twice continuously differentiable utility function. The dotted horizontal line is the graph of the matching
function of an expected utility function, which represents a benchmark preference
for the two ambiguity-averse utility functions.
The graphs of matching probabilities can help us distinguish the nature ofambiguity aversion of twice continuously diﬀerentiable utility functions from thatof biseparable preferences. For a biseparable preference, at every constant act,
there is a range of probabilities of an ambiguous event that is the decision makerdeems possible or relevant. When it comes to betting on an event (which induces a
non-constant act), however, his assessment of the likelihood of the event goes, discontinuously, down to the minimum of these probabilities, with no further decreasein the assessment when the prize goes up. In contrast, for a twice continuously
diﬀerentiable utility function, at every constant act, the decision maker believes
in a single probability, which coincides with the benchmark probability. As theprize goes up, his assessment of the likelihood of the event keeps, smoothly, going down. Therefore, for a suﬃciently small prize, a biseparable preference is, interms of matching probabilities, more ambiguity-averse than a twice continuously32

diﬀerentiable utility function; but an increase in prizes may well cause the former
less ambiguity-averse than the latter.

8.5

Minimum over twice continuously diﬀerentiable functions

We would now like to study ambiguity premiums for the non-diﬀerentiability case,but, as we will elaborate on in the next subsection, the analysis is more subtle
than that of matching probabilities because the second-order impact of risk aversion, v ′′ < 0, is harder to evaluate than in the analysis of matching probabilities.
To ease the analysis, we concentrate on the case where the non-diﬀerentiable aggregator I is the minimum of the twice continuously diﬀerentiable functions. This
class of aggregators I is of particular interest, as it includes maximin expectedutility functions of Gilboa and Schmeidler (1989) and more general one for which
the matching probability jumps down for any, however small, positive prize anddecreases further as the value of the prize increases.
Assumption 1 There is a set J of twice continuously diﬀerentiable (C 2 ), increasing, normalized, and concave real-valued functions deﬁned on v(T )S thatis compact with respect to the C 2 compact-open topology and satisﬁes I(g) =minJ∈J J(g) for every g ∈ v(T )S .
The set of C 2 real-valued functions deﬁned on v(T )S is endowed with the weak
topology in Hirsch (1976, Section 1 of Chapter 2), which is also referred to as theC 2 compact-open topology in Mas-Colell (1985, Section K.1 of Chapter 1). With
respect to this topology, a sequence of functions Jn (n = 1, 2, . . . ) converges toa function J if and only if, on every compact subset B of v(T )S , the sequencesof the Jn and all of the derivatives up to the second order of the Jn converge,
uniformly over B, to J and all of the derivatives up to the second order of J. The
compactness assumption guarantees that for every g ∈ v(T )S , minJ∈J J(g) indeed
exists. The assumption implies that I is continuous, increasing, normalized, andconcave.
Assumption 1 requires every element of J to be normalized. This is intended to make the class of aggregators under consideration a generalization of
the class of maximin expected utility functions. This should be contrasted withthe dual expression of utility functions by Cerreia-Vioglio, Maccheroni, Marinacci,and Montrucchio (2011, Theorem 3). They showed that every uncertain-averse
33

aggregator can be written as I(g) = minp∈∆ G(E p [g], p) for some grounded function G : R × ∆ → R (that is, minp∈∆ G(y, p) = y for every y). But, the (partial)
functions g 7→ G(E p [g], p) in this expression need not be normalized.
Equally important for the subsequent analysis is that the topology with respect
to which J is required to be compact is the C 2 compact-open topology. If J
were required to be compact only with respect to the C 1 compact-open topology,
every increasing and concave I could satisfy this assumption with a set J thatconsists of functions with zero Hessians at a given point,15 which indicates that
there is no way to relate the second Peano derivatives of I with the Hessians of
the elements of J .
We now give a suﬃcient condition for I to be twice Hadamard right-diﬀerentiable
under Assumption 1. Note that ∆yI = conv {∇J(ye) | J ∈ J } for each y ∈ v(T ).
For each y ∈ v(T ) and each z ∈ RS , deﬁne J (y, z) as the set of all J ∈ J thatsatisfy ∇J(ye)z ≤ ∇K(ye)z for all K ∈ J . Then, J (y, z) is nonempty andcompact, because J is compact, and ∆yI (z) = conv {∇J(ye) | J ∈ J (y, z)}.
Proposition 4 Let y ∈ v(T ) and z ∈ RS . Under Assumption 1, I is Hadamard
right-diﬀerentiable at ye in the direction z, and dH I(ye; z) = ∇J(ye)z for any
J ∈ J (y, z). If, in addition, z ⊤ ∇2 J(ye)z = z ⊤ ∇2 K(ye)z for all J ∈ J (y, z)
and K ∈ J (y, z), then I is twice Peano-Hadamard right-diﬀerentiable at ye inthe direction z, and d2H I(ye; z) = z ⊤ ∇2 J(ye)z for any J ∈ J (y, z).
Since the suﬃcient condition in this proposition for twice Peaon-Hadamard
right-diﬀerentiability will be used in the sequal, we put it up as an assumption.
Assumption 2 Let y ∈ v(T ) and z ∈ RS . The aggregator I satisﬁes Assumption
1 and z ⊤ ∇2 J(ye)z = z ⊤ ∇2 K(ye)z for all J ∈ J (y, z) and K ∈ J (y, z)
The following two examples satisfy Assumption 2 and, hence, twice PeanoHadamard right-diﬀerentiability.
Example 1
1. Let Λ a strictly convex and compact subset of ∆, where thestrict convexity is with respect to the relative topology on ∆. Let J : RS ×
Λ → R be twice continuously diﬀerentiable. Assume that for every p ∈ ∆,the partial function J( · , p) : RS → R is increasing, normalized, and concave,and that ∇g J(ye, p) = p⊤ for every y, where ∇g denote the partial derivative
Think of J as consisting of convolutions of piecewise-linear utility functions that approximate I as in Afriat’s (1967) construction of utility functions.
15

34

(Jacobian) with respect to the ﬁrst coordinate. Deﬁne J = {J( · , p) | p ∈Λ}.
2. Denote by Π the set of all permutations on S. For each p ∈ ∆ and π ∈ Π,
denote (p(π(s)))s ∈ ∆ by p◦π. Let p∗ ∈ ∆ and Λ = {p∗ ◦π | π ∈ Π}. Let γ >0. For each p ∈ Λ, let Jp : RS → R be a twice continuously diﬀerentiable,
increasing, normalized, and concave function that satisfy Jp (g) = E p [g] −(γ/2)Varp [g] for every g in some open set that includes the diagonal {ye |y ∈ R}. Let J = {Jp | p ∈ Λ}.
The ﬁrst example diﬀers from maximin expected utility functions as it allows
the Hessians ∇2g J(ye, p) to be non-zero and, yet, requires the set Λ of relevant
probabilities to be strictly convex. The strict convexity is obtained when Λ is theset of all probabilities whose distance from some reference probability is at mostsome small threshold, and the distance is a strictly convex function, as in the caseof the Euclidean norm and the relative entropy. The second example is, locally,a mean-variance utility function under ambiguity, in the sense that the decision
maker has a mean-variance utility function but is unsure of which probability to use
to evaluate mean and variance. The set of relevant probabilities, Λ, is symmetric, inthe sense that any probability distribution that the decision maker deems relevant
is still relevant after swapping the probabilities of any two states.

8.6

Ambiguity premium

In this subsection, we extend the characterization result (Theorem 4) of our ambiguity measure in terms of the ambiguity premium under Assumption 2. Let x ∈ T ,v(x)z ∈ RS , and p ∈ ∆I . For each ε ≥ 0, denote by fpx (ε, z) ∈ Π(T ) the lottery
that coincides with the distribution of the monetary act x + εz on T under the
probability p. Regarding fpx (ε, z) as a lottery act, deﬁne the ambiguity premiumκxp (ε, z) under p byx
V f x−κp (ε,z) (ε, z) = V (xe + εz).
Let I¯p be the expectation operator under p and deﬁne V̄p by (v, I¯p ). Then, κxp (ε; z)can equivalently deﬁned byV̄p (x − κxp (ε; z))e + εz = V (xe + εz).

35

This deﬁnition of the ambiguity premium is diﬀerent from the original deﬁnition(10) in that we specify the benchmark probability p under which the lottery act
fpx (ε, z) is deﬁned. We need to do so because there are multiple benchmark probabilities when I is not (twice continuously) diﬀerentiable but concave.
Theorem 6 Let x ∈ T and z ∈ RS .
v(x)1. Let p ∈ ∆I . Under Assumption 1, the function ε 7→ κxp (ε; z) is rightv(x)diﬀerentiable at 0 and dH κxp (0; z) = p · z − q · z for any q ∈ ∆I (z).
v(x)

2. Let p ∈ ΛI (z). Under Assumption 2, the function ε 7→ κxp (ε; z) is twice
Peano right-diﬀerentiable at 0, dH κxp (0; z) = 0, and d2H κxp (0; z) = −v ′ (x)d2H I(v(x)e; z).
In a terminology analogous to that of Segal and Spivak (1990), Part 1 pins
down the source of the ﬁrst-order ambiguity aversion, and Part 2 relates, in theabsence of the ﬁrst-order ambiguity aversion, the second-order ambiguity aversion
to our extended measure of ambiguity aversion. More speciﬁcally, part 1 shows thatthe ﬁrst-order ambiguity aversion emerges as the positive ﬁrst right-derivative of
the ambiguity premium only if the benchmark probability p used in its evaluationv(x)lies outside ∆I (z). This would be the case if there is a “natural” benchmark
v(x)
v(x)
probability (in ∆I but) outside ∆I (z), as we explained after Theorem 5. If,v(x)instead, we use a benchmark probability in ΛI (z) to evaluate the ambiguity
premium, then the ﬁrst-order ambiguity aversion disappears and the second-orderambiguity aversion can be related to the ambiguity aversion via
1κxp (ε, z) ≈ − v ′ (x)d2H I(v(x)e; z)
2for a small ε > 0.
It is worth noting here that the condition needed for I in Theorem 5 was much
weaker than that needed for I in Theorem 6. In Theorem 5, we only required I to
be twice Gateaux right-diﬀerentiable, while, in Theorem 6, we impose Assumption
2. The diﬀerence is due to the ease with which to assess the impact of the deviationin the direction the indicator function eA of any event A in Theorem 5. Speciﬁcally,
the vector of changes in realized utility levels caused by εeA , v ◦(xe+εeA )−v ◦(xe),is equal to (v(x + ε) − v(x))eA , which is a scalar multiple of eA regardless of thevalue of ε. Thus, the directions of these changes are invariant, which makes thediﬀerence between twice Hadamard and Gateaux right-diﬀerentiability irrelevant
as far as the change in utility, V (xe + εeA ) − V (xe), is concerned. This is not the36case for Theorem 6, because the direction of the vector v ◦ (xe + εz) − v ◦ (xe) in
RS varies as ε → 0+.
Denote by κ̄xp (ε; z) the risk premium of the benchmark utility function V̄p . We
then obtain the following characterization of H x (z) in the non-diﬀerentiable casefrom Theorem 6 and (12).
v(x)

Corollary 7 Let x ∈ T , z ∈ RS , and p ∈ ΛI
Under Assumption 2,(z). Suppose that Varp [z] > 0.

d2H κxp (0; z)
H (z) = 2 x.
∂ κ̄
(0, z)∂ε2x

The asymptotic result (13) can be extended to the non-diﬀerentiable case,thanks to the deﬁnition of the second Peano right-derivative.
v(x)Corollary 8 Let x ∈ T , z ∈ RS , and p ∈ ΛI
Under Assumption 2,(z). Suppose that Varp [z] > 0.

κxp (ε, z)→ H x (z)xκ̄p (ε, z)as ε → 0+.

9Conclusion

In this paper, we introduced an Arrow-Pratt-type measure of ambiguity aversion
for a class of twice right-diﬀerentiable utility functions. The notion of twice rightdiﬀerentiability we employed is so weak that the class include not only smoothambiguity models but also maximin and Choquet expected utility functions andother one that are neither diﬀerentiable in the standard sense nor biseparable.
While we assumed in the main body of the paper that the risk attitude can be represented by an expected utility function, the deﬁnitions and results are extended,in an appendix, to the case where it cannot.
This type of measure is particularly useful for a quantitative analysis where themagnitude of ambiguity aversion is estimated, inferred, or calibrated from laboratory ﬁndings or market data. Our measure makes it possible to compare ambiguity
attitudes of two decision makers even when they apparently have diﬀerent risk at37

titudes. Moreover, when the estimated/inferred/calibrated values of our measureare signiﬁcantly diﬀerent between two settings, such as laboratory experiments versus asset markets, they indicate that there is room for further improvement in thesettings, such as the contents of questionnaires in experiments and the assumptionof complete markets on asset markets.

A

Lemmas and proof for the diﬀerentiable case

Proof of Theorem 1 For each w ∈ RS , denote by [w] the S × S matrix of which
the s-th diagonal entry is equal to the s-th coordinate of w and the oﬀ-diagonal
entries are all equal to zero. By the chain rule diﬀerentiation,∇V (f ) = ∇I(v ◦ f )[v ′ ◦ f ],(21)

for every f ∈ T S , where the gradients are row vectors. Thus,∇2 V (f ) = [v ′ ◦ f ]∇2 I(v ◦ f )[v ′ ◦ f ] + [∇I(v ◦ f )][v ′′ ◦ f ](22)

for every f ∈ T S . Thus, if f = xe for some x ∈ T , then v ◦f = v(x)e, v ′ ◦f = v ′ (x)eand ∇V (f ) = p. Since [e] coincides with the I × I identity matrix,
V (xe) = v(x),∇V (xe) = v ′ (x)p⊤ ,(23)∇2 V (xe) = (v ′ (x))2 ∇2 I(v(x)e) + v ′′ (x)[p].

(24)

Since v(V ∗ (f )) = V (f ), v ′ (V ∗ (f ))∇V ∗ (f ) = ∇V (f ) andv ′′ (V ∗ (f ))∇V ∗ (f )⊤ ∇V ∗ (f ) + v ′ (V ∗ (f ))∇2 V ∗ (f ) = ∇V 2 (f )for every f ∈ T S . Thus, if f = xe for some x ∈ T , thenV ∗ (xe) = x,(25)

∇V ∗ (xe) = p⊤ ,(26)v ′′ (x)pp⊤ + v ′ (x)∇2 V ∗ (xe) = ∇2 V (xe).

38

(27)

By (24) and (27),v ′′ (x)∇ V (xe) = v (x)∇ I(v(x)e) + ′([p] − pp⊤ )
v (x)
2

∗

′

2

(28)

¯
Thus, (2) follows from z ⊤ ([p] − pp⊤ )z = Varp [z]. Since ∇2 I(v(x)e)= 0, (3) can besimilarly proved.
///
Proof of Theorem 2
For each ε ∈ R with xe + εz ∈ T S , deﬁne λ(ε)
P
1/2
by V ∗ (xe + λ(ε)S −1/2 e + εkzk−1 z) − V ∗ (xe) = 0, where kzk = ( s (z(s))2 ) ,which is also equal to (Varp [z])1/2 because E p [z] = 0. Then, λ(0) = 0 and thecurve f ∈ T S ∩ L(z) | V (f ) = V (xe) is parameterized by ε 7→ xe + λ(ε)S −1/2 e +εkzk−1 z. By the implicit function theorem, λ(ε) is a twice continuously diﬀerentiable function of ε, λ′ (0) = 0, and′′λ (0) = −S

1/2


 ⊤ ′
z (v (x)∇2 I(v(x)e)) z v ′′ (x)+ ′.
Varp [z]v (x)Since the curve f ∈ T S ∩ L(z) | V (f ) = V (xe) is parameterized by ε 7→ xe +λ(ε)S −1/2 e + εkzk−1 z on the plane L(z), cx (z) is equal to its curvature when it isregarded as a subset of the plane L(z) with an orthonormal basis (S −1/2 e, kzk−1 z).
Thuscx (z) =|λ′′ (0)|((λ′ (0))2 + 1)

= |λ′′ (0)| = S 1/2
3/2

z ⊤ (v ′ (x)∇2 I(v(x)e)) z v ′′ (x)+ ′Varp [z] .
Varp [z]v (x)Since I is quasi-concave and E p [z] = 0, z ⊤ (v ′ (x)∇2 I(v(x)e)) z ≤ 0. Since v ′′ <0 < v ′ , v ′′ (x)/v ′ (x) < 0. Thus,c (z) = −S
x

1/2

 ⊤ ′

z (v (x)∇2 I(v(x)e)) z v ′′ (x)+ ′.
Varp [z]v (x)We can similarly show that′′
1/2 v (x).
c̄ (z) = −Sv ′ (x)x///
The proof of Theorem 3 is based on the following lemma. It can be proved
based on Taylor’s theorem, and we omit the proof.

39

Lemma 1 Let D be an open interval in R that contains 0. Let F : D → R be
twice diﬀerentiable and satisfy F (0) = 0. Deﬁne G : D → R by
 F ′ (0) if δ = 0,
G(δ) =
 F (δ) otherwise.
δThen G is diﬀerentiable at 0 and G′ (0) = (1/2)F ′′ (0).
Proof of Theorem 3 Let D be the set of all δ ∈ R such that v(x) + δ ∈ v(T ),then D is an open interval that contains 0. For each δ ∈ D, deﬁne F : D → R by
F (δ) = I(v(x)e + δeA ) − v(x). Then F (0) = 0, F ′ (0) = ∇I(v(x)e)eA , and2F ′′ (0) = e⊤
A ∇ I(v(x)e)eA .

(29)

Deﬁne G : D → R as in Lemma 1, thenI (v(x)e + (v(x + ε) − v(x))eA ) − v(x)v(x + ε) − v(x)F (v(x + ε) − v(x))== G(v(x + ε) − v(x)).
v(x + ε) − v(x)ρx (ε, A) =By Lemma 1, the right-hand side is a function of ε that is diﬀerentiable at ε = 0,
and the derivative at ε = 0 is equal to (1/2)F ′′ (0)v ′ (x). By (29), this completes
the proof///
The proof of Corollary 3 requires the following lemma.
Lemma 2 Let A ⊂ S and x ∈ T , then
v ′′ (x)
′
2
⊤
′
2
(x)∇
I(xe)
e⊤
v
e
=
e
v
(x)∇
I(v(x)e)e+
Varp [eA ].
AAAAv ′ (x)(30)

Proof of Lemma 2 Since v(I(f )) = V (f ) for every f ∈ T S , v ′ (I(f ))∇I(f ) =∇V (f ) for every f ∈ T S . Hence v ′′ (I(f ))∇I(f )⊤ ∇I(f ) + v ′ (I(f ))∇2 I(f ) =∇2 V (f ) for every f ∈ T S . When f = xe, these equalities can be rewritten asv ′ (x)∇I(v(x)e) = ∇V (xe) and v ′′ (x)∇I(xe)⊤ ∇I(xe) + v ′ (x)∇2 I(xe) = ∇2 V (xe).
By (23) and (24), ∇I(xe) = ∇I(v(x)e) = p andv ′′ (x)pp⊤ + v ′ (x)∇2 I(xe) = (v ′ (x))2 ∇2 I(v(x)e) + v ′′ (x)[p].

40

Thus,∇2 I(xe) = v ′ (x)∇2 I(v(x)e) +v ′′ (x)
⊤
[p]
−
pp
.
v ′ (x)Hence,
v ′′ (x)2⊤′2∇I(xe)e=ev(x)∇
I(v(x)e)e+e⊤
Varp [eA ].
AAAAv ′ (x)Since v ′ (x) = 1 by deﬁnition, this establishes (30).
Proof of Corollary 3///

Apply Theorem 3 to (v, I) and (v, V ), then, by (30),∂ρx∂ρx
1 v ′′ (x)(0, A) =(0, A) +Varp [eA ].
∂ε∂ε
2 v ′ (x)(31)

By rearranging this, we obtain the denominator of H x (eA ) and complete the proof.
///
Proof of Theorem 4 Deﬁne V ∗ = v −1 ◦ V and V̄ ∗ = v −1 ◦ V̄ . By (25), (10) canbe rewritten as V̄ ∗ ((x − κx (ε, z)) e + εz) = V ∗ (xe + εz). By diﬀerentiating bothsides with respect to ε, we obtain∂κx(ε, z)e + z∇V̄ ((x − κ (ε, z)) e + εz) −
∂ε
∗

x


= ∇V ∗ (xe + εz)z.

(32)

By letting ε = 0 and using (26), we obtain∂κx(0, z) = 0
∂ε

(33)

By diﬀerentiating both sides of (32) with respect to ε, we obtain∂κx−(ε, z)e + z
∂ε

⊤



∂κx
∇ V̄ ((x − κ (ε, z)) e + ze) −(ε, z)e + z
∂ε
 2 x

∂ κ
∗
x
+ ∇V̄ ((x − κ (ε, z)) e + ze) − 2 (ε, z)e
∂ε
2

∗



x

= z ⊤ ∇2 V ∗ (xe + εz)z.

(34)

41

By letting ε = 0 and using (26) and (33), we obtainz ⊤ ∇2 V̄ ∗ (xe) z −∂ 2 κx(0, z) = z ⊤ ∇2 V ∗ (xe)z.
2
∂ε

Thus,

∂ 2 κx
2 ∗
⊤
2 ∗
V̄
(xe)
−
∇
V
(xe)
z.
(0,
z)
=
z
∇
∂ε2

(35)

¯
By using (28) and an analogous result for ∇2 V̄ (xe) (for which ∇2 I(v(x)e)= 0),
we complete the proof.
///

42

Online Appendix
B

Proofs on the smooth ambiguity model

Proof of Proposition 1 The twice continuous diﬀerentiability of I follows fromthat of v and ϕ. If g = ye for some y ∈ v(T ), then p · g = y = I(g) for every p ∈ ∆.
R
Thus, I is normalized. By diﬀerentiating both sides of ϕ(I(g)) = ∆ ϕ (p · g) dµ(p)with respect to g, we obtainZ′ϕ′ (p · g) p⊤ dµ(p)ϕ (I(g))∇I(g) =

(36)

∆

R
S
S
Since ϕ′ > 0 and ∆ p dµ(p) ∈ R++, ∇I(g) ∈ R++. If g = ye for some y ∈ v(T ),R ⊤the equality is reduced to ∇I(ye) = ∆ p dµ(p) = (pI )⊤ .
By diﬀerentiating both sides of (36) with respect to g, we obtain
′′

⊤

′

Z

⊤

ϕ′′ (p · g) pp⊤ dµ(p).

ϕ (I(g))∇I(g) ∇I(g) + ϕ (I(g))∇ I(g) =
2

∆

Thus,
1
∇ I(g) = ′ϕ (I(g))



Z

′′

⊤

′′

⊤

ϕ (p · g) pp dµ(p) − ϕ (I(g))∇I(g) ∇I(z) .

2

(37)

∆

In particular, for every y ∈ v(T ),ϕ′′ (y)∇ I(ye) = ′
ϕ (y)

Z


⊤

I ⊤

pp dµ(p) − p (p )

2

I

∆

ϕ′′ (y)
= ′
ϕ (y)

Z

(p − pI )(p − pI )⊤ dµ(p).
∆(38)
Thus, for every x ∈ T and every z ∈ RS ,ϕ′′ (v(x))
z ∇ I(v(x)e)z = ′ϕ (v(x))⊤Z(p − pI ) · z22dµ(p).

(39)∆

Since E µ [E · [z]] = E p [z] by the law of iterated expectation, the law of total variance implies thatIZ
(p − p ) · z
I

∆

2

Z 
dµ(p) =

I

E p [z] − E p [z]

∆

43

2

dµ(p) = Varµ [E · [z]].

(40)Recall that w(x) = ϕ(v(x)) for every x ∈ T . By diﬀerentiating both sides withrespect to x, we obtainw′ (x) = ϕ′ (v(x))v ′ (x).

(41)

By diﬀerentiating both sides with respect to x, we obtainw′′ (x) = ϕ′′ (v(x))(v ′ (x))2 + ϕ′ (v(x))v ′′ (x).

(42)

By dividing each side of (42) by the same side of (41), we obtainw′′ (x)ϕ′′ (v(x)) ′v ′′ (x)
=
v
(x)
+
,
w′ (x)ϕ′ (v(x))
v ′ (x)that is,ϕ′′ (v(x))
w′′ (x)−ϕ′ (v(x)) ′w′ (x)
v (x) =
− 1.
′′
v (x)
v ′′ (x)− ′v ′ (x)v (x)Thus, by (39) and (40),−v ′ (x)H x (z) =
ϕ′′ (v(x)) Varµ [E · [z]]w′′ (x) − w′ (x) Varµ [E · [z]]ϕ′ (v(x)) VarpI [z]

=
− 1
 VarpI [z] .
v ′′ (x)v ′′ (x)
− ′
− ′
v (x)
v (x)
///

Proof of Proposition 2
absolute risk aversion,

Since v and w have constant coeﬃcients θ and γ ofw′′ (x)γw′ (x)− 1 = − 1 = η.
′′
v (x)
θ
− ′
v (x)

−

Second, since the second-order belief and the reduced probability are N (R̄1, ΣM )44

and N (R̄1, ΣR ), Varµ [E · [R1 ]] = 0 and
Varµ [E · [R2 ]]pI

Var [R2 ]

=

τ σ2
= τ.
σ2

By Proposition 1, H x (R1 ) = 0 and H x (R2 ) = ητ .
Hara and Honda (2022, equation (8)) showed that the solution to (17) is given
by
x=

x1
x2

!

R̄ − Rf=(τ η + 3/4)σ 2 θ

1/2 + τ η
1/2!

and x0 = W − (x1 + x2 ). Thus, the share of the wealth invested in the foreign
stock in the wealth invested in the two stocks is
1/2
1
=
.
(1/2 + τ η) + 1/2
2(1 + τ η)///C

Extension to the non-expected utility caseWe assumed throughout this paper that the decision maker has an expected utility
function on the set Π(T ) of lotteries. In this appendix, we extend the deﬁnitionof the measure of ambiguity aversion to the case where he does not, in threesteps. First, we present a general approach to a measure of ambiguity aversion
once an ambiguity-neutral preference against which we evaluate the measure isgiven. When the decision maker has an expected utility function over lottery
acts, this benchmark ambiguity-neutral preference is represented by a subjective
expected utility function. When he does not, however, the choice of a benchmark
ambiguity-neutral preference requires careful consideration. In the second step,
we construct such a benchmark ambiguity-neutral preference, which would respect
his preference over lottery acts (not representable by expected utility functions) if
there were no ambiguity on the state space. Third, we take this ambiguity-neutral
preference as the benchmark preference in the general approach to arrive at our
extended measure of ambiguity aversion. We then touch on the nature of the twice
continuous diﬀerentiability that is needed in this argument. Finally, we discuss
how the results on the relation to matching probabilities and ambiguity premiums
45

can be extended to the non-expected utility case.
Suppose that there are a utility function u : Π(T ) → R that is continuous withrespect to the weak topology and strictly increasing with respect to the ﬁrst-orderstochastic dominance, and an aggregator I : u(Π(T ))S → R that is increasing
and normalized. For each f = (f (s))s ∈ Π(T )S , with f (s) ∈ Π(T ) for every s,
write u ◦ f = (u(f (s)))s ∈ u(Π(T ))S . Deﬁne a utility function V : Π(T )S → R
by letting V (f ) = I(u ◦ f ) for every f ∈ Π(T )S . This deﬁnition of V is the sameas the deﬁnition of V that we have used so far, except that there need not be an
expected utility function over lottery acts, that is, there need not be a BernoulliRutility function v : T → R such that u(P ) = T v(x) dP (x) for every P ∈ Π(T ) (orany monotone transformation of the right-hand side).
Let ū, I¯ and V̄ be just as u, I, and V . We assume that V is at least as
ambiguity-averse as V̄ in the sense of Epstein (1999). Then, u and ū represents
the same preference over lottery acts and, thus, we can assume that u = ū. Assume
that for each p ∈ ∆, the function f 7→ u(p−1 ◦ f ) deﬁned on the set T S of monetaryacts is twice continuously diﬀerentiable and its derivative in the direction e isstrictly positive. Deﬁne v : T → R by equating v(x) equal to the value of u atthe degenerate probability on x. Then, v is twice continuously diﬀerentiable andv ′ > 0. By the inverse function theorem, it has an inverse v −1 . If u were an expectedutility function, then v would the corresponding Bernoulli utility function, but it
is not so in the case of non-expected utility functions.
Let x ∈ T and p = ∇I(v(x)e). Since V (xe) = v(x) = V̄ (xe) and V is at least as¯
ambiguity-averse as V̄ , p = ∇I(v(x)e).
Let z ∈ RS and assume that Varp [z] > 0.
Then, the measure of ambiguity aversion for V against V̄ is deﬁned as

⊤
′
2
⊤
′
2¯
−z
(v
(x)∇
I(v(x)e))
z
+
z
v
(x)∇
I(v(x)e)z
H x (z) =
.
′′

v (x)
p
′
2
⊤
¯
− ′
Var [z] − z v (x)∇ I(v(x)e) z
v (x)

(43)


¯
This diﬀers from the original deﬁnition (1) in that the new term z ⊤ v ′ (x)∇2 I(v(x)e)
zis added and subtracted in the numerator and the denominator on the right-handside. The original deﬁnition corresponds to the case where (v is a Bernoulli utility¯
function and) I¯ is the expectation operator under p. Hence, ∇2 I(v(x)e)= 0 andthe original deﬁnition is a special case of the above deﬁnition.
We saw that the original measure (1) of ambiguity aversion is invariant to any
increasing and aﬃne transformation of v when it is coupled with an increasing46

transformation of I in such a way that the resulting aggregator is normalized andthe resulting utility function represents the same preference as V . We skip the
proof, but this extended measure is invariant to any increasing transformation ofu when it is coupled with an increasing transformation of I in such a way that theresulting aggregator is normalized and the resulting utility function represents thesame preference as V . Note here that as u is a utility function over the set Π(T ) oflotteries, we should not limit ourselves to aﬃne transformations but accommodate
all increasing transformations to check the invariance of the extended measure.
Having deﬁned the measure of ambiguity aversion of V against an arbitrary V̄ ,¯ = u(p◦(v −1 ◦we now deﬁne an ambiguity-neutral utility function V̄ via letting I(g)g)−1 ) for every g ∈ v(T )S , where v −1 ◦ g is the function deﬁned on S that takes
values v −1 (g(s)) at each s and p ◦ (v −1 ◦ g)−1 is the probability on T induced from¯ ◦ f ).
the probability p on the state space S by v −1 ◦ g. We then deﬁne V̄ (f ) = I(uOur choice of I¯ and V̄ can be understood as follows. For each monetary act
f = (f (s))s ∈ T S , with f (s) ∈ T for every s, write v ◦ f = (v(f (s)))s ∈ v(T )S .
For each Anscombe-Aumann act f ∈ Π(T )S , let f¯ ∈ T S be the monetary act
of certainty equivalents of f , that is, f¯(s) = v −1 (u(f (s))) for every s. Then,¯ ◦ f¯) = u(p ◦ f¯−1 ) for every f ∈ Π(T )S . That is,v ◦ f¯ = u ◦ f and, thus, V̄ (f ) = I(vV̄ represents a ﬁctitious preference on the set of two-stage lotteries that shares thesame preference as V on the set of simple lotteries, regardless of whether they are
given at the ﬁrst stage (in which case they are monetary acts under p) or the second
stage (in which case they are lottery acts). In the terminology of Segal (1990), V̄satisﬁes the time-neutrality. It also satisﬁes the compound independence axiom,
because V̄ (f ) depend on f only through f¯, just as V (f ) = I(v ◦ f¯).16 For thesereasons, it is appropriate to deﬁne the measure of ambiguity aversion for V as adeviation from V̄ .
¯ the Hessian ∇2 I(v(x)e)¯
For this choice of I,
can be written as¯∇2 I(v(x)e)
=


1
2
−1
′′
∇
u(p
◦
(xe)
)
−
v
(x)[p],
f(v ′ (x))2

where ∇2f u(p ◦ (xe)−1 ) is the Hessian of the function f 7→ u(p ◦ f ) deﬁned on the
set T S of monetary acts at the constant act xe. By plugging this into (43), we can
16

There are other utility functions deﬁned on the set Π(T )S of Anscombe-Aumann acts that
represents the same preference over lotteries as V . Segal (1990) gave such an example thatsatisﬁes the reduction of compound lotteries axiom but violates the compound independence
axiom.

47

also write
1v ′′ (x) p 22−1−z (v (x)∇ I(v(x)e)) z + z
∇
u(p
◦
(xe)
)
z
−
E [z ]v ′ (x) fv ′ (x)x
H (z) =.
v ′′ (x)1v ′′ (x) p 2
p
2
⊤
−1
− ′
Var [z] − z∇ u(p ◦ (xe) ) z + ′E [z ]v (x)v ′ (x) f
v (x)
(44)


⊤

′

2

⊤

This is the extended measure of ambiguity aversion to the case where the preference
over lottery acts may not be represented by any expected utility function.
We simply assumed in the preceding argument that the function f 7→ u(p ◦ f )deﬁned on the set T S of monetary acts is twice continuously diﬀerentiable. Machina(1982) gave a notion of Fréchet diﬀerentiability of utility functions over cumulative distribution functions based on the L1 norm. Allen (1987), instead, used L2
norm and Wang (1993) used Lp norm with p ≥ 1. The utility function v overconsumption levels (constant acts) that we deﬁned is diﬀerent from the local expected utility function (Riesz representation of the Fréchet derivative) of Machina(1982), but, at every consumption level x, our function v and the local expectedutility function of Machina at the cumulative distribution function degenerate onx share the same derivative. Segal and Spivak (1997) showed that the risk aversion exhibited by a local expected utility function of Machina is of the ﬁrst order(that is, the derivative of the risk premium at 0 is strictly positive) if and only if
the local utility function at the cumulative distribution function degenerate at x
is non-diﬀerentiable at x. No suﬃcient condition for twice diﬀerentiability of thefunction f 7→ u(p ◦ f ), however, seems to have been given in the literature.
We now explain how the results on the measure of ambiguity aversion in the
expected utility case can be extended to the non-expected utility case. As in theﬁrst step of deﬁning the measure of ambiguity aversion in the non-expected utility¯ Deﬁne v,
case, take a non-expected utility function u and two aggregators I and I.
V , and V̄ as before. Deﬁne, also, V ∗ = v −1 ◦V and V̄ ∗ = v −1 ◦ V̄ . Then, (28) is valid¯ V̄ ∗ ). This implies that Corollary 1 can be extended tofor (I, V ∗ ) and also for (I,the non-expected utility case. As for the ambiguity premium, deﬁne κx (ε, z) andκ̄x (ε, z) (the risk premium) as in Section 6. Then, (35) holds. Moreover, sincex − κ̄x (ε, z) = V̄ ∗ (xe + εz), by diﬀerentiating both sides twice and evaluating at48

ε = 0, we obtain∂ 2 κ̄x(0, z) = −z ⊤ ∇2 V̄ ∗ (xe)z.
2∂εCombining these two results and applying the extension of Corollary 1, we can
extend Corollary 4 to the non-expected utility case.
As for the matching probabilities, note that Theorem 3 shows that the derivative of the matching probability is equal to the numerator, multiplied by −1/2,of the measure (1) of ambiguity aversion in the expected utility case. In the nonexpected utility case, we aim at proving that∂ρx
1(0, A) =
∂ε
2


e⊤
A

′

2

v (x)∇ I(v(x)e)

eA − e⊤A
1v ′′ (x)−12∇ u(p ◦ (xe) ) eA + ′p(A) ,v ′ (x) f
v (x)
(45)

because E p [e2A ] = p(A). To do so, deﬁne ρx (ε, A) as in (5). Then, (6) is replacedby u(f x (ε, ρx (ε, A))) = V (xe + εeA ). For each (ε, r), write wx (ε, r) = u(f x (ε, r)).
Then, wx is twice continuously diﬀerentiable, and satisﬁes wx (0, r) = v(x) for everyr ∈ [0, 1] andwx (ε, ρx (ε, A)) = V (xe + εeA )(46)
for every ε. Assume that ∂ 2 wx (0, p(A))/∂ε∂r = v ′ (x). This assumption is met
by expected utility functions and also by utility functions that are quadratic inprobabilities in the sense of Machina (1982). By diﬀerentiating both sides of (46)with respect to ε twice and evaluating at ε = 0, we obtain∂ 2 wx∂ρx∂ 2 wx
x
x
(0,
ρ
(0,
A))
+
2
(0,
ρ
(0,
A))
(0, ρx (0, A))
2
∂ε
∂r∂ε
∂ε
 x
2
2 x
∂ w
∂ρ
∂wx
∂ 2 ρxxxx+(0, ρ (0, A))(0, ρ (0, A)) +(0, ρ (0, A)) 2 (0, ρx (0, A))
2
∂r
∂ε
∂r
∂ε
2
= e⊤
A ∇ V (xe)eA .

(47)

Note that ρx (0, A) = p(A), ∂wx (0, p(A))/∂r = 0, ∂ 2 wx (0, p(A))/∂r2 = 0, and∂ 2 wx (0, p(A))/∂ε∂r = v ′ (x). Also, by the deﬁnition of wx (ε, r),∂ 2 wx−1(0, p(A)) = e⊤A ∇f u(p ◦ (xe) )eA .
∂ε2
Thus, by applying (24) to the right-hand side of (47) and noting that E p [e2A ] =49

p(A), we obtain−1′e⊤
A ∇f u(p ◦ (xe) )eA + 2v (x)

∂ρx
2
′′
(0, ρx (0, A)) = (v ′ (x))2 e⊤
A ∇ I(v(x)e)eA + v (x)p(A).
∂εBy rearranging the above equality, we obtain (45).
The presence of the last two terms on the right-hand side of (45) distinguishes(45) from Theorem 3. It is easy to see that some sort of additional terms isnecessary because the matching probability may be diﬀerent from the benchmark
probability even for lottery acts when the preference over lottery acts cannot be
represented by any expected utility function. What is more intriguing is that thelast two terms are equal to the change in matching probabilities of lottery acts inthe following sense: Deﬁne ρ̄x (ε, r) byρ̄x (ε, r)v(x + ε) + (1 − ρ̄x (ε, r))v(x) = wx (ε, r),
then
∂ ρ̄x
1
(0, p(A)) =
∂ε
2




e⊤
A



v ′′ (x)12−1∇ u(p ◦ (xe) ) eA − ′p(A) .
v ′ (x) f
v (x)

Thus,

∂ρx
1
∂ ρ̄x
′
2
(0, A) = e⊤v(x)∇
I(v(x)e)e−(0, p(A)).
A
∂ε
2 A
∂ε
Thus, if ∂ ρ̄x (0, p(A))/∂ε < 0, then the ﬁrst term of the right-hand side, which
would be equal to ∂ρx (0, A)/∂ε in the expected utility case, overestimates theimpact of ambiguity aversion on the reduction of matching probabilities as theprize ε goes up, and the second term ∂ ρ̄x (0, A)/∂ε needs to be subtracted to correctit.

D

A theorem and proofs for the non-diﬀerentiable
case

Proof of Theorem 5
ρx (ε, A) =By the deﬁnition (5),I (v(x)e + (v(x + ε) − v(x))eA ) − I(v(x)e)
v(x + ε) − v(x)50Since I is Gateaux right-diﬀerentiable at x in the direction eA , ρx (ε, A) → dG I(v(x)e; eA )v(x)as ε → 0+. Since dG I(v(x)e; eA ) = p(A) = ρx (0, A) for every p ∈ ∆I (eA ), thisshows that ρx ( ·, A) is right-continuous at 0. Moreover,ρx (ε, A) − ρx (0, A)εI (v(x)e + (v(x + ε) − v(x))eA ) − I(v(x)e)− dG I(v(x)e; eA )v(x + ε) − v(x)=εI (v(x)e + (v(x + ε) − v(x))eA ) − I(v(x)e) − (v(x + ε) − v(x))dG I(v(x)e; eA )=(v(x + ε) − v(x))2v(x + ε) − v(x)×.
ε
As ε → 0+, the far right-hand side converges to (1/2)v ′ (x)d2G I(v(x)e; eA ). This
completes the proof.
///
We now establish a chain rule for the second derivative of the composite of two
twice Peano-diﬀerentiable functions. To do so, we need to introduce some notation.
For a function F that is Hadamard right-diﬀerentiable at a point a in all directions
near a point b, if the Hadamard right-derivative dH F (a; ·) is Hadamard rightdiﬀerentiable at b in a direction c, then we write the Hadamard right-derivativedH (dH F (a; ·))(b; c) of dH F (a; ·) at b in the direction c as dH F (a; b; c). If dH F (a, ·)is linear (as in the case where F is diﬀerentiable), thenεdH F (a; z)
dH F (a; b + εz) − dH F (a; b)== dH F (a; z) → dH F (a; c)εε
as (ε → 0 and) z → c. Thus, dH F (a; b; c) = dH F (a; c). The need for the deﬁnition of dH F (a; b; c), therefore, arises from the nonlinearity of dH F (a, ·), which isoften the case for Hadamard right-derivatives of non-diﬀerentiable functions. Fora mapping G taking values in RM with M ≥ 1, if, for each m, the m-th coordinate
function Gm is twice Peano-Gateaux right-diﬀerentiable at a point a in a directionb, then we denote by dG G(a; b) the vector in RM of which the m-th row is equal
to dG Gm (a; b) and by d2G G(a; b) the vector in RM of which the m-th coordinate isequal to d2G Gm (a; b).
Theorem 7 Let N and M be positive integers, A be an open subset of RN , C be
an open subset of RM , F : C → R, and G : A → C. Let a ∈ A and b ∈ RN .
Suppose that the m-th coordinate function Gm of G is twice Peano-Gateaux right51

diﬀerentiable at a in the direction b for every m. Suppose also that F is twice
Peano-Hadamard right-diﬀerentiable at G(a) in all directions near dG G(a; b), anddH F (a; ·) is Hadamard right-diﬀerentiable at dG G(a; b) in the direction d2G G(a; b).
Deﬁne H = F ◦ G. Then, H is twice Peano-Gateaux right-diﬀerentiable at a in
the direction b andd2G H(a; b) = d2H F (G(a); dG G(a; b)) + dH F (G(a); dG G(a; b); d2G G(a; b)).
While the function F is assumed to be twice Peano-Hadamard right-diﬀerentiable,
the composite function H is shown to be only twice Peano-Gateaux right-diﬀerentiable.
The assumption cannot be weakened to twice Peano-Gateaux right-diﬀerentiability.
Indeed, Delfour (2019, Example 3.9 in Section 3 of Chapter 3) gave an example ofa Gateaux diﬀerentiable F and an inﬁnitely diﬀerentiable function G such that H
does not even satisfy one-sided continuity.
The assumption imposed on dG F (a; ·) in Theorem 7 is satisﬁed whenever F is
diﬀerentiable, concave, or convex. Indeed, if F is diﬀerentiable, then dH F (a, ·) :RM → R is linear and, hence, diﬀerentiable. If F is concave or convex, then, by
Theorem 4.6 of Section 4 of Chapter 3 of Delfour (2019), it is Hadamard rightdiﬀerentiable and the Hadamard right-derivative, d H F (a, ·) : RM → R, is concave
or convex. Thus, it is Hadamard right-diﬀerentiable at every point in all directions.
The following proof is based on the proof of Proposition 3.1 of Ren and Sen(2001) but diﬀers from it in that the limit operation is one-sided (ε > 0) and neither
the linearity of the ﬁrst derivative nor the quadraticity of the second derivative isassumed or implied.
Proof of Theorem 7 By Theorem 3.5 in Section 3 of Chapter 3 of Delfour(2019), H is Gateaux right-diﬀerentiable at a in the direction b, and dG H(a; b) =dG F (G(a), dH G(a; b)). For each suﬃciently small ε > 0, write1Ψ(ε) = 2εε2
G(a + εb) − G(a) − εdG G(a, b) − d2G G(a, b)
2

∈ RM .

By the deﬁnition of the second Peano-Gateaux right-diﬀerentiability, Ψ(ε) → 0 as
ε → 0. Note thatH(a + εb) − H(a) − εdG H(a; b)
K1 (ε) K2 (ε)
=
+
,
2
ε
ε2
ε2

52

(48)

whereε
K1 (ε) = H(a + εb) − H(a) − εdH F G(a); dG G(a; b) + d2G G(a; b) + εΨ(ε) ,2ε 2K2 (ε) = εdH F G(a); dG G(a; b) + dG G(a; b) + εΨ(ε) − εdG H(a; b).
2
Since
ε
dG G(a; b) + d2G (a; b) + εΨ(ε) → dG G(a; b)2
as ε → 0+ and since F is twice Peano-Hadamard right-diﬀerentiable at G(a) in
the direction dG G(a; b),
K1 (ε)
1 
ε 2
=
F
G(a)
+
ε
d
G(a,
b)
+
d
G(a;
b)
+
εΨ(ε)
− F (G(a))
G
ε2
ε2
2 G


ε
− εdH F G(a); dG G(a; b) + d2G G(a; b) + εΨ(ε)
21 2→ dH F (G(a); dG G(a; b))2
as ε → 0+. Since dH F (a; ·) is Hadamard right-diﬀerentiable at dH G(a; b) in the
direction d2H G(a; b),K2 (ε) dH F G(a); dG G(a; b) + ε 12 d2G (a; b) + Ψ(ε)
=
ε2
ε
1
2
→ dH F (G(a); dG G(a; b); dG G(a; b))2

− dH F (G(a); dG G(a; b))as ε → 0+. Hence,
K1 (ε) + K2 (ε)
1 2
2
→
d
F
(G(a);
d
G(a;
b))
+
d
F
(G(a);
d
G(a;
b);
d
(a;
b))
G
H
G
H
G
ε2
2
as ε → 0+. By (48), this completes the proof.
Proof of Corollary 5///

Note, ﬁrst, that

I (v(x)e + (v(x + ε) − v(x))eA ) − I(v(x)e) v(x + ε) − v(x)V (xe + εz) − V (xe)=.
εv(x + ε) − v(x)εSince I is Gateaux right-diﬀerentiable at v(x)e in the direction eA , as ε → 0+,
the right-hand side converges to v ′ (x)dG I(v(x)e; eA ). Hence, V is Gateaux rightdiﬀerentiable at xe in the direction eA and dG V (xe; eA ) = v ′ (x)dG I(v(x)e; eA ).
53

Write1
ψ(ε) =
ε




ε2 ′′
v(x + ε) − v(x) − εv (x) − v (x) .
2′

By Taylor’s theorem, ψ(ε) → 0 as ε → 0. Note, in addition, that
V (xe + εeA ) − V (xe) − εdG V (xe; eA )K1 (ε) K2 (ε)
=
+
,
2
ε
ε2
ε2
where


ε
K1 (ε) = V (xe + εeA ) − V (xe) − ε v ′ (x) + v ′′ (x) + εψ(ε) dH I (v(x)e; eA ) ,
2


ε ′′
′
K2 (ε) = ε v (x) + v (x) + εψ(ε) dH I (v(x)e; eA ) − εdG V (xe; eA ).
2
Then, K1 (ε)/ε2 is equal to 
I v(x)e + ε v ′ (x) + 2ε v ′′ (x) + εψ(ε) eA − I(v(x)e) − ε v ′ (x) + 2ε v ′′ (x) + εψ(ε) dH I (v(x)e; eA )2
ε v ′ (x) + 2ε v ′′ (x) + εψ(ε)
 !2
ε v ′ (x) + 2ε v ′′ (x) + εψ(ε)×ε
Since I is twice Peano-Gateaux right-diﬀerentiable at v(x)e in the direction eA ,this converges to (1/2)(v ′ (x))2 d2G I(v(x)e; eA ) as ε → 0+. Note also that
ε v ′ (x) + 2ε v ′′ (x) + εψ(ε) dH I (v(x)e; eA ) − εv ′ (x)dG I(v(x)e; eA )K2 (ε)
=
ε2
ε2


1 ′′
1
=
v (x) + ψ(ε) dH I (v(x)e; eA ) → v ′′ (x)dH I (v(x)e; eA )22
as ε → 0+. Hence,
V (xe + εeA ) − V (xe) − εdG V (xe; eA )11
→ (v ′ (x))2 d2G I(v(x)e; eA ) + v ′′ (x)dG I(v(x)e; eA )2ε22as ε → 0+. Thus, V is twice Peano-Gateaux right-diﬀerentiable at xe in the
direction eA andd2G V (xe; eA ) = (v ′ (x))2 d2G I(v(x)e; eA ) + v ′′ (x)dG I(v(x)e; eA ).
Since v −1 is (twice continuously) diﬀerentiable and V is twice Peano-Gateaux54right-diﬀerentiable at xe in the direction eA on the set T S of monetary acts, byapplying Theorem 7 to v −1 and V in place of F and G, we can show that I¯ is twice
Peano-Gateaux right-diﬀerentiable at xe in the direction eA , and2dG
I(xe; eA )= d2H v −1 (V (xe); dG V (xe; eA )) + dH v −1 (V (xe), dG V (xe; eA ); d2G V (xe; eA ))= (v −1 )′′ (v(x))(v ′ (x)dG I(v(x)e; eA ))2+ (v −1 )′ (v(x)) (v ′ (x))2 d2G I(v(x)e; eA ) + v ′′ (x)dG I(v(x)e; eA )

v ′′ (x)v ′′ (x)2′2(dI(v(x)e;
e
))
+
v
(x)d
I(v(x)e;e)+dG I(v(x)e; eA )GAAGv ′ (x)v ′ (x)v ′′ (x)= v ′ (x)d2G I(v(x)e; eA ) + ′dG I(v(x)e; eA )(1 − dG I(v(x)e; eA ))
v (x)

=−

v(x)

Since dG I(v(x)e; eA ) = p(A) for any p ∈ ∆I(eA ) and v ′ (x) = 1, this implies thatd2G I(xe; eA ) = v ′ (x)d2G I(v(x)e; eA ) +v ′′ (x)p(A)(1 − p(A)),
v ′ (x)that is,v ′′ (x)2′2v (x)dG I(v(x)e; eA ) − v (x)dG I(xe; eA ) = − ′p(A)(1 − p(A)).
′

v (x)

By applying Theorem 5 to ρ and ρ, we obtain (20).
Proof of Proposition 3///

Since V is biseparable,
ρ(A) =V (xe + εeA ) − V (xe)V ((x + ε)e) − V (xe)for every suﬃciently small ε > 0. The right-hand side of this equality is equal toI(v(x)e + (v(x + ε) − v(x))eA ) − I(v(x)e).
v(x + ε) − v(x)Thus,
ρ(A) =

I(v(x)e + δeA ) − I(v(x)e)
.
δ

(49)

for every suﬃciently small δ > 0. Hence, I is Gateaux right-diﬀerentiable at v(x)e55

in the direction eA , and dG I(v(x)e; eA ) = ρ(A). Thus,
I(v(x)e + δeA ) − (I(v(x)e) + dG I(v(x)e; δeA ))δ2
I(v(x)e + δeA ) − (I(v(x)e) + ρ(A)δ)==0
δ2
by (49). Hence, I is twice Peano-Gateaux right-diﬀerentiable at v(x)e in the direction eA , and d2G I(v(x)e; eA ) = 0. Since 0 < ρ(A) < 1 and ρ(A) = p(A) for any
v(x)
v(x)
p ∈ ΛI (eA ), Varp [eA ] > 0 for any p ∈ ΛI (eA ). Thus, H x (eA ) = 0.
///
Proof of Proposition 4 First, we introduce some deﬁnitions. For each positive
integer n, deﬁne Kn (y, z) as the set of all J ∈ J for which there are an ε > 0
and a w ∈ RS such that ε ≤ 1/n, kw − zk ≤ 1/n, and J(ye + εw) ≤ K(ye + εw)
for every K ∈ J (y, z). Then, Kn (y, z) is nonempty and compact for every n,Tbecause so is J . Deﬁne K (y, z) = n Kn (y, z). Then, K (y, z) is nonempty
and compact because so is Kn (y, z) for every n. Moreover, K (y, z) ⊆ J (y, z).
Indeed, let J ∈ K (y, z) and K ∈ J . Then, for each n, there are an εn and azn ∈ RS such that εn ≤ 1/n, kzn − zk ≤ 1/n, and J(ye + εn zn ) ≤ K(ye + εn zn ).
Since J(ye) = y = K(ye),K(ye + εn zn ) − K(ye)J(ye + εn zn ) − J(ye)
≤
.
εn
εn
By taking the limits of both sides as n → ∞, we obtain ∇J(ye)z ≤ ∇K(ye)z.
Hence J ∈ J (y, z).
Let (εn )n and (zn )n be sequences such that εn → 0+ and zn → z as n → ∞.
To show the Hadamard right-diﬀerentiability, it suﬃces to show that for any J ∈J (y, z),J(ye + εn zn ) − J(ye)→ ∇J(ye)zεn
as n → ∞. Then, for each n, there is a Jn ∈ J such that I(ye + εn zn ) =Jn (ye + εn zn ). Since I(ye) = y = Jn (ye),I(ye + εn zn ) − I(ye)Jn (ye + εn zn ) − Jn (ye)
=
.
εn
εn

56

By the mean-value theorem, for each n, there is a θn ∈ [0, 1] such thatJn (ye + εn zn ) − Jn (ye)= ∇Jn (ye + θn εn zn )zn .
εnSince ye + θn εn zn → ye and J is compact with respect to the C 1 compact-open
topology, the set {∇Jn (ye + θn εn zn )zn | n = 1, 2, . . . } is bounded. Thus, it suﬃces
to prove that every convergent subsequence of (∇Jn (ye + θn εn zn )zn )n converges
to ∇J(ye)z for any J ∈ J (y, z). To ease notation, assume that (∇Jn (ye +θn εn zn )zn )n is itself convergent. We can further assume, without loss of generality,that (Jn )n is convergent, because J is compact. Denote its limit by J. Then J ∈J and, since ∇Jn → ∇J uniformly on every compact set, ∇Jn (ye + θn εn zn )zn →∇J(ye)z.
It now remains to prove that J ∈ J (y, z). Since εn → 0+ and zn → z,
for every m, there is an Nm such that Jn ∈ Km (y, z) for every n > Nm . Since
Km (y, z) is compact, J ∈ Km (y, z). Since this is true for every m, J ∈ K (y, z).
Since K (z) ⊆ J (y, z), J ∈ J (y, z).
We now move on to the proof of the twice Peano-Hadamard right-diﬀerentiability.
Let (εn )n and (zn )n be sequences such that εn → 0+ and zn → z as n → ∞. It
suﬃces to show that for any J ∈ J (y, z),1
I(ye + εn zn ) − (I(ye) + dH I(ye; z)εn )→ z ⊤ ∇2 J(ye)z2εn2as n → ∞. Then, for each n, there is a Jn ∈ G such that I(ye + εn zn ) =Jn (ye + εn zn ). Since J(ye) = y = Jn (ye) and, by the preceding result on the
Hadamard right-derivative, dH I(ye; zn ) ≤ ∇Jn (ye)zn ,I(ye + εn zn ) − (I(ye) + dH I(ye; zn )εn )ε2nJn (ye + εn zn ) − (Jn (ye) + ∇Jn (ye)zn εn )≥.
ε2nBy Taylor’s theorem, for every n, there is a θn ∈ [0, 1] such thatJn (ye + εn zn ) − (Jn (ye) + ∇Jn (ye)zn εn )1
= zn⊤ ∇2 Jn (ye + θn εn zn )zn .
2εn2Since J is compact, the set of the values on the right-hand side over all n is57

bounded. Thus, it has a convergent subsequence. Moreover, every subsequence
has a further subsequence of which the corresponding Jn ’s constitute a convergent
sequence. Let J be the limit of such a subsequence. Then, as in the previous
paragraph, we can show that J ∈ J (y, z). Moreover, the limit of the further
subsequence of the values on the right-hand side is equal to (1/2)z ⊤ ∇2 J(x)z, whichis, by assumption, independent of the choice of J ∈ J (y, z). Thus, the values on
left-hand side is bounded and the limit of every convergent subsequence of thesevalues is equal to (1/2)z ⊤ ∇2 J(ye)z for any J ∈ J (y, z). Thus,
lim infn

I(ye + εn zn ) − (I(ye) + dH I(ye; zn )εn )1≥ z ⊤ ∇2 J(ye)z2εn2for any J ∈ J (y, z).
It now remains to prove thatlim supn

1 ⊤ 2
I(ye + εn zn ) − (I(ye) + dH I(ye; zn )εn )≤z ∇ J(ye)z
ε2n
2

(50)

for any J ∈ J (y, z). To do so, for each n, we now let Jn ∈ J (y, zn ), that is,dH I(ye; zn ) = ∇Jn (ye)zn . Since I(ye + εn zn ) ≤ Jn (ye + εn zn ) and I(ye) = y =Jn (ye),I(ye + εn zn ) − (I(ye) + dH I(ye; zn )εn )ε2nJn (ye + εn zn ) − (Jn (x) + ∇Jn (ye)zn εn )≤.
ε2nBy Taylor’s theorem, for every n, there is a θn ∈ [0, 1] such thatJn (ye + εn zn ) − (Jn (ye) + ∇Jn (ye)zn εn )1
= zn⊤ ∇2 Jn (ye + θn εn zn )zn .
2εn2Since J is compact, the set of the values on the right-hand side over all n isbounded. Thus, it has a convergent subsequence. Moreover, every subsequence
has a further subsequence of which the corresponding Jn ’s constitute a convergent
sequence. Let J be the limit of such a subsequence. Then, it can be easily shownthat J ∈ J (z). Moreover, the limit of the further subsequence of the values on theright-hand side is equal to (1/2)z ⊤ ∇2 J(ye)z, which is, by assumption, independentof the choice of J ∈ J (y, z). Thus, the values on left-hand side is bounded and the58

limit of every convergent subsequence of these values is equal to (1/2)z ⊤ ∇2 J(x)z
for any J ∈ J (y, z). Thus, (50) holds.
///
Proof of Example 1 1. Since J is twice continuously diﬀerentiable on RS × Λ,
if pn → p, then J( · pn ) → J( · , p) with respect to the C 2 compact-open topology.
Since Λ is compact, so is J . If z is a scalar multiple of e, then z ⊤ ∇2 Jp (ye)z = 0 for
every p ∈ Λ because Jp is normalized. If not, then J (y, z) is a singleton becauseΛ is strictly convex. Thus, Assumption 2 is met.
2. Since Π is ﬁnite, J is compact. To show that Assumption 2 is met, lety ∈ R and z ∈ RS . Deﬁne a partition {S1 , S2 , . . . , SN } of S as follows. There are
z1 , z2 , . . . , zN such that z(s) = zn for all s ∈ Sn and n, and z1 < z2 < · · · < zN .
For each p, ∇Jp (ye) = p⊤ and, hence, ∇Jp (ye)z = E p [z]. Let p ∈ Λ. If there
are an s and t such that z(s) < z(t) and p(s) < p(t), let τ be the transposition ofswapping s and t. Then ∇Jp◦τ (ye)z < ∇Jp (ye)z. Hence, for every p ∈ J (y, z), ifz(s) < z(t), then p(s) ≥ p(t). We now show that the converse also holds, that is, ifp(s) ≥ p(t) whenever z(s) < z(t), then p ∈ J (y, z). Indeed, then, p(s) ≥ p(t) for
all n = 1, 2, . . . , N , s ∈ S1 ∪ S2 ∪ · · · ∪ Sn and t ∈ Sn+1 ∪ Sk+2 ∪ · · · ∪ SN . Thus, the|S1 |+|S2 |+· · ·+|Sn | largest coordinates of p must coincide with (p(s))s∈S1 ∪S2 ∪···∪Sn .
Thus,
X

X

p(s) ≥s∈S1 ∪S2 ∪···∪Snp(π(s))s∈S1 ∪S2 ∪···∪Snfor every π ∈ Π. Since every q ∈ Λ can be written as q = p ◦ π for some π ∈ Π, this
implies that the distribution of z under p is ﬁrst-order stochastically dominated bythat under q ∈ Λ for every q ∈ Λ. Since E p [z] = ∇Jp (ye)z, Jp ∈ J (y, z).
Note that the above characterization of J (y, z) implies that Jp ∈ J (y, z) if
and only if, for each n, (p(s))s∈Sn coincides with the (|S1 | + · · · + |Sn−1 | + 1)-th tothe (|S1 | + · · · + |Sn−1 | + |Sn |)-th largest coordinates of p. Since
E p [z] =Xp(s)z(s) =s

E p [z 2 ] =

X

X

X

n

s∈Sn

p(s)(z(s))2 =

s

59

!
p(s) zn ,

X

X

n

s∈Sn

!
p(s) zn2 ,they do not depend on the choice of Jp ∈ J (y, z). Sincez ⊤ ∇2 Jp (ye)z =γ ⊤z [p] − pp
2


⊤


z=

X

γ
p(s)(z(s))2 −
2
s

X

!2 
p(s)z(s),sz ⊤ ∇2 Jp (ye)z does not depend on the choice of Jp ∈ J (y, z). Hence, Assumption
2 is met.
///
The following lemma characterizes the second Peano-Gateaux derivative of
V under Assumption 2, and will be used in the proof of Theorem 6. For eachy ∈ v(T ), z ∈ RS , and w ∈ RS , denote by J (y, z, w) the set of all J ∈ J (g, z)
such that ∇J(ye)w ≤ ∇K(ye)w for all K ∈ J (g, z). Note that J (y, z, w) is
diﬀerent from J (y, w) in that the latter requires the last inequality to hold for all
K ∈ J , not just for K ∈ J (y, z).
Lemma 3 Under Assumption 2,
d2G V (xe; z) = (v ′ (x))2 z ⊤ ∇2 J(v(x)e)z + v ′′ (x)∇J(v(x)e)z 2
for all x ∈ T , z ∈ RS , and J ∈ J (v(x), v ′ (x)z, v ′′ (x)z 2 ), where z 2 = (z(s)2 )s ∈RS .
In the ﬁrst term of the right-hand side, J can indeed be any element of J (v(x), z).
Hence, the additional constraint that J is an element of J (v(x)e, v ′ (x)z, v ′′ (x)z 2 )is needed to correctly evaluate the second term.
Proof of Lemma 3
By Proposition 4, I is twice Peano-Hadamard rightdiﬀerentiable at v(x)e in the direction z. By applying Theorem 7 to I and v
in place of F and G and using the fact that the ﬁrst and second derivatives of the
mapping ε 7→ v ◦ (xe + εz) = (v(x + εz(s)))s of R into RS are equal to v ′ (x)z andv ′′ (x)z 2 , we can show that V is twice Peano-Gateaux right-diﬀerentiable at xe in
the direction z andd2G V (xe; z) = d2H I(v(x)e; v ′ (x)z) + dH I(v(x)e; v ′ (x)z; v ′′ (x)z 2 ).
By Proposition 4,
d2H I(v(x)e; v ′ (x)z) = (v ′ (x)z)⊤ ∇2 J(v(x))(v ′ (x)z) = (v ′ (x))2 z ⊤ ∇2 J(v(x))z60

for every J ∈ J (v(x)e, v ′ (x)z). It thus remains to show thatdH I(v(x)e; v ′ (x)z; v ′′ (x)z 2 ) = ∇J(v(x)e)(v ′′ (x)z 2 ) = v ′′ (x)∇J(v(x)e)z 2
for every J ∈ J (v(x)e; v ′ (x)z; v ′′ (x)z 2 ). For this, it suﬃces to establish the following, more general, fact: dH I(ye; z; w) = ∇J(ye)w for all y ∈ v(T ), z ∈ RS ,w ∈ RS , and J ∈ J (y, z, w).
Denote by K (y, z, w) the set of all J ∈ J for which there are sequences (εn )nand (wn )n such that εn → 0+ and wn → w as n → ∞, and J ∈ J (y, z + εn wn ) for
every n. We claim that K (y, z, w) ⊆ J (y, z, w). (The reverse inclusion also holds
but it is not necessary in the subsequent proof.) Indeed, let J ∈ K (y, z, w). Then,
there are sequences (εn )n and (wn )n such that εn → 0+ and wn → w as n → ∞,and J ∈ J (x; y+εn zn ) for every n. Hence, ∇J(ye)(z +εn wn ) ≤ ∇K(ye)(z +εn wn )for all K ∈ J and n. As n → ∞, we obtain ∇J(ye)z ≤ ∇K(ye)z for all K ∈ J .
Thus, J ∈ J (y, z). Hence, ∇J(ye)z = ∇K(ye)z for all K ∈ J (y, z). Since∇J(ye)(z + εn wn ) ≤ ∇K(ye)(z + εn wn ), ∇J(ye)wn ≤ ∇K(ye)wn for all K ∈J (y, z) and n. As n → ∞, we obtain ∇J(ye)w ≤ ∇K(ye)w for all K ∈ J (y, z).
Thus, J ∈ J (y, z, w). Hence, K (y, z, w) ⊆ J (y, z, w).
Let two sequences (εn )n and (zn )n be such that εn → 0+ and zn → z as
n → ∞. By Proposition 4, for every n, there is a Jn ∈ J (y, z + εn wn ) such thatdH I(ye; z + εn wn ) = ∇Jn (ye)(z + εn wn ). By taking a subsequence if necessary, we
can assume that the sequence (Jn )n is convergent, and denote its limit by J. As in
the proof of Proposition 4, we can show that J ∈ K (y, z, w). Since K (y, z, w) ⊆J (y, z, w), J ∈ J (y, z, w). Moreover, since ∇Jn → ∇J uniformly on every
compact set, ∇Jn (ye)(z + εn wn ) → ∇J(ye)z. Hence, dH I(ye; z) = ∇J(ye)z. On
the other hand, by Proposition 4, dH I(ye; z) = ∇K(ye)z for every K ∈ J (y, z).
In particular, dH I(ye; z) = ∇J(ye)z. Thus,
1
1(dH I(ye; z + εn wn ) − dH I(ye; z)) =(∇J(ye)(z + εn wn ) − ∇J(ye)z) = ∇J(ye)wnεnεnfor every n. As n → ∞, the right-hand side converges to ∇J(ye)w. Thus, so does
the left-hand side. Hence, dH I(ye; z; w) = ∇J(ye)w.
///
Proof of Theorem 6 1. Let (εn )n and (zn )n be sequences such that εn → 0+v(x)and zn → z as n → ∞. Let q ∈ ∆I (z). To show the right-diﬀerentiability of61κx (·, z) at 0, and its diﬀerential is equal to p · z − q · z, it is suﬃcient to prove that
κxp (εn , z) − κxp (0, z)→p·z−q·zεn(51)

as n → ∞. Suppose thatv(x)lim sup

κp

n

v(x)

(εn , z) − κp
εn(0, z)> p · z − q · z.

Let δ satisfyv(x)lim sup

κp

n

v(x)

(εn , z) − κp
εn(0, z)> δ > p · z − q · z.

By taking a subsequence if necessary, we can assume that
v(x)

κp

v(x)

for every n. Since κpv(x)(εn , z) − κpεnv(x)(0, z) = 0, κp(0, z)>δ(εn , z) ≥ δεn for every n. Thus,V̄p ((x − δεn )e + εn z) > V (xe + εn z),
and, hence,
V (xe + εn z) − V (xe)V̄p ((x − δεn )e + εn z) − V̄p (xe)
>
.
εn
εn
As n → ∞, the left-hand side converges to v ′ (x)(p · z − δ) and the right-hand side
converges to dH V (xe; z), which is equal to v ′ (x)(q · z) by Proposition 4. Thus,p · z − δ ≥ q · z, but this is a contradiction. Hence,v(x)lim sup

κp

n

v(x)

(εn , z) − κp
εn(0, z)≤ p · z − q · z.

(0, z)≥ p · z − q · z.

We can analogously show thatv(x)lim inf
n

κp

v(x)

(εn , z) − κpεnThis complete the proof of part 1.

62

2. The statement on the right-diﬀerentiability and the right-derivative folv(x)lows from part 1 and the assumption that p ∈ ∆I (z). To show the twice
right-diﬀerentiability of κx (·, z) at 0, and its second right-derivative is equal to−v ′ (x)d2H I(v(x)e; z), let (εn )n and (zn )n be sequences such that εn → 0+ and
v(x)
v(x)
zn → z as n → ∞. Since κp (0, z) = 0 and (∂κp /∂ε)(0, z) = 0, it is suﬃcientv(x)to prove that for any J ∈ J (v(x), z) and p ∈ ∆I (z),(εn , z)
1 ⊤ ′
2
→
−
z
v
(x)∇
J(v(x)e)
z
ε2n
2

v(x)

κp

as n → ∞. Suppose that(εn , z)
1
> − z ⊤ v ′ (x)∇2 J(v(x)e) z.
2
εn
2

v(x)

lim supκpn

Let δ satisfy1 ⊤ ′(εn , z)
2
>
δ
>
−
z
v
(x)∇
J(v(x)e)
z.
ε2n
2

v(x)

lim supκpn

By taking a subsequence if necessary, we can assume that
v(x)

κp

v(x)

for every n. Then, κp(εn , z)>δε2n(εn , z) > δε2n for every n. Thus,V̄p ((x − δε2n )e + εn z) > V (xe + εn z).
By the chain rule for derivatives,dV̄p ((x − δε2 )e + εz) ε=0 = v ′ (x)p · z.
dε
By the chain rule for Hadamard right-derivatives,
dH V (xe; z) = dH I(v(x)e; v ′ (x)z) = v ′ (x)p · z.

63

(52)

Hence,V̄p ((x − δε2n )e + εn z) − V̄p (xe)V (xe + εn z) − V (xe)= v ′ (x)p · z = lim
.
n
εn
εn

lim
n

Furthermore, for z 2 = ((z(s))2 )s ∈ RS ,d2
V̄p ((x − δε)e + εz) ε=0 = v ′ (x)p · z 2 − 2δv ′ (x),2dεd2H V (xe; z) = (v ′ (x))2 d2H I(v(x)e; z) + v ′′ (x)p · z 2
by the chain rule for derivatives and Lemma 3. By Taylor’s theorem,limnV̄p ((x − δε2n )e + εn z) − (V̄p (xe) + v ′ (x)p · z)
1 ′′=v (x)p · z 2 − 2δv ′ (x) .
2εn2By deﬁnition,
lim
n


1
V (xe + εn z) − (V (xe) + v ′ (x)p · z)
′
2 ⊤ 2
′′
2
=
(v
(x))
z
∇
J(v(x)e)z
+
v
(x)p
·
z
.
ε2n
2

Since
V̄p ((x − δε2n )e + εn z) − (V̄p (xe) + v ′ (x)p · z)nε2n
V (xe + εn z) − (V (xe) + v ′ (x)p · z)
≥ lim
,
n
ε2n
lim

this implies thatv ′′ (x)p · z 2 − 2δv ′ (x) ≥ (v ′ (x))2 z ⊤ ∇2 J(v(x)e)z) + v ′′ (x)p · z 2 ,that is,1δ ≤ − z ⊤ v ′ (x)∇2 J(v(x)e) z.
2This is contradiction. Hence,(εn , z)
1
≤ − z ⊤ v ′ (x)∇2 J(v(x)e) z.
2
εn
2

v(x)

lim supnκp64

We can analogously show that(εn , z)
1
≥ − z ⊤ v ′ (x)∇2 J(v(x)e) z.
2
εn
2

v(x)

lim infnκpThus, (52) follows.

///

65

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