Economic Theory 20, 159–181 (2002)
Subjective ambiguity, expected utility
and Choquet expected utility
Jiankang Zhang
Department of Economics, Carleton University, 1125 Colonel By Drive,
Ottawa, Ontario K1S 5B6, CANADA (e-mail: Jiankang [email protected])
Received: March 23, 2000; revised version: April 24, 2001
Summary. Using the Savage set up, this paper provides a simple axiomatization
of the Choquet Expected Utility model where the capacity is an inner measure.
Two attractive features of the model are its specificity and the transparency of its
axioms. The key axiom states that the decision-maker uses unambiguous acts to
approximate ambiguous ones. In addition, the notion of ‘ambiguity’ is subjective
and derived from preferences.
Keywords and Phrases: Ambiguity, Expected utility, Choquet expected utility,
Capacity, Inner measure, λ-system.
JEL Classification Numbers: C69, D81.
1 Introduction
1.1 Motivation
Much empirical evidence, inspired by Ellsberg [3], shows that the Subjective
Expected Utility (SEU) model, axiomatized by Savage [15], cannot accommodate
aversion to uncertainty or ambiguity. Recently, a number of generalizations of
the standard model have been developed that are axiomatic and can accommodate
the noted aversion as well. Most notable from the perspective of this paper is
the Choquet Expected Utility (CEU) model, or expected utility with respect to
‘nonadditive probabilities’ or ‘capacities,’ due to Schmeidler [16] and Gilboa [8].
I am deeply indebted to Larry Epstein for his valuable suggestions and important comments. I
am grateful to the financial support from the Canadian SSHRC, to Chew Soo Hong, M. Marinacci,
U. Segal, and P. Wakker for valuable discussions and suggestions, to the anonymous referee for
suggesting substantial improvements.
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J. Zhang
Schmeidler uses the Anscombe-Aumann [1] set-up to deliver the CEU model,
but this approach presumes the existence of objective lotteries. Gilboa avoids
this drawback by using the Savage set-up, but his axioms are hard to interpret.
Furthermore, it is not clear from these models why and how aversion to ambiguity
leads a decision maker (DM) to use a nonadditive probability measure to represent
her beliefs about the likelihoods of events and a CEU model to represent her
preferences over acts.
Using the Savage set up, this paper provides an axiomatic generalization of
SEU that is more restrictive than the general CEU model, but that can nevertheless accommodate Ellsberg type behavior. In particular, our model amounts to
Choquet expected utility theory with the added restriction that the capacity is an
inner measure, and the inner measure is generated from a probability measure
over λ-systems which will be defined in Section 2. The greater specificity of
our model is advantageous from the perspective of proper model development;
one wants models that are as close as possible to SEU and can still explain the
evidence at hand. The freedom within the general CEU model to choose any
capacity means that one can explain ‘almost anything’ by a suitable choice of
capacity. This embarrassment of riches is evident particularly in applications to
market data, where discipline similar to that derived within the SEU framework
from the rational expectations hypothesis is lacking (see [12]). Another important
attractive feature of the model is the simplicity of its axioms, which facilitates
understanding of the situations where they do or do not have appeal.
There are two classical extreme approaches to modeling decision making under uncertainty — the Savage probability–based model, and the other extreme
of complete ignorance and criteria such as the maximin. (See [11, Ch. 13] for
clarification of the meaning of ‘complete ignorance’ and for an axiomatic characterization.) Our axiomatic model provides an intermediate approach by combining these two extremes and delivering thereby a utility function representing
preferences that has two principal features. The first is that the DM has sufficient information about the set of payoff-relevant states to form probabilistic
beliefs about the likelihoods of events in A, a collection of events interpreted
as “unambiguous”. That is, there exists an additive probability measure p on A
representing beliefs about likelihoods and such that an expected utility function
with p represents preferences over unambiguous prospects; technically, over acts
that are A-measurable. The second principal feature of the utility function is
that there is complete ignorance of the state space besides what is modeled by
the measure p on A; a rough interpretation is that the information underlying p
is the only information available to the DM. These two features together deliver
Choquet expected utility with capacity given by the inner measure generated by
A and p. The two classical extremes appear as special cases of our model corresponding to alternative specifications for A. One obtains the Savage model if
A is the collection of all events and the other extreme of pure ignorance if A
consists of only the empty set and the full state space.
An important aspect of the model is the subjective nature of A; it is derived
from preferences rather than being specified exogenously. For this purpose, the
Subjective ambiguity, expected utility and Choquet expected utility
161
paper adopts a definition of ‘ambiguity’ and subsequent analysis that are related
to but distinct from those developed in Epstein and Zhang [5]. Clarification of
the similarities and differences is provided there. We emphasize that [5] is silent
on the nature of preferences over ambiguous acts, which is the focus and main
contribution of this paper.
1.2 Two examples
1.2.1 The Ellsberg Paradox
The Ellsberg Paradox is described here. There are 90 balls in an urn, 30 red ones
and the rest of 60 either black or yellow. One ball will be drawn at random from
the urn. The following preferences over acts are typical,




$100 if s ∈ R
$0
if s ∈ R
if s ∈ B   $100 if s ∈ B  = g and
f =  $0
$0
if s ∈ Y
$0
if s ∈ Y

$100
f =  $0
$100

if s ∈ R
if s ∈ B 
if s ∈ Y

≺
$0
 $100
$100

if s ∈ R
if s ∈ B  = g ,
if s ∈ Y
where R, B and Y denote the events corresponding to the chosen ball being red
or black or yellow and s refers to the ball that is drawn.
The DM picks f rather than g, presumably because the chance of getting
$100 in act f is precisely known to be 1/3, but the chance of getting $100 in
g is ambiguous since any number between 0 and 2/3 is possible. Similarly, she
prefers g to f , because the chance of getting $100 in g is precisely known to
be 2/3, whereas the chance of getting $100 in f is ambiguous since any number
between 1/3 and 1 is possible. It can be proved easily that the above preferences
can not be represented by an expected utility function Ep u(·) with respect to any
strictly increasing vNM utility indexes u and any additive probability measures p.
The key point of the Ellsberg Paradox is that there are two kinds of events—
unambiguous and ambiguous to the DM sometimes. Intuitively, an event is unambiguous if it has precisely known probability. But the SEU model cannot
distinguish them since there is only one additive probability measure defined
on all the events. For example, the set of unambiguous events in the Ellsberg
Paradox is
A = {∅, {R, B , Y }, {R}, {B , Y }}.
The following probability measure p on A = {∅, {R, B , Y }, {R}, {B , Y }} is
natural:
p(∅) = 0, p({R, B , Y }) = 1, p({R}) = 1/3 and p({B , Y }) = 2/3.
(1.1)
It is also intuitive that she uses an expected utility model Ep u(·) to evaluate
her unambiguous acts F ua which are A-measurable (defined precisely below),
where u is a vNM index.
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J. Zhang
After evaluating the unambiguous acts, a natural and simple way to evaluate
the ambiguous acts she can use is to approximate them by the unambiguous ones
from below.1 That is, the utility of act h is defined by U (h) = sup{Ep u(h ) :
h ≥ h ∈ F ua }, where h ≥ h means h(s) ≥ h (s) for all s ∈ {R, B , Y }.
Accordingly,
U (f ) > U (g) and U (f ) < U (g ).
That is, our model resolves the Ellsberg Paradox.
1.2.2 The four color example2
The set of unambiguous events in Section 1.2.1 is an algebra. However, this is
not true in general, because the set of unambiguous events cannot be expected
to be closed with respect to intersections. The following example will show this.
Suppose there are 100 balls in an urn and that a ball’s color may be black
(B), red (R), grey (G) or white (W). The sum of black and red ball is 50 and
the sum of black and grey ball is also 50. Similarly, one ball will be drawn at
random from the urn. We consider the following preferences:




$1
if s ∈ B
$100 if s ∈ B
 $100 if s ∈ R 

if s ∈ R 
 f2 =  $0
,
f1 = 

 $0

if s ∈ G
$0
if s ∈ G 
$0
if s ∈ W
$0
if s ∈ W




$1
if s ∈ B
$100 if s ∈ B
 $100 if s ∈ R 
 $0
if s ∈ R 



g1 = 
 $100 if s ∈ G  ≺ g2 =  $100 if s ∈ G  , and
$0
if s ∈ W
$0
if s ∈ W




$1
if s ∈ B
$100 if s ∈ B
 $100 if s ∈ R 
 $0
if s ∈ R 



h1 = 
 $100 if s ∈ G  h2 =  $100 if s ∈ G  .
$100 if s ∈ W
$100 if s ∈ W
The DM picks f1 rather than f2 , mainly because the chance of getting $100 in
f1 is the same as in f2 , but also with additional chance to get $1 in act f1 . The only
difference between pairs {g1 , g2 } and {f1 , f2 } is the payoff at state G. Though
both events {B } and {G} are ambiguous, the combination of them, however,
leads to {B , G} being unambiguous since it has precisely known probability of
one half. Thus, picking act g2 leads her to get $100 with probability of one half,
while picking act g1 leads her only to get $1 with probability of one half and no
idea with how much probability to get $100. Thus, she prefers g2 to g1 . Finally,
the same reasons as with the pair of acts {f1 , f2 }, she will prefer h1 to h2 .
An important and interesting phenomenon has happened: Changing the common outcome on event {G} in the pair of acts {f1 , f2 } leads the DM to change
1
2
Here ‘below’ reflects her attitude towards ambiguity. In other words, she is ambiguity averse.
This example is based on a suggestion by U. Segal.
Subjective ambiguity, expected utility and Choquet expected utility
163
the ranking, while changing the common outcome on event {G, W } in the pair
of acts {f1 , f2 } leaves the ranking of acts unchanged. What is the ‘key’ difference between events {G} and {G, W }? Intuitively, event {G} is ‘ambiguous’,
since the probability of event {G} is unknown to the DM. While event {G, W }
is ‘unambiguous’, since the probability of event {G, W } is known to her. This
example suggests that unambiguous and ambiguous events can be derived from
preferences. Roughly, an event A is unambiguous if replacing a common outcome for all states in A by any other ones does not change the ranking of the
pair of acts being compared (for formal expression, see Section 3).
Intuitively, the set of unambiguous events is
A = {φ, {B , R, G, W }, {B , G}, {R, W }, {B , R}, {G, W }}.
However, A is not an algebra, since it is not intersection-closed. For example,
both {B , G} and {B , R} are unambiguous, but {B } = {B , G} ∩ {B , R} is ambiguous. Though A is not an algebra, she still can assign probabilities over it.
The following are natural to her:
p(S ) = 1, p(φ) = 0, p({B , G}) = p({R, W }) = p({B , R}) = p({G, W }) = 1/2.
(1.2)
If she conforms to our model, then she has an expected utility function Ep u(·)
to represent her preferences over F ua and for other acts, she uses the utility
function U (h) = sup{Ep u(h ) : h ≥ h ∈ F ua }. As a result, U (f1 ) > U (f2 ),
U (g1 ) < U (g2 ), and U (h1 ) > U (h2 ).
1.3 Related literature
The papers in the literature that are most closely related to ours are Sarin and
Wakker [14], Jaffray and Wakker [10] and Mukerji [13]3 . Sarin and Wakker also
employ unambiguous events, but they do not explicitly derive them and simply
assume that there exist the two kinds of events. In addition, Sarin and Wakker’s
key axiom differs substantially from ours; this is by necessity since the general CEU model is axiomatized in Sarin and Wakker [14], while we axiomatize
the special case corresponding to an inner measure. Jaffray and Wakker exploit
and adapt the well-known Dempster-Shafer representation of belief functions
to show that CEU with a capacity that is a belief function captures precisely
the information structure described earlier – a class of unambiguous events and
complete ignorance otherwise. In comparison to the classic Savage model and
the model we present, the formal model in Jaffray & Wakker [10] employs two
added primitives taken from the Dempster-Shafer representation. These are an
auxiliary space of states and a (multi-valued) mapping that links it with S and
delivers a definition for ‘unambiguous’ events in S . As a result, verification
of the axioms requires that the modeler observe or hypothesize these auxiliary
3
The Jaffray-Wakker and Mukerji are similar and we discuss only the former here.
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J. Zhang
primitives; such verification is not possible through observation exclusively of
choices between alternative acts over S . This is in contrast to our model, where
all axioms are expressed in terms of preferences over Savage style acts over the
(payoff-relevant) state space S , without reliance on auxiliary state spaces or ad
hoc assumptions regarding A. Another difference from our paper is that the two
papers deliver different classes of preferences. Jaffray and Wakker admit any
belief function while our axioms deliver inner measures which are not necessary
belief functions.
The use of inner measures as a way of modeling uncertainty is due to Fagin
and Halpern [6], but their notion of inner measure is more restrictive than ours.
Ours is generated from a probability measure on a λ-system, while Fagin and
Halpern begin with a probability measure on a σ-algebra (or algebra). Fagin and
Halpern take the perspective of the artificial intelligence literature where beliefs
are primitive rather than derived from preferences. Our contribution is to show
how their analysis may be adapted to a decision theoretic framework with a more
general notion of inner measures.
This paper proceeds as follows. λ-systems, inner measures and inner integrals
are defined next. The axioms and the representation results are described in
Section 3. Some examples are provided in Section 4. All proofs are collected in
appendices.
2 λ-systems, inner measures and inner integrals
Let S be a state space with associated σ-algebra given by the power set. The
usual way of representing a decision maker’s beliefs about the likelihoods of
events is by means of an (additive) probability measure on 2S . This structure
is too restrictive as described in the examples in Section 1.2. In this paper, we
generalize this standard approach in two ways: (i) we relax the additivity of
beliefs on all events to additivity on a subset A of 2S ; (ii) we require that A
be a λ-system but not necessarily an algebra.
2.1 λ-Systems
Definition 2.1. A nonempty class of subsets A of S is a λ–system if
λ.1. S ∈ A;
λ.2. A ∈ A =⇒ Ac ∈ A; and
λ.3. An ∈ A, n = 1, 2, . . . and Ai ∩ Aj = ∅, ∀i =
/ j =⇒ ∪n An ∈ A.
This definition and terminology appear in [2, p. 36].
Think of A as the collection of DM’s “unambiguous” events and that she
can assign probability to each event in A. λ.1 is a usual assumption, since
the DM knows that event S will happen without ambiguity. Assumption λ.2 is
natural. If she can assign probability to event A, it is natural for her to assign
the difference between the probabilities of S and A to Ac . The intuition for λ.3
Subjective ambiguity, expected utility and Choquet expected utility
165
is that if she can assign probabilities to two disjoint events A and B , then it is
natural for her to assign the sum of probabilities of events A and B to the union
A ∪ B . On the other hand, even if she can assign probabilities to both events A
and B , she still cannot assign probability to the intersection A ∩ B when A ∩ B
is “ambiguous”; recall the example in the Section 1.2.2.
The key difference between a σ–algebra and a λ–system is that the latter is
not required to be intersection-closed. It seems evident that a λ–system is more
natural for modeling unambiguous events.
2.2 Probabilities and integrals
Though A is not a σ–algebra, a probability measure can still be defined on it.
Say that a function p : A −→ [0, 1] is a countably additive probability measure
∞
over A if: 1. p(∅) = 0, p(S ) = 1; and 2. p(∪∞
i =1 Ai ) = Σi =1 p(Ai ), ∀Ai ∩ Aj = φ,
i=
/ j , i , j = 1, 2, . . ..
Denote by (S , A, p) a λ–system probability space. Say that a probability p
is convex-ranged if for all A ∈ A and 0 < r < 1, there exists B ⊂ A, B ∈ A,
such that p(B ) = r p(A).
Given a λ-system probability space (S , A, p), say that a finite-ranged function f : S −→ R 1 is A—measurable, if for any x ∈ R 1 , {s ∈ S : f (s) ≤ x } ∈
A.
Denote by
F
F
ua
The integral of f ∈ F
Σi xi p({s : f (s) = xi }).
= {f : f
has finite range} and
= {f ∈ F : f is A − measurable}.
ua
with respect to p is defined as follows:
fdp =
2.3 Inner and outer measures and integrals
Think of a λ-system A as consisting of all ‘unambiguous’ events and of the
probability measure p as representing the DM’s beliefs about the likelihoods of
events in A. Suppose that the information underlying p is all that is available to
her and that she is extremely pessimistic. Then one way to assess likelihoods for
‘ambiguous’ events, that is, events not in A, is by means of the inner measure
p∗ corresponding to (S , A, p), defined by: For all events B ,
p∗ (B ) ≡ sup {p(A) : A ∈ A , A ⊆ B }.
(2.1)
Note that p∗ = p is additive on A, but generally non-additive outside A.
Similarly, if she is extremely optimistic, then she might assess likelihoods for
‘ambiguous’ events by means of the outer measure p ∗ corresponding to (S , A,
p), defined by: For all events B ,
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J. Zhang
p ∗ (B ) ≡ inf {p(A) : A ∈ A , A ⊇ B }.
(2.2)
The inner, outer measures p∗ (B ), p ∗ (B ) provide lower, upper bounds on the
likelihood of event B respectively. The use of a lower, rather than upper bound,
reflects aversion to ambiguity. Similarly, the use of an upper, rather than lower
bound, reflects an affinity for ambiguity. Evidently, the upper and lower bounds
agree on A.
Corresponding to inner and outer measures, we introduce inner and outer
integrals as follows:
fdp = sup{ gdp : f ≥ g ∈ F A }, ∀f ∈ F and
∗∗
fdp = inf{ gdp : f ≤ g ∈ F A }, ∀f ∈ F ,
where f ≥ g means that f (s) ≥ g(s), ∀s ∈ S . Their intuitive meanings are similar
to those for inner and outer measures.
2.4 Capacities and Choquet integrals
Inner and outer measures are special cases of capacities. Say that ν : 2S −→
[0, 1] is a capacity if: 1. A ⊂ B =⇒ ν(A) ≤ ν(B ); and 2. ν(∅) = 0, ν(S ) = 1.
Call ν the conjugate of ν, where ν(A) = 1 − ν(Ac ).
A capacity is superadditive if: For all disjoint events A and B in 2S , ν(A∪B ) ≥
ν(A) + ν(B ). A capacity is subadditive if: For all disjoint events A and B in 2S ,
ν(A ∪ B ) ≤ ν(A) + ν(B ). A capacity is convex if: For all events A and B in 2S ,
ν(A ∪ B ) + ν(A ∩ B ) ≥ ν(A) + ν(B ).
It is not hard to verify the following:
Lemma 2.2. Let p∗ and p ∗ be the inner and outer measures defined in (2.1) and
(2.2). Then (a): p ∗ and p∗ are conjugates and (b): p∗ is superadditive. However,
the outer measure p ∗ is not subadditive in general.
The inner measure p∗ is not convex in general. For example, consider the p∗
generated from A and p in example 1.2.2. But p∗ is convex if A is an algebra.
See Proposition 3.1 in [6] for an elementary proof for the case of σ-algebra.
Convexity has been interpreted by Schmeidler [16] as reflecting ambiguity
aversion of the DM. But the preferences given in example 1.2.2 are obviously
ambiguity averse and the capacity p∗ corresponding to them is not convex. For
more detailed explanation, see Epstein [4] and Epstein and Zhang [5].
Applications of capacities to decision theory employ the notion of Choquet
integration. For any capacity ν and integrand f : S −→ R 1 , the Choquet integral
is defined by
f dν ≡
0
∞
ν({s : f (s) ≥ t}) dt +
0
−∞
[ν({s : f (s) ≥ t}) − 1] dt.
(2.3)
Subjective ambiguity, expected utility and Choquet expected utility
167
When f is a finite-ranged function, we can suppose that f assumes the values
x1 < · · · < xn on the events E1 , . . . , En respectively. In this case, fd ν =
Σin=1 xi [ν(∪nj=i Ej ) − ν(∪nj=i +1 Ej )], where ν(∪nn+1 Ej ) ≡ 0.
Since any inner measure is a capacity, Choquet integration is defined also for
inner and outer measures. In these cases, Choquet integrals are related to inner
and outer integrals, as shown next.
Theorem 2.3. Let p∗ , p ∗ be the inner, outer measures defined in (2.1) and (2.2).
Then:
∗
fdp; and
1. For any function f ∈ F , ∗ fdp ≤ fdp∗ ≤ fdp ∗ ≤
2. If p∗ (A ∪ B ) = p(A) + p∗ (B ) for all A ∈ A and B ⊂ Ac , then
fdp = fdp∗ and
∗
(2.4)
∗
fdp = fdp ∗ .
Under the condition in part 2, (2.4) provide a representation and novel intuition for Choquet integrals that are based on inner and outer measures. This
representation is used to prove our main result, Theorem 3.3.
3 The model and main results
3.1 Unambiguous events
Savage’s set-up is adopted throughout. Denote by S the set of states of the world,
by 2S the set of all events and by X the set of outcomes. Prospects are modeled
via simple acts, mappings from S to X having finite range. The set of acts is
F = {. . . , f , . . .}. The DM has a preference relation on the set of acts.
The following definition is central:
Definition 3.1. An event A is unambiguous if: (i) For all acts f , f and outcomes
x, y ∈ X ,
f if s ∈ Ac
f if s ∈ Ac
=⇒
x if s ∈ A
x if s ∈ A
(3.1)
f if s ∈ Ac
f if s ∈ Ac
;
y if s ∈ A
y if s ∈ A
and (ii) The condition obtained if A is everywhere replaced by Ac in (i) is also
satisfied. Otherwise, A is called ambiguous.
Define
A = {A ∈ 2S : A is unambiguous}
F ua = {f ∈ F : f is A—measurable}.
Both sets A and F
ua
are nonempty because ∅ and S are unambiguous.
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J. Zhang
It merits emphasis that ambiguity is subjective; it is endogenously derived
from the DM’s preference order. This distinguishes the present analysis from
that of Fishburn [7], who takes ambiguity as a primitive, and from [14], where
the issue of the derivation of the class of unambiguous events is not explicitly
addressed and an exogenously specified class of unambiguous events is used.
To understand definition 3.1, recall Savage’s key axiom.
Sure-Thing Principle (STP): For all events A and all acts f , f , g and g ,
f if s ∈ Ac
f if s ∈ Ac
=⇒
g if s ∈ A
g if s ∈ A
f
g
if s ∈ Ac
if s ∈ A
f
g
if s ∈ Ac
if s ∈ A
(3.2)
.
An implication of STP is that for all events A and all acts f , f , and outcomes
x , y,
f if s ∈ Ac
f if s ∈ Ac
=⇒
x if s ∈ A
x if s ∈ A
(3.3)
f if s ∈ Ac
f if s ∈ Ac
.
y if s ∈ A
y if s ∈ A
Therefore, if two acts imply different subacts (f (·) versus f (·)) over an event Ac ,
but the same outcome over event A, the ranking of these acts will not depend
on this common outcome. This axiom implies that preferences are separable
across mutually exclusive events, which is the key property of expected utility
preferences, either over objective probability distributions or over acts. However,
this principle is contradicted by the typical choices in the Ellsberg Paradox. We
interpret such choices as evidence that the separability required by the STP
between outcome in A and those in Ac ≡ S \A is descriptively (and perhaps
even normatively) problematic when the event A is “ambiguous.” In such cases,
changing a common outcome in for some states in Ac can cast an entirely new
light on the pair of acts being compared.
Though the STP is no longer appealing in similar situations, it is still of use
in building an alternative formal model. If one views “ambiguity” as the only
source of violation of the Sure-Thing Principle, then one is led to use STP to give
formal meaning to “ambiguity.” This leads to our definition 3.1. It is immediate
that preferences satisfy STP if and only if all events are unambiguous.
Another alternative to define unambiguous events is to fully use STP. Denote
by
A = {A : A and Ac satisfies (3.2) for all f , f , g , g}.
(3.4)
But A is intersection-closed, making it unsuitable as the class of unambiguous
events. In addition, A is ‘too small’ in general. For instance, A = {∅, S } in
example 1.2.2.
Subjective ambiguity, expected utility and Choquet expected utility
169
3.2 Preferences over unambiguous acts
A natural and interesting question is that without assuming Savage’s STP, can
we still find ‘reasonable’ axioms such that the DM’s beliefs are represented by an
additive probability measure over A and restricted over F ua is represented
by a SEU? The answer is ‘yes.’ For example, Axioms 1-6 in Epstein and Zhang
[5] will do, but we omit their statement for the sake of brevity. To state our
main result Theorem 3.3 concisely and clearly, we define Savage-representable
preference order over F ua as follows:
Definition 3.2. A preference order is Savage-representable on F ua if there
exist a unique countably additive convex-ranged probability measure p on A
and a (nonconstant) utility index u : X −→ R 1 such that
(3.5)
f g ⇐⇒ u(f )dp ≥ u(g)dp,
∀f , g ∈ F ua .
As a result, is Savage-representable on F ua if it satisfies Axioms 1-6 in [5].
While Savage-representable preference order describes the nature of preferences
over unambiguous acts, our primary concern is the nature of the ordering of
ambiguous acts. We turn to this aspect of preferences next.
3.3 Ambiguous acts and Choquet expected utilities
The remaining task is to relate the ordering of ambiguous acts to the ordering
of unambiguous ones. The next axioms restrict both the set of outcomes X and
. The following notation is adopted:
f ≥ g means that f (s) g(s)
∀s ∈ S .
Axiom 1 (Weak Monotonicity). For any two acts f , g with f ∈ F
F ua , f g if f ≥ g.
(3.6)
ua
or g ∈
Axiom 2 (Certainty Equivalent). For any f ∈ F , there exists x ∈ X such
that f ∼ x .
Axioms 1 and 2 are self-explanatory and common. The final axiom is the
central one.
Axiom 3 (Approximation from Below). For any f ∈ F , x ∈ X , if f x ,
then there exists an act g ∈ F ua such that f ≥ g and g x .
The interpretation of Axiom 3 is that in order to evaluate an arbitrary act
f , the DM approximates f from below by an unambiguous act g. Given Weak
Monotonicity, any such g provides a lower bound for the utility derived from f .
Roughly speaking, Approximation from Below requires further that f be ranked
only as highly as the most preferred of such approximating unambiguous acts g.
In other words, the DM facing the ambiguous act f asks “what is the worst that
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J. Zhang
can happen if I choose f ?” and answers this question by relying exclusively on
the acts that she understands well, namely on unambiguous acts. This algorithm
seems inappropriate for situations where the distinction between ‘slightly ambiguous’ and ‘highly ambiguous’ events (and acts) is important. Our DM thinks
in terms of black or white rather than in terms of shades of grey.
Note that Axiom 3 does not require unambiguous acts to be rich. The DM
ranks f x because she can find g ∈ F ua such that f ≥ g and g x . Now we
can state our main result – the representation for preferences on F .
Theorem 3.3. Let be a preference relation on F and A the corresponding
set of unambiguous events. Then is Savage-representable on F ua and satisfies
axioms 1-3 iff A is a λ-system and there exist a unique convex-ranged, countably
additive probability measure p on A, and a nonconstant utility index u : X −→
R 1 having convex range such that
f, g ∈ F ,
(3.7)
f g ⇐⇒ u(f )dp∗ ≥ u(g)dp∗ ,
where p∗ is the inner measure generated from the λ-probability space (S , A,
p) and integration is in the sense of Choquet.
Theorem 3.3 tells us that it is complete ignorance outside A and approximation from below that lead the DM to use a capacity p∗ to represent her beliefs
and a Choquet Expected Utility to represent her preferences. Accordingly, complete ignorance outside A and approximation from below capture the heuristic
meaning of ambiguity aversion.
The interpretation of the utility representation, particularly the subjective nature of A, warrants emphasis. It cannot be argued that exclusive reliance (via
p∗ ) on events in A to compute likelihoods of other events reflects an extreme or
unreasonable degree of ignorance or ambiguity aversion.4 At a formal level, that
is because A is a component of the utility representation that is inseparable from
the use of inner measure. Less formally, the above argument is unsupportable
because A is subjective; whether or not there is a large degree of ignorance or
ambiguity aversion implied depends on the size of A. Indeed, a primary role of
A is to model the degree of ambiguity aversion of the DM.
Finally, there is an obvious mirror image of this analysis in which ‘optimism,
uncertainty affinity and Approximation From Above’ replace ‘pessimism, uncertainty aversion and Approximation From Below’. That is, replace axiom 3 by
the following axiom:
Axiom 4 (Approximation from Above). For any f ∈ F , x ∈ X , if f ≺ x ,
then there exist an act g ∈ F ua such that f ≤ g and g ≺ x .
This delivers the CEU model of preferences with capacity given by the outer
measure p ∗ .
4 There is a parallel with the question of how to interpret the set of priors and the minimization
over that set that appears in the multiple-priors model of preference in Gilboa & Schmeidler [9].
Subjective ambiguity, expected utility and Choquet expected utility
171
We close this section with a brief discussion of the extreme case of a Savage
preference, which judges all events as unambiguous. Intuitively, such a preference
should satisfy both approximation from below and above. The following Theorem
proves this is true:
Theorem 3.4. Let be a preference relation on F and A the corresponding
set of unambiguous events. Then is Savage-representable on F ua and satisfies
axioms 1—4 iff A = 2S , F = F ua and there exist a unique convex-ranged,
countably additive probability measure p on 2S , and a nonconstant utility index
u : X −→ R 1 having convex range such that for all acts f , g ∈ F ,
f g ⇐⇒ u(f )dp ≥ u(g)dp.
4 Examples
4.1 Choquet expected utility model
Now suppose that the DM’s preference order is represented by CEU with a
monotone, superadditive capacity ν. That is,
f g ⇐⇒ u(f )d ν ≥ u(g)d ν,
∀f , g ∈ F ,
(4.1)
where u : X −→ R 1 is a nonconstant utility index with a convex range. Since
the attitude towards ambiguity is included in the capacity ν, the precise relation
between ν and A is of interest.
Define
Σ∗
Σ
∗∗
= {A ∈ 2S : ν(A ∪ B ) = ν(A) + ν(B ), ∀B ⊂ Ac }
= {A ∈ 2S : ν(A) + ν(Ac ) = 1}.
Theorem 4.1. Let on F be defined in 4.1. Then:
1. If Σ ∗ is symmetric in the sense that A ∈ Σ ∗ ⇐⇒ Ac ∈ Σ ∗ , then Σ ∗ = A
and Σ ∗ is closed with respect to disjoint unions.
2. If A ∈ Σ ∗∗ , then Ac ∈ Σ ∗∗ . Σ ∗∗ is an algebra if ν is convex, but not more
generally.
3. Σ ∗ ⊆ Σ ∗∗ and they are not equal in general.
4. If ν is convex, then Σ ∗ = Σ ∗∗ .
Collection Σ ∗∗ represents an extreme possible approach to defining unambiguous events directly in terms of the capacity ν. But Σ ∗∗ is not closed with
respect to disjoint unions. For example:
Example 4.2. Let S = {ω1 , ω2 , ω3 } and ν be defined on (S , 2S ) as follows:
ν(S ) = 1, ν(∅) = 0 = ν({ω1 })
ν({ω2 }) = ν({ω3 }) = 1/3
ν({ω1 , ω2 }) = ν({ω1 , ω3 }) = ν({ω2 , ω3 }) = 2/3.
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J. Zhang
In this example, ν is superadditive but not convex. And
Σ ∗∗ = {∅, S , {ω1 , ω2 }, {ω1 , ω3 }, {ω2 }, {ω3 }}
is not closed with respect to disjoint unions since the union {ω2 , ω3 } is not in
Σ ∗∗ .
4.2 A ‘counterexample’
This example illustrates violation of our model. It is a variation of the example
in subsection 1.2.2.
Let B + R = 80 and B + G = 10. One ball is to be drawn at random. The
following preferences are intuitive:




$100 if s ∈ B
$0
if s ∈ B

 $100 if s ∈ R 
if s ∈ R 
=f .
  $0
f1 = 


 $0
$100 if s ∈ G  2
if s ∈ G
$100 if s ∈ W
$0
if s ∈ W
The DM picks f1 rather than f2 mainly because she knows from B + R = 80
and B + G = 10 that R = G + 70 and W = B + 10. Though the events {R}
and R c = {B , G, W } are both ambiguous to the DM, however, the probability of
{R} is objectively at least 0.7, and therefore greater than two times that of its
complement {B , G, W }.
It is intuitive that all unambiguous events in this example are:
A = {φ, S , {B , G}, {R, W }, {B , R}, {G, W }},
where S = {B , R, G, W }. And the DM’s beliefs over A can be represented by
the following probability measure
p(S )
p({B , R})
However,
= 1, p(φ) = 0 and p({B , G}) = 0.1, p({R, W }) = 0.9
= 0.8, p({G, W }) = 0.2
u(f1 )dp∗ <
u(f2 )dp∗ ,
where p∗ is the inner measure induced from p and the integral is in the sense of
Choquet.
What is wrong with the inner measure model? The problem is that the
decision-maker has more information than just the probabilities of events in
A, e.g., she knows that R = G + 70 and W = B + 10, even though {R, G} and
{W , B } are ambiguous.
One suggestion to solve the problem is as follows: The DM knows the true
law is an additive probability measure p on 2S and both p and p agree on A,
but she does not know what p looks like outside A. Thus, any one probability
measure in
Subjective ambiguity, expected utility and Choquet expected utility
Π(p) = {p : p is an extension of (S , A, p) to 2S }
is possible. Due to ambiguity aversion, she may use minp ∈Π(p)
evaluate acts f . As a result,
min
u(f
u(f2 )dp .
)dp
>
min
1
p ∈Π(p)
173
u(f )dp to
p ∈Π(p)
Extension of our model to accommodate such ‘partial information’ is the subject
of current research.
A. Appendix A
Theorem 2.3 is proved here. To prove it, the following Lemma is needed:
Lemma A.3. Let (S , A, p) be a λ-system probability space. If p∗ (A ∪ B ) =
p(A) + p∗ (B ), ∀A ∈ A and B ⊂ Ac , then:
(i) For any A ∈ 2S , there exists a sequence {An } in A satisfying An ⊂ An+1 ⊂ A,
n = 1, 2, . . . , such that
p∗ (A) = lim p(An ) = p(∪n An ).
n→∞
(ii) For any chain ∅ = D0 ⊂ D1 ⊂ D2 ⊂ · · · ⊂ Dk −1 ⊂ Dk = S , there exists a
chain ∅ = B0 ⊂ B1 ⊂ B2 ⊂ · · · ⊂ Bk −1 ⊂ Bk = S in A with Bi ⊆ Di for
i = 0, 1, 2, . . . , k such that p(Bi ) = p∗ (Di ) for i = 0, 1, 2, . . . , k .
Proof.
Proof of (i). If p∗ (A) = 0, the conclusion is obviously true. Next, assume p∗ (A) >
0. Take 0 < < p∗ (A). By the definition of p∗ , there exists A1 ∈ A with A1 ⊂ A
such that
p(A1 ) > p∗ (A) − > 0.
If p(A1 ) = p∗ (A), then let An = A1 , n = 1, 2, . . . , and the conclusion is proved.
Next, let p(A1 ) < p∗ (A). Then p∗ (A\A1 ) > 0 follows from p∗ (A) = p∗ (A1 ∪
(A\A1 )) = p(A1 ) + p∗ (A\A1 ). Similarly, there exists A ∈ A with A ⊂ A\A1
such that
p(A ) > p∗ (A\A1 ) − /2.
Denote A2 = A1 ∪ A ∈ A. As a result,
p(A2 )
= p(A1 ∪ A ) = p(A1 ) + p(A )
= p∗ (A) + (p(A ) − p∗ (A\A1 )) > p∗ (A) − /2.
By induction, there exists an increasing sequence {An }∞
n=1 in A such that
p(An ) > p∗ (A) − /n, n = 1, 2, . . . .
Define B = ∪An . Therefore,
(A.1)
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J. Zhang
p∗ (A) − /n < p(An ) ≤ p(B ) ≤ p∗ (A), for all n.
That is,
p∗ (A) = lim p(An ) = p(B ).
n→∞
Proof of (ii). We only need to prove that for any two events D1 ⊂ D2 , there
exist two events B1 , B2 in A with B1 ⊂ B2 and Bi ⊂ Di , i = 1, 2 such that
p(Bi ) = p∗ (Di ), i = 1, 2. By part (i), there exists event B1 in A with B1 ⊂ D1
such that p(B1 ) = p∗ (D1 ). Accordingly,
p∗ (D2 ) = p∗ (B1 ∪ (D2 \B1 ) = p(B1 ) + p∗ (D2 \B1 ).
Again by part (i), there exists event C1 in A with C1 ⊂ D2 \B1 such that
p(C1 ) = p∗ (D2 \B1 ). Let B2 = B1 ∪ C1 , then B2 ∈ A, B2 ⊂ D2 , B1 ⊂ B2 and
p(B2 ) = p∗ (D2 ).
Proof of Theorem 2.3.
(1): Since p(A) ≤ p∗ (A), ∀A ∈ A,
gdp = gdp∗ ≤ fdp∗ , f ≥ g ∈ F
And this implies
fdp = sup{
∗
Similarly, we can prove
A
.
gdp : f ≥ g ∈ F
fdp ∗ ≤
F
}≤
fdp∗ .
∗
fdp.
fdp ∗ directly follows from p∗ (A) ≤ p ∗ (A), ∀A ∈ 2S .
(2): We prove only ∗ fdp = fdp∗ here, the other is similar. By part (1), we
need only prove that
sup{ gdp : f ≥ g ∈ F F } ≥ fdp∗ , ∀f ∈ F .
Finally,
fdp∗ ≤
Let
and x1 < x2 < · · · < xn .

x1



x2
f (s) =
···



xn
if s ∈ E1
if s ∈ E2
······
if s ∈ En.
Subjective ambiguity, expected utility and Choquet expected utility
175
By the definition of Choquet integral,
fdp∗ = Σin=1 xi [p∗ (∪nj=i Ej ) − p∗ (∪nj=i +1 Ej )],
where ∅ = ∪nj=n+1 Ej ⊂ ∪nj=n Ej ⊂ · · · ⊂ ∪nj=2 Ej ⊂ ∪nj=1 Ej = S is a chain. By
Lemma A.3 (ii), there exists a chain ∅ = B0 ⊂ B1 ⊂ · · · ⊂ Bn = S in A with
Bi ⊂ ∪nj=n+1−i Ej such that p∗ (∪nj=n+1−i Ej ) = p(Bi ), i = 0, 1, 2, . . . , n. Therefore,
fdp∗
= Σin=1 xi [p∗ (∪nj=i Ej ) − p∗ (∪nj=i +1 Ej )]
= Σin=1 xi [p(Bn+1−i ) − p(Bn−i )]
f dp,
= Σin=1 xi p(Bn+1−i \Bn−i ) = 
x1




 x2
···
f (s) =


x


 n−1
xn
where
if s ∈ Bn \Bn−1
if s ∈ Bn−1 \Bn−2
······
.
if s ∈ B2 \B1
if s ∈ Bn
Obviously, f ∈ F A and f ≤ f.
Therefore,
fdp∗ = f dp ≤ sup{ gdp : g ∈ F
A
, g ≤ f }.
B. Appendix B
To prove Theorem 3.3, we need to prove the following two Lemmas first:
Lemma B.1. If an act f has the following form
x If s ∈ A
f =
,
y If s ∈ Ac
then
sup{ u(h)dp : h ∈ F
ua
,h ≤ f}
=
u(f )dp∗ ,
(B.1)
where x y and the integral of left side of (B.1) is in the sense of Choquet.
Proof. For any integer n > 1, by the definition of inner measures, there is
An ∈ A with An ⊂ A such that p(An ) > p∗ (A) − 1/n. Let
x if s ∈ An
gn (s) =
y if s ∈ Acn .
Obviously, gn ∈ F
ua
and gn ≤ f . Therefore,
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J. Zhang
u(gn )dp
= u(x )p(An ) + u(y)p(Acn ) = (u(x ) − u(y))p(An ) + u(y)
> (u(x ) − u(y))p∗ (A) + u(y) − (u(x ) − u(y))/n.
Accordingly,
sup{ u(h)dp : h ∈ F ua , h ≤ f }
≥ limn→∞ [(u(x ) − u(y))p∗ (A)+ u(y) − (u(x ) − u(y))/n]
= (u(x ) − u(y))p∗ (A) + u(y) = u(f )dp∗ .
The conclusion follows from (B.2) and part (1) of Theorem 2.3.
(B.2)
Lemma B.2. For any A ∈ A,
p∗ (A ∪ B ) = p(A) + p∗ (B ), ∀B ⊂ Ac .
Proof. For any two disjoint events A and B , p∗ (A ∪ B ) ≥ p∗ (A) + p∗ (B ) follows
from the superadditivity of p∗ . Thus, we only need to prove that for any event
A ∈ A,
p∗ (A ∪ B ) ≤ p(A) + p∗ (B ), ∀B ⊂ Ac .
(B.3)
Suppose (B.3) is not true, then there exists an unambiguous event A ∈ A and
event B ⊂ Ac such that
p∗ (A ∪ B ) > p(A) + p∗ (B ).
Since u is nonconstant and convex ranged, there exist outcomes x1 , y1 , y2 , and
x2 such that
u(x1 ) > u(y1 ) > u(y2 ) > u(x2 ).
(B.4)
Let




x1 if s ∈ B
y1 if s ∈ B
f1 (s) =  x2 if s ∈ Ac \B  , g1 (s) =  y2 if s ∈ Ac \B  and
x1 if s ∈ A
x1 if s ∈ A




x1 if s ∈ B
y1 if s ∈ B
f2 (s) =  x2 if s ∈ Ac \B  , g2 (s) =  y2 if s ∈ Ac \B  .
y1 if s ∈ A
y1 if s ∈ A
Therefore, by (B.9) and Lemma B.1,
f1 g1 ⇐⇒ sup{ u(h)dp
: h ∈ F ua , h ≤ f1 }
ua
>
sup{ u(h)dp : h ∈ F , h ≤ g1 }ua
⇐⇒ u(f1 )dp∗ > sup{ u(h)dp : h ∈ F , h ≤ g1 }
and
f2 ≺ g2 ⇐⇒ sup{ u(h)dp
: h ∈ F ua , h ≤ f2 }
ua
< sup{
, h ≤ g2 }.
u(h)dp : h ∈ F
ua
⇐⇒ sup{ u(h)dp : h ∈ F , h ≤ f2 } < u(g2 )dp∗ .
(B.5)
Subjective ambiguity, expected utility and Choquet expected utility
177
If
outcomes
{x
,
x
,
y
,
y
}
satisfy
u(f
)dp
>
u(g
)dp
and
u(f2 )dp∗ <
1
2
1
2
1
∗
1
∗
u(g2 )dp∗ , then
sup{ u(h)dp : h ∈ F ua
)dp∗
, h ≤ f1 } = u(f1ua
> u(g1 )dp∗ ≥ sup{ u(h)dp : h ∈ F , h ≤ g1 }
and
sup{ u(h)dp : h ∈ F ua , h ≤ f2 } ≤ u(f2 )dp∗
< u(g2 )dp∗ = sup{ u(h)dp : h ∈ F ua , h ≤ g2 }.
By direct computations,
u(f1 )dp∗ > u(g1 )dp∗ ⇐⇒
[u(x2 ) − u(y2 )][1 − p∗ (A ∪ B )] + [u(x1 ) − u(y1 )][p∗ (A ∪ B ) − p∗ (A)] > 0
(B.6)
and
u(f2 )dp∗ < u(g2 )dp∗ ⇐⇒
[u(x2 ) − u(y2 )][1 − p∗ (A ∪ B )] + [u(x1 ) − u(y1 )]p∗ (B ) < 0.
(B.7)
Since p∗ (A ∪ B ) > p∗ (A) + p∗ (B ) and the range of u is convex, there exist
outcomes x1 , x2 , y1 , y2 satisfying (B.4), (B.6) and (B.7). This implies that f1 g1 ,
but f2 ≺ g2 . This contradicts that A is unambiguous.
Proof of Theorem 3.3.
(=⇒): It is enough to prove (3.7) assuming (i) the representation (3.5) where
A satisfies λ.1 and λ.2 (not necessarily λ.3) and (ii) Axioms 1-3 for .
Since constant acts are in F ua , (3.5) implies x y ⇐⇒ u(x ) ≥ u(y). Since
p is convex-ranged, it follows from (3.5) and Certainty Equivalent for F ua –
measurable acts that u has convex range.
Lemma B.3. (i) For any act f ∈ F , f ∼ x implies that
u(x ) = sup { u(g)dp : f ≥ g ∈ F ua }.
(ii) The ordering on F is represented by U where, for any f ∈ F ,
U (f ) = sup{ u(g)dp : f ≥ g ∈ F ua }.
(B.8)
Proof.
(i) By Weak Monotonicity, f ≥ g implies x ∼ f g. By (3.5), u(g)dp ≤ u(x ),
∀g ∈ F ua , this means that
sup{ u(g)dp : f ≥ g ∈ F ua } ≤ u(x ).
For the other direction, it is true if x is the worst outcome in X . Next, suppose
that there exists x ≺ x . Since u has convex range, for any sufficiently small
> 0, there exists x such that u(x ) − < u(x ) < u(x ). Now suppose that
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J. Zhang
u(x ) − > sup{ u(g)dp : f ≥ g ∈ F
Then x g for all g ∈ F
Below.
ua
ua
} for some > 0.
. This contradicts f x and Approximation from
(ii) follows from (i), Certainty Equivalent and transitivity of .
It remains to show that the utility function defined in (B.8) is the required
Choquet integral with respect to the inner measure p∗ , that is,
(B.9)
U (f ) ≡ u(f )dp∗ .
And this directly follows from Lemma B.2 and part (ii) of Theorem 2.3.
To prove Theorem 3.4, we need to prove some Lemmas first.
Lemma B.4. Let (S , A, p) be a λ-system probability space. If p∗ (A ∪ B ) =
p(A) + p∗ (B ) for any A ∈ A and B ⊂ Ac , then
{A : p∗ (A ∪ B ) = p∗ (A) + p∗ (B ), ∀B ⊂ Ac } = {A : p∗ (A) + p∗ (Ac ) = 1}.
Proof. Let
Σ 2 = {A : p∗ (A ∪ B ) = p∗ (A) + p∗ (B ), ∀B ⊂ Ac } and
Σ 3 = {A : p∗ (A) + p∗ (Ac ) = 1}.
Obviously, Σ 2 ⊆ Σ 3 . Next, we prove that Σ 3 ⊆ Σ 2 .
First, we prove that Σ 2 is closed with respect to disjoint unions. Take A1 , A2 ∈
2
Σ are disjoint. For any event B ⊂ (A1 ∪ A2 )c ,
p∗ ((A1 ∪ A2 ) ∪ B ) = p∗ (A1 ∪ (A2 ∪ B )) = p∗ (A1 ) + p∗ (A2 ∪ B )
= p∗ (A1 ) + p∗ (A2 ) + p∗ (B ) = p∗ (A1 ∪ A2 ) + p∗ (B ).
Next, we prove if A ∈ Σ 2 , p∗ (A) = 0 and A1 ⊂ A, then A1 ∈ Σ 2 . For any
B ⊂ Ac1 ,
p∗ (A1 ∪ B ) ≤ p∗ (A ∪ B ) = p∗ (A ∪ (B \A))
= p∗ (A) + p∗ (B \A) ≤ p∗ (A1 ) + p∗ (B ).
p∗ (A1 ∪ B ) ≥ p∗ (A1 ) + p∗ (B ) directly follows from superadditivity of p∗ . This
completes the conclusion.
Now let A ∈ Σ 3 . From Lemma A.3 (i), there exist two events A1 and A2 in
A satisfying A1 ⊂ A and A2 ⊂ Ac such that p∗ (A) = p(A1 ) and p∗ (Ac ) = p(A2 ).
As a result, (A1 ∪A2 )c ∈ A ⊂ Σ 2 and p((A1 ∪A2 )c ) = 0. Since A\A1 ⊂ (A1 ∪A2 )c
and (A1 ∪ A2 )c ∈ Σ 2 , therefore, A\A1 ∈ Σ 2 . Since Σ 2 is closed with respect to
disjoint unions, A = A1 ∪ (A\A1 ) ∈ Σ 2 .
Lemma B.5. If p∗ (A) = p ∗ (A) for event A, then p∗ (A) + p∗ (Ac ) = 1.
Subjective ambiguity, expected utility and Choquet expected utility
179
Proof. Suppose, p∗ (A) = p ∗ (A) for event A. Then, we have p∗ (Ac ) = 1 − p ∗ (A) =
1 − p∗ (A) = p ∗ (Ac ). If p∗ (A) + p∗ (Ac ) =
/ 1, then p∗ (A) + p∗ (Ac ) < 1 by the
superadditivity of p∗ . Accordingly,
p ∗ (A) + p ∗ (Ac )
[1 − p∗ (Ac )] + [1 − p∗ (A)]
2 − [p∗ (A) + p∗ (Ac )] > 1 > p∗ (A) + p∗ (Ac ).
=
=
But this contradicts the assumption.
Proof of Theorem 3.4.
Obviously from Theorem 3.3, we have for any acts f , g ∈ F
∗
f g ⇐⇒ u(f )dp∗ ≥ u(g)dp∗ ⇐⇒ u(f )dp ≥ u(g)dp ∗ .
Claim 1 : For any act f ,
u(f )dp∗ =
u(f )dp ∗ .
(B.10)
(B.11)
1. If ∗B.11 is not true, then there exists an act f such that
Proof of Claim
u(f )dp∗∗ < u(f )dp . Now pick outcome x such that u(f )dp∗ < u(x ) <
u(f )dp . But this contradicts B.10.
Claim 2 : For any event A, p∗ (A) = p ∗ (A).
Proof of Claim 2. Suppose this is not true, then there exists an event A such that
p ∗ (A) > p∗ (A). Let
∗
if s ∈ A
x
,
f =
x
if s ∈ Ac
where x ∗ x . Then
u(f )dp∗ = u(x )[1 − p∗ (A)] + u(x ∗ )p∗ (A) = u(x ) + [u(x ∗ ) − u(x )]p∗ (A)
< u(x ) + [u(x ∗ ) − u(x )]p ∗ (A) = u(f )dp ∗ .
But this contradicts Claim 1.
Then A = 2S directly follows from Lemmas B.4, B.5, B.2 and part 1 of
Theorem 4.1.
C. Appendix C
Proof of Theorem 4.1 is provided here:
Proof of (1). We only prove A ⊂ Σ ∗ here. The other parts can be proved by
routine verification.
Suppose it is not true. Without loss of generality, then there exists an unambiguous event A ∈ A and B ⊂ Ac such that
ν(A ∪ B ) > ν(A) + ν(B ).
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J. Zhang
Since u is nonconstant and convex ranged, there exist outcomes satisfying
u(x1 ) > u(y1 ) > u(y2 ) > u(x2 ).
Let
(C.1)




x1 if s ∈ B
y1 if s ∈ B
f1 (s) =  x2 if s ∈ Ac \B  , g1 (s) =  y2 if s ∈ Ac \B  and
x1 if s ∈ A
x1 if s ∈ A




x1 if s ∈ B
y1 if s ∈ B
f2 (s) =  x2 if s ∈ Ac \B  , g2 (s) =  y2 if s ∈ Ac \B  .
y1 if s ∈ A
y1 if s ∈ A
By direct computations,
u(f1 )d ν >
u(g1 )d ν ⇐⇒
[u(x2 ) − u(y2 )][1 − ν(A ∪ B )] + [u(x1 ) − u(y1 )][ν(A ∪ B ) − ν(A)] > 0
and
u(f2 )d ν <
(C.2)
u(g2 )d ν ⇐⇒
[u(x2 ) − u(y2 )][1 − ν(A ∪ B )] + [u(x1 ) − u(y1 )]ν(B ) < 0.
(C.3)
Since ν(A ∪ B ) > ν(A) + ν(B ) and the range of u is convex, there exist x1 , x2 , y1 ,
y2 satisfying (C.1), (C.2) and (C.3). Thus, f1 g1 and f2 ≺ g2 . This contradicts
that A is unambiguous.
Proof of 2. Obviously A ∈ Σ ∗∗ =⇒ Ac ∈ Σ ∗∗ and Ω, φ are in Σ ∗∗ . To prove
that Σ ∗∗ is an algebra when ν is convex, it suffices to show that if A1 , A2 are
in Σ ∗∗ , then A1 ∩ A2 and A1 ∪ A2 are also in Σ ∗∗ . That ν is convex implies that
ν is concave. Therefore,
ν(A1 ) + ν(A2 ) ≤ ν(A1 ∩ A2 ) + ν(A1 ∪ A2 ) ≤ ν(A1 ∩ A2 ) + ν(A1 ∪ A2 )
≤ ν(A1 ) + ν(A2 ) = ν(A1 ) + ν(A2 ).
Because
ν(A1 ∪ A2 ) ≤ ν(A1 ∪ A2 ) and
ν(A1 ∩ A2 ) ≤ ν(A1 ∩ A2 ), then
ν(A1 ∩ A2 ) = ν(A1 ∩ A2 ) and ν(A1 ∪ A2 ) = ν(A1 ∪ A2 ).
Proof of 3. The set inclusions follow from the definitions of Σ ∗ and Σ ∗∗ . Example
4.2 shows that the sets differ in general.
Proof of 4. We only need to prove that Σ ∗∗ ⊆ Σ ∗ . Since ν is convex,
ν(A) = inf{p(A) : p is in the core of ν}.
Let ν(A) = 1 − ν(Ac ). Then we prove p(A) = q(A), for any p, q in the core
of ν. Suppose this is not true. That is, there are p, q in the core of ν such
that p(A) > q(A) and p(Ac ) < q(Ac ). Thus, ν(A) ≤ q(A), ν(Ac ) ≤ p(Ac ) and
ν(A) + ν(Ac ) ≤ q(A) + p(Ac ) < q(A) + q(Ac ) = 1, a contradiction. If p(A) = q(A)
for any p, q in the core of ν, then A is in Σ ∗ .
Subjective ambiguity, expected utility and Choquet expected utility
181
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