A Model of Ambiguity Aversion†
Pavlo Blavatskyy
Abstract:
Abstract This paper presents a new model for describing decision making under
uncertainty. In the proposed model a decision maker behaves as if maximizing
subjective expected utility while also caring about whether the states of the world that
belong to relatively less ambiguous events yield the same consequence or not. The
proposed model can rationalize Allais (1953) and Ellsberg (1963) paradoxes and
Machina (2009) reflection example. Behavioral characterization of the model is
provided.
Keywords:
Keywords Subjective Uncertainty, Ambiguity, Ambiguity Aversion, Ellsberg Paradox,
Allais Paradox
JEL Classification Codes:
Codes D81
†
This is a substantially revised version of the earlier draft “Modeling Ambiguity
Aversion as Aversion to Utility Dispersion Caused by Ambiguous Events” that
considered only two sources of uncertainty (events with known and unknown objective
probabilities). I am grateful to Aurélien Baillon and Peter Wakker for helpful comments.
1
Introduction
Models of ambiguity aversion are usually motivated by Ellsberg (1961) paradox.
An example of this paradox is illustrated on Figure 1. A decision maker chooses between
two acts. An act f1 yields $4000 if heads come up on a toss of a fair coin; and nothing if
tails come up. An act g1 yields $4000 if a black ball is drawn from an urn containing only
black and white balls (in unknown proportion); and nothing if a white ball is drawn.
Most people prefer f1 over g1 and this preference holds for any permutation of rows
and/or columns on Figure 1. Such preference falsifies subjective expected utility theory
(Savage, 1954).
Act f1
BLACK
WHITE
HEADS
$4000
$4000
TAILS
$0
$0
Act g1
BLACK
WHITE
HEADS
$4000
$0
TAILS
$4000
$0
Figure 1 Ellsberg (1961) twotwo-color example
Recently, Machina (2009) presented two examples falsifying Choquet expected
utility (Gilboa, 1987; Schmeidler, 1989) and cumulative prospect theory (Tversky and
Kahneman, 1992). L’Haridon and Placido (2010) provided experimental evidence on
Machina (2009) reflection example. Baillon et al. (2011) showed that this evidence
falsifies the model of uncertainty averse preferences (Cerreia-Vioglio et al., 2009), which
includes as special cases: maxmin expected utility or multiple priors (Gilboa and
Schmeidler, 1989), multiplier preferences (Hansen and Sargent, 2001; Strzalecki, 2011),
Klibanoff et al. (2005) smooth model with concave φ, variational preferences
(Maccheroni et al., 2006), the model of Ahn (2008) with concave φ and the model of
Chateauneuf and Faro (2009).
Baillon et al. (2011) also showed that α-maxmin theory (Ghirardato et al., 2004)
can accommodate either Ellsberg (1961) paradox or Machina (2009) reflection example
but not both at the same time. This conclusion also holds for the model of Olszewski
(2007).
Act f2
BLACK
WHITE
HEADS
$4000
$4000
TAILS
$8000
$0
Act g2
BLACK
WHITE
HEADS
$4000
$8000
TAILS
$4000
$0
Figure 2 Blavatskyy (2012) simplification of Machina (2009) reflection example
2
Blavatskyy (2012) proposed a simplification of Machina (2009) reflection example
that is illustrated on Figure 2 (random events are the same as in the Ellsberg paradox
presented on Figure 1). Blavatskyy (2012) showed that people preferring f2 over g2 for
any permutation of rows and/or columns on Figure 2 cannot simultaneously exhibit the
Ellsberg paradox illustrated on Figure 1 if their preferences are described by vector
expected utility (Siniscalchi, 2009), the model of Nau (2006) or Neilson (2010), linear
utility theory for belief functions (Jaffray, 1989) or expected uncertain utility theory
(Gul and Pesendorfer, 2010).
Two examples illustrated on Figures 1 and 2 appear as plausible and even natural.
It is unfortunate that many established models of ambiguity aversion cannot rationalize
the preference for f1 over g1 on Figure 1 together with the preference for f2 over g2 on
Figure 2. Such preferences can be intuitively explained as follows. In both examples
illustrated on Figures 1 and 2 a decision maker derives extra satisfaction from the fact
that a relatively less ambiguous event HEADS (or TAILS) contains states of the world
that yield the same outcome. Thus, acts f1 and f2 bring extra utility because the decision
maker is certain to receive $4000 in the event HEADS. At the same time, the decision
maker derives no extra satisfaction from the fact that a relatively more ambiguous event
BLACK (or WHITE) contains states of the world that yield the same outcome. Thus, acts
g1 and g2 bring no extra utility.
The idea that a decision maker enjoys extra utility from a constant payoff
contingent on a relatively less ambiguous event can be further supported by examples of
decision under objective uncertainty (risk). One famous example is the Allais (1953)
paradox illustrated on Figure 3. There are three states of the world s1, s2, and s3 and
their objective probabilities are 0.01, 0.1 and 0.89 correspondingly. Many people prefer
f3 over g3 when outcome x is ₣100m but they prefer g3 over f3 when outcome x is ₣0.
If the universal event {s1}∪{s2}∪{s3} is viewed as a relatively less “ambiguous”
event, then the Allais paradox can be intuitively explained like examples shown on
Figures 1 and 2. A decision maker derives extra satisfaction from the fact that a
relatively less “ambiguous” event {s1}∪{s2}∪{s3} contains states of the world that yield
the same outcome. Thus, act f3 brings extra utility when outcome x is ₣100m because the
decision maker then receives ₣100m for sure. On the other hand, act f3 brings no extra
utility when outcome x is ₣0.
3
State s1, π(s1)=0.01
State s2, π(s2)=0.1
State s3, π(s3)=0.89
Act f3
₣100m
₣100m
x
Act g3
₣0
₣500m
x
Figure 3 The Allais (1953) paradox
According to this line of reasoning, the switching choice pattern typically revealed
in the Allais paradox may disappear when choice problems do not involve a degenerate
constant act. Indeed, there is strong experimental evidence that the common
consequence effect diminishes in the interior of the Marschak-Machina probability
triangle (e.g., Conlisk, 1989; Camerer, 1992).
The main objective of this paper is to present a new model of ambiguity aversion
that formalizes the simple intuition that people may care about the fact whether states
of the world belonging to less ambiguous events yield the same outcome or not. As a
starting point it is natural to assume that there is a collection of events that a decision
maker may perceive as relatively less ambiguous. One such event can be the universal
event that includes all states of the world (as in the Allais paradox above). A constant act
that yields the same outcome in all states of the world is associated with no uncertainty
at all. A decision maker may value sure outcomes that she can receive with certainty.
Another example of an event that a decision maker may perceive as relatively
unambiguous is an event with a known objective probability (as in examples on Figures
1 and 2). An act that yields the same outcome in all states of the world belonging to an
event with a known objective probability is associated with a relatively lower subjective
uncertainty and may have only objective uncertainty (risk). Again, a decision maker may
value such acts (see also a classical discussion in Knight, 1921).
Yet another example of an event that a decision maker may perceive as relatively
less ambiguous is an event associated with familiar sources of subjective uncertainty. An
act that yields the same outcome in all states of the world belonging to such an event has
a relatively lower exposure to unfamiliar sources of uncertainty and may involve only
familiar sources of uncertainty. There is strong experimental evidence that people value
such acts, a phenomenon known as the source preference (e.g., Tversky and Fox, 1995,
pp. 279-281; Abdellaoui et al., 2011). In fact, the Ellsberg paradox itself can be viewed as
a source preference for a fair coin over an urn with an unknown composition of black
and white balls.
4
Once we fix a collection of events that a decision maker perceives as relatively less
ambiguous, we construct an utility function with the following property. Ceteris paribus,
an act that yields the same outcome in all states of the world belonging to one of
relatively less ambiguous events has a relatively higher utility than an act that yields
different outcomes across these states. In fact, our utility function constitutes a tradeoff
between subjective expected utility and a measure of utility dispersion across events
that a decision maker perceives as relatively less ambiguous. Thus, our proposed model
of ambiguity aversion is in the spirit of the mean-variance approach in finance
(Markowitz, 1952). In fact, our model can be viewed as a generalization of the theory of
disappointment without prior expectation proposed by Delquié and Cilo (2006) in the
following sense. If we restrict our attention only to events with known objective
probabilities (i.e., we consider decision under risk) and the set of events that a decision
maker perceives as relatively unambiguous contains only one element—the universal
event, then our model coincides with Delquié and Cilo (2006).
The paper is organized as follows. Section 1 introduces notation, a new model of
ambiguity aversion and several illustrative examples. Section 2 provides a behavioral
characterization of the model. Section 3 concludes.
5
1. Notation,
Notation, Model and Examples
There is a non-empty set S (that can be finite or infinite). The elements of set S
are called states of the world. Only one state of the world is true but a decision maker
does not know which one. There is a sigma-algebra ⅀ of the subsets of S that are called
events. There is a connected set X. The elements of X are called outcomes. An act f :S →X
is a ⅀-measurable function from S to X. Acts may be unbounded. The set of all acts is
denoted by ℱ. A constant act that yields outcome x ∊X in all states of the world is
denoted by x ∊ ℱ.
A decision maker has a preference relation ≽ on ℱ. As usual, the symmetric part of
≽ is denoted by ∼ and the asymmetric part of ≽ is denoted by ≻. An interval B⊂X is a
preference interval when for all x, y, z ∊X if x, z ∊B and x ≽y ≽z then y ∊B. We assume
that set X is endowed with an algebra that contains all preference intervals.
Preferences are represented by utility function U :ℱ→Թ when f ≽g if and only if
U(f )≥U(g ) for all f, g ∊ℱ. In this paper we consider utility function (1).
(1)
U ( f ) = ∫ u f ( s ) π ( ds ) + 0.5∑ E∈A ∫
S
E
∫ ϕ ( u f ( s ) − u f ( t ) ) π ( ds ) π ( dt )
E
E
In formula (1), standard (Bernoulli) utility function u : X →Թ is continuous and
determined up to an increasing linear transformation. Probability measure π : ⅀ →ሾ0,1ሿ
is unique. Integral is a standard Lebesgue integral with respect to measure π.
Collection of events A⊂⅀ that a decision maker perceives as relatively less
ambiguous may be an empty set. Function φE :Թ+→Թ is continuous and, in general,
determined up to an increasing linear transformation. Yet, we use a convenient
normalization φE (0)=0 for all E ∊A throughout the paper. Thus, function φE :Թ+→Թ
remains determined up to a multiplication by a positive constant. Function φE :Թ+→Թ
captures the attitude of a decision maker to utility dispersion across states of the world
that belong to event E ∊A.
Subjective expected utility theory (Savage, 1954) is a special case of
representation (1) when either set A is empty or φE (v)=0 for all E ∊A and all v ∊Թ+.
The mean-variance approach (Markowitz, 1952) is a special case of representation (1)
when X ⊆Թ, u(x)=x for all x ∊X, A={S}, φS(v)=av2, for some a=const and all v ∊Թ+, and
probability measure π:⅀→ሾ0,1ሿ is exogenously given. The theory of disappointment
without prior expectation (Delquié and Cilo, 2006) is a special case of representation (1)
when A={S} and probability measure π:⅀→ሾ0,1ሿ is exogenously given.
6
Example 1 Consider Ellsberg (1961) two-color example presented on Figure 1. A
natural assumption is that A={HEADS, TAILS, HEADS∪TAILS}. For simplicity, let us also
assume that π(BLACK)=π(HEADS)=0.5. A decision maker then prefers f1 over g1 when
ϕ HEADS ( u ( $4000 ) − u ( $0 ) ) + ϕTAILS ( u ( $4000 ) − u ( $0 ) ) < 0 . This example suggests that
ambiguity aversion is captured by a negative function φ (.) and ambiguity seeking
(preference for g1 over f1) is captured by a positive function φ (.) in representation (1).
Example 2 Consider a modification of the reflection example (Machina, 2009)
proposed by Blavatskyy (2012) and presented on Figure 2. We maintain the same
assumptions as in Example 1. A decision maker then prefers f2 over g2 when
ϕTAILS ( u ( $8000 ) − u ( $0 ) ) > ϕHEADS ( u ( $8000 ) − u ( $4000 ) ) + ϕTAILS ( u ( $4000 ) − u ( $0 ) ) . Thus,
preference for f2 over g2 can be captured by a convex function φ (.) and preference for g2
over f2 can be captured by a concave function φ (.)in representation (1) . Both
preferences can coexist with either ambiguity averse or ambiguity seeking preferences
revealed in Example 1 since a convex/concave function can be simultaneously
negative/positive.
Example 3 Consider the Allais (1953) paradox presented on Figure 3. A natural
assumption is that set A contains only one event—the universal event {s1}∪{s2}∪{s3}.
Since set A is singleton, we write function φ (.) without any subscripts. We normalize
utility function so that u(₣0)=0 and u(₣500m)=1. Furthermore, let u(₣100m)=v. A
decision maker then prefers f3 over g3 when x=₣100m if inequality (2) is satisfied.
(2)
0.11v > 0.1+0.001φ (1)+0.0089φ (v)+0.089φ (1-v)
The same decision maker prefers g3 over f3 when x=₣0 if inequality (3) is satisfied.
(3)
0.11v < 0.1+0.09φ (1) — 0.11*0.89φ (v)
Inequalities (2) and (3) can hold simultaneously only if φ(1)>1.2φ(v)+φ(1-v).
Thus, a decision maker may reveal the Allais paradox when φ (.) is convex and negative.
Example 4 Consider an illustration of the common ratio effect (Kahneman and
Tversky, 1979). Lottery S yields ₪3000 with probability α ∊(0,1ሿ (and nothing with
probability 1—α). Lottery R yields ₪4000 with probability 0.8α (and nothing with
probability 1—0.8α). A decision maker prefers S over R when α=1 but switches to
preferring R over S when α=0.25. As in Example 3, it is natural to assume that set A
contains only one element—the universal event. We normalize utility function so that
u(₪0)=0 and u(₪4000)=1. Furthermore, let u(₪3000)=w. A decision maker then
prefers S over R when α=1 if inequality (4) is satisfied.
7
(4)
w>0.8+0.16φ(1)
The same decision maker prefers R over S when α=0.25 if inequality (5) holds.
(5)
w<0.8+0.64φ (1)—0.75φ (w)
Thus, the decision maker may exhibit the common ratio effect only if
0.64φ(1)>φ(w). This condition is satisfied when either function φ(.) is negative and w
is sufficiently close to one or function φ(.) is positive and w is sufficiently close to zero.
On the other hand, condition 0.64φ(1)>φ(w) is violated when either function φ(.) is
negative and w is sufficiently close to zero or function φ(.) is positive and w is
sufficiently close to one. Thus, model (1) predicts that the common ratio effect may be
reversed by changing the middle outcome ₪3000 from very unattractive (so that w is
close to zero) to very attractive (so that w is close to one). Blavatskyy (2010) presented
experimental evidence confirming this prediction.
Example 5 Consider two acts presented on Figure 4 (random events are the same
as on Figures 1 and 2). Both acts yield $4000 with probability 0.5 and $0 with
probability 0.5. Hence, acts f4 and g4 are consequentially equivalent. Yet, a decision
maker may not be indifferent between f4 and g4. Model (1) can rationalize a strict
preference between acts f4 and g4 if set A contains events HEADS and TAILS but it does
not contain “diagonal” events on Figure 4. This example allows a test of model (1)
against a general class of second-order probabilistically sophisticated preferences
(Ergin and Gul, 2009), which can simultaneously rationalize Examples 1 and 2 but
predicts an indifference between f4 and g4.
Act f4
BLACK
WHITE
HEADS
$4000
$4000
TAILS
$0
$0
Act g4
BLACK
WHITE
HEADS
$4000
$0
TAILS
$0
$4000
Figure 4 Two consequentially equivalent acts
8
2. Behavioral characterization of the model
The purpose of this section is to characterize subjective parameters of
representation (1) in terms of observed behavioral properties of the preference relation
≽. In particular, we identify a list of properties that the preference relation ≽ must
satisfy in order to admit representation (1). We follow the so-called connected topology
approach (e.g., Debreu, 1960; Chapter 6 in Kranz et al. (1971); Wakker 1989) rather
than the approach of Savage (1954) or Anscombe-Aumann (1963). The preference
relation ≽ is assumed to be a weak order (Axioms 1 and 2).
Axiom 1 (Completeness) For all f, g ∊ℱ either f ≽g or g ≽f (or both).
Axiom 2 (Transitivity) For all f, g, h ∊ℱ if f ≽g and g ≽h then f ≽h.
Conceptually, we first derive utility representation of preferences over step acts.
Then we extend this representation to all other acts. Consider a partition {E1,…,En} of the
state space S into n events (i.e., Ei ∊⅀ for all i ∊{1,…,n}, E1⋃…⋃En =S and Ei ⋂Ej =∅ for
all i,j ∊{1,…,n}, i≠j). Let {x1,E1;…;xn,En} denote a step act that yields outcome xi ∊X in a
state s ∊Ei, i ∊{1,…,n}. Let ॲ⊂ℱ denote the set of all step acts (ॲ is a subset of the set of
measurable simple acts).
Axiom 3 (Step-continuity) For any partition {E1,…,En} of set S into n events and any
step act {x1,E1;…;xn,En} the sets {( y1 , …, yn ) ∈ X n : { y1 , E1 ;…; yn , En } { x1 , E1 ;…; xn , En }}
and {( y1 , …, yn ) ∈ X n : { x1 , E1 ;…; xn , En } { y1 , E1 ;…; yn , En }} are closed with respect to the
product topology on Xn.
Proposition 1 (Debreu, 1954) Preference relation ≽ satisfies Axioms 1-3 if and only
if it is represented by a continuous utility function U : ॲ →Թ.
For compact notation, let xEif ∊ॲ denote a step act that yields outcome x ∊X in a
state s ∊Ei and outcome f (s) in a state s ∊S \Ei , i ∊{1,…,n}. An event Ei ∊⅀ is inessential
(or null) if xEif ≽yEif for all x , y ∊X and f :S \Ei →X. Otherwise, an event is essential (or
nonnull).
An event Ei ∊⅀ is separable if Thomsen-Blaschke condition (Blaschke, 1928) holds:
xEif ≽zEih for all x, y, z ∊X and all f, g, h : S \Ei →X such that xEig ≽yEih and yEif ≽zEig.
Otherwise, an event is inseparable. Note that if event Ei ∊⅀ is (in)separable then its
complement S \Ei is also (in)separable. Events S and ∅ are (in)separable if and only if ≽
is (in)transitive on ॲ. Thus, Axiom 2 (Transitivity) can be alternatively formulated as
“events S and ∅ are separable”. In subjective expected utility theory all events are
9
separable. In Choquet expected utility (Gilboa, 1987; Schmeidler, 1989) only
comonotonic events are separable. In model (1) an event Ei ∊⅀ is separable if for every
E ∊A either Ei⋂E =∅ or Ei⋃E=Ei. Without a loss of generality, let us assume that the first
m ∊{0,…,n} events in partition {E1,…,En} are separable.
Proposition 2 (Debreu, 1960) Preference relation ≽ satisfies Axioms 1-3 if and only
m
if U ({ x1 , E1 ;…; xn , En } ) = ∑ ui ( xi ) + V ( xm +1 ,..., xn ) , where ui : X →Թ and V : Xn-m →Թ are
i =1
continuous functions. Moreover, functions ui : X →Թ and V : Xn-m →Թ are unique up to a
positive affine transformation if there are at least two essential separable events.
Model (1) evaluates outcomes contingent on separable events exactly as in
subjective expected utility theory. Thus, we can weaken the usual tradeoff consistency
condition (cf. Chapter 4 in Wakker, 2010; Chapter 6 in Wakker 1989) so that it holds
only for separable events.
Axiom 4 (Tradeoff-Consistency for Separable Events) If xEf ≽yEg, zEg≽wEf, and
yOh≽xOk then zOh≽wOk for all x, y, z, w ∊X; f, g : S \E →X; h, k : S \O →X, any essential
separable event E ∊⅀ and any separable event O ∊⅀.
Proposition 3 (Wakker, 1984, Theorem 2.1) Preference relation ≽ satisfies axioms
1-4 if and only if it admits representation (6), where pi ≥0 for all i ∊{1,…,m} and
functions u : X →Թ and V : Xn-m →Թ are continuous. Moreover, these functions are unique
up to a positive affine transformation if there are at least two essential separable events.
m
(6)
U ({ x1 , E1 ;…; xn , En }) = ∑ pi ⋅ u ( xi ) + V ( xm +1 ,..., xn )
i =1
For any inseparable event Ei ∊⅀ let Êi ∊⅀ denote an event such that Ei ⋃Êi is
separable but any O ⊂Ei ⋃Êi , O ∊⅀, is inseparable. Clearly, such an event Êi exists for
any inseparable event Ei if Axiom 2 holds (i.e., if S is separable). Intuitively, event Êi is
the smallest possible event that is needed to create a separable event when united with
Ei. The next axiom states that Thomsen-Blaschke condition holds if acts have the same
utility dispersion across Ei and Êi.
Axiom 5 (Thomsen-Blaschke condition for Inseparable Events) If xEig ≽yEih and
yEif ≽zEig then xEif ≽zEih for all inseparable events Ei ∊⅀ as well as all x, y, z ∊X and all
f, g, h : S \Ei →X such that:
a) there exist a separable event E ∊⅀ and a, b : S \E →X such that xEa ~g(s)Eb
and yEa ~ h(s)Eb for all s ∊Êi ;
10
b) there exist a separable event O ∊⅀ and c, d: S \O →X such that yOc ~f (s)Od
and zOc ~ g(s)Od for all s ∊Êi .
Proposition 4 Preference relation ≽ satisfies axioms 1-5 if and only if it admits
representation (7), where pi ≥0 for all i ∊{1,…,m} and functions u : X →Թ, ui : X →Թ and
φij : Թ→Թ are continuous. Moreover, these functions are unique up to a positive affine
transformation if there are at least two essential separable events and at least two
essential inseparable events.
m
(7)
U ({ x1 , E1 ;…; xn , En } ) = ∑ pi ⋅ u ( xi ) +
i =1
n
∑
n
ui ( xi ) +
i = m +1
n
∑ ∑ ϕ (u ( x ) − u ( x ))
ij
i
j
i = m +1 j = m +1
E j ⊂ Eˆi
Proof is presented in the Appendix.
Next, we impose a condition, known as cardinal independence, which rules out
state-dependent preferences. In other words, it is desirable to restrict all utility
functions ui : X →Թ in (7) so that they are positive affine transformations of the same
function u : X →Թ.
Axiom 6 (Cardinal Independence) If xEif ≽yEig, xEig ≽zEif, and yOh≽xOk then
xOh≽zOk for all essential events Ei ∊⅀, all events O ∊⅀ as well as all outcomes x, y, z ∊X;
all g, h : S \Ei →X; and all f, k : S \O →X such that:
a) if Ei ∊⅀ is inseparable then there exist a separable event E ∊⅀ and a, b :S \E →X
such that xEa ~f (s)Eb and yEa ~ g(s)Eb for all s ∊Êi ;
b) if Ei ∊⅀ is inseparable then there exist a separable event E ∊⅀ and c, d :S \E →X
such that xEc ~g(s)Ed and zEc ~ f (s)Ed for all s ∊Êi ;
c) if O ∊⅀ is inseparable then there exist a separable event E ∊⅀ and q, r :S \E →X
such that yEq ~h(s)Er and xEq ~ k(s)Er for all s ∊Ȏ;
d) if O ∊⅀ is inseparable then there exist a separable event E ∊⅀ and t, w :S \E →X
such that xEt ~h(s)Ew and zEt ~ k(s)Ew for all s ∊Ȏ.
Proposition 5 Preference relation ≽ satisfies axioms 1-6 if and only if it admits
representation (8), where pi ≥0 for all i ∊{1,…,n} and functions u : X →Թ and φij : Թ→Թ
are continuous. Moreover, these functions are unique up to a positive affine
transformation if there are at least two essential separable events and at least two
essential inseparable events.
n
(8)
U ({ x1 , E1 ;…; xn , En }) = ∑ pi ⋅ u ( xi ) +
i =1
n
n
∑ ∑ ϕ (u ( x ) − u ( x ))
ij
i = m +1 j = m +1
E j ⊂ Eˆi
11
i
j
Proof follows immediately from Proposition 4 and Theorem 15 in Krantz et al.
(1971, Section 6.11.2).
In Proposition 5 the restriction that constants pi ≥0 sum up to one can be always
arranged since utility function is unique up to a positive affine transformation.
Axiom 8 (Nontriviality) f ≽g holds not for all f, g ∊ℱ.
Axiom 9 (Monotonicity) For all f, g ∊ℱ if f (s)≽g (s) for all s ∊ S then f ≽g.
Axiom 10 (Step-equivalence) For all f ∊ℱ there exists g ∊ॲ such that f ~g.
For any f ∊ℱ and any x ∊X let f <x ∊ℱ denote an act that yields outcome x in any
state s ∊ S such that f (s)≻x and outcome f (s) in all remaining states. Furthermore, let
f >x ∊ℱ denote an act that yields outcome x in any state s ∊ S such that x ≻f (s) and
outcome f (s) in all remaining states.
Axiom 11 (Truncation continuity) For all f ∊ℱ and all g ∊ॲ if f ≻g then there exists
x ∊X such that f <x ∊ℱ, f <x ≽g and if g ≻f then there exists y ∊X such that f >y ∊ℱ, g ≽f >y.
Axioms 9, 10 and 11 are required for extending an integral representation from ॲ
to ℱ (cf., Theorem 2.5 in Wakker, 1993). Axiom 8 is required to insure that the range of
an utility function is a non-degenerate interval.
3. Conclusion
This paper proposes a novel way of modeling the phenomenon of ambiguity
aversion. The preferences of a decision maker are represented by a standard separable
utility function that includes extra terms for utility dispersion across ambiguous states.
This modeling approach captures attitudes to ambiguity through subjective functions on
the set of outcomes (not the state space). Such modeling approach avoids the necessity
of considering mixture acts and extending the preferences of a decision maker over the
set of mixture acts.
Model (1) evaluates an act through a linear tradeoff between its subjective
expected utility and its utility dispersion across certain states that a decision maker
perceives as relatively less ambiguous. The double integral on the right hand side of
equation (1) can be interpreted as a subjective measure of ambiguousness of an act.
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