Journal of Economic Theory 105, 531539 (2002)
doi:10.1006jeth.2001.2824
Completing the State Space with Subjective States 1
Emre Ozdenoren
Department of Economics, University of Michigan, Ann Arbor, Michigan 48109-1220
emreoumich.edu
Received May 25, 2000; final version received February 8, 2001;
published online November 7, 2001
We define an opportunity act as a mapping from an exogenously given objective
state space to a set of lotteries over prizes, and consider preferences over opportunity acts. We allow the preferences to be possibly uncertainty averse. Our main
theorem provides an axiomatization of the maxmin expected utility model. In the
theorem we construct subjective states to complete the objective state space. As in
E. Dekel et al. (Econometrica, in press), we obtain a unique subjective state space.
We also allow for preference for flexibility in some of the subjective states and commitment in others. Journal of Economic Literature Classification Number: D81.
2001 Elsevier Science (USA)
Key Words: subjective states; unforeseen contingencies; preference for flexibility;
uncertainty aversion; ambiguity.
1. INTRODUCTION
In a seminal paper, Kreps [9] provides a characterization of preference
for flexibility. As a primitive he takes preferences over opportunity sets,
which are nonempty subsets of prizes, and derives a set of subjective states.
Dekel et al. [5], on the other hand, take as primitive preferences over sets
of lotteries over prizes. Using the richness of this choice set, they derive a
unique subjective state space. 2 , 3
In many decision problems, however, there is a natural objective state
space that we expect the decision maker to use. Indeed in Kreps's [10]
interpretation, objective states correspond to contingencies that are
1
I thank an associate editor, Nabil Al-Najjar, Ramon Casadesus-Masanell, Ehud Kalai,
and especially Peter Klibanoff for helpful comments. All errors are mine.
2
For an unforeseen contingencies interpretation of the model, and a discussion on the
importance of uniqueness of the state space see Dekel et al. [5].
3
In Dekel et al. [5] the decision maker may prefer flexibility in some subjective states and
commitment in others. Gul and Pesendorfer [8], in a closely related framework, also provide
a representation theorem where commitment is valued in some states.
531
0022-053101 35.00
2001 Elsevier Science (USA)
All rights reserved.
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EMRE OZDENOREN
foreseen by the decision maker. The decision maker is also aware that there
may be unforeseen contingencies, and that her preferences may change in
the future. Subjective states reflect this aspect of the problem. Therefore, it
is useful to have a framework that takes the objective state space as part
of the primitives. In contrast to Kreps [9] and Dekel et al. [5], our model
explicitly specifies an objective state space.
Having an objective state space also allows us to consider uncertainty
averse preferences. Ellsberg [6] and many afterward demonstrated that
uncertainty averse preferences seem common and are incompatible with
the standard expected utility theory. 4 In this paper, we use the maxmin
expected utility model, 5 which is a common way to model uncertainty
averse preferences, and derive a set of subjective states within this
framework.
Formally, we define an opportunity act as a mapping from the set of
objective states into the set of lotteries over prizes. We take as primitive
preferences over opportunity acts, and we allow these preferences to be
uncertainty averse.
Our main theorem provides an axiomatization of the maxmin expected
utility model. In the theorem we construct subjective states derived from
preferences to complete the objective state space. As in Dekel et al. [5], we
obtain additivity over subjective states and a unique subjective state space.
We also allow for preference for flexibility in some of the subjective states
and commitment in others. 6
2. NOTATION AND DEFINITIONS
Let 0 be the nonempty set of states of nature and O be an algebra of
subsets of 0. We call 0 as the objective states of nature, and we allow 0
to be of any cardinality. Let B=[b 1 , ..., b n ] be a finite set of prizes. Let 2B
be the set of all (simple) probability distributions on B. Let X be the set
of nonempty subsets of 2B. Elements of X are called opportunity sets. An
opportunity act is a function from 0 into X. A simple opportunity act is
an opportunity act which gives finitely many distinct opportunity sets.
Denote by H* the constant opportunity acts. We sometimes denote the
constant opportunity act that gives the opportunity set x for all | # 0 by x*.
4
See, for example, Camerer and Weber [1] for a survey.
Gilboa and Schmeidler [7] first provided an axiomatization of maxmin expected utility
model in an AnscombeAumann framework. In a Savage framework there are two axiomatizations by Casadesus-Masanell et al. [2, 3].
6
Nehring [11] looks at preferences over opportunity acts in a Savage framework. He does
not consider uncertainty averse preferences, and his model allows for only preference for
flexibility. For more on related literature see the survey by Dekel et al. [4].
5
SUBJECTIVE STATES
533
A simple opportunity act h is O-measurable if [| # 0 | h(|) # W ] # O, where
WX. Let H be the set of all O-measurable opportunity acts defined as the
closure of the set of all O-measurable simple acts. 7 The interpretation of these
acts is as follows. First the agent will learn which objective state | has
occurred. Then she will pick a lottery from the set h(|) # X. Finally the
objective uncertainty will resolve and the agent will receive a prize.
For x, x$ # X and for * # [0, 1], let *x+(1&*) x$ be the set of all
;" # 2B such that there exists ; # x and ;$ # x$ with ;"=*;+(1&*) ;$.
This is the natural way to think about taking the convex combination of
two opportunity sets. That is, the convex combination of an element of the
first set and an element of the second set are taken and these are put into
the set of convex combinations, and this procedure is repeated for all
possible pairs. The convex combination of two acts is obtained by taking
the convex combination of opportunity sets for each state. Formally, for h
and g in H and * # [0, 1], define *h+(1&*) g by (*h+(1&*) g)(|)=
*h(|)+(1&*) g(|) for all | # 0.
3. AXIOMS
Suppose p is a binary relation on H. Next we consider axioms on this
binary relation. The first four axioms are standard and are used throughout
the analysis.
Axiom 1 (Weak Order). p on H is a weak order.
p on H induces a relation also denoted by p on X. Specifically, if
x, y # X, x py if and only if x* p y*.
7
The distance between two opportunity sets is defined as follows. Let d denote any distance
on 2B. For any pair x, x$ # X, we define
d(:, x$)# inf d(:, ;)
; # x$
and
e(x, x$)#sup d(:, x$)
:#x
the excess of x over x$. Now, let the distance between x and x$ be defined as
d hausdorff (x, x$)#max[e(x, x$), e(x$, x)].
Finally, we define the distance between any two opportunity acts h and h$ by
d(h, h$)# sup d hausdorff (h(|), h$(|)).
|#0
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EMRE OZDENOREN
Axiom 2 (Continuity). For any h # H, the sets L(h)=[h$ # H | h oh$]
and U(h)=[h$ # H | h$ o h] are both open.
Axiom 3 (Monotonicity).
| # 0, then f p g.
For any f, g # H, if f (|) p g(|) for all
The monotonicity axiom implicitly requires the preference relation to be
state independent. That is the agent believes that her ex post preferences do
not depend on which objective state occurs.
The next axiom rules out preferences with complete indifference.
Axiom 4 (Nondegeneracy).
There exists h, g # H such that h o g.
Next, we introduce an independence condition for opportunity acts.
Axiom 5 (Independence). For any f, g, h # H and * # [0, 1], f pg if
and only if * f +(1&*) h p *g+(1&*) h.
To see how this axiom works, suppose that the decision maker is comparing * f +(1&*) h with *g+(1&*) h, and that an objective state | has
occurred. Then the decision maker is faced with * f (|)+(1&*) h(|) and
*g(|)+(1&*) h(|). Suppose once she learns about her preferences, the
decision maker chooses ; from f (|), ;$ from g(|) and ;" from h(|). Since
she receives ;" with the same probability in both cases, her preferences
should be determined only by how she feels about ; and ;$ which only
depends on f (|) and g(|). Therefore her overall preference should depend
only on how she feels overall about f and g.
Axiom 6 (C-Independence). For any f, g # H, h # H* and * # [0, 1],
f p g if and only if * f +(1&*) h p*g+(1&*) h.
Here, we are relaxing the previous axiom by requiring independence to
hold only when h is a constant act. C-Independence was introduced in
Gilboa and Schmeidler [7], and the idea is that it is easier for the decision
maker to visualize the mixtures of f and g with constant acts, and therefore
she is less likely to violate C-independence than independence.
Axiom 7 (Set-monotonicity). Suppose x, x$ # H* then x$x$ O xpx$.
Set-monotonicity says that, the agent faced with two constant acts
prefers the act that gives her a bigger opportunity set. We say the decision
maker has preference for flexibility if for some acts the preference is strict.
This is the analogue of Kreps's [9] monotonicity axiom in our framework.
Without this axiom the decision maker may prefer commitment in some
subjective states and flexibility in other states.
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SUBJECTIVE STATES
Axiom 8 (Uncertainty Aversion). For any f, g # H and * # [0, 1],
ftg implies * f +(1&*) g p f.
This is Gilboa and Schmeidler's uncertainty aversion axiom, and it
roughly says that the decision maker likes hedging. 8
4. A REPRESENTATION THEOREM
First we define additive maxmin expected utility (AMMEU) representation of p . In this representation, ex ante, the decision maker is a maxmin
expected utility maximizer. She has subjective states determining her
preferences in the implicitly modelled ex post stage which are in the expected
utility form, and which enter the representation additively. Ex post for each
subjective state the decision maker picks the best lottery from the set that
is available to him, and her utility at any objective state is computed by
adding her expected utility over all subjective states from each such lottery
using some measure. The formal definition follows:
Definition 4.1. An AMMEU representation of p, (C, S, +, U ), is a set
C of finitely additive probability measures on O, a set S, a finitely additive
measure + with full support on S, and a continuous utility function
U: 2B_S Ä R such that V is continuous and represents p where
V( f )=min
P#C
| _|
0
&
sup U(;, s) d+ dP
S ; # f (|)
and where U(. , s) is an expected-utility function, i.e.,
U(;, s)= : U(b, s) ;(b).
b#B
AMMEU allows for uncertaintyuncertainty aversion about the objective
states while simultaneously allowing for preference for flexibility or commitment. The complete state space is 0_S, and this is the sense in which
state space is completed with subjective states. Now, we are ready for the
main representation theorem:
Theorem 4.2. p satisfies weak order, C-independence, continuity, monotonicity, uncertainty aversion and non-degeneracy if and only if p has an
AMMEU representation, (C, S, +, U ) and the set C is unique. If in addition
to the previous axioms p also satisfies set-monotonicity then the measure +
8
See Casadesus-Masanell et al. [2, Section 4.2] for a detailed discussion of this axiom.
536
EMRE OZDENOREN
is positive. If (C, S, +$, U$) is another AMMEU representation of p , then,
there exists a>0, a$>0, and b such that +$=a$+ and U$=aU+b.
See Section 7 for the proof of Theorem 4.2. It is clear from Theorem 4.2
that by assuming different versions of the independence axiom, we can
obtain different representation theorems. In particular by replacing the Cindependence axiom with the independence axiom, we would obtain a
representation theorem where maxmin expected utility is replaced with
expected utility. 9 See the remark at the end of Section 7 for the proof of
this claim.
So far we only stated uniqueness results for a fixed subjective state space.
Next we look at the question of uniqueness of the subjective state space.
5. UNIQUENESS OF THE SUBJECTIVE STATE SPACE
In this section we provide a uniqueness result for the subjective state
space which follows from a similar result in Dekel et al. [5]. Given an
AMMEU representation (C, S, +, U ), define os as,
; os ;$ U( ;, s)>U(;$, s).
Let [ os : s # S] be the subjective state space for (C, S, +, U ).
We need the following definition to state the result:
Definition 5.1. Given a sequence [ ok ] of expected-utility preferences
over 2B, we say that o is a limit of the sequence if it is a nontrivial
expected utility preference such that
; o ;$ O _K such that ; ok ;$, \kK.
The closure of a set of expected utility preferences adds to the set all such
limit points.
We focus only on subjective state spaces with relevant states. A subjective state s is relevant if for any neighborhood of s, there are two constant
acts among which the agent is not indifferent, even though the particular
menus that these constant acts yield are evaluated identically at all subjective states that are not in this neighborhood. 10
9
Similarly if we assume comonotonic independence rather than C-independence, we would
obtain Choquet expected utility. For a discussion of the comonotonic independence axiom
and Choquet expected utility see Schmeidler [12].
10
For a detailed discussion of the topology on the set of all expected-utility preferences see
Dekel et al. [5].
SUBJECTIVE STATES
537
Definition 5.2. Suppose (C, S, +, U ) and (C$, S$, +$, U$) are AMMEU
representations of a preference relation p (where S and S$ contain only
relevant states). We say that these representations have essentially equivalent subjective state spaces if the closures of [ os : s # S] and [ os : s # S$]
are the same.
Theorem 5.3. Suppose p has an AMMEU representation, then all
AMMEU representations of p have essentially equivalent subjective state
spaces.
Proof.
Follows from Theorem 1 part C of Dekel et al. [5]. K
6. CONCLUSION
Dealing with unforeseen contingencies has been problematic in decision
theory while it is widely agreed that it is an important aspect of most decision making problems. In this paper following Kreps [10] we tackle this
problem by allowing for objective states (or foreseen states) as well as subjective states (or unforeseen states) in the representation. Formally we do
this by generalizing the model by Dekel et al. [5] to opportunity acts. Our
representation allows for different attitudes towards uncertainty about the
objective states. This approach also pins down the subjective state space
uniquely, which is a necessary condition for any fruitful application of such
models. The representation theorem in this paper is general enough to
allow for both preference for flexibility and commitment.
7. PROOFS
We only prove sufficiency part of Theorem 4.2. Necessity is easy to show,
and uniqueness follows from the uniqueness part of Lemma 7.1. Let Y be
the set of closed and convex sets in X.
Lemma 7.1. There exists a linear function v: Y Ä R such that for all
x, x$ # Y, x px$ iff v(x)v(x$). For * # [0, 1],
v(*x+(1&*) x$)=*v(x)+(1&*) v(x$).
v is unique up to affine transformations and continuous.
Proof. Our axioms restricted to constant acts imply the axioms of
Dekel et al. [5]. Therefore this lemma follows from their Proposition 2. K
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EMRE OZDENOREN
Lemma 7.2.
Proof.
For every x # X, xtconv(x) and xtcl(x).
Similarly follows from lemmas 1 and 2 in Dekel et al. [5].
K
By the previous lemma we can restrict our attention to Y.
Lemma 7.3. Given a v: Y Ä R from Lemma 7.1, there exists a unique
J: H Ä R such that:
1. f pg iff J( f )J( g), for all f, g # H.
2. for f =x* # H*, J( f )=v(x).
Proof. On H*, J is uniquely defined by 2. We extend J to all acts
as follows. Given f # H, by monotonicity there exits x, x # X such that
x p f px. By continuity there exists a unique * # [0, 1] such
that f t*x +
(1&*) x. Define J( f )=J(*x +(1&*) x ). By construction J satisfies 1,
hence is also unique.
By the nondegeneracy axiom there exists x 1 , x 2 # X such that x 2 o x 1 .
Choose a specific v: X Ä R such that, v(x 1 )=&2 and v(x 2 )=2. Denote by
B the space of all bounded O measurable real valued functions on 0. Let
B 0 be the space of functions in B which assume finitely many values. For
# # R, let #* # B 0 be the constant function on 0 with the value #.
Lemma 7.4.
There exists a functional I: B 0 Ä R such that:
(i)
For all f # H, I(v b f )=J( f ). (Hence I(1*)=1.)
(ii)
I is monotonic (i.e., for a, b # B 0 , ab O I(a)I(b)).
(iii)
I is superadditive and homogenous of degree 1.
(iv) I is C-independent (i.e., for any a # B 0 and # # R, I(a+#*)=
I(a)+I(#*)).
Proof. The proofs follow by using arguments that are similar to those
in Gilboa and Schmeidler's proof of Lemma 3.3. K
By a fundamental lemma from Gilboa and Schmeidler [7, Lemma 3.5],
there exists a closed and convex set C of finitely additive probability
measures on O such that for all b # B,
I(b)=min
P#C
| b dP.
(7.1)
SUBJECTIVE STATES
539
Restricting our axioms to constant acts, from Dekel et al. [5, Theorem
4], we know that there is a finitely additive measure +, a set S and
functions U: 2B_S Ä R such that for all x # X,
v(x)=
|
sup U( ;, s) d+,
(7.2)
S ;#x
where each U(., s) is an expected utility function. If preferences also satisfy
set-monotonicity + is a positive measure.
The desired result follows by combining 7.1 and 7.2.
We claimed in Section 4 that by replacing the C-independence axiom
with the independence axiom, we obtain a representation where maxmin
expected utility is replaced with expected utility. The proof of this claim
follows by recognizing that independence axiom makes I additive, in this
case there is a finitely additive probability measure P on O such that for
all b # B, I(b)= b dP, and the result follows by combining this with 7.2. K
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