A GENERALIZED NOTION OF PROBABILISTIC
SOPHISTICATION THAT APPLIES TO AMBIGUOUS
BELIEFS
1
For the latest version click on the title
FATEMEH BORHANI 2
THE PENNSYLVANIA STATE UNIVERSITY
fzb5027@psu.edu
11/14/2015
Abstract. Capturing the beliefs of a decision maker who cannot assign precise odds to every event with a set of priors, rather than with a single prior,
has become the dominant approach in the ambiguity literature. Examples of
such representations include “maxmin expected utility” axiomatized by Gilboa
and Schmeidler [1989] and the “smooth model of ambiguity” axiomatized by
Klibanoff et al. [2005]. Each of these preference representations, and most
other representations in the ambiguity literature, combines such multiple-prior
beliefs with some specific functional form for tastes. This two-part structure
suggests that the specific classes of preferences corresponding to those various
representations are all sub-classes of a more fundamental class: the class of all
preferences induced by multiple-prior beliefs.
Theorem 1 of this study provides necessary and sufficient conditions , in a
version of Anscombe and Aumann’s framework, for a preference relation to be
consistent with beliefs in the form of multiple priors. A controversial axiom,
state monotonicity, is not imposed on the preference relation. It is shown in
theorem 6 that, when beliefs are modeled using multiple priors, the set of
acts over which probabilistic sophistication has a force (clear acts) typically is
strictly larger than the set of unambiguous acts studied by Epstein and Zhang
[2001]. A behavioral definition of such acts is provided.
1. Introduction
Expected utility is the paradigm example of a theory in which preferences under
uncertainty reflect the interaction of two, conceptually distinct, features: tastes
and beliefs. In response to objections that have been raised to specific features of
expected utility—such as the “Allais paradox” and “Ellsberg paradox”—dozens of
generalized decision-theoretic models have subsequently been formulated. Broadly
speaking, the two paradoxes just mentioned address the two respective features
that interact to form preferences. Allais paradox is a criticism of the assumption
that tastes are separable across states of nature, and Ellsberg paradox is a criticism
of the assumption that the representation of beliefs need not distinguish between
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2
This research began during my visit to the Center for Mathematical Studies in Economics
and Management Science at Northwestern University. I’m specially grateful to Willemien Kets,
Marciano Siniscalchi and Peter Klibanoff for discussions that introduced me to this field. I gratefully acknowledge the Human Capital Foundation for research support through the Economics
Department at the Pennsylvania State University. This paper has subsequently benefited from
many discussions with Ed Green, Kalyan Chatterjee and Henrique de Oliveira and other faculty
and graduate students in my department.
1
2
“objective” and “subjective” uncertainty. Generalized decision theories that address Allais paradox posit various models of tastes, such as quadratic utility model1
of Chew et al. [1991], but retain the assumption that beliefs are represented by a
single probability measure. Generalized theories that address Ellsberg paradox also
relax the EU assumption regarding beliefs. The most prevalent way to do so, has
been to represent the decision maker’s beliefs by a set (typically not a singleton) of
priors (probability measures).
Let’s consider the class of single-prior models. Even if a preference relation cannot be represented by expected utility model, we cannot rule out that the decision
maker may have beliefs induced by a single probability distribution over the state
space. That is, an expected utility representation has two components; beliefs, represented by a single probability distribution over states; and tastes, represented by a
specific utility function. So the reason expected utility model is violated might have
to do with tastes rather that beliefs. The same would be true of a preference relation that violates any other specific single-prior representation, such as quadratic
utility. Probabilistic sophistication, axiomatized by Machina and Schmeidler [1992,
1995], isolates tastes from beliefs and provide a general test of whether there exists
some single-prior representation. Those researchers asked: what behavioral axioms2 characterize all preferences that can be represented by a single-prior model,
allowing that any representation of tastes that respects stochastic dominance may
be adopted? Machina and Schmeidler used the term probabilistically sophisticated
to refer to a preference relation that can be so represented, and they formulated
behavioral axioms that they proved are necessary and sufficient for probabilistic
sophistication.
Now let’s consider the class of multiple-prior representations. The goal of this
research is to accomplish for multiple-prior models what Machina and Schmeidler
have done for the single-prior models. Here I am going to ask: what axioms characterize all preferences that can be represented by a multiple-prior model, allowing
any representation of tastes that respects first order stochastic-dominance condition whenever it is applicable. I will provide a formal definition of this generalized
notion of probabilistic sophistication in the next section after introducing the setup
and notations used in this study. Roughly speaking, a preference relation satisfies
generalized probabilistic sophistication, if beliefs are representable with a set of
priors and when comparing two acts, each one with a single implied lottery, the
implied lotteries are a sufficient statistic for the preference between the respective
acts. Please note that, with multiple priors, in general each act implies a set of
lotteries and not every act implies a single lottery.
1.1. Literature review. In the ambiguity literature, several criteria have been
offered for reasonable behavior. Gilboa et al. [2010] propose two distinct criteria,
which they develop in Anscombe and Aumann’s framework. Using these criteria, they draw a connection between the representation theorems of Bewley [2002]
and Gilboa and Schmeidler [1989]. Both of these representation theorems assume
Anscombe and Aumann’s axiom of state monotonicity.
1
P
P
u(x, p) = xi ∈X xj ∈X T (xi , xj )pi pj
2
An important component of the idea of a behavioral axiom is that it is formulated in a
restricted language in which preferences among acts can be discussed, but in which probability
measures over the state space cannot be discussed explicitly. In particular, it is impossible to state
within that language that a preference relation is induced by a single probability distribution.
3
Another important research project is due to Epstein and Zhang [2001]. They
model a decision maker who assigns determinate probabilities to some, and not
all, events. Such events are called unambiguous. Acts measurable with respect to
unambiguous events are called unambiguous acts. Epstein and Zhang characterize
preference relations that respect first order stochastic dominance when comparing unambiguous acts. Their characterization is applicable only to preferences
that satisfy Savage’s axiom P 3, which is the counterpart in Savage’s framework of
Anscombe and Aumann’s state monotonicity axiom.
State monotonicity and P 3 share a common implication which is state separability of preferences. State separability will be defined in section 6. It has become a
controversial assumption to impose in contexts where beliefs are not representable
with a single prior. An example to be presented in section 6 will show why it is
problematic. It will also be shown there that state separability cannot be derived
from the axioms used in this paper’s characterization.
Another major difference between the present study and Epstein and Zhang
[2001] is that here, respect for stochastic dominance is derived for comparisons
within a class of acts that is larger, and strictly larger in general, than Anscombe
and Aumann’s counterpart of Epstein and Zhang’s unambiguous acts, although it
coincides with that class when the the theory is specialized to the Savage framework
that they consider.
Chew and Sagi [2007] is concerned with collections of events that they call small
worlds. They provide sufficient conditions such that conditional on an event that is
the union of all events in some small world, the decision maker satisfies probabilistic
sophistication on acts that are measurable with respect to the events within that
small world. This can be considered as another direction of generalizing Epstein
and Zhang [2001]. Similar to present study, Chew and Sagi [2007] relax Savage’s
P 3.
2. Formal setting
Adopting the traditional setting of Anscombe and Aumann [1963], I consider
a finite state space S = {1, ..., S} and a set of prizes X ⊆ R.3 Set X can be
interpreted as monetary prizes. Let ∆(X) be the set of probability distributions
over X with finite support. Such a probability distribution over X is called a roulette
lottery. Generic roulette lotteries are shown by lowercase letters f, g, . . . . An act
is a function from set of states to the set of roulette lotteries. Generic acts are
shown by capital letters F, G, ... The interpretation of act F is that ,if state s is
realized, then the decision maker receives a prize according to distribution F (s).
Constant acts are acts with the same roulette lottery in any state. According to
standard abuse of notation, a constant act is shown by its objective lottery. A
preference relation on acts is shown by . As usual, ∼ and respectively denote
the symmetric and asymmetric parts of . I end this section by definitions of
implied lottery, set of implied lotteries and affine combination of acts.
3The condition X being a subset of real numbers can be relaxed, however it wouldn’t make
any conceptual change.
4
Definition (implied lottery). Act F and prior p ∈ ∆(S) imply lottery mF,p ∈ ∆(X)
defined for any x ∈ X as
X
mF,p (x) =
p(s)[F (s)(x)]
s∈S
Let MF,P be the set of implied lotteries by F and P ⊆ ∆(S):
MF,P = {mF,p |p ∈ P }
If given P ⊆ ∆(S) the set of implied lotteries of act F is a singleton then mF,P
is used to show that lottery.
Definition (affine combination of acts). For any α ∈ R affine combination of two
acts is defined as below:
[αF + (1 − α)H](s) = αF (s) + (1 − α)H(s)
Please note that, although convex combination of acts is always an act, an affine
combination might not be an act if the constraint that a probability must lie between
zero and one were violated.
3. Probabilistic Sophistication and Generalized Probabilistic
Sophistication
If a preference relation cannot be represented by expected utility representation,
we cannot rule out that the decision maker may have beliefs in the form of a
single probability distribution over the state space. That is, an expected utility
representation has two components; beliefs, represented by a single probability
distribution over states; and tastes, represented by a specific utility function. So
the reason expected utility model is violated might have to do with tastes rather
that beliefs. The same would be true of a preference relation that violates any
other specific single-prior representation, such as quadratic utility. Probabilistic
sophistication, axiomatized by Machina and Schmeidler [1992, 1995], isolates tastes
from beliefs and provide a general test of whether there exists some single-prior
representation.
Definition (Probabilistic Sophistication). A preference relation is said to satisfy
probabilistic sophistication if there exists a probability distribution over the state
space, p ∈ ∆(S), and a continuous real valued function V on roulette lotteries (not
necessarily expected utility) that is strictly monotone with respect to first order
stochastic dominance, such that for any acts F and G,
F G ⇔ V (mF,p ) ≥ V (mG,p )
Probabilistic sophistication expresses the idea that a decision maker in comparison between any two acts, treats objective and subjective uncertainty equally and
therefore evaluates each act according to its implied lottery.
The Ellsberg paradox represents a systematic violation of probabilistic sophistication that intuitively suggests multiple-prior beliefs. That is even though the
decision maker believes that the realized state of nature is governed by a probability distribution, she is uncertain which distribution it is. Parallel to what was
noted earlier about single-prior representations, there are numerous multiple-prior
models. Just because a preference relation is incompatible with one particular such
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representation, does not mean that it is incompatible with all multiple-prior representations. The goal of the present research is to identify when beliefs of a decision
maker are compatible with some multiple prior representation that respects first
order stochastic dominance where applicable.
Definition (Generalized Probabilistic Sophistication). A preference relation is said
to satisfy generalized probabilistic sophistication if there exists a set of probability
distributions over state space P ⊆ ∆(S) and a mixture continuous real valued
function V on roulette lotteries (not necessarily expected utility) that is strictly
monotone with respect to first order stochastic dominance such that for any pair
of acts F and G such that the set of implied lotteries with respect to P for each is
a singleton.
F G ⇔ V (mF,P ) ≥ V (mG,P )
3.1. Clear acts. As mentioned in the introduction, if it were possible to observe
directly a decision maker’s belief set, we would expect her preferences to satisfy
probabilistic sophistication on the class of acts that imply a single lottery. We do
not have direct access to beliefs so we cannot identify such acts directly. Here I
propose a behavioral definition of a class of acts, which I call clear acts. Later, in
theorem 3, I prove that an act implies a single lottery if and only if it is a clear act.
Definition (prize substitution). A prize substitution function is a function µ : X →
X.
Definition (prize substitution in a lottery). Let m be a lottery and µ a prize substitution map. mµ is the lottery defined below:
X
mµ (x) =
m(y)
{y∈X|µ(y)=x}
Definition (prize substitution in an act). Consider act F and prize substitution
function µ. Define F µ to be the act defined as:
[F µ ](s) = [F (s)]µ
Definition (indistinguishable acts). Acts F and G are indistinguishable if for any
prize substitution function µ, F µ ∼ Gµ .
Definition (clear act). Act F is a clear act if the exists a constant act g such that
F and g are indistinguishable.
c
Notation F = g (read F and g are clearly equivalent) is used if and only if act F
and constant act g are indistinguishable.
The intuition behind this definition is that, if a preference relation satisfies generalized probabilistic sophistication and an act implies a single lottery, then that
act under any prize substitution should still imply a single lottery. Later, theorem
3 proves that in fact, if act F is indistinguishable from constant act g, not only
does act F imply a single lottery but also the implied lottery is g.
3.2. Axioms. This section states the axioms to be used.
Axiom 1 (Ordering): is a complete, reflexive and transitive binary relation on
clear acts.
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Axiom 2 (continuity of preference relation on constant acts): If constant acts
F, G, H satisfy F H G then there exists α ∈ (0, 1) such that H ∼ αF +(1−α)G.
Axiom 3 (FOSD monotonicity of preference relation restricted to constant acts):The
restriction of to constant acts satisfies strong monotonicity with respect to first
order stochastic dominance.
c
c
Axiom 4 (convex combination preserves clearness): If F1 = g1 , F2 = g2 , θ ∈ (0, 1)
c
then θF1 + (1 − θ)F2 = θg1 + (1 − θ)g2 .
Axiom 5 (convex combination does not create clearness): If F and G are clear
acts and G = θF + (1 − θ)H for some θ ∈ (0, 1) and act H, then H is also a clear
act.
4. Representation of generalized probabilistic sophistication
A sequence of theorems are stated in this section and the next two sections.
Proofs are presented in the appendix.
Theorem 1 is about necessary and sufficient conditions for generalized probabilistic sophistication:
Theorem 1. The following conditions are equivalent:
(i) Preference relation satisfies axioms 1-5
(ii) There exists a unique maximum convex set P ⊆ ∆(S) and and a continuous
function V : ∆(X) → R which is strictly monotone with respect to first order
stochastic dominance such that, for any pair of acts that with respect to P each
imply a single lottery,
F G ⇔ V (mF,P ) ≥ V (mG,P ).
Theorem 2 spells out the relation between this generalized notion and Machina
and Schmeidler [1995] notion of probabilistic sophistication. Clear acts play a
crucial role in the connection between the two.
Theorem 2. The following two statements are equivalent:
• satisfies probabilistic sophistication.
• satisfies generalized probabilistic sophistication and all acts are clear.
5. Unambiguous Acts vs. Clear Acts
Theorem 3 justifies the intuition behind the definition of clear acts.
Theorem 3. If satisfies generalized probabilistic sophistication with P ⊆ ∆(S)
c
being the belief set from theorem 1, then F = g if and only if MF,P = {g}.
Unambiguous acts are defined and well studied in the ambiguity literature 4.
Theorem 3 may give the impression that clear and unambiguous acts are the same.
In the following, I will show that this is not correct and will clarify the relation
4For example look at Klibanoff et al. [2011]
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between the two. To begin, I need formally to define unambiguous events and unambiguous acts. To define unambiguous events, I need to define bets:
Definition (bet on an event). Let m, n be two lotteries and E ⊆ S an event. Then
the following act is a bet on event E
(
m if s ∈ E
F =
n if s ∈
/E
Note that by definition, a bet on an event is also a bet on its complement.
Definition (unambiguous event). Event E ⊆ S is called unambiguous iff all bets on
E are clear.
Theorem 4. The set of unambiguous events is a λ-system.5
Next theorem shows that the definition of an unambiguous event satisfies the
intuition that all priors agree on the probability of such events.
Theorem 5. An event is unambiguous iff all priors from theorem 1 agree on its
probability.
Now that unambiguous events are defined, I can define unambiguous acts:
Definition (unambiguous act). An act is unambiguous iff it is measurable6 with
respect to unambiguous events.
Theorem 6. Every unambiguous act is clear.
Even though every unambiguous act is clear, not every clear act is unambiguous.
The following is an example of an ambiguous clear act.
Example (A clear act that is not unambiguous): Consider state space and priors
as defined below
7 6 2
4 3 2
S = {a, b, c}, P = co{( , , ), ( , , )}
9 9 9 15 15 15
where co is the convex hull operator and each element in the belief, shows probability
of states a, b, c respectively. Let’s consider act F defined by


if a
(1, $0)
1
2
F = ( 3 , $0; 3 , $1) if b

 1
( 2 , $0; 12 , $1) if c
5K ⊂ 2S is a λ-system iff
• S∈K
• A ∈ K ⇒ Ac ∈ K
• A ∩ B = ∅ and A, B ∈ K ⇒ A ∪ B ∈ K
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the set of unambiguous events is a λ-system which in general is not an algebra. Nonetheless
a probability measure can be defined on λ-systems. For more details look at Epstein and Zhang
[2001]
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The set of implied lotteries by act F is
1
4
MF,P = {( , $0; , $1)}.
5
5
This is a singleton and by theorem 3, F is a clear act. Please note that S and
∅ are the only events that all priors agree upon their probability and hence are
unambiguous. Hence constant acts are the only unambiguous acts. Therefore F is
non an unambiguous act. Nonetheless F is clear.
The following theorem states that, if a clear act maps every state to a deterministic lottery, then it must be an unambiguous act.7
Theorem 7. The set of clear acts with deterministic lotteries is the set of unambiguous acts.
This theorem shows that generalized probabilistic sophistication nests the notion
of probabilistic sophistication in Epstein and Zhang [2001].
6. State monotonicity, an axiom that is avoided
The state monotonicity axiom of Anscombe-Aumann has been a standard assumption in the ambiguity literature. Machina [2009] provides an example of
counter intuitive preferences resulting from tail-separability, a consequence of this
axiom. Bommier [2015] also argues, and shows in an example, that state monotonicity is not as innocuous as it may seem. After stating state monotonicity, I will
next present and discuss Bommier [2015]’s example.
Axiom 5 (state monotonicity): Suppose acts F and G are such that in any state
s, F (s) G(s) then F G.
Example (Bommier-2015): Consider three urns. Each is filled in a specific manner. Urn one N1 black balls and N1 white balls. Urn two contains N2 red balls
and N2 green balls. Urn three is constructed by putting all the balls from urn
one and urn two into a third urn. Numbers N1 and N2 are the only unknowns.
Therefore a state of nature can be shown by a pair of numbers (N1 , N2 ). Suppose N1 and N2 can take values from N = {1, . . . , 1000}.Therefore state space is
S = N 2 . In Bommier’s thought experiment, the decision maker has to choose one
of the three urns. A ball will be drawn from the chosen urn. The color of the
drawn ball will determine her payoff. payoff(W ) is the payoff if the drawn ball is
white. Payoffs of other ball colors are shown in a similar manner. Let Fi be the act
associated with choosing urn i. Acts F1 is a constant act because the lottery of this
is m1 = ( 21 , payoff(W ); 12 , payoff(B)) in any state (N1 , N2 ). Similarly act F2 is also
a constant act with lottery m2 = ( 12 , payoff(G); 12 , payoff(R)) in any state (N1 , N2 ).
Act F3 is not a constant act:
N1
N2
F3 (N1 , N2 ) =
m1 +
m2
N1 + N2
N1 + N2
Assume payoffs satisfy
7A lottery that assigns probability one to one prize
9
payoff(W ) < payoff(G) < payoff(R) < payoff(B).
Suppose payoffs are chosen in a way that makes the decision maker indifferent
between urns one and two.
Suppose that the decision maker’s risk preferences satisfy the condition that,
for any two roulette lotteries between which the decision maker is indifferent, she
is indifferent between any convex combination of the two as well. Then, state
monotonicity implies that this decision maker is indifferent between act F3 and the
other two acts, because in any state, the lottery of act F3 is a convex combination
of lotteries of the other two acts. However, the theories with which this paper is
concerned, are intended to model decision makers who are more averse to unknown
probabilities than known probabilities. In the context of such theories, it is most
natural to assume that the decision maker strictly prefers urns one and two to
urn three. But to be consistent with this assumption, state monotonicity must be
avoided.
Here I suggest a preference representation that evaluates urn one and urn two
as being equally good and as being strictly preferred over urn three.
P = ∆(S),
U (F ) =
min E(m) − [ max var(m) −
m∈MP,F
m∈MP,F
min var(m)]
m∈MP,F
Consider payoffs to be
payoff(W ) = $1,
payoff(G) = $2,
payoff(R) = $4,
payoff(B) = $5
Then the set of implied lotteries for each of these acts are given by:
MF1 ,P = {m1 },
MF3 ,P = {
X
N ∈N 2
p(N )[
MF2 ,P = {m2 }
N2
N1
m1 +
m2 ]|p ∈ P }
N1 + N2
N1 + N2
As mentioned before, F1 and F2 each one implies a unique lottery. Among lotteries
implied by F3 , the maximum and the minimum variances belong to 1000
1001 m1 +
1
1000
1
8
9
1001 m2 ' m1 and 1001 m1 + 1001 m2 ' m2 respectively.
U (F1 ) = E(m1 ) = 3, U (F2 ) = E(m2 ) = 3,
U (F3 ) ' minm∈MF3 ,P E(m) − [var(m1 ) − var(m2 )] ' 3 − [4 − 1] = 0
Therefore according to this preference relation, urns one and two are equally good
and strictly preferred over urn three. As explained before, being indifferent between
urns one and two and strictly preferring them over urn three requires the state
monotonicity axiom to be violated. Therefore the preference relation defined by
this representation violates state monotonicity. However, it satisfies all the axioms
in this study. This proves that these axioms does not imply state monotonicity.
Theorem 8. Axioms 1-4 do not imply axiom 5 (state monotonicity).
The state monotonicity axiom is defined in the Anscombe and Aumann setting.
Its counterpart in Savage setting is Savage’s P 3.
Definition (null event). Event E ⊆ S is null iff any pair of acts F and G such that
for any s ∈
/ E, F (s) = G(s) satisfy F ∼ G.
8m
F3 ,p
9m
F3 ,p
for degenerate p ∈ P that puts all the probability on (1000, 1))
for degenerate p ∈ P that puts all the probability on (1, 1000))
10
Call an event non-null if it is not null.
Axiom 6 (Savage’s P 3): For all non-null events E and for all x, y ∈ X and Savage
act F ,
(
(
δy
if s ∈ E
δy
if s ∈ E
δy δx ⇔
F (s) if s ∈
/E
F (s) if s ∈
/E
This axiom imposes state independence and state separability of preferences. By
state independence, it is meant that preferring one prize over another is independent of the common event in which they accrue. The axioms used in this study
impose state independence on preferences over clear acts. In addition to state independence, both state monotonocisity and also P 3 impose state separability of
preferences, defined as:
Definition (state separability). For any non-null state s∗ and any Savage acts F, G
and prizes x, y ∈ X,
(
δy
if s = s∗
F (s) if s 6= s∗
(
δx
if s = s∗
⇔
F (s) if s 6= s∗
(
(
δy
if s = s∗
δx
if s = s∗
∗
G(s) if s 6= s
G(s) if s 6= s∗
That is, if two Savage acts differ only in one state, then the preference between
them is only dependent on their respective prizes in this state.
In the preceding example, state separability was violated by a preference relation
over acts that are not Savage acts. One may be concerned that state separability
might be entailed by axioms 1-4 over preferences among Savage acts. Here I am going to address that concern by giving an example of a preference relation satisfying
axioms 1-4 but violating state separability of preferences even among Savage acts.
Since P 3 implies state separability, this example also shows that P 3 is not provable
from axioms 1-4. Recall that Epstein and Zhang [2001] theorem is contingent on P 3
holding. Thus, the next example also shows that theorem 1 strictly extends that of
Epstein and Zhang [2001], even in the Savage framework withing which they work.
Consider S = {s1 , s2 } and Savage acts G, H defined by:
(
(
δ1 if s1
δ0 if s1
G=
, H=
δ0 if s2
δ0 if s2
State separability requires that G should be preferred to H. Let preference
relation in this example be induced by the same utility funcstion as in previous
example. Contrary to state separability, this preference relation ranks H above G.
Thus we have proved:
Theorem 9. Axioms 1-4 do not impose state separability, even for preferences
among Savage acts. Savage’s P 3 is not implied by axioms 1-4.
7. Concluding remarks
Theorem 1 has provided necessary and sufficient conditions for a preference relation to be consistent with beliefs in the form of multiple priors. In the axiomatization, state monotonicity is not imposed on the preference relation. It is shown
that, when beliefs are modeled using multiple priors, the set of acts over which
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probabilistic sophistication has a force (clear acts) typically is strictly larger than
the set of unambiguous acts. A behavioral definition of such acts is provided.
It is important to note that characterisations of generalized probabilistic sophistication determines a unique maximum set of possible priors (theorem 1) and does
not claim a unique set of priors. This cannot be avoided when axioms have a bite
only on clear acts. To see this note the next example.
Example: Let state space and set of prizes to be
S = {s1 , s2 , s3 }
X = {0, 1}
Consider P = ∆(S). For any p ∈ P , probability of si is shown by pi . Since there
are only two prizes, each lottery can be shown by its probability of 0. For a generic
act F the set of implied lotteries is
MF,P = {p1 [F (s1 )(0)] + p2 [F (s2 )(0)] + (1 − p1 − p2 )[F (s3 )(0)]|p ∈ P }
If act F implies a single lottery, then MF,P is a singleton and mF,p = mF,q for
all p, q ∈ P . This is equivalent to
(p1 − q1 )[F (s1 )(0)] + (p2 − q2 )[F (s2 )(0)] + (q1 + q2 − p1 − p2 )[F (s3 )(0)] = 0
This property should hold for all choices of p, q from P = ∆(S). Consider p =
(0.3, 0.7, 0), q = (0.7, 0.3, 0). For this pair, above equation is satisfied if and only if
F (s1 )(0) = F (s2 )(0). Now consider another pair p = (0.3, 0, 0.7), q = (0.7, 0, 0.3).
For this pair, above equation holds if and only if F (s1 )(0) = F (s3 )(0). Therefore
F (s1 )(0) = F (s2 )(0) = F (s3 )(0). This means that F (s1 ) = F (s2 ) = F (s3 ). Thus
act F is a constant act. I showed that if an act implies a single lottery, it must be
a constant act. Each constant act trivially implies a single lottery. Thus the set of
acts that imply a single lottery is equal to the set of constant acts.
Now suppose a preference relation satisfies axioms 1-4 and theorem 1 results in
P = ∆(S) as the maximum set of priors. P is called the maximum convex set
of priors because there could be a convex set Q ⊂ P such that theorem 1 cannot
differentiate a decision maker with belief set Q and a decision maker with belief
set P . Here I am going to explain that Q = {q ∈ P |q(s1 ) > 0.1} is such a subset
of P . Theorem 3 states that the set of clear acts is equal to the set of acts with a
single implied lottery given P . Therefor to show that theorem 1 cannot differentiate
decision makers with belief sets P and Q, I should show that the set acts implying
a single lottery given G is also the set of constant acts. Above equations and
reasoning of the previous paragraph can be repeated exactly for G to show that the
set of acts implying a single lottery is equal to the set of constant acts.
One might suspect that a unique set of priors in theorem 1 could be achieved
by imposing stronger axioms that were not restricted (as axiom 4 is) to clear acts.
This suspicion is not correct. Siniscalchi [2006] provides an example of a preference
relation that can be represented by maxmin model with a set of priors and also
represented by α-maxmin model using a different set of priors. That is, even though
the axioms of each of this representations are much stronger than axioms 1-4, they
are not sufficient for a unique set of priors.
References
F. Anscombe and Robert J. Aumann. A definition of subjective probability. The
Annals of Statistics, 34:199–205, 1963.
12
T. Bewley. Knightian decision theory: Part I. Decisions in Economics and Finance,
25:79–110, 2002.
Antoine Bommier. A dual approach to ambiguity aversion. 2015.
S.H. Chew, L. G. Epstein, and U. Segal. Mixture symmetry and quadratic utilily.
Econometrica, 59:139–163, 1991.
Soo Hong Chew and Jacob S. Sagi. Small worlds: Modeling attitudes toward sources
of uncertainty. 2007.
Larry G. Epstein and Jiankang Zhang. Subjective probabilities on subjectively
unambiguous events. Econometrica, 69:265–306, 2001.
Itzhak Gilboa and David Schmeidler. Maxmin expected utility with non-unique
prior. Econometrica, 18:141–153, 1989.
Itzhak Gilboa, Fabio Maccheroni, Massimo Marinacci, and David Schmeidler. Objective and subjective rationality in a multiple prior model. Econometrica, 78:
755–770, 2010.
Peter Klibanoff, Massimo Marinacci, and Sujoy Mukerji. A smooth model of decision making under ambiguity. Econometrica, 73:18491892, 2005.
Peter Klibanoff, Massimo Marinacci, and Sujoy Mukerji. Definitions of ambiguous
events and the smooth ambiguity model. Economic Theory, 48(2-3):399–424,
2011.
M. Machina. Risk, ambiguity, and the rank-dependence axioms. The American
Economic Review, 99(1):385392, 2009.
Mark J. Machina and David Schmeidler. A more robust definition of subjective
probability. Econometrica, 60:745–780, 1992.
Mark J. Machina and David Schmeidler. Bayes without Bernoulli: Simple conditions for probabilistically sophisticated choice. Journal of Economic Theory, 67:
106–128, 1995.
Marciano Siniscalchi. A behavioral characterization of plausible priors. Journal of
Economic Theory, 128, 2006.
13
Appendix A. Proofs
A.1. Proof of theorem 1. By Debreu’s theorem, axioms 1-3 are necessary and
sufficient for existence of a continuous utility function on constant acts that is
strictly monotone with respect to first order stochastic dominance. In the following,
I show that a maximal convex set of priors P ⊆ ∆(S) exists such for any act F if
MF,P = {g} then F ∼ g. If such a P exists then trivially if MF,P = {g}, for any
c
choice of prize substitution function µ, MF µ ,P = {g µ } and therefor F = g. Thus
it is natural to begin with investigating clearly equivalent relations. The role of
axiom 4 is to put some regularity on clearly equivalence relations. The rest of the
proof is broken into a series of lemmas.
c
Lemma 10. If F = g then, supp(g) ⊆ supp(F ). Where supp(.) is the support of:
supp(F ) = ∪s∈S supp(F (s)).
Proof. There are two cases, X being finite or infinite. Let’s first consider X be finite.
Let X and 1 be such that for any y ∈ X, 1 ≤ y ≤ X. Suppose for some y 6= X,
y ∈ supp(g) \ supp(F ). Then gy→X ∼ Fy→X = F ∼ g. By transitivity, gy→X ∼
g. But this contradicts axiom 3 because gy→X strictly first order stochastically
dominates g. Now suppose X ∈ supp(g) \ supp(F ). Then gX→1 ∼ FX→1 = F ∼ g.
By transitivity, gX→1 ∼ g. But this contradicts axiom 3 because gX→1 is strictly
first order stochastically dominated by g.
For the case of X being an infinite set. Suppose y ∈ supp(g) \ supp(F ). Choose
z ∈ X such that z > y. Then gy→z ∼ Fy→z = F ∼ g. By transitivity, gy→z ∼
g. But this contradicts axiom 3 because gy→z strictly first order stochastically
dominates g.
Lemma 11. For any clear act there exists a unique constant act such that they are
clearly equivalent.
Proof. Existence comes from definition of clearly equivalent. Now to see uniqueness,
c
c
suppose F = f , F = g but f 6= g. Then there exists y ∈ X such that without loss of
generality f (y) < g(y). Define µ : X → X by
(
1 if x = y
µ(x) =
2 if x 6= y.
c
c
By definition of clearly equivalence F µ = f µ , F µ = g µ . By transitivity f µ ∼ g µ .
But f µ first order stochastically dominates g µ and f µ ∼ g µ is a contradiction to
axiom 3.
c
For any pair of acts F and f that satisfy F = f define ΓF and γF as follows:




F (1)(1) F (2)(1) . . . F (S)(1)
f (1)
 F (1)(2) F (2)(2) . . . F (S)(2) 
 f (2) 




ΓF = 
, γF =  . 

..
..
.
.
.



.
.
.
...
. 
F (1)(X) F (2)(X) . . . F (S)(X)
f (X)
Note that by lemma 11 for any clear act there is only one constant act clearly
equivalent to it. Therefor there is no need to subscript constant act in the above
definitions. I’m looking for probability distributions that give f as unique implied
14
lottery by F . These probability distributions satisfy the following system of linear
equations.
 
Y1
 Y2 
 
ΓF  .  = γF .
 .. 
YS
Next claim shows that any finite number of such linear systems have non-negative
solutions.
Lemma 12. The intersection of non-negative solutions of linear systems of any
finite number of clear acts is non empty.
c
Proof. Consider {Fi = fi }Ii=1 for finite I. Define Γ and γ as follows:




f1 (1)
F1 (1)(1)
F1 (2)(1) . . .
F1 (S)(1)
 f1 (2) 
 F1 (1)(2)
F1 (2)(2) . . .
F1 (S)(2) 




..
..
 .. 


..
 . 


.
.
.
.
.
.




 F1 (1)(X) F1 (2)(X) . . . F1 (S)(X) 
 f1 (X) 




. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .




 f2 (1) 
 F2 (1)(1)
F2 (2)(1) . . .
F2 (S)(1) 




 f2 (2) 
 F2 (1)(2)
F2 (2)(2) . . .
F2 (S)(2) 




 .. 


..
..
..




.
...
.
.
 . 

Γ=
 F2 (1)(X) F2 (2)(X) . . . F2 (S)(X)  , γ =  f2 (X) 




. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .




 . 


..
..
..
.




.
.
.
...

 . 

. . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .




 fN (1) 
 FN (1)(1) FN (2)(1) . . . FN (S)(1) 




 fN (2) 
 FN (1)(2) FN (2)(2) . . . FN (S)(2) 




 . 


..
..
..
.



.
. 
...
.
.
fN (X)
FN (1)(X) FN (2)(X) . . . FN (S)(X)
I should show the following system of linear equations has non negative solutions.
 
Y1
 Y2 
 
Γ .  = γ
 .. 
YS
By Farkas lemma, it suffice to show {W ∈ RN X |W T Γ ≥ 0, W T γ < 0} = ∅. Suppose
W ∈ RN X is such that W T Γ ≥ 0. there exists large enough k ∈ R such that
T
1 ≥ Wk Γ ≥ 0.10 Define K = k1 W .
(
1 if x = y
µi,y (x) =
2 if x 6= y
Define H to be:
10 W T Γ is a vector and by 1 ≥ W T Γ ≥ 0, it is meant that each element of 1 ≥ W T Γ ≥ 0
k
k
k
belongs to [0, 1].
15
H=
N X
X
X
µi,x
K(i−1)+x Fi
+ (1 −
i=1 x=1
NX
X
Ki )δ2
i=1
µ
δ2 is a constant act that pays 2 in any state. Note that all Fi i,x are clear acts.
By axiom 5, if H is an act then it is a clear act. Some simple algebra shows that
H can be rewritten as

T

if x = 1
(K Γ)(s)
T
H(s)(x) = 1 − (K Γ)(s) if x = 2


0
if x =
6 1, 2
Since 1 ≥ K T Γ ≥ 0, H is an act. Define h as follows:
h=
X
N X
X
µi,x
K(i−1)+x fi
i=1 x=1
+ (1 −
NX
X
Ki )δ2
i=1
With some simple algebra we can see that

T

if x = 1
K γ
T
h(x) = 1 − K Γ if x = 2


0
if x 6= 1, 2
c
By definition of clear acts and axiom 6, H = h. Thus h ∈ L and K T γ ≥ 0. Which
intern means that W T γ ≥ 0. Therefore there exists no W such that W T Γ ≥ 0 and
W T γ < 0.
Lemma 13. The intersection of non-negative solutions of linear systems of all
clear acts is non empty and is a set of distributions over states.
Proof. Previous lemma shows for any finite number of linear systems of clear acts,
the intersection of non-negative solutions is non-empty. Remember that constant
acts are clear. Non-negative solutions of linear system of a constant act is ∆(S).
Therefore for any finite number of clear acts that include a constant act, the intersection of nun-negative solutions is a nonempty set of probability distributions.
Additionally this is convex and compact in Euclidean topology. Since this is true
for any finite set of clear acts, the intersection of non-negative solutions for all acts
is non-empty, and forms a convex set of probability distributions.
A.2. Proof of theorem 2.
Proof. Firstly, let satisfies probabilistic sophistication with p ∈ ∆(S) being the
subjective belief. Then trivially satisfies generalized probabilistic sophistication
with {p} being the belief set. The set of implied lotteries of each act is a singleton
given that belief set is a singleton. Now to show each act is clear, I show that
each act is clearly equivalent to its unique implied lottery. Since preferences are
c
probabilistically sophisticated F ∼ mF,p . To show F = mF,p , I need to show for
µ
µ
any µ : X → X, F ∼ [mF,p ] . Given that satisfies probabilistic sophistication,
16
to show F µ ∼ [mF,p ]µ it is enough to show that [mF,p ]µ is the implied lottery of
[mF,p ]µ .
X
mF,p (x) =
p(s)[F (s)(x)]
s∈S
then
[mF,p ]µ (x) = [
X
f (y)] = [
y∈{y∈X|µ(y)=x}
=
X
p(s)[
s∈S
X
X
X
p(s)[F (s)(y)]].
y∈{y∈X|µ(y)=x} s∈S
[F (s)(y)]] =
X
p(s)[F µ (s)(x)] = mF µ ,p
s∈S
y∈{y∈X|µ(y)=x}
Secondly, suppose that satisfies generalized probabilistic sophistication with
P being the belief set and all acts are clear but P is not a singleton. I show there
exists an act such that the set of its implied lotteries is not a singleton. By theorem
3, this is a contradiction to that act being clear. Since P is not a singleton, there
are r, q ∈ P such that for some s∗ ∈ §, r(s∗ ) 6= q(s∗ ). Consider act F defined as
F (s∗ )δ1 and F (s) = δ2 for any s 6= s∗ . Note that mr,F 6= mq,F .
A.3. Proof of theorem 3.
c
Proof. suppose F = f . Then from proof of theorem 1, P is a subset of solutions of
the following system of linear equations:
  


Y1
f (1)
F (1)(1) F (2)(1) . . . F (S)(1)
 F (1)(2) F (2)(2) . . . F (S)(2)   Y2   f (2) 
  


 .  =  . 

..
..
..

.
. . .   ..   .. 
.
.
F (1)(X) F (2)(X) . . .
F (S)(X)
YS
f (X)
Therefore given any q ∈ P , mq,F (x) = f (x) is satisfied for all x ∈ X. Thus
MP,F = f . Now suppose MF,P = f . For any µ : X → X trivially MF µ ,P = f µ . Since
preference relation satisfies generalized probabilistic sophistication, by definition of
c
clear acts, F = f .
A.4. Proof of theorem 4.
Proof. Since any bet on S is a constant act and therefore a clear act, S is an unambiguous event. By definition of a bet, if an event is unambiguous its compliment is
also unambiguous.
Suppose disjoint A, B ⊂ S are unambiguous. I should show A ∪ B is also unambiguous. Trivial from theorem 5.
A.5. Proof of theorem 5.
Proof. Suppose satisfies generalized probabilistic sophistication with P being the
belief set.
Suppose all priors in P agree on the probability of E ⊆ S. Let rE be that
probability. Consider lotteries n1 , n2 and a bet on E.
(
n1 if s ∈ E
F =
n2 if s ∈
/E
Then the set of implied lotteries by this act is
MF,P = {rE n1 + (1 − rE )n2 }
17
which is a singleton. Theorem 3 implies that F is a clear act. Since F is a general
bet on E and is clear, event E is unambiguous.
Now suppose event E is unambiguous. I should show all the priors agree on its
probability. Suppose q1 , q2 ∈ P and q1 (E) > q2 (E). Consider the following bet on
E:
(
δ1 if s ∈ E
F =
δ2 if s ∈
/E
Trivially mF,q1 6= mF,q2 and as a result the set of implied lotteries of this act is not
a singleton which is a contradiction to it being clear by theorem 3.
A.6. Proof of theorem 6.
Proof. suppose act F is an unambiguous act. Then there exists a partition of
unambiguous events {E1 , . . . , EM } such that F is measurable with respect to it.
By theorem 5 probability of each of these events is well defined. Let ri be the
probability of Ei . Then for any set of lotteries {m
P1 , . . . mM } act F defined as
F (s) = mi , s ∈ Ei implies a single lottery: MP,F = i ri mi . By theorem 3, F is a
clear act.
A.7. Proof of theorem 7.
Proof. Let F be a Savage act. Then for any prior q the implied lottery is
X
mF,q (x) =
q(s∗ ) = q({s ∈ S|F (s) = δx })
s∗ ∈{s∈S|F (s)=δx }
If F is clear then by theorem 3 the set of implied lotteries is a singleton.
q1 , q2 ∈ P ⇒ mF,q1 = mF,q2
The above two equations imply that
q1 , q2 ∈ P ⇒ [∀x ∈ X,
q1 ({s ∈ S|F (s) = δx }) = q2 ({s ∈ S|F (s) = δx })]
Let E be a partition on state space that F measurable with respect to. Above
equation implies that all priors agree on the probability of any block of E. By
theorem 5, E is an unambiguous partition. Therefore F is unambiguous.