--- Page 1 ---
CHAPTER 8
Choice Under Uncertainty
In this chapter we turn to models of choice under uncertainty. We consider an
agent who makes choices without fully knowing the consequences of those
choices, and focus on models in which uncertainty can be quantiﬁed and
formulated probabilistically. The most important such model is, of course,
expected utility.
8.1
OBJECTIVE PROBABILITY
There are times when probabilities can be thought to be objective and known,
or observable. This is the case, for example, when outcomes are randomized
according to some known physical device—such as a game in a casino, or a
randomization device used by an experimenter in the laboratory.
We consider two basic environments. In one the primitive objects of
choice are lotteries. In the other, the objects of choice are state-contingent
consumption.
8.1.1
Notation
Let X be a ﬁnite set. We denote by (X) = {p ∈RX : p ≥0;
x∈X p(x) = 1} the
set of all probability distributions over X.
8.1.2
Choice over lotteries
Given is a ﬁnite set X of possible prizes. (X) is the set of all lotteries over X.
We imagine an agent who chooses a lottery. The agent understands that there
is uncertainty over the realization of the lottery: over which prize the lottery
will result in. But the probabilities speciﬁed in the lottery are accurate (or at
least useful) representations of that uncertainty.
We investigate a very basic result on revealed preference in this
environment.


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8.1 Objective probability
115
An expected utility preference ⪰is a binary relation for which there exists
u : X →R such that for all p,q ∈(X),
p ⪰q iff

x∈X
p(x)u(x) ≥

x∈X
q(x)u(x).
The classical axiomatization of expected utility preferences relies on the
independence axiom of decision theory; namely, that for all p,q,r ∈(X) and
all α ∈(0,1], p ⪰q iff αp + (1 −α)r ⪰αq + (1 −α)r.
Most experimental studies refuting the expected utility model are direct
refutations of the independence axiom. The best-known such refutation is
through a thought experiment, known as the Allais paradox. Instead of setting
up a thought experiment, we are going to assume that we are given data on
choices among pairs of lotteries.
The data can be organized into a revealed preference pair ⟨⪰R,≻R⟩, where
each of ⪰R and ≻R are ﬁnite sets. The idea is that an agent makes “binary
choices:” choices from budgets with two alternatives, and these choices deﬁne
⟨⪰R,≻R⟩. We ask when there exists an expected utility preference ⪰such that
for all p,q ∈(X), p ⪰R q implies p ⪰q and p ≻R q implies p ≻q.
The following example demonstrates that observed data can be incompatible
with the expected utility model without directly violating the independence
axiom.
Example 8.1
Let X = {x,y,z}, and consider the rankings: (1,0,0) ≻R
(1/3,1/3,1/3), (0,1,0) ≻R (1/3,1/3,1/3), (0,0,1) ≻R (1/3,1/3,1/3). This
is clearly incompatible with the expected utility model: namely, the rankings
would imply that u(x),u(y),u(z) > u(x)+u(y)+u(z)
3
, which is impossible. However,
there is no direct refutation of the independence axiom. There is, instead, a
refutation of the joint hypotheses of the independence axiom and transitivity.
(Note, incidentally, that the example is not a direct refutation of transitivity
either.) Our aim is to uncover all implications of the joint hypotheses of
independence and transitivity for ﬁnite datasets.
Theorem 8.2
Suppose that each of ⪰R and ≻R are ﬁnite, and without loss of
generality let ⪰R= {(pi,qi)}K
i=1 and ≻R= {(pi,qi)}L
i=K+1. There is an expected
utility preference ⪰such that for all p,q ∈(X), p ⪰R q implies p ⪰q and
p ≻R q implies p ≻q iff there is no λ ∈(L) for which λ({K + 1,...,L}) > 0
and L
i=1 λipi = L
i=1 λiqi.
It is worth remarking on what Theorem 8.2 says. The λ ∈(L) can
be understood as a lottery over lotteries (a “ﬁrst stage” lottery), which is
compounded either with the lotteries {pi}L
i=1 or the lotteries {qi}L
i=1. When λ
is compounded with the lotteries {pi}L
i=1, the reduced lottery is L
i=1 λipi, and
when compounded with {qi}L
i=1, it is L
i=1 λiqi. The condition in Theorem 8.2
requires that, for weights λ, it is impossible that the corresponding p lotteries
are preferred ex-post, yet ex-ante the two compound lotteries are identical.
Figure 8.1 illustrates the condition in Theorem 8.2.


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116
Choice Under Uncertainty
p1
1 – λ
1 – λ
p1  ≥R  q1
p2  >R  q2
p2
q1
q2
=
λ
λ
Fig. 8.1 An illustration of Theorem 8.2.
Proof. The proof is almost a direct translation of Lemma 1.12 to this
environment. Namely, we seek the existence of u ∈RX such that for all
i = 1,...,K, (pi −qi) · u ≥0, and for all i = K + 1,...,L, (pi −qi) · u > 0.
Non-existence of u is therefore equivalent to existence of λ ∈RL
+ such that for
some i ∈{K + 1,...,L}, λi > 0 and L
i=1(pi −qi) = 0. Since L
i=1 λi > 0, we
can normalize λ so that L
i=1 λi = 1.
8.1.3
State-contingent consumption
Many applications of choice under uncertainty in economic models involve
a state-contingent environment. Suppose that there is a ﬁnite set  of states
of the world. A state-contingent consumption bundle is modeled as a vector
in R
+. An agent chooses x ∈R
+: If the state of the world is ω ∈ then the
agent obtains a monetary payment of xω. The vectors x ∈R
+ are referred to as
monetary acts in decision theory.
The focus of our discussion will be expected utility theory with risk
aversion. Consider an agent with a known prior probability measure π ∈()
describing her beliefs over the possible states of the world. We suppose that for
all ω ∈, πω > 0. The prior is known. This means that it is observable; possibly
it has been induced by some experimental design (for example, experiments in
economics often use a randomization device).
Expected utility theory says that the choice of x ∈R
+ is determined by
maximizing a utility function of the form
U(x) =

ω∈
πωu(xω),
where u : R+ →R is a strictly monotonically increasing and concave function.
The function u is a utility function over money, and the concavity of u means
that the agent in question is risk averse.
Suppose we are given data on the behavior of our agent in the market. When
faced with prices p = (pω)ω∈ ∈R
++ and an income I > 0, the agent chooses
x to maximize U(x) over the x that she can afford. It is important to emphasize
what the meaning of prices are here. In the development of the theory, we take


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8.1 Objective probability
117
prices for state-contingent consumption as given, but to have access to such
prices in real data requires the relevant ﬁnancial markets to be complete.1
A dataset is a collection (xk,pk)k∈K of pairs of a consumption xk ∈R
+ chosen
at prices pk ∈R
++ and budget pk · xk. Here K denotes the set {1,...,K}, an
instance of inconsequential notational abuse.
We are given a probability distribution π ∈(). A function u : R+ →R
weakly expected-utility rationalizes a dataset (xk,pk)k∈K if u is strictly
increasing and concave, and if pk · y ≤pk · xk implies that

ω∈
πωu(yω) ≤

ω∈
πωu(xk
ω)
for all k ∈K. We should emphasize that:
I) The prior π is given and known. We are interested in testing whether
the agent behaves according to expected utility with respect to prior
π, rather than any other (possibly subjective) prior.
II) The exercise is restricted to concave utility functions. Concavity
means that the agent is risk averse.
The existence of a known prior π allows us to compute risk-neutral prices,
deﬁned as follows: for k ∈K and ω ∈, let
ρk
ω = pk
ω
πω
.
Risk-neutral prices turn out to be the relevant prices one needs to use to test
for expected utility theory.
We can gain some intuition for how the problem can be solved by
considering the case when u : R+ →R is differentiable. The ﬁrst-order
condition for utility maximization demands that, for any k and ω, we have
πωu′(xk
ω) = λkpk
ω, where λk is the Lagrange multiplier for the maximization
problem in which xk is chosen, and we have assumed an interior optimum.
Using the deﬁnition of risk-neutral prices, we obtain that u′(xk
ω) = λkρk
ω. It
follows from concavity of u, then, that xk
ω > xk
ω′ implies ρk
ω/ρk
ω′ ≤1. This
property can be thought of as downward-sloping demand.
More generally, if we have xk
ω > xk′
ω′ and xk′
ω′′ > xk
ω′′′ then we obtain that
1 ≥u′(xk
ω)
u′(xk′
ω′)
· u′(xk′
ω′′)
u′(xk
ω′′′) = ρk
ω
ρk′
ω′
· ρk′
ω′′
ρk
ω′′′
,
as u is concave, and Lagrange multipliers cancel out. The implication is not
that two pairs of quantities and prices are inversely related, but in some sense
that (xk
ω,xk′
ω′) and (xk′
ω′′,xk
ω′′′) are “inversely related” to (ρk
ω,ρk′
ω′) and (ρk′
ω′′,ρk
ω′′′).
The preceding calculation suggests that one should consider sequences of
pairs (xkiωi,x
k′
i
ω′
i)n
i=1 for which each k appears in ki (on the left of the pair) the
1 Prices p can be calculated (uniquely) from an array of asset prices under two assumptions: that
the markets are complete and that there is no arbitrage.


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118
Choice Under Uncertainty
same number of times it appears in k′
i (on the right). Such sequences will be
called balanced.
A dataset {(xk,pk)}K
k=1 satisﬁes the Strong Axiom of Revealed Objective
Expected Utility (SAREU) if, for any balanced sequence of pairs (xkiωi,x
k′
i
ω′
i)n
i=1
with the property that xkiωi > x
k′
i
ω′
i for all i, the product of relative risk-neutral
prices satisﬁes
n
"
i=1
ρkiωi
ρ
k′
i
ω′
i
≤1.
It is possible to write SAREU so that it rules out certain kinds of cycles. We
use the syntax above to make the comparison with subjective probabilities in
Section 8.2.2 (below) easier.
Theorem 8.3
The following statements are equivalent:
I) {(xk,pk)}K
k=1 is weakly expected-utility rationalizable.
II) For all k ∈K and s ∈S there exist λk > 0 and vk
ω > 0 such that
πωvk
ω = λkpk
ω
and xk
ω > xk′
ω′ implies that vk
ω ≤vk′
ω′.
III) {(xk,pk)}K
k=1 satisfy SAREU.
Remark 8.4
Statement (II) asserts the existence of Afriat inequalities for this
problem. The numbers vk
ω are meant to be the marginal utilities (for money) at
the quantity xk
ω, and λk is meant to be the Lagrange multiplier.
We could equivalently have written Statement (II) as: For all k ∈K and ω ∈
there exist λk > 0 and uk,ω such that
uk,ω′ ≤ul,ω + λl pl
ω
πω
[xk
ω′ −xl
ω],
for all k,l ∈K and all ω,ω′ ∈.
Proof. To begin with, we establish the equivalence between statements (II)
and (III), the more interesting aspect of the proof of Theorem 8.3. The reader
should note the similarities between the construction in the proof that (III)
implies (II), and the construction in Theorem 1.9. The proof of the equivalence
of (I) and (II) is standard, and is included below for completeness’ sake.
Let Y = {xk
ω : k ∈K;ω ∈}; enumerate the elements of Y in increasing order,
as follows:
y1 < y2 < ... < yn.
First we show that (III) implies (II). Suppose that {(xk,pk)}K
k=1 satisﬁes
SAREU. We shall construct a solution to the system
πωvk
ω = λkpk
ω
(8.1)
xk
ω > xk′
ω′ ⇒vk
ω ≤vk′
ω′,
(8.2)


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8.1 Objective probability
119
where λk > 0 and vk
ω > 0 for each k, thereby proving (II).
Let k0 be such that xk0
ω = yn for some ω, and such that
ρk0
ω = max{ρk′
ω′ : xk′
ω′ = yn}.
For k,l ∈K, deﬁne η(k,l) as follows. Let
η(k,l) = max

ρk
ω
ρl
ω′
: xk
ω > xl
ω′

if there is ω,ω′ ∈ such that xk
ω > xl
ω′, and η(k,l) = 0 otherwise.
Note that SAREU implies that
η(k1,k2)η(k2,k3)···η(km,k1) ≤1.
Therefore, for any sequence k1,...,km ∈K, in which ki = kj for i < j, we have
that
η(k0,k1)η(k1,k2)···η(ki−1,ki)η(ki,ki+1)···η(kj−1,kj)η(kj,kj+1)···
η(km−1,km) ≤η(k0,k1)η(k1,k2)···η(ki−1,ki)η(kj,kj+1)···η(km−1,km).
(8.3)
Let λl = 1 whenever l is such that there is ω ∈ such that xl
ω = yn. For any
other l, let
λl = max{η(k0,k1)η(k1,k2)···η(km,l) : k1,...,km ∈K and m ≥0.}
Observe that λl is well deﬁned, as Equation (8.3) implies that one can restrict
attention to a ﬁnite set of sequences k1,...,km; observe also that λl > 0.
The deﬁnition of (λl)l∈K implies that, when yn > xk
ω > xl
ω′ we have
λl ≥η(k,l)λk ≥ρk
ω
ρl
ω′
λk;
(8.4)
and when yn = xk
ω > xl
ω′ we have that, for some ˆω,
λl ≥η(k0,l) ≥ρk0
ˆω
ρl
ω′
≥ρk
ω
ρl
ω′
λk;
where the ﬁrst inequality is by deﬁnition of λl; the second by deﬁnition of η
(as yn = xk0
ˆω > xl
ω′); the third by deﬁnition of k0, and where we have used that
λk = 1 in this case. Therefore we have that λl ≥ρkω
ρl
ω′ λk holds whenever xk
ω > xl
ω′,
regardless of whether xk
ω = yn or xk
ω < yn.
If we let vl
ω = λlρl
ω for all l ∈K and ω ∈, then we have solutions to
system (8.1)–(8.2).
Conversely, suppose that (II) is true. Let there be strictly positive numbers
vk
ω, λk, for ω ∈ and k = 1,...,K, solving system (8.1)–(8.2).


--- Page 7 ---
120
Choice Under Uncertainty
Let (xkiωi,x
k′
i
ω′
i)n
i=1 be a sequence of pairs under the assumptions of SAREU.
Since xkiωi > x
k′
i
ω′
i we have that vkiωi ≤v
k′
i
ω′
i. Hence
1 ≤
n
"
i=1
v
k′
i
ω′
i
vkiωi
=
n
"
i=1
λk′
iρ
k′
i
ω′
i
λkiρkiωi
=
n
"
i=1
ρ
k′
i
ω′
i
ρkiωi
,
as each k appears as ki the same number of times it appears as k′
i in the
sequence, and therefore the λki and λk′
i cancel out. So the data satisfy SAREU,
and thus we establish (III).
We shall now prove that (I) implies (II). Let (xk,pk)K
k=1 be weakly
rationalizable by an expected utility preference with prior π. Let u : R+ →R
be a strictly increasing and concave rationalizing utility function. We can then
consider the ﬁrst-order conditions for a maximizing utility: see, for example,
Theorem 28.3 of Rockafellar (1997) for a formulation that does not require
u to be smooth. The ﬁrst-order conditions say that there are numbers λk ≥0,
k = 1,...,K such that if we let
vk
ω = λkpk
ω
πω
then vk
ω ∈∂u(xk
ω) if xk
ω > 0, and there is w ∈∂u(xk
ω) with vk
ω ≥w if xk
ω = 0.
In fact, since u is strictly increasing it is easy to see that λk > 0, and therefore
vk
ω > 0.
By the concavity of u, and the consequent monotonicity of ∂u(xk
ω)
(Theorem 1.9), if xk
ω > xk′
ω′ > 0, vk
ω ∈∂u(xk
ω), and vk′
ω′ ∈∂u(xk′
ω′), then vk
ω ≤vk′
ω′.
If xk
ω > xk′
ω′ = 0, then w ∈∂u(xk′
ω′) with vk′
ω′ ≥w. So vk
ω ≤w ≤vk′
ω′.
Next, we show that (II) implies (I). Suppose that the numbers vk
ω, λk, for
s = 1,...,S and k = 1,...,K, are as in (II).
Let
yi = min{vk
ω : xk
ω = yi} and ¯yi = max{vk
ω : xk
ω = yi}.
Let zi = (yi + yi+1)/2, i = 1,...,n −1; z0 = 0, and zn = yn + 1. Let f be a
correspondence deﬁned as follows:
f(z) =
⎧
⎪⎨
⎪⎩
[yi, ¯yi]
if z = yi,
max{¯yi : z < yi}
if yn > z and ∀i(z ̸= yi),
yn/2
if yn < z.
By the assumptions placed on vk
ω, and by construction of f, y < y′, v ∈f(y)
and v′ ∈f(y′) imply that v′ ≤v. Then the correspondence f is monotone, and
there exists a concave function u for which f(z) ⊆∂u(z) (see Corollary 1.10 of
Theorem 1.9). Given that vk
ω > 0 for all k and ω, all the elements in the range
of f are positive, and therefore u is a strictly increasing function.


--- Page 8 ---
8.2 Subjective probability
121
Finally, for all (k,s), pk
ω/πω = vk
ω ∈∂u(vk
ω) and therefore the ﬁrst-order
conditions to a maximum choice of x hold at xk
ω. Since u is concave the
ﬁrst-order conditions are sufﬁcient. The data are therefore rationalizable.
8.2
SUBJECTIVE PROBABILITY
The expected utility model of Section 8.1 assumes that an agent’s behavior
can be represented probabilistically, and in Section 8.1 probabilities are in fact
observable. However, in most situations of interest to economists, probabilities
are not observable. Economic agents are assumed that assign subjective
probabilities to the states of the world. It is important to understand when
agents’ behavior is consistent with the use of subjective probabilities.
8.2.1
The Epstein Test
One of the most basic questions in the theory of choice under uncertainty
is whether individuals perceive uncertainty probabilistically. One way of
formalizing this idea is due to Machina and Schmeidler (1992), and is
called probabilistic sophistication. A preference ⪰over R
+ is said to be
probabilistically sophisticated if there is a probability measure π on  such
that for all x,y ∈R, if the random variable x ﬁrst-order stochastically
dominates y on the probability space (,π), then x ⪰y; and x ≻y when the
ﬁrst-order stochastic dominance is strict.2
Consider a dataset {(xk,pk)}K
k=1, as in 8.1. The dataset is weakly ratio-
nalizable by a probabilistically sophisticated preference if there is a proba-
bilistically sophisticated preference ⪰such that xk ⪰y, for all y ∈R
+ with
pk · y ≤pk · xk.
The following test of the probabilistic sophistication hypothesis is due to
Larry Epstein. The idea is as follows. For two states, ω and ω′, what type of
behavior could reveal that πω ≥πω′? Epstein’s idea was that if prices in state
ω are higher than they are in state ω′, and the individual demands more in state
ω, then if the probabilistic sophistication hypothesis were true, the only reason
this could occur would be if πω > πω′.
Theorem 8.5
Suppose the dataset {(xk,pk)}K
k=1 contains observations l,m for
which pl
ω > pl
ω′ and pm
ω ≤pm
ω′, and xl
ω > xl
ω′ and xm
ω′ > xm
ω. Then there is no
probabilistically sophisticated preference which weakly rationalizes the data.
Proof. Suppose by way of contradiction that there is a probabilistically
sophisticated rationalization with associated probability π. We know that
either πω > πω′ or πω′ ≥πω. In the ﬁrst case, because pm
ω ≤pm
ω′, we know
that
pm
ωxm
ω′ + pm
ω′xm
ω ≤pm
ωxm
ω + pm
ω′xm
ω′.
2 We say a random variable x ﬁrst-order stochastically dominates y on probability space (,π)
if for all a ∈R, π({ω ∈ : X(ω) ≥a}) ≥π({ω ∈ : Y(ω) ≥a}).


--- Page 9 ---
122
Choice Under Uncertainty
But the bundle which results by switching consumption in states ω and ω′ in
xm strictly ﬁrst-order stochastically dominates xm, a contradiction. So it must
be the case that πω′ ≥πω. Now, we have
pl
ωxl
ω′ + pl
ω′xl
ω < pl
ωxl
ω + pl
ω′xl
ω′,
so the bundle which results from switching consumption in states ω and
ω′ for bundle xl ﬁrst-order stochastically dominates xl, and is strictly
cheaper. By increasing consumption in every state, there is a bundle which
strictly ﬁrst-order stochastically dominates xl and is feasible at prices pl, a
contradiction to the fact that xl is demanded.
8.2.2
Subjective expected utility
The benchmark model of decisions under uncertainty is the model of subjective
expected utility (SEU). This model postulates that agents’ choices are governed
by expected utility calculations, as in Section 8.1.3, but where the prior π is
not given, or observable. Instead, the agents’ choices are as if there were some
prior, and some utility function, that could explain them.
As in Section 8.1.3, a dataset is a collection {(xk,pk)k∈K} of pairs of a
consumption xk ∈R
+ chosen at prices pk ∈R
++ and income pk · xk.
A utility function u : R+ →R, together with a prior π ∈(), weakly
subjective expected-utility rationalizes a dataset (xk,pk)k∈K if u is strictly
increasing and concave, and if pk · y ≤pk · xk implies that

ω∈
πωu(yω) ≤

ω∈
πωu(xk
ω),
for all k ∈K. We say that such a dataset is weakly SEU rationalizable.
Viewed in this light, it is clear that the SEU model is a special case
of both the additive separable model considered in Section 4.2.2, and the
probabilistically sophisticated model of Section 8.2.1.
Now one can reason along the same lines of 8.1.3 to obtain an implication
from quantities on prices. The complication is that, since the prior π is not
observable, one cannot subsume probabilities into risk-neutral prices. Instead,
unknown probabilities need to be accounted for in the analysis. The details are
omitted; but sufﬁce it to say that we need the same notion of balancedness as in
8.1.3. In fact, we need more: say that a sequence of pairs (xkiωi,x
k′
i
ω′
i)n
i=1 is doubly
balanced if it is balanced and if, moreover, each ω ∈ appears as ωi (on the
left of the pair) the same number of times it appears as ω′
i (on the right).
A dataset {(xk,pk)}K
k=1 satisﬁes the Strong Axiom of Revealed Subjective
Expected Utility (SARSEU) if, for any sequence of pairs (xkiωi,x
k′
i
ω′
i)n
i=1 for which


--- Page 10 ---
8.3 Complete class results
123
xkiωi > x
k′
i
ω′
i for all i, the product of relative prices satisﬁes
n
"
i=1
pkiωi
p
k′
i
ω′
i
≤1.
SAREU in 8.1.3 can be written as ruling out certain kinds of cycles. With
SARSEU this is not possible because the axiom involves, in a sense, pairs of
cycles (one cycle in k’s and one in ω’s).
Theorem 8.6
The following statements are equivalent:
I) {(xk,pk)}K
k=1 is weakly SEU rationalizable.
II) For all k = {1,...,K}, there exist λk > 0 and π ∈() such that for
all ω ∈, πω > 0, and for each pair k,ω, there exists uk,ω such that
for all k,l and all ω,ω′,
uk,ω′ ≤ul,ω + λl pl
ω
πω
[xk
ω′ −xl
ω].
III) {(xk,pk)}K
k=1 satisﬁes SARSEU.
We will not offer a proof of Theorem 8.6, but note that it rests on familiar
ideas. Basically, (II) in Theorem 8.6 are the relevant Afriat inequalities for
the problem at hand. We only care to construct a utility index for a single
commodity, which should be the same across states after rescaling; πω here
acts as a scaling factor. Importantly, these Afriat inequalities deﬁne a nonlinear
polynomial system in the variables u,λ,π, which presents a signiﬁcant
complication. The proof that SARSEU characterizes SEU rationalizability
rests on linearizing the Afriat inequalities, and then using an approximation
argument.
8.3
COMPLETE CLASS RESULTS
The above discussion has focused on consumption data, but one could ask the
same type of question in an abstract environment of choice. We now assume as
given a single observation of a choice in an abstract environment: the set X is
some abstract payoff space;  is a ﬁnite set of states; and the objects of choice
are acts, mappings f :  →X. The set of all acts is X.
Choice is modeled by a preference relation ⪰over acts. The important aspect
of ⪰will be that it has a maximal element in some “budget,” or set of available
acts, F; so we know that there is f ∗∈F such that f ∗≻g for all g ∈F\{f ∗}. The
existence of f ∗can be obtained from a single observation of a choice at F.3
3 Of course, information on the other comparisons contained in ⪰would have to entail additional
observations.


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124
Choice Under Uncertainty
A constant act is one whose payoff is independent of the state. We
identify an element x ∈X with the constant act that takes the value x for all
states.
A null state is a state ω with the property that the agent does not care about
what obtains on ω. Formally, ω ∈ is null if for all x,y ∈X and f ∈X, xωf ∼
yωf, where xωf denotes the act which pays x if ω obtains and f otherwise.
A preference ⪰on X satisﬁes monotonicity if for all f,g ∈X, f(ω) ⪰g(ω)
for all ω ∈ implies f ⪰g, with strict preference if there is non-null ω ∈ for
which f(ω) ≻g(ω).
In this context, a preference ⪰is a subjective expected utility preference if
there is a function u : X →R and a probability measure π on  for which f ⪰g
iff 
ω u(f(ω))πω ≥
ω u(g(ω))πω.
The following result (due to T. B¨orgers) is related to the results in
statistical decision theory known as “complete class theorems,” except that no
convexiﬁcation via randomization is required.
Theorem 8.7
Suppose that F ⊆X is ﬁnite and that ⪰satisﬁes monotonicity.
If f ∗∈F satisﬁes f ∗≻g for all g ∈F\{f ∗}, then there exists a subjective
expected utility preference ⪰∗for which f ∗≻∗g for all g ∈F\{f ∗}.
Proof. Because f ∗∈F is strictly better than all g ̸= f ∗where g ∈F, it follows
that there must exist at least one non-null state. In the rest of the proof we
ignore null states; it is easy to assign them probability zero at the end.
The proof is by induction on the size of . We actually prove the slightly
stronger induction hypothesis: if G is a ﬁnite set of acts, ⪰is a preference
on G, and g∗∈G has the property that for all g ∈G\{g∗}, there is ω ∈
for which g∗(ω) ≻g(ω), then there is u : X →R and π on (,2) for
which 
ω u(g∗(ω))πω > 
ω u(g(ω))πω for all g ∈G\{g∗}. (The proof of
Theorem 8.7 will then be done, as monotonicity implies that for all f ∈F\{f ∗},
there is ω ∈ for which f ∗(ω) ≻f(ω).)
The result is trivial if || = 1. Suppose then that || > 1, and that the result is
true whenever the size of the state space is strictly smaller than ||. Let X∗=
{x ∈g∗() : g∗(ω) ⪰x for all ω ∈}; that is, X∗is the set of worst possible
outcomes occurring with g∗(recall that g∗() is ﬁnite). Consider ∗= {ω ∈
 : g∗(ω) ∈X∗} and G∗= {g ∈G : ∃ω ∈ such that for all x ∈X∗,x ≻g(ω)}:
∗is the set of states which lead to one of the worst outcomes, and G∗is the
set of acts which realize outcomes worse than any in X∗.
Suppose that \∗̸= ∅. Clearly, |\∗| < ||. Further, g∗∈G\G∗. Finally,
there is no g ∈G\G∗for which g(ω) ⪰g∗(ω) for all ω ∈\∗. If there were,
then by deﬁnition, since g ̸∈G∗, g never realizes a worse outcome than an
outcome in X∗; consequently, we would have g(ω) ⪰g∗(ω) for all ω ∈∗as
well, so that g(ω) ⪰g∗(ω) for all ω ∈, contradicting the hypothesis.
Therefore, by the induction hypothesis, there exists u∗: X →R and π∗on
 \ ∗for which 
ω∈\∗u∗(g∗(ω))π∗
ω > 
ω∈\∗u∗(g(ω))π∗
ω for all g ∈
G \G∗. For δ > 0, let u∗
δ : X →R be deﬁned by u∗(x) if x ⪰x∗for some x∗∈X∗,


--- Page 12 ---
8.3 Complete class results
125
and u∗
δ(x) = u∗(x) −δ otherwise. Further, for ε > 0, deﬁne π∗
ε ({ω}) =
ε
|∗| if
ω ∈∗, otherwise, π∗
ε ({ω}) = (1 −ε)π∗({ω}).
For
ε
small
enough,
g∗
satisﬁes

ω∈ u∗(g∗(ω))π∗
ε ({ω})
>

ω∈ u∗(g(ω))π∗
ε ({ω}) for all g ∈G\G∗. Now, for all δ > 0, and for all
g ∈G\G∗, we have 
ω∈ u∗(g(ω))π∗
ε ({ω}) = 
ω∈ u∗
δ(g(ω))π∗
ε ({ω}).
By
choosing
δ > 0
large,
we
can
ensure
that
for
any
g ∈G∗,

ω∈ u∗
δ(g(ω))π∗
ε ({ω}) can be made arbitrarily small. This completes
the induction step.
On the other hand, if \∗= ∅, we know that g∗(ω) ∼g∗(ω′) for all ω,ω′.
In this case, let π be arbitrary, and let u∗: X →R so that for any x ∈X∗and
g ∈G, if x ≻g(ω), then u∗(x) > u∗(g(ω)). Again by considering u∗
δ for δ > 0
large, the result follows.
Theorem 8.7 establishes that, with one observation, the empirical content
of monotonicity coincides with the empirical content of subjective expected
utility maximization. With more than one observation, such a result does not
hold in general.
Given the conclusion of Theorem 8.7, it is useful to have an understanding of
the empirical content of monotonicity. We shall explain this empirical content
in the case of one observation.
Proposition 8.8
Suppose that F ⊆X is ﬁnite, and let f ∗∈F. Then there
exists a monotonic preference relation ⪰for which f ∗≻g for all g ∈F \
{f ∗} iff ωg ∈ can be chosen for all g ∈F\{f ∗} such that the binary relation
{(f ∗(ωg),g(ωg))}g∈F\{f ∗} on X is acyclic.
Proof. If there exists a monotonic and rational ⪰, we know that for each g ∈
F\{f ∗}, there is ωg for which f ∗(ωg) ≻g(ωg); if not, then g(ω) ⪰f ∗(ω) for all
ω ∈, whereby monotonicity dictates that g ⪰f ∗, contradicting f ∗≻g. The
relation {(f ∗(ωg),g(ωg))}g∈F\{f ∗} is clearly acyclic, as it is a subrelation of ≻
deﬁned on X.
On the other hand, suppose that there are states ωg ∈ for which the
relation {(f ∗(ωg),g(ωg))}g∈F\{f ∗} on X is acyclic. By Szpilrajn’s Theorem
(Theorem 1.4) there is an extension of this binary relation to a linear order
⪰∗on X. Deﬁne ⪰′ on F by f ⪰′ g if and only if f(ω) ⪰∗g(ω) for all ω ∈:
note that by deﬁnition ⪰′ is reﬂexive, transitive, and antisymmetric. Further, if
g ⪰′ f ∗, then g = f ∗. Let
⪰′′ =⪰′ ∪

g∈F\{f ∗}
{(f ∗,g)}.
There can be no ⟨⪰′′,≻′′⟩cycles as, by construction, if g ⪰′ f ∗, then g = f ∗.
As there are no ⪰′′ cycles, we can use Theorem 1.5 again to extend ⪰′′ to a
preference relation ⪰on F such that for all g ∈F\{f ∗}, f ∗≻g. The preference
relation is, by construction, monotonic.


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126
Choice Under Uncertainty
8.4
SUBJECTIVE EXPECTED UTILITY WITH AN
ACT-DEPENDENT PRIOR
Many modern theories of decisions under uncertainty are predicated
on an assumption that probability can depend on the act chosen; for
example,
the
theory
of
maxmin
expected
utility
supposes
a
utility
of the form U(f) = 
ω∈ u(f(ω))πf ({ω}), where πf
is chosen from
argminπ∈

ω∈ u(f(ω))πf ({ω}) for some  ⊆().
We end the chapter by considering the testable implications of the notion that
choices are guided by an expected utility calculation in which the probability
measure over states can depend on the act chosen. The notion of a dataset will
correspond to that in Chapter 2, namely abstract choice.
Let X and  be ﬁnite sets. Say that a preference ⪰on X is an act-dependent
probability representation if there is u : X →R and, for all f ∈X, there is
πf ∈() such that the function U(f) = 
ω∈ u(f(ω))πf ({ω}) represents ⪰.
It is quite easy to characterize act-dependent probability preferences. Say
that a preference ⪰satisﬁes uniform monotonicity if f ⪰g whenever f,g ∈X
are such that for all ω,ω′ ∈, f(ω) ⪰g(ω′).4
Proposition 8.9
If X is ﬁnite, a preference ⪰has an act-dependent
probability representation iff it satisﬁes uniform monotonicity.
Proof. Suppose ⪰is a preference satisfying uniform monotonicity. Since X is
ﬁnite and ⪰is a preference relation, there exists U : X →R which represents
⪰. Deﬁne u(x) = U(x), where U(x) is the value of U applied to the constant
act taking outcome x. Note that for every f, by uniform monotonicity,
min
ω∈ u(f(ω)) ≤U(f) ≤max
ω∈ u(f(ω)).
Therefore, U(f) can be expressed as a convex combination of minω∈ u(f(ω))
and maxω∈ u(f(ω)). Choose the probability πf so as to obtain the result of this
convex combination.
The other direction is equally simple; suppose that for all ω,ω′ ∈, f(ω) ⪰
g(ω′): U(f) = 
ω∈ u(f(ω))πf ({ω}) ≥minω∈ u(f(ω)) ≥maxω∈ u(g(ω)) ≥

ω∈ u(f(ω))πg({ω}) = U(g).
We can now speak of a choice function deﬁned on a collection of “budgets,” or
sets of feasible acts , each element of  being a subset of X. We assume that
 is certainty inclusive, in the sense that for all x,y ∈X, {x,y} ∈. Thus, choice
from every pair of certain outcomes can be observed. According to this choice
function, we have our standard revealed preference pair, ⟨⪰c,≻c⟩, as deﬁned
in Chapter 2. We introduce a new relation ⪰′ deﬁned by f ⪰′ g if f(ω) ⪰c g(ω′)
for all ω,ω′ ∈.
With the property of certainty inclusiveness, the empirical content of
uniform monotonicity is quite easy to describe using previous results.
4 With a slight abuse of notation, an element of X is identiﬁed with a constant act returning that
element.


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8.5 Chapter references
127
Theorem 8.10
A choice function on a certainty inclusive domain is strongly
rationalized by a preference relation satisfying uniform monotonicity iff
⟨⪰c ∪⪰′,≻c⟩is acyclic.
Proof. Follows easily from Corollary 1.6 and Theorem 2.5.
8.5
CHAPTER REFERENCES
Allais’ paradox is due to Allais (1953). He presents a classic test of the
expected utility hypothesis. Typical choices in Allais’ experiment directly
violate the independence axiom. Theorem 8.2 appears in Fishburn (1974),
among other related results. Bar-Shira (1992) also investigates the set of
linear inequalities arising in this problem, and uses it to bound risk aversion
in the context of monetary lotteries. Border (1992) extends this idea to
a choice-theoretic approach, assuming monetary lotteries and a strictly
increasing utility index. Border’s approach is closer to the ideas in Afriat’s
Theorem. Kim (1996) also provides a generalization of this result.
The equivalence between (I) and (III) in Theorem 8.3 is essentially taken
from Kubler, Selden, and Wei (2014). The equivalence between (I) and (II) in
Theorem 8.3 is a version of Afriat inequalities for this problem: it appears
in Green and Srivastava (1986), Varian (1983b), Varian (1988b), Bayer,
Bose, Polisson, Renou, and Quah (2012), and Diewert (2012). The recent
paper by Chambers, Liu, and Martinez (2014) provides a revealed preference
axiom for the case of multiple goods in each state (the setup of Green and
Srivastava, 1986). Green and Osband (1991) study a version of the objective
probability problem in which the probability measure over states is changing,
and “demand” as a function of the objective probability measure over states
is observed. Park (1998) conducts a related investigation of the weighted
expected utility model of Hong (1983).
The Epstein Test is due to Epstein (2000), who viewed his test as a market
counterpart of Ellsberg’s paradox (Ellsberg, 1961). Theorem 8.6 is based on the
work of Green and Srivastava (1986), who take π as an observable, and also
provide a cyclic monotonicity-style test; Kim (1991) presents related results
in this direction. Bayer, Bose, Polisson, Renou, and Quah (2012) investigate
related conditions which arise in the context of ambiguity models.
Axiomatizations of subjective expected utility have a long history, but the
most important are due to Savage (1954) and Anscombe and Aumann (1963).
The latter of these provides an axiomatization for ﬁnite states of the world.
Theorem 8.6 is due to Echenique and Saito (2013), which also presents
a characterization of state-dependent utility (i.e. additively separable utility
across states). The equivalence between rationalizability and the version of
Afriat inequalities in Theorem 8.6 is the same as obtained by Green and
Srivastava (1986) (presented in Theorem 8.3 for the case of objective expected
utility, only existentially quantiﬁed over the prior). The papers by Bayer, Bose,
Polisson, Renou, and Quah (2012), Ahn, Choi, Gale, and Kariv (2014), and


--- Page 15 ---
128
Choice Under Uncertainty
Hey and Pace (2014) apply revealed preference tests to experimental data on
uncertainty and subjective probabilities.
Polisson, Renou, and Quah (2013) also develop a system of Afriat
inequalities for the model of subjective expected utility, and for other models.
One important difference between the work of Polisson and Quah and other
papers is that they do not require the utility over money to be concave.
Chambers, Echenique, and Saito (2015) give revealed preference axioms
for translation invariant and homothetic models of choice under uncertainty,
including the maxmin model of Gilboa and Schmeidler (1989), and the
expected utility model when the utility function takes the speciﬁc “constant
absolute risk aversion” (CARA) or “constant relative risk aversion” (CRRA)
form.
The paper by Richter and Shapiro (1978) should be mentioned as well. They
study the design of a set of pairwise comparisons so that, given the outcome
of such a pair of comparisons, a deﬁnitive statement can be made about the
agents’ subjective probabilities. An example of such a statement is whether
the probability of state ω1 is at least twice that of state ω2.
Theorem 8.7 is due to B¨orgers (1993), and the interpretation offered here
is due to Lo (2000). It can be shown that for the primitive ⪰, one can choose
⪰∗to have the same null states and the same ranking over constant acts as
⪰. This result is related to a class of results in statistical decision theory
known as “complete class theorems,” which establish related results when
the outcome space has a convex structure due to randomization and linearity
of payoffs in randomization. These results are originally due to Wald (1950,
1947a,b); Dvoretzky, Wald, and Wolfowitz (1951) use the fact that the range
of a nonatomic vector measure is convex and compact to establish a similar
result without any required randomization. A classic reference on these topics
is Ferguson (1967).
Bossert and Suzumura (2012) establish Theorem 8.10. The question of the
empirical content of preferences satisfying uniform monotonicity on domains
which do not satisfy certainty inclusiveness is open, as is the same question for
monotonic preferences.
Finally, there is a literature investigating the empirical content of different
updating rules for probabilistic beliefs. Shmaya and Yariv (2012) characterize
the empirical content of Bayes’ rule, and show that it is equivalent to many
apparently more general classes of updating rules.
