Another proof, which is more algebraic and less geometric, builds on the state-dependent utility
representation of Theorem 10.3 and appears in Kreps, pp. 110–111, .
Exercise 10.16.
Prove the necessity (
⇐
) part of Theorem 10.14:
if there exists a vNM index
v
:
X
→
R
and a probability
μ
∈
ΔΩ such that
∑
s
μ
(
s
) [
∑
x
v
(
x
)
h
s
(
x
)] is a utility representation
of
%
, then
%
is independent, Archimedean, and state-independent.
Exercise 10.17.
Prove the uniqueness part of Theorem 10.14: if there exist
h
g
and
U
(
h
) =
∑
s
μ
(
s
) [
∑
x
v
(
x
)
h
s
(
x
)] represents
%
, then
U
0
(
h
) =
∑
s
μ
0
(
s
) [
∑
x
v
0
(
x
)
h
s
(
x
)] represents
%
if and
61

only if
μ
0
=
μ
and
v
0
=
av
+
b
for some
a >
0 and
b
∈
R
. Prove that if
h
∼
g
for all
h, g
∈
H
, then
the uniqueness result fails.
Exercise 10.18.
Suppose Ω =
{
s
1
, s
2
, s
3
}
and
X
=
{
x
1
, x
2
, x
3
}
.
The preference relation
%
is
independent, Archimedean, and state-independent.
The following are known:
δ
x
1
δ
x
2
δ
x
3
and (
δ
x
1
, δ
x
2
, δ
x
3
)
∼
(
δ
x
1
,
(0
,
1
2
,
1
2
)
,
(0
,
1
2
,
1
2
))
∼
((
2
3
,
1
3
,
0)
,
(
2
3
,
1
3
,
0)
, δ
x
3
).
Find the decision maker’s
subjective belief
μ
.
Exercise 10.19.
Suppose there exists a vNM index
v
:
X
→
R
such that
%
is represented by
U
(
h
) = min
s
∈
Ω
X
x
v
(
x
)
h
s
(
x
)
.
Prove or provide counterexamples to the following statements:
%
is independent;
%
is Archimedean;
%
is state-independent.
Exercise 10.20.
Suppose there exists a strictly positive probability
μ
∈
Δ
S
and a vNM index
v
:
X
→
R
such that
%
is represented by
U
(
h
) =
X
s
μ
(
s
) [min
{
v
(
x
) :
h
s
(
x
)
>
0
}
]
.
Prove or provide counterexamples to the following statements:
%
is independent;
%
is Archimedean;
%
is state-independent.
Exercise 10.21.
Suppose there exists a closed set of strictly positive probabilities
C
⊆
Δ
S
and a
vNM index
v
:
X
→
R
such that
%
is represented by
U
(
h
) = min
μ
∈
C
X
s
μ
(
s
)
"
X
x
v
(
x
)
h
s
(
x
)
#!
.
Prove or provide counterexamples to the following statements:
%
is independent;
%
is Archimedean;
%
is state-independent.
In the Anscombe–Aumann Expected Utility Theorem, the preference identifies two things: the
utility index over consequences
v
:
X
→
R
and the probability on states
μ
∈
ΔΩ. This contrasts
with the von Neumann–Morgenstern Expected Utility Theorem, which only identifies the utility
index. Different preferences may imply different beliefs on Ω.
62

11
Basics of Savage expected utility
This entire section is totally optional.
It is only included to introduce interested students to what
Kreps calls the “crowning achievement of singe-person decision theory.”
In both the von Neumann–Morgenstern and Anscombe–Aumann models, we assumed the existence of
some objective randomizing device. Ideally, all uncertainty in the model would be subjective: no probabilities
are assumed
a priori
and all probabilities are identified by preferences over basic objects that are not
contaminated with any assumed objective probability.