10
Anscombe and Aumann Expected Utility
Let Ω be a finite set of
m
states of the world, with generic elements
s, t
∈
Ω. Let
X
be a finite
set of
n
consequences with a generic element
x
∈
X
. It will often be useful to identify a state
s
with its index and enumerate Ω =
{
1
,
2
, . . . , m
}
. Let
H
= (Δ
X
)
Ω
denote the space of all functions
from Ω to Δ
X
; such functions are also called (Anscombe–Aumann) acts.
So a generic element
h
∈
H
= (Δ
X
)
Ω
is a function
h
: Ω
→
Δ
X
from Ω to Δ
X
, hence assigns each state
s
∈
Ω a lottery
h
(
s
)
∈
Δ
X
. For ease of notation, we let
h
s
=
h
(
s
)
∈
Δ
X
, so
h
s
(
x
) = [
h
(
s
)](
x
)
∈
[0
,
1]. One way
to think of
h
s
(
x
) is as the probability of the consequence
x
conditional on the state
s
, i.e.
h
s
(
x
)
represents Pr(
x

s
), given the act
h
.
For any act
h
∈
(Δ
X
)
Ω
, state
s
∈
Ω, and lottery
π
∈
Δ
X
, define the new act (
h

s
, π
) : Ω
→
Δ
X
by (
h

s
, π
) = (
h
1
, . . . , h
s

1
, π, h
s
+1
, . . . , h
m
). So
[(
h

s
, π
)](
t
) =
(
π
if
t
=
s
h
(
t
)
if
t
6
=
s
;
or, in our subscript notation, (
h

s
, π
)
s
=
π
and (
h

s
, π
)
t
=
h
t
for all states
t
6
=
s
. In words (
h

s
, π
)
replaces
h
s
, the lottery that the act
h
assigns to state
s
, with the lottery
π
; the other lotteries are
kept the same.
In a slight abuse of notation, for any lottery
π
∈
Δ
X
, we let
π
also denote the constant act
f
: Ω
→
Δ
X
such that
f
(
s
) =
π
for all
s
∈
Ω.
H
is a convex subset of the space of functions
from Ω to
R
n
,
H
= (Δ
X
)
Ω
⊂
(
R
n
)
Ω
, where if
f, g
: Ω
→
R
n
are functions from Ω to
R
n
, then the
function
αf
+
βg
: Ω
→
R
n
is defined by the [
αf
+
βg
](
s
) =
αf
(
s
) +
βg
(
s
).
The classic interpretation of
H
is as follows. We can consider elements of Δ
X
as bets on an
objective roulette, where the probabilities of outcomes are physically determined. Consider each
of the world as the event that a specific horse, named
s
, wins a race in a field of horses Ω, where
different people can have different assessments of each horse’s strength. Our aim is to identify the
decision maker’s personal assessment of the probability that horse
s
will win the race. To do so, we
allow the payout on horse
s
to be another lottery that depends on the outcome of a roulette spin.
So, we first let the horses run and then we spin the roulette. The payoff on the roulette depends on
which horse wins. There are at least three natural mathematical representations of the space
H
:
•
H
= (Δ
X
)
Ω
. This is the original mathematical interpretation. For example, suppose Ω =
{
s
1
, s
2
, s
3
}
and
X
=
{
x
1
, x
2
, x
3
}
. Then a particular
h
: Ω
→
Δ
X
would be the following:
h
(
s
1
)
=
(0
.
3
,
0
.
2
,
0
.
5)
h
(
s
2
)
=
(0
.
4
,
0
.
6
,
0)
h
(
s
3
)
=
(0
,
1
,
0)
56
If
π
= (0
.
5
,
0
.
4
,
0
.
1), then (
h

s
2
, π
) would be:
[(
h

s
2
, π
)] (
s
1
)
=
(0
.
3
,
0
.
2
,
0
.
5)
[(
h

s
2
, π
)] (
s
2
)
=
(0
.
5
,
0
.
4
,
0
.
1)
[(
h

s
2
, π
)] (
s
3
)
=
(0
,
1
,
0
,
0)
•
Another way to represent
H
is as a set of compound lotteries, where the subjective first
stage lottery is over which state
s
∈
Ω obtains, and the objective second stage lottery, which
is conditional on
s
, is over which
x
∈
X
finally obtains.
These compound lotteries can be
written as probability trees. For example,
h
and can be denoted:
•
•
s
1
•
s
2
•
s
3
.