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be a family of normalized non-negative conditional concordant utilities associated with
all pairs of disjoint subgroups of
X
n
. These utilities are endogenously aggregated if
and only if, for every pair of disjoint subgroups
X
m
and
X
k
,
U
X
m
X
k
(
α
m
,α
k
)=
F
[
U
X
k
(
α
k
)
,U
X
m
X
k
(
α
m
|
α
k
)]
(8)
=
U
X
m
X
k
(
α
m
|
α
k
)
U
X
k
(
α
k
)
.
(9)
This theorem was originally introduced by [5] as an alternative development of the
mathematical syntax of probability theory. The proof below follows [8].
Proof of the Aggregation Theorem.
Let
X
i
,
X
j
,and
X
k
be arbitrary pairwise dis-
joint subgroups of
X
n
,andlet
U
X
i
X
j
X
k
,
U
X
i
|X
j
X
k
,
U
X
i
X
j
|X
k
,
U
X
i
X
j
,
U
X
i
|X
j
,and
U
X
i
be
endogenously aggregated concordant utilities. That is,
U
X
i
X
j
X
k
(
α
i
,α
j
,α
k
)=
F
U
X
j
X
k
(
α
j
,α
k
)
,U
X
i
|
X
j
X
k
(
α
i
|α
j
,α
k
)
(10)
=
F
U
X
k
(
α
k
)
,U
X
i
X
j
|
X
k
(
α
i
,α
j
|α
k
)
.
(11)
But
U
X
j
X
k
(
α
j
,α
k
)=
F
U
X
k
(
α
k
)
,U
X
j
|
X
k
(
α
j
|
α
k
)
(12)
and
α
k
)=
F
U
X
j
|
X
k
(
α
j
|
α
j
,α
k
)
.
U
X
i
X
j
|
X
k
(
α
i
,α
j
|
α
k
)
,U
X
i
|
X
j
X
k
(
α
i
|
(13)
Substituting (12) into (10) and (13) into (11) yields
F
F
U
X
k
(
α
k
)
,U
X
j
|
X
k
(
α
j
|
α
j
,α
k
)
=
F
U
X
k
(
α
k
)
,F
U
X
j
|
X
k
(
α
j
|α
k
)
,U
X
i
|
X
j
X
k
(
α
i
|α
j
,α
k
)
.
(14)
α
k
)
,U
X
i
|
X
j
X
k
(
α
i
|
In terms of general arguments, this equation becomes
F
[
F
(
x, y
)
,z
]=
F
[
x, F
(
y,z
)]
,
(15)
called the
associativity equation
. By direct substitution it is easy to see that (15) is
satisfied if
f
[
F
(
x, y
)] =
f
(
x
)
f
(
y
)
(16)
for any function
f
. It has been shown by [5] that if
F
is differentiable in both arguments,
then (16) is the general solution to (15). Taking
f
as the identity function,
F
(
x, y
)=
xy
,
and
U
X
i
X
j
(
α
i
,α
j
)=
F
U
X
i
(
α
i
)
,U
X
j
|
X
i
(
α
j
|
α
i
)
=
U
X
i
(
α
i
)
U
X
j
|
X
i
(
α
j
|
α
i
)
.
(17)
To prove the converse, we note that
F
given by (9) is nondecreasing in both arguments
since
U
X
k
X
k
are
arbitrary, (9) holds if we reverse the roles of
m
and
k
. Thus, consistency is satisfied and
the aggregation is endogenous.
X
m
and
and
U
X
m
|
X
k
are nonnegative. Also, since the subgroups
