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Principle 2 (Endogeny).
If a concordant ordering for a group of agents exists, it must
be determined by the social interactions among the subsets of the group.
This principle precludes the exogenous imposition of aggregation structures. For ex-
ample, a common conventional aggregation procedure is to form the weighted sum of
individual utilities. Such a structure, however, is appropriate only under conditions of
preferential independence (e.g., see [6,9]). When preferential dependencies exist, we
seek an aggregation structure that naturally emerges from within the group.
Given the existence of a concordant utility
U
X
m
and a conditional concordant utility
U
X
k
|
X
m
, our goal is to compute the concordant utility of the union of the two subgroups;
i.e., to form
U
X
m
X
k
, the concordant utility for
X
m
∪X
k
.
Definition 5.
Let
X
k
=
{
X
i
1
,...,X
i
k
}
and
X
m
=
{
X
j
1
,...,X
j
m
}
be disjoint sub-
groups of
X
n
such that
U
X
m
and
U
X
k
|
X
m
are defined. These utilities are
endogenously
aggregated
if there exists a function
F
such that
U
X
m
X
k
(
α
m
,α
k
)=
F
[
U
X
m
(
α
m
)
,U
X
k
|
X
m
(
α
k
|
α
m
)]
,
(1)
When social relationships exist among the members of a group, there may not be a
unique way to represent them mathematically. Consider the following example.
Example 2.
Consider the two-agent system
{
X
1
,X
2
}
and let us suppose that
X
1
pos-
A
sesses a categorical utility
u
X
1
,butthat
X
2
possesses a conditional utility of the
form
u
X
2
|
X
1
;thatis,
X
2
conditions its preferences on the preferences of
X
1
. Our desire
is to define a function
F
such that
over
U
X
1
X
2
(
a
1
,
a
2
)=
F
[
u
X
1
(
a
1
)
,u
X
2
|
X
1
(
a
2
|
a
1
)]
.
(2)
Now let us suppose that there is a well-defined social relationship between
X
1
and
X
2
such that, when defining their preferences, they both take into consideration that, ulti-
mately, they will be operating in a group environment, and not in isolation. Under these
conditions, it is possible to re-frame the scenario by
X
2
defining a categorical utility
u
X
2
and
X
1
defining a conditional utility
u
X
1
|
X
2
. Under this framing, the aggregation
problem requires
U
X
2
X
1
(
a
2
,
a
1
)=
F
[
u
X
2
(
a
2
)
,u
X
1
|
X
2
(
a
1
|
a
2
)]
.
(3)
Principle 3 (Consistency).
If a multiagent decision problem can be framed in more
than one way using exactly the same information, all such framings should yield the
same aggregated concordant ordering.
Definition 6.
Let
X
n
and suppose there exist two
framings of the preferences and relationships between the two subgroups of the forms
{
X
k
and
X
m
be disjoint subgroups of
U
X
k
,U
X
m
|
X
k
}
and
{
U
X
m
,U
X
k
|
X
m
}
. The endogenous aggregation is
consistent
if
F
[
U
X
k
(
α
k
)
,U
Xm
|
X
k
(
α
m
|
α
k
)] =
F
[
U
X
m
(
α
m
)
,U
X
k
|
Xm
(
α
k
|
α
m
)]
.
(4)
