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In-Depth Information
3
Aggregation of Preferences
Our aim is to define an aggregation mechanism by which a notion of group prefer-
ence ordering can emerge from the aggregation of conditional individual preference
orderings. An essential characteristic of any such mechanism is that it must possess the
following property.
Principle 4 (Monotonicity).
If a subgroup prefers one alternative to another and the
complementary subgroup is indifferent with respect to the two alternatives, then the
group as a whole must not prefer the latter alternative to the former one.
Principle 4 invokes the common sense concept that, in the absence of opposition, the
group must not arbitrarily override the wishes of individuals. Thus, if
X
1
prefers
a
to
a
and
X
2
is indifferent between the two profiles, the group
should not prefer
a
to
a
. In terms of utilities, this condition means that
F
must be nondecreasing in both
arguments.
When modeling influence relationships, it is critical that we delimit generality to
ensure computational tractability. We thus propose the following principle.
{
X
1
,X
2
}
Principle 5 (Acyclicity).
No cycles occur in the influence relationships among the
agents.
X
k
and
X
m
of
X
n
, acyclicity means that it cannot hap-
Given two disjoint subgroups
X
m
.
The fact that cycles are not permitted does reduce the generality of the model. Never-
theless, restricting to one-way influence relationships is a significant generalization of
the neoclassical approach, which assumes that all utilities are categorical and, hence,
are trivially acyclical.
X
m
directly influences
X
k
and
X
k
directly influences
pen that, simultaneously,
3.1
The Aggregation Theorem
It remains to define a function
F
that complies with the above-mentioned principles.
Since positive affine transformations preserve the mathematical integrity of von
Neumann-Morgenstern utilities, we may assume, without loss of generality, that all
utilities are non-negative and normalized to sum to unity; that is,
U
X
k
(
α
k
)
≥
0
∀
α
k
, U
X
m
|
X
k
(
α
m
|
α
k
)
≥
0
∀
α
m
,α
k
,
(5)
U
X
k
(
α
k
)=1
,
α
m
U
X
m
|
X
k
(
α
m
|
α
k
)=1
α
k
.
∀
(6)
α
k
Theorem 1 (The Aggregation Theorem.).
Let
X
n
=
{
X
1
,...,X
n
}
be an
n
-member
multiagent system and let
B
m
denote the set of all
m
-element subgroups of
X
n
. That is,
X
m
∈B
m
if
X
m
=
X
m
∈
{
X
i
1
,...,X
i
m
}
with
1
≤
i
1
<
···
<i
m
≤
n
.Let
{
U
X
m
:
B
m
,m
=1
,...,n
}
be a family of normalized non-negative concordant utilities and let
X
m
∩X
k
=
X
m
∈B
m
,
X
k
∈B
k
,m
+
k
{
U
X
m
|
X
k
:
∅
,
≤
n
}
(7)
