Information Technology Reference
In-Depth Information
Definition 9.
A
conjecture subprofile
,for
X
i
k
, denoted
a
i
k
, is the subprofile compris-
ing the elements of of
a
i
k
that influence
X
i
. We then have
u
X
i
|
pa (
X
i
)
[
a
i
|
pa(
a
i
)] =
u
X
i
|
pa (
X
i
)
[
a
i
|
pa(
a
i
)]
,
(34)
where
pa(
a
i
)=(
a
i
1
,...,
a
i
p
i
)
.
is
completely conjecture sociated
if
a
i
k
=
a
i
k
for
k
=1
,...,p
i
and
i
=1
,...,n
.Itis
completely conjecture dissociated
if
a
i
k
=
a
i
k
for
k
=1
,...,p
i
and
i
=1
,...,n
,inwhichcase,
pa(
a
i
)=(
a
i
1
,...,a
i
p
i
)
.
Otherwise, the group is
partially conjecture sociated
.
{X
1
,...,X
n
}
Definition 10.
A
utility subprofile
, denoted
a
i
, comprises the subprofile of
a
i
that af-
fects
X
i
's utility. We then have
u
X
i
|
pa (
X
i
)
[
a
i
|
pa(
a
i
)] =
u
X
i
|
pa (
X
i
)
[
a
i
|
pa(
a
i
)]
,
(35)
where
u
denotes
u
with the dissociated arguments removed.
is
com-
pletely utility sociated
if
a
j
=
a
j
for
i
=1
,...,n
.Itis
completely utility dissociated
if
a
i
=
a
i
for
i
=1
,...,n
,inwhichcase
{
X
1
,...,X
n
}
pa (
a
i
)]
.
u
X
i
|
pa (
X
i
)
[
a
i
|
pa(
a
i
)] =
u
X
i
|
pa (
X
i
)
[(
a
i
|
(36)
Otherwise, the group is
partially utility sociated
.
Definition 11.
A group
is
completely dissociated
if it is both completely
conjecture dissociated and completely utility dissociated, in which case,
pa (
a
i
)=
pa(
a
i
)=(
a
i
1
,...,a
i
p
i
)
, the profile of conjecture actions of the members of
pa(
X
i
)
.
For a partially sociated system, the concordant utility assumes the form
{X
1
,...,X
n
}
U
X
1
···
X
n
(
a
1
,...,
a
n
)=
U
X
1
···
X
n
(
a
1
,...,
a
n
)
(37)
n
=
u
X
i
|
pa (
X
i
)
[
a
i
|
pa (
a
i
)]
,
(38)
i
=1
where
U
is
U
with the dissociated arguments removed. For a completely dissociated
group, the concordant utility coincides with the group welfare function and assumes the
form
n
w
X
1
···
X
n
(
a
1
,...,a
n
)=
u
X
i
|
pa (
X
i
)
[
a
i
|
pa(
a
i
)]
.
(39)
i
=1
Example 3.
Let us now reconsider the automobile buying example introduced in Ex-
ample 1. We shall assume that the influence flows are as depicted in Figure 1, with the
corresponding concordant utility of the form expressed by (23), yielding
U
X
1
X
2
X
3
[(
a
11
,a
12
,a
13
)
,
(
a
21
,a
22
,a
23
)
,
(
a
31
,a
32
,a
33
)] =
u
X
1
(
a
11
,a
12
,a
13
)
u
X
2
|
X
1
(
a
21
,a
22
,a
23
|a
11
,a
12
,a
13
)
u
X
3
|
X
1
X
2
[
a
31
,a
32
,a
33
|
(
a
11
,a
12
,a
13
)
,
(
a
21
,a
22
,a
23
)]
.
(40)
