Information Technology Reference
In-Depth Information
If all agents have opinions about the model, manufacturer, and color attributes, then
each agent must specify utility valuations for each of the eight possible outcomes. Thus,
X
1
would specify eight utility values.
X
2
, however, would need to define eight condi-
tional utilities, each with eight valuations, and thus would make 64 utility specifications.
Finally,
X
3
would define 64 conditional utilities, each with eight valuations, and thus
would make 512 utility specifications — a formidable task.
Now let us suppose that
X
1
is concerned only about the model and manufacturer,
but has no opinion about the color. Also, we assume that
X
2
is concerned only about
the manufacturer given
X
1
's conjecture about the model and manufacturer. Finally, let
us assume that
X
3
is concerned only about the color given
X
1
's conjecture about the
model and
X
2
's conjecture about the manufacturer.
As a result of these simplifications, we see that
X
1
is partially utility sociated, thus
a
1
=(
a
11
,a
12
)
. We also see that
X
2
is completely utility dissociated and partially
conjecture sociated, hence
a
2
=
a
22
and
pa (
a
22
)=(
a
11
,a
12
)
. Finally,
X
3
is also
completely utility dissociated and partially conjecture sociated, thus
a
3
=
a
33
and
pa(
a
33
)=(
a
11
,a
22
)
. The concordant utility simplifies to
U
X
1
X
2
X
3
(
a
11
,a
12
,a
22
,a
33
)=
u
X
1
(
a
11
,a
12
)
u
X
2
|
X
1
(
a
22
|
a
11
,a
12
)
u
X
3
|
X
1
X
2
(
a
33
|
a
11
,a
22
)
.
(41)
Under these simplifications, we see that
X
1
need only make four specifications when
defining
u
X
1
,
X
2
need make two specifications for each of
X
1
's four specifications,
resulting in eight specifications when defining
u
X
2
|
X
1
,and
X
3
need make two specifi-
cations for each of the four joint specifications of
X
1
and
X
2
, resulting in eight spec-
ifications when defining
u
X
3
|
X
1
X
2
, yielding a grand total of 20 utility specifications
— a considerable reduction in complexity from the 584 specifications required under
the condition of complete sociation (and even less than the 24 specifications needed to
define categorical utilities).
Tables 1(a), (b), and (c) respectively tabulate
X
1
's categorical utility,
X
2
's condi-
tional utilities given
X
1
's conjectures, and
X
3
's conditional utilities given the conjec-
tures for
X
1
and
X
2
.
Ta b l e
1 .
(a)
The
categorical
utility
u
X
1
(
a
11
,a
12
),
(b)
the
conditional
utility
u
X
2
|
X
1
(
a
22
|
a
11
,a
12
), and (c) he conditional utility
u
X
3
|
X
1
X
2
(
a
33
|
a
11
,a
22
)
(a)
a
12
a
11
DF
C
0.1 0.4
S
0.3 0.2
(b)
(
a
11
,a
12
)
a
22
(
C,D
)(
C,F
)(
S, D
)(
S, F
)
D
(c)
(
a
11
,a
22
)
a
33
(
C,D
)(
C,F
)(
S, D
)(
S, F
)
R
0.3
0.5
0.6
0.4
0.1
0.3
0.5
0.8
F
0.7
0.5
0.4
0.6
G
0.9
0.7
0.5
0.2
The individual and group welfare functions are illustrated in Tables 2 (a) and (b).
The group negotiation set is
N
X
1
X
2
X
3
=
{
(
C,D,G
)
,
(
C,F,G
)
,
(
S, F, R
)
}
and the
negotiation rectangle is
R
X
1
X
2
X
3
=
{
(
C,F,G
)
,
(
S, F, G
)
}
, yielding the compromise
set
C
X
1
X
2
X
3
=
{
(
C,F,G
)
}
: a green foreign-made convertible.
