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where
pa (
a
i
)=
{
a
i
1
,...,
a
i
p
i
}
is the joint conjecture corresponding to
pa(
X
i
)=
{
.If
p
i
=0
,then
u
X
i
|
pa (
X
i
)
=
u
X
i
, a categorical utility.
To illustrate, consider the network illustrated in Figure 1.
X
1
is a root vertex, and
possesses a categorical utility
u
X
1
,
pa (
X
2
)=
X
i
1
,...,X
i
p
i
}
{
X
1
}
,and
pa(
X
3
)=
{
X
1
,X
2
}
.The
concordant utility is
U
X
1
X
2
X
3
(
a
1
,
a
2
,
a
3
)=
u
X
1
(
a
1
)
u
X
2
|
X
1
(
a
2
|
a
1
)
u
X
3
|
X
1
X
2
(
a
3
|
a
1
,
a
2
)
.
(23)
X
1
u
X
2
|
X
1
u
X
3
|
X
1
X
2
X
2
X
3
Fig. 1.
The DAG for a three-agent system
If the utilities of all agents are categorical, then no social influence exists, the corre-
sponding DAG has no edges, and, hence, no social bonds are generated. The aggrega-
tion formula defined by (22) becomes analogous to the creation of the joint distribution
of independent random variables as the product of the marginal distributions, and ag-
gregation sheds no additional light on group behavior.
4
Conditional Games
A
conditional game
is a triple
{X
n
,
A
,U
X
n
}
where
X
n
=
{
X
1
,...,X
n
}
is a group
of
n
agents with product action space
A
1
×···×A
n
and
U
X
n
=
U
X
1
···
X
n
is
a concordant utility. Equivalently, by application of (22), a conditional game can be
defined in terms of the conditional utilities
u
X
i
|
pa (
X
i
)
,
i
=1
,...,n
. If all utilities are
categorical, a conditional game becomes a conventional game.
With a conditional game, the possibility exists for an expanded notion of rational
behavior. To proceed, we observe that, since each agent can control only its own actions,
what is of interest is the utility for the group if all agents
make conjectures over, and
only over, their own action spaces
.
Definition 8.
Consider the concordant utility
U
X
1
···
X
n
(
a
1
,...,
a
n
)
.Let
a
ij
denote the
j
th element of
a
i
; that is,
a
i
=(
a
i
1
,...,a
in
)
is
X
i
's conjecture. Next, form the action
profile
(
a
11
,...,a
nn
)
by taking the
i
th element of each
X
i
's conjecture,
i
=1
,...,n
.
Now let us sum the concordant utility over all elements of each
a
i
except the
ii
-th
elements to form the
group welfare function
for
A
=
{
X
1
,...,X
n
}
, yielding
w
X
1
···
X
n
(
a
11
,...,a
nn
)=
∼
a
11
···
U
X
1
···
X
n
(
a
1
,...,
a
n
)
,
(24)
∼
a
nn
where
∼
a
ii
means the sum is taken over all
a
ij
except
a
ii
.The
individual welfare
function
of
X
i
is the
i
-th marginal of
w
X
1
···
X
n
, that is,
w
X
i
(
a
ii
)=
∼
a
ii
w
X
1
···
X
n
(
a
11
,...,a
nn
)
.
(25)
