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Principle 1 represents an important shift in perspective from conventional game theory.
With the conventional approach, the utility of an individual defines its self-interest with
respect to the instantiation of actions taken by all players. By contrast, we view the util-
ity of an individual as the consequent of a hypothetical proposition whose antecedent
is the assumption that those who influence it have identified their most preferred out-
comes.
Definition 1.
A
conjecture
for
X
j
is an action profile, denoted
a
j
, that is hypothesized
as
X
j
's most preferred outcome. Let
X
m
=
{
X
j
1
,...,X
j
m
}
be a subgroup of
X
n
.
A
joint conjecture
for
X
m
, denoted
α
m
=
{
a
j
1
,
···
,
a
j
m
}
, is a collection of action
m
=
profiles in
A
A×···×A
(
m
times), where
a
j
l
is a conjecture for
X
j
l
,
l
=1
,...,m
.
Now suppose
X
i
is influenced by a subgroup
X
m
. Given a joint conjecture
α
m
,the
consequent of the hypothetical proposition is a conditional preference ordering for
X
i
.
X
m
=
X
n
that influences
X
i
Definition 2.
Let
{
X
j
1
,...,X
j
m
}
be a subgroup of
and let
α
m
=
X
m
.A
conditional utility
{
a
j
1
,
···
,
a
j
m
}
be a joint conjecture for
α
m
)
is a real-valued function defined over
A
u
X
i
|
X
m
(
that specifies the preference or-
dering for
X
i
given the joint conjecture
α
m
. That is,
u
X
i
|
X
m
(
a
·|
α
m
)
>u
X
i
|
X
m
(
a
|
α
m
)
|
means that
X
i
prefers
a
to
a
, given that
X
j
l
conjectures
a
j
l
,
l
=1
,...,m
.
Each
X
i
must define a conditional utility for the joint conjectures of the subgroup that
influences it. This requirement increases the complexity of a problem statement over
the conventional requirement of specifying only one categorical utility for each
X
i
.
However, as we shall explore in Section 5, there often will be ways to simplify the
specification that keeps the complexity under control. Nevertheless, the inclusion of
social influence will generally result in increased complexity.
2.3
Group Preference
Once we extend beyond self-interest via social bonds induced by conditional prefer-
ences, it becomes possible to consider a more general notion of group preference. It
may happen that the social bonds are so strong that unanimity will result, but that situ-
ation will not generally obtain. If agents disagree regarding what outcome is best, then
some degree of conflict, or discord, will exist within the group. Thus, when designing
a system whose members must coordinate, a critical issue is the
concordance
,orthe
degree of harmony, among its members. Accordingly, a meaningful notion of group
preference is for its members to function concordantly.
Definition 3.
Let
X
k
=
{
X
i
1
,...,X
i
k
}
be a subgroup of
X
n
.A
concordant utility
k
such that, for each joint conjecture
α
k
=
U
X
k
is a real-valued function defined over
A
k
,
U
X
k
(
α
k
)
defines the concordance of
α
k
.When
k
=1
, the con-
cordant utility becomes a conventional categorical utility for
X
k
, that is,
U
X
k
≡
(
a
i
1
,...,
a
i
k
)
∈A
u
X
k
.
When
k>
1
, the concordant utility is a generalization of individual utility which,
rather than providing a preference ordering for a single agent over
A
, provides a con-
A
k
of joint conjec-
cordant ordering for a
k
-member subgroup over the product space
tures. When
a
i
1
=
···
=
a
i
k
, the concordant utility measures the degree of harmony
