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if all members conjecture the same action profile. When the conjectures are different,
U
X
k
(
a
i
1
,...,
a
i
k
)
measures the degree of concordance that exists among the members
of the subgroup if each
X
i
l
were to view
a
i
l
as its most-preferred outcome. The expres-
sion
U
X
k
(
α
k
)
>U
X
k
(
α
k
)
means that
α
k
causes a more severe conflict for the group
than does
α
k
.
Definition 4.
Let
X
k
=
{
X
i
1
,...,X
i
k
}
and
X
m
=
{
X
j
1
,...,X
j
m
}
be disjoint sub-
m
,a
conditional concordant utility
given
α
m
is a
groups of
X
n
.Foreach
α
m
∈A
k
that defines a concordant utility
real-valued function
U
X
k
|
X
m
(
·|
α
m
)
defined over
A
for
X
m
jointly conjectures
α
m
.When
k
=1
, the concordant utility
becomes a conditional utility for
X
k
, that is,
U
X
k
|
Xm
≡
X
k
given that
u
X
k
|
Xm
(see Definition 2).
Example 1.
Suppose the group
is to purchase an automobile.
X
1
is to
choose the model, either a convertible (
C
)orasedan(
S
),
X
2
is to choose the manufac-
turer, either domestic (
D
)orforeign(
F
), and
X
3
is to choose the color, either red (
R
)
or green (
G
). The action spaces are
{
X
1
,X
2
,X
3
}
A
1
=
{
C,S
}
,
A
2
=
{
D,F
}
,
A
3
=
{
R,G
}
.
U
X
1
X
2
[(
S, D, G
)
,
(
C,F,R
)]
means that concordance
is higher (i.e., it is less conflictive) for the subgroup
U
X
1
X
2
[(
C,F,R
)
,
(
S, D, G
)]
≥
if
X
1
were to most-prefer
a foreign made red convertible and, simultaneously,
X
2
were to most-prefer a domestic-
made green sedan, than if
X
1
were to most-prefer a domestic green sedan and, simul-
taneously,
X
2
were to most-prefer a foreign-made red convertible. Thus, even though
the two stakeholders do not have the same preferences in either case, the severity of
the differences in opinion is less for the
(
C,F,R
)
,
(
S, D, G
)
combination than for the
(
S, D, G
)
,
(
C,F,R
)
combination.
U
X
1
|
X
2
(
C,F,R
{
X
1
,X
2
}
C,F,R
)
means that
X
1
prefers a forei-
gn made red convertible to a domestic-made green sedan, given the hypothesis that
X
2
most-prefers a foreign-made red convertible. Notice that, since the consequent involves
only one stakeholder, the conditional joint conjecture ordering becomes a conditional
ordering, and we may more properly rewrite this expression as
u
X
1
|
X
2
(
C,F,R|C,F,R
)
≥ u
X
1
|
X
2
(
S, D, G
)
|C,F,R
)
.
u
X
1
|
X
2
X
3
[
S, D, G
|
C,F,R
)
≥
U
X
1
|
X
2
(
S, D, G
|
(
C,F,R
)
,
(
S, D, G
)]
means that
X
1
prefers a domestic-made green sedan to a foreign-made red convert-
ible, given the hypothesis that
X
2
most-prefers a foreign-made red convertible and that
X
3
most-prefers a domestic-made green sedan.
U
X
2
X
3
|
X
1
[(
C,F,R
)
,
(
S, F, R
)
|
(
C,F,R
)
,
(
S, D, G
)]
≥
u
X
1
|
X
2
X
3
[
C,F,R
|
|
S, D, G
]
≥
U
X
2
X
3
|
X
1
[(
S, D, G
)
,
(
C,D,G
)
|
S, D, G
]
means that the subgroup
is less conflicted, given that
X
1
most-prefers
(
S, D,
G
)
,for
X
2
and
X
3
to prefer
(
C,F,R
)
and
(
S, F, R
)
, respectively, than respectively to
most-prefer
(
S, D, G
)
and
(
C,D,G
)
.
{
X
2
,X
3
}
Computing the concordant utility of a group
is a key component
of our approach, since that function captures all social relationships that exist in the
group. Using this function, we can define notions of rational behavior both for the group
as a whole and for each of its members. Rather than define such a function from first
principles, however, we propose to synthesize it from more elementary relationships.
We begin by imposing the following principle.
X
n
=
{
X
1
,...,X
n
}
