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Obviously, for all members of
C
2
(that is, agent
A
1
in this case),
x
1
=(movie
,
movie
,
movie
,
movie)
is a better consequence. So we have found a consequence
x
1
∈ V
(
C
2
)
⊆ X
and a coalition
C
2
⊆ N
, such that
x
1
1
,{
Good
}
x
2
holds but not
x
2
1
,{
Good
}
x
1
.
By Definition 3.7,
x
2
is not in the weak core.
We now consider a situation with a stronger condition. That is,
while we insist that all members in
C
should consider
x
to be better
than
x
,
however
, we do allow some members of
C
to also consider that
x
is better than
x
. Note that this is possible, as we have demonstrated
in Example 3.12. We can see that with such a seemingly slight revision,
in general a number of consequences in the weak core will be found not
to satisfy the new condition any more. For each of these consequences
x
, although we cannot find any coalition
C
, the member of which will
unanimously agree that
x
is better than
x
but
x
is not better than
x
,
maybe it is just the case that some members of
C
actually consider
that both
x
is better than
x
and
x
is better than
x
. Therefore, if we
remove these consequences from the weak core because they no longer
satisfy the revised condition, then we have a new set of consequences
that satisfy a stronger condition for membership. This new set of con-
sequences is named the
strong core
of the game. We formally define
strong core in the following definition.
Definition 3.8 (Strong Core: Unconditionally Stable Solu-
tions)
GivenanNTU-IUgame
g
=
N,E,
(
I
i
)
,H,X,
(
P
i
)
,
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