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is in the weak core, but it is not in the strong core. This is because for
agents
A
3
and
A
4
, whose opinions are
B
3
=
B
4
=
{
Good
,
Average
}
,
seeing movie is better than playing tennis, but playing tennis is also
better than seeing movie. On the other hand,
x
2
= (tennis
,
tennis
,
tennis
,
tennis)
is in neither the weak core or the strong core, because agents
A
1
and
A
2
will definitely object to it.
Thus we see that the (
N,x
1
) solution is very stable, and is there-
fore a good candidate solution that should be considered by coalition
formation mechanisms. On the other hand, (
N,x
2
) is not stable and
should be rejected.
In general, we have the following result:
Theorem 3.1
The weak core is a superset of the strong core.
Proof
For an NTU-IU game
g
=
N,E,
(
I
i
)
,H,X,
(
P
i
)
,
let
S
denote its strong core, and
x
S
∈
S
be a consequence in
S
.By
Definition 3.8, for
x
S
, there does not exist another consequence
x
∈
V
(
C
)
⊆
X
and a coalition
C
⊆ N
, such that
x
i,B
i
x
S
holds but not
x
S
i,B
i
x
C
,where
B
i
⊆
for each member
i
V
H
is agent
i
's opinion. Obviously
x
S
satisfies the weaker condition that there does not exist another
consequence
∈
x
∈ V
(
C
)
⊆ X
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