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3.5 The Strong Core and the Weak Core
We can now define the coalition stability criteria. For the sake of com-
pleteness, we first define the definition of the core for NTU-IU game,
which is only meaningful if the prevailing state
s
∗
∈
V
H
is known. This
serves as a baseline reference in the subsequent discussions.
Definition 3.6 (The Core of an NTU-IU Game)
Given
an
NTU-IU game
g
=
N,E,
(
I
i
)
,H,X,
(
P
i
)
,
the
core
core(
g, s
∗
)
of g for a given prevailing state s
∗
∈
V
H
is a subset
of
X
, such that for each
core(
g, s
∗
)
x
∈
⊆
X,
there does not exist another feasible consequence
x
∈
V
(
C
)
⊆
X
and
a coalition
C
⊆
N
, such that
x
i,{s
∗
}
x
holds but not
x
x
i,{s
∗
}
C
,where
s
∗
∈
for each member
i
∈
V
H
is the prevailing state.
Example 3.17
Consider scenario depicted in Section 3.2, which is
formalised as an NTU-IU game in Example 3.5. Assume that the pre-
vailing state is
s
∗
= Good, then the consequence
x
=(movie
,
movie
,
movie
,
movie)
is in the core. This is easy to prove by contradiction. Suppose
x
is not
in the course, then there must be a consequence
x
=(
o
1
,o
2
,o
3
,o
4
)
∈
X
such that at least one of
o
1
,
o
2
,
o
3
,and
o
4
is tennis instead of movie.
Without loss of generality, suppose
o
1
= tennis. Then according to
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