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in
S
1
, then we know that both John and Mary in the coalition in
α
genuinely prefer
S
1
to
S
2
. Formally, we write
{
(
{
John
,
Mary
}
, Yung Kee
)
}
dom
(
{
John
,
Mary
},Yung Kee
)
,α
)
{
(
{
John
,
Mary
}
}
.
In general, it is not di
cult to see that if
S
1
dominates
S
2
through
α
,then
S
1
is an object to
S
2
, and the 'ground' of such objection is the
coalitional act
α
. Hence, the definition of the core of NTU-PB games
can be rephrased using domination relation as follows.
Definition 4.5 (Core of NTU-PB Games)
The
core of an NTU-
PB game N,A,
(
i
)
,B
is the set of all those coalitional act profile
S
, such that there does not exist a coalitional act
α
and an alternative
coalitional act profile
S
, such that
S
dom
α
S
.
Example 4.15
By Definition 4.4, a belief-based domination rela-
tion exists if there is at least one agent who believes the domination
relation to be real. In Example 4.4, for instance, we have
{
(
{
a, b, c
}
,
movie)
}
b-dom
(
{a,b,c},
movie)
{
(
{
a
}
,
movie)
,
(
{
b, c
}
,
movie)
}
because this is believed to be true by all agents. In fact, according to
Definition 4.4, as long as any one of the agents
a
,
b
or
c
believes that
this is true, that is, it is true that all agents like the coalitional act
(
,
movie) the most, the belief-based domination relation will
hold, no matter whether or not the other two agents believe this to be
true.
{
a, b, c
}
Now, we are ready to define a new belief-based stability criterion,
the b-core:
Definition 4.6 (The b-Core of NTU-PB Game)
The
b-core of
an NTU-PB game g
is defined as follows. Given an NTU-PB game
g
=
N,A,
(
i
)
,B
,
a coalitional act profile
S
=
{
(
C
1
,a
1
)
,...,
(
C
k
,a
k
)
},
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