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The definitions of weak core and strong core of NTU-BU games are
similar to that of NTU-IU games, as follows:
Definition 6.6 (w-Core:Weak Core of NTU-BU Game)
The
w-core of NTU-BU game is defined as follows. A coalition structure
CS
=
{
C
1
,C
2
,...,C
k
}
in the weak core (w-core) if there does not exist another coalition
C
C
,wehave
C
i
coal
i
(
CS
).
∈
/
CS
such that for all agents
i
∈
Definition 6.7 (s-Core: Strong Core of NTU-BU Game)
The
s-core of NTU-BU game is defined as follows. A coalition structure
CS
=
{
C
1
,C
2
,...,C
k
}
in the strong core (s-core) if there does not exist another coalition
C
∈ CS
such that for all agents
i ∈ C
,wehave
C
i
coal
i
(
CS
)or
C
i
coal
i
(
CS
).
We represent the w-core and s-core of a NTU-BU game
g
as w-
core(
g
)ands-core(
g
) respectively.
Example 6.3
Continuing with Example 6.1, the strong core (s-core)
is empty, and the coalition structures
{{
a, b, c
}}
and
{{
a
}
,
{
b
}
,
{
c
}}
are
in the weak core (w-core).
6.3.2.1
Objections in NTU-BU Games
At this point, we are ready to discuss the external belief-based equiva-
lence of the above concepts. We call them the external-belief-based
weak core (wb-core) and the external-belief-based strong core (sb-
core). Before we get into the formal definitions, let us first review
the underlying rationales.
The intuitive idea is straightforward: for any objection to be con-
sidered as valid, apart from having to fulfil the above conditions given
in Definitions 6.6 and 6.7, we also require the objection to be consis-
tent with the belief of at least one member of the deviating coalition.
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