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Definition 6.9 (sb-Core: External-belief-based Strong Core
of NTU-BU Game)
A coalition structure
CS
=
{
C
1
,C
2
,...,C
k
}
is in the external-belief-based strong core (sb-core) if there does not
exist another coalition
C
∈
CS
that satisfy the following two condi-
tions:
1) for all agents
i ∈ C
,wehave
C
i
coal
i
(
CS
)
orC
i
coal
i
(
CS
)
,
C
such that, for all agents
k
C
,we
2) there exists an agent
j
∈
∈
do not have
k
C
)
.
B
j
(coal
k
(
CS
)
Intuitively, the wb-core requires every objection to fulfill two con-
ditions.
First, as in w-core, every agent in the deviating coalition must
prefer the new coalition to the coalition of which he is currently a
member.
Second, the objection must not be in conflict with the external-
beliefs of all members of the coalition simultaneously. That is, at least
one agent in the coalition can see that the objection is a viable option
and has a chance to succeed, and this agent will therefore be willing to
raise the objection, which will then be accepted by all other members
because of the first condition.
The idea behind the sb-core is similar, except that potential objec-
tions are also allowed in the first condition. (The second condition, on
the other hand, needs not be extended to cover potential objections
for reasons already discussed above).
The four proposed concepts, namely, s-core, w-core, sb-core, and
wb-core, are labelled collectively as the belief-based cores.
{{a}, {b}, {c}}
Example 6.4
Continuing with Example 6.1,
and
{{a, b, c}}
are in the wb-core, whereas only
{{a}, {b}, {c}}
is in the
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