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Given an NTU-PB game
g
, we use the notation core(
g
)torepre-
sent the set of coalitional act profiles that is in the core of the game
g
. Intuitively, that a coalitional act profile
S
is in the core means
that there is not a subset
C ⊂ N
of agents such that all members
of
C
prefer deviating from their respective coalitions in
S
and joining
the alternative coalitional act
α
=(
C, a
)whereeachmemberofthe
coalition
C
will be better off. If such a condition is satisfied, then the
coalitional act profile
S
is said to be in the core. It should be noted
that here all agents are assumed to compare the alternative coalitional
act
α
=(
C, a
) and the current one
α
i
(
S
) using their respective real
preference
i
. Private beliefs are not involved in the definition.
Example 4.10
In Example 4.5, the core contains the coalitional act
profile
{
(
{
John
,
Mary
}
, Yung Kee
)
}
.
This is because when we look for an alternative coalitional act (
C, a
)
such that all members of
C
prefer (
C, a
)to(
, Yung Kee
),
we know that
C
cannot be a singleton, as singleton coalitions are
always least preferred. On the other hand, if
{
John
,
Mary
}
C
=
{
John
,
Mary
}
,
then we also know that both John and Mary actually prefer
Yung Kee
most (note that here we consider the real preferences, not the believed
preferences). Hence such a coalitional act (
C, a
) cannot be found, and
(
{
John
,
Mary
}, Yung Kee
)isinthecore.
Example 4.11
Continuing the discussion in Example 4.10, it is
easy to show that the core contains only one coalitional act profile
{
(
{
John
,
Mary
}
, Yung Kee
)
}
. If we consider another coalitional act
profile
{
(
{
John
,
Mary
}
,α
)
}
where
α
is not
Yung Kee
, then immediately we know that it cannot
be in the core as (
{
John
,
Mary
},α
) is always less preferred by both
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