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John and Mary than
.Furthermore,itis
not possible that the core contains any coalitional act profile with
singleton coalitions, i.e., a coalitional act profile in the form of
{
(
{
John
,
Mary
}
, Yung Kee
)
}
{
(
{
John
}
,α
1
)
,
(
{
Mary
}
,α
2
)
}
,
because singleton coalitions are always least preferred.
Example 4.12
As discussed in Example 4.3, the core of the game in
that example is an empty set. This is because, as illustrated in Figure
4.3, for any coalitional act we can always find another coalitional act
that is more preferred by the agents involved.
The definition of the core of an NTU-PB game in Definition 4.2
is based on the classical definition of the core. However, as discussed
in Examples in the previous section, there are situations in NTU-PB
games that are not well handled by the core. Intuitively, these are
situations in which stable coalitions are formed with misunderstand-
ing. The reason is that, unlike in traditional NTU games, the agents'
preferences in NTU-PB games are private information represented by
beliefs, which is not reflected by the traditional concepts.
In the following, we will define our main stability criterion for this
chapter. However, before we do this, we shall define two more concepts:
Definition 4.3 (Domination Relation
dom
)
The
Domination Re-
lation
,
dom
, is defined as follows. Given any two coalitional act profiles
S
1
and
S
2
,wesay
S
1
is dominated by
S
2
through a coalitional act
α
=(
C, a
)
∈
S
2
, written
S
2
dom
α
S
1
, if and only if for each agent
i
∈
C
⊆
N
,wehave
α
i
α
1
(
S
1
).
Definition 4.4 (Belief-based Domination Relation
b-dom
)
The
Belief-based Domination Relation
,
b-dom
, as follows. Given any two
coalitional act profiles
S
1
and
S
2
,wesay
S
1
is dominated based on be-
lief by
S
2
through a coalitional act
α
=(
C, a
)
S
2
, written
S
2
b-dom
α
S
1
,
if there exists an agent
i ∈ C
such that, for each agent
k ∈ C
,wehave
bel
i
(
α
k
α
k
(
S
1
)).
∈
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