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We define a coalitional act profile to be a tuple
S
of coalitional acts
that corresponds to a coalition structure, i.e.,
S
=
{
α
1
,α
2
,...,α
k
}
where
k
is the number of possible coalitions in the coalition structure,
and each
α
i
=(
C
i
,a
i
), 1
k
represents the coalitional act of the
i
th
coalition. We use
C
i
(
S
) to denote the coalition in
S
which the agent
i
is a member of, and
α
i
(
S
) to denote its corresponding coalitional act.
That is,
α
i
(
S
)=(
C
i
,a
i
) such that
i
i
∈
C
i
.
Example 4.9
Consider the dating game in Example 4.3 again. The
coalition structure that agents
a
and
b
go to see the movie together,
while agent
c
goes alone, corresponds to the following coalitional act
profile:
S
=
{
(
{a, b},
movie)
,
(
{c},
movie)
}
for which we have
α
a
(
S
)=
α
b
(
S
)=
α
1
=(
{
a, b
}
,
movie)
,
α
c
(
S
)=
α
2
=(
{
c
}
,
movie)
.
Given Definition 4.1 of NTU-PB game, it is obvious that the clas-
sical solution concept of the core corresponds to the situation when all
agents are omniscient. That is, every agent knows correctly the pref-
erences of all other agents, and hence does not use their private beliefs
to determine the stability of coalitions.
Definition 4.2 (Core of NTU-PB Games)
The
core of an NTU-
PB game
N,A,
(
i
)
,B
is the set of coalitional act profile
S
=
{
(
C
1
,a
1
)
,
(
C
2
,a
2
)
,...,
(
C
k
,a
k
)
}
,
where
C
1
,C
2
,...,C
k
⊆
for any
i
=
j
, such that there does not exist an alternative coalitional act
α
=(
C, a
),
C ⊂ N
,
a ∈ A
, such that
α
i
α
i
(
S
) for all
i ∈ C
.
N
,
a
1
,a
2
,...,a
k
∈
A
,
C
i
∩
C
j
=
∅
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