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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 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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