Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 60

the strong core strong-core( g ) of g is a subset of X , such that for each

x ∈ strong-core( g ) ⊆ X,

there does not exist another consequence x ∈ V ( C )

⊆ X and a

coalition C ⊆

N , such that

x i,B i x

C ,where B i ⊆ holds for each member i ∈ V H is agent i 's opinion.

What is not explicitly mentioned in Definition 3.8 is that for the
consequence x ∈ X and coalition C ⊆ N , although

x i,B i x

C , whether or not holds for each member i ∈

x i,B i x

holds is irrelevant. That is to say, when we consider whether a conse-
quence x is in the strong core, we must not find a defecting coalition
C , and another consequence x , such that all members of C unani-
mously agree that x is better than x , even though some members of
C might concurrently hold the opinion that x is also better than x .
Intuitively, if such C and x can be found, then the defecting coali-
tion C can be successfully formed with the consequence x that every
member of C prefers.

Example 3.21 Consider again the coalition

n 1 = { A 1 ,A 2 }

and the consequence

x 1 =(movie , movie , movie , movie)

in Example 3.19. We can show that x 1 is also in the strong core of the
NTU-IU game that n 1 plays. This is because with the evidences

E = { (Director = Spielberg) ∧ (Type = S.F.) },