Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 64

where both B i and B i are the opinions of agent i ,wesay O is more
refined than O if there exists agent i such that B i is more refined than
B i , and either B j is more refined than B j = B j or B j for all other

agents j ∈ N .

Theorem 3.2 Given two games,

g = N,E, ( I i ) ,H,X, ( P i )

and

g = N,E , ( I i ) ,H,X, ( P i ) ,

with agent's opinions

O =( B 1 ,B 2 ,...,B n )

and

O =( B 1 ,B 2 ,...,B n )

weak-core( g )and

respectively, we have weak-core( g ) ⊆

strong-core( g ) ⊆ strong-core( g )

if O is more refined than O .

Proof Consider the objections in the two games g and g .Wesee
that any definite objection in g is also a definite objection in g .In
other words, any solution that has no objection in g also has no ob-
jection in g , meaning that any solution that is in the weak-core of
g is also in the weak-core of g . Similarly, we see that any potential
objection in g is also a potential objection in g , or, in other words,
any solution that has no potential objection in g also has no potential
objection in g , meaning that any solution that is in the strong-core of
g is also in the strong-core of g . This is depicted in Figure 3.2.

We end this section with a study of the relationships between the
core and the concepts of strong core and weak core. The core assumes
common knowledge, which means that each agent is assumed to know