Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 63

and a coalition C ⊆ N , such that

x i,B i x S

C . Hence by Definition 3.7, x S holds for each member i ∈ is a conse-

quence in weak core.

Corollary 3.1 For any NTU-IU game, the strong core is empty if
theweakcoreisempty.

Proof By Theorem 3.1, the strong core is a subset of the weak core.

3.6 Effect of Refinedness of Agent Opinions

The reason that, in general, the weak core of an NTU-IU game differs
from the strong core is that the opinions of the agents, which are based
on their (generally limited) experiences, are often uncertain, meaning
that there are often more than one states that the agents consider to
be the possible current prevailing state. To formally capture the effects
of such uncertainty, we have the followings definitions and theorems.

Definition 3.9 (Refinedness Relation of Opinions) Given two

agent opinions B and B , B is more refined than B if B B . ⊂

In other words, B is more refined than B if it is based on B ,with
some of possible states in B removed. Intuitively, B is an refinement
of B after some states are known to be impossible as the current
prevailing state.

Definition 3.10 (Refinedness Relation of Opinion Profiles)
Given two agent opinion profiles

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

and

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