Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 57

If we compare Definition 3.6 and Definition 3.7, we see that the
main difference is that in Definition 3.7, each agent determines its
preferences using its own opinion, instead of the prevailing state s ∗ .
We should note that generally speaking, an agent's opinion is a set
of states instead of a single state. Intuitively, these are all 'possible
states' concluded from the agent's experience, and the agent does not
have any information to determine which of these 'possible states' is
the actual prevailing state. However, by the Definition 3.1 of NTU-IU
games, an agent can still form his preferences using his preference rules
when there are multiple possible states in his opinion. However, as we
do not require s ∗ to be a common knowledge, the agents will determine
their respective preferences based on their own (generally conflicting)
opinions.
If we read the definition of weak core careful, we will agree that
the condition for a particular consequence x to be in the weak core is
that there does not exist another consequence x ∈ X and a coalition

C ⊆ C , N , such that for each member i ∈

1. x i,B i x holds and;
2. x i,B i x does not hold.

In other words, for x to be in the weak core, we should not be able to
find a defecting coalition C , such that there is another consequence
x which all members of C will unanimously agree to be better than
x and they also unanimously agree that x is not better than x .That
is to say, we should not be able to find a defecting coalition C ,the
members of which unanimously agree that x is strictly better than x .
This is because if such C can x can be found, that x is not attractive
enough to hold everyone (at least the members of C )tobeinthe
grand coalition N .

Example 3.19 In Example 3.5 in Section 3.2, for an NTU-IU game

that the coalition

n 1 = {A 1 ,A 2 }