Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 55

For the two memb ers of C , the consequence

x =(movie , movie , tennis , tennis)

is more attractive than x , as according to Table 3.1, both

x 3 ,{ Average }

x

and

x 4 ,{ Average } x

should hold. Therefore, by Definition 3.6, x is not in the core.

Again, the fact that a consequence is not in the core core( g, s ∗ )
does not prevent it from being the actual consequence of a stable
coalition. To illustrate this, consider again the scenario of Example
3.18. Although we assume that the prevailing state is

s ∗ = Average ,

the grand coalition N with the consequence

x =(movie , movie , movie , movie)

is still stable if the agents, from their own experiences, somehow con-
clude that the quality of the movie is good. If this is the case, every
one of them will be happy to see the movie.
It is possible that the core core( g, s ∗ ) is empty. Intuitively, this
means that the grand coalition N is always unstable if the prevailing
state s ∗ ∈ V H is a common knowledge. We have to note again that even
if the core is empty, this does not imply that stable grand coalition
cannot be formed. All that matter are the opinions of agents and their
preference rules. This mimics the real life situations and, as a matter
of fact, is more realistic than the 'ideal' situation in which every piece
of information is a common knowledge, as assumed in conventional
game theoretic analysis.
As seen from Definition 3.6, the core requires the knowledge of the
value of the prevailing state s ∗ , which makes it generally not always