Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 67

Fig. 3.3 Illustration of Theorem 3.3.

Theorem 3.4 Given a collections of NTU-IU games

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

where the opinion B i of all agents i N in all games are correct, which

differ only in the agents' experience ( I i ) k and hence the corresponding
opinions profile O k , the core and strong core of all games in the collec-
tion are empty if the weak core of any of the games in the collection
is empty.

∈

Proof Assume the weak core of any game g in the collection is
empty. Since the opinion all agents are correct, it follows from Theorem
3.3 that the core of g is also empty. Since all games in the collection
differ in the agents' experience and hence their opinions only, all games
must have the same core, and so the core of all games are empty, and
hence the strong cores are also empty.

The idea is illustrated in Figure 3.4, where the strong cores (solid
lines) and weak cores (dashed lines) of three games are shown. The
opinions of all three games are assumed to be correct so they share a
common core (shaded area). It is obvious from the diagram that the