Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 66

in the opinion B i , together with other states. Intuitively, agent i does
not need to be certain that s ∗ is the prevailing state. Instead, agent i 's
opinion is correct if it include s ∗ as one of the uncertain possibilities.

Theorem 3.3 GivenanNTU-IUgame

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

where the opinion B i of all agents i ∈ N are correct, we have

core( g, s ∗ ) , strong-core( g ) ⊆

and

core( g, s ∗ ) ⊆ weak-core( g ) ,

where s ∗ is the prevailing state.

Proof Since each agent's opinion contains the prevailing state if his
opinion is correct, and that the core for the prevailing state in this case
is simply the same as the strong core, and weak core for a modified
game where all agent's opinions consist only of the prevailing state. It
follows from Theorem 3.2 and Definition 3.11 that the strong core of
a game, where each agent's opinion is correct, is a subset of the core,
which in turn is a subset of the weak core. This is depicted in Figure
3.3.

Example 3.24 We revisit Example 3.5. Suppose the prevailing state
is s ∗ = Good. If we assume that the agents A 1 , A 2 , A 3 ,and A 4 are
more experienced now (than those shown in Tables 3.2 to 3.5), that
their opinions all become

s ∗ } B 1 = B 2 = B 3 = B 4 = { = { Good } .

In this case, according to Table 3.1, all agents should prefer seeing
movie to playing tennis. It is not di cult to show that in this case the
strong core, the core for the prevailing state s ∗ = Good, and the weak
core, of the game are identical, which is the set { movie , movie , movie,
movie } .