Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 58

plays, the consequence

x 1 =(movie , movie , movie , movie)

is in the weak core. To see this, we first list the opinions of the agents
in n 1 . Given the evidences

E = { (Director = Spielberg) ∧ (Type = S.F.) } ,

we have

B 1 = (see Table 3.2);

B 2 = { Good } (see Table 3.3).

{ Good }

Let C be the defecting coalition in Definition 3.7. Without loss of
generality, let

C = { A 1 } .

By the agent preferences shown in Table 3.1, we see that both agents
A 1 and A 2 prefer to see the movie. Therefore, we shall not be able to
find an consequence x , such that x is better than x 1 for A 1 and x 1 is
not better than x for A 1 , because seeing movie is the best consequence
for him. For the same reason, there is no better consequence than x 1
for A 1 .Hence,

x 1 =(movie , movie , movie , movie)

is in the weak core for the coalition n 1 .

Example 3.20 Continuing from Example 3.19, we consider the con-

sequence

x 2 = (tennis , tennis , tennis , tennis) .

This consequence is not in the the weak core. For example, consider
the coalition

C 2 = {A 1 }.