Two Classes of Stability — Strong Core and Weak Core - Belief-based Stability in Coalition Formation with Uncertainty - page 61

we have

B 1 = (see Table 3.2);

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

{ Good }

We assume that there is a defecting coalition C (see Definition 3.8).

If

C = ,

then we have to find a consequence x , such that x is better than x 1
for A 1 (and we forget about A 2 for the moment). However, we know
from Table 3.1 that this is not possible. For the same reason, C cannot
be

{ A 1 }

{ A 2 } . Hence,

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

is in the strong core of the NTU-IU game that n 1 plays.

Example 3.22 It is trivial to show that the consequence

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

is not in the strong core. As discussed in Example 3.20, we consider
the coalition

C 2 = {A 1 },

which has only one member agent A 1 , who prefers

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

to x 2 , or, formally,

x 1 1 ,{ Good } x 2 .

By Definition 3.8, x 2 is not in the strong core.

Example 3.23 We now consider the grand coalition N of all four
agents. Suppose the preference rule P 4 of agent A 4 are exactly the
same as those of rule P 3 in Example 3.14. For this case, we can see
that

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