The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 107

More precisely, you believe that Mary prefers both eating at Yung
Kee to Spasso , and Mary also believes that you prefer that the two of
you eat at Yung Kee to Spasso . However, all of your other beliefs are
inaccurate. In this case, we can follow the logic of reasoning presented
in Examples 4.21 and conclude that any coalition with any act is stable,
except that the coalitional act that both of you eat at Spasso is no
longer stable. This can be understood as follows. When Mary (or you)
considers the proposal that both of you eat at Spasso , she (or you) will
surely raise the objection that both of you should have a better option
of eating at Yung Kee . Mary (and you) believes that both of you will
agree that this is a valid objection, so Mary (and you) will actually
raise this objection because she (and you) believes that this objection
will be agreed by the other person. Actually, even if only you have the
correct belief of this preference (and Mary does not), then you will
raise this objection, which will be gladly accepted by Mary. Hence the
proposal that both eat at Spasso will not be stable even if only you
have accurate belief of this particular preference.

Example 4.24 The discussion in the previous example can be pre-
sented in a more formal manner. In the new setting, we still have the
assumption that for any two coalitional act profiles S 1 and S 2 we have

( S 1 b-dom α S 2 ) ⇔¬ ( S 1 dom α S 2 )

for some α , except that when

S 1 = { ( { John , Mary } , Yung Kee ) } ,

we do not have

{ ( { John , Mary } , Spasso ) } dom α { ( { John , Mary } , Yung Kee ) }

for any α . This is because both Mary and you now have the correct
belief about this particular preference, that is,

bel John ( { ( { John , Mary }, Yung Kee ) } Mary { ( { John , Mary }, Spasso ) } ) ,