The Complete Picture: sb-core and wb-core - Belief-based Stability in Coalition Formation with Uncertainty - page 159

Otherwise, if the objection is inconsistent with every member's be-
liefs, then no agent will realise there is an objection and it will never
be raised.
Now, the remaining problem is that how to define an objection as
being consistent with the agents' beliefs. There are four options:
Option 1: Raisable definite objections. We can require at least one
member of the deviating coalition certainly believes that every member
of the coalition prefers that to his previous coalition.
That is, let CS be a coalition structure and let C

CS be a
coalition (an objection), we call C a raisable objection if there exists
agent j

∈ /

C such that for all agent k C ,wehave

∈ ∈

B j ( C k coal k ( CS )) .

Option 2: Raisable potential objections. Alternatively, we can re-
quire that at least one member of the deviating coalition believes that
every member of the coalition either prefers the deviating coalition to
his previous coalition or is uncertain about it.
That is, let CS be a coalition structure and let C CS be a

coalition (an objection), we call C a raisable objection if there exists
agent j in C such that for all agent k ∈ C ,wehaveeither

∈ /

B j ( C k coal k ( CS ))

or

B j ( C k coal k ( CS )) .

Option 3: No conflicting raisable definite objections. Alternatively,
we can require that for at least one member of the deviating coalition,
the objection does not conflict with his external beliefs.
That is, let CS be a coalition structure and let C ∈ CS be a
coalition (an objection), we call C a raisable objection if there exists
agent j in C such that for all agent k C ,

∈

B j (coal k ( CS ) k C )