The Complete Picture: sb-core and wb-core - Belief-based Stability in Coalition Formation with Uncertainty - page 161

any such cases are always in the external belief-based cores. Yet this
contradicts Assumption 1 as obviously not every coalition structure is
inthew-coreorthes-core.
Thus, we are left with Options 3 and 4. We argue for Option 3 by
the following rationale:
Suppose that an agent i is considering to raise an objection to a
coalition structure CS by proposing a new deviating coalition C , but
he notices that this objection conflicts with one of his external beliefs
that involves uncertain preferences.
For example, there exists a member j C such that agent i is

not really sure whether j prefers the original coalition structure to the
deviating coalition. In such a case, the objection does not necessarily
conflict with agent i 's beliefs as the possibility of the objection be-
ing acceptable to agent j is not ruled out completely. However, such
objections are ruled out by Option 4.

∈

6.3.2.2 wb-core and sb-cores

We now define the following external-belief-based core, in accordance
to Option 3, as follows.

Definition 6.8 (wb-Core: External-belief-based Weak Core
of NTU-BU Game) A coalition structure

CS = { C 1 ,C 2 ,...,C k }

is in the external-belief-based weak core (wb-core) if there does not
exist another coalition C ∈ CS that satisfy the following two condi-

tions:
1) for all agents i

C ,wehave

∈

C i coal i ( CS );

2) there exists an agent j ∈ C such that, for all agents k ∈ C ,we
do not have

k C ) .

B j (coal k ( CS )