Example Application of the b-Core: Buyer Coalition Problem with Beliefs - Belief-based Stability in Coalition Formation with Uncertainty - page 127

Problem with Beliefs

It is not di cult to see that each of the agents in coalition structure
CS must be enjoying a consequence satisfactory enough if no one of
them is ever motivated to deviate and form a new coalition structure
CS . Such suciently satisfactory consequences are generally differ-
ent in different elements of the coalition structure CS .Ifweinsistin
strictly follow Definition 2.4, then there will be one separate core for
each of the coalition

C i ∈ CS = { C 1 ,C 2 ,...,C m } ,

such that any members of such a core will be able to keep the corre-
sponding coalition in CS stable.
As pointed out in Chapter 4, there is a drawback when applying the
classical concepts to games where the agents' actions are determined
by private beliefs instead of public knowledge: a buyer coalition should
also be viewed as stable as long as every agent in that coalition believes
that there are no better alternatives, no matter whether those beliefs
are accurate or not. For this reason, the b-core is a more suitable
criterion for this problem:

Definition 5.5 (The b-Core in NTU-Buyer Game) Given age-

nts

N = { a 1 ,a 2 ,...,a n } ,

and a coalition structure

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

we say CS is in the b-core of an NTU-Buyer game if there does not
exist another coalition structure CS and a subset of agents C ⊆ N
such that 1) for each agent a ∈ C ,wehave

b a|CS a b a|CS ,

and 2) there exists at least one agent j ∈ C such that for each agent
k ∈ C ,wehave

bel j ( b k|CS k b k|CS ) .