Example Application of the b-Core: Buyer Coalition Problem with Beliefs - Belief-based Stability in Coalition Formation with Uncertainty - page 128

Again here we abuse the term b-core , and define it to be a set
of stable coalition structures based on private beliefs, instead of a
set of consequences. However, the principles for deciding stability of
coalitions are the same as those we use in Definition 4.6 of the b-core
of NTU-PB game.

5.2.4 Properties of the b-Core in NTU-Buyer Games

For completeness of our discussion, we are listing out the following
properties regarding the b-core of NTU-buyer games. These properties
are the counterparts of the theorems of Chapter 4.
The first theorem below states that the core is included in the
b-core of an NTU-Buyer game, which should not be a surprise.

Theorem 5.1 The core of an NTU-Buyer game is a subset of the

b-core.

Proof In Definition 5.5, part of the requirements for a coalition
structure CS to be in the b-core of an NTU-Buyer game is that there
does not exist another coalition structure CS and a subset of agents
C

⊆ N such that for each agent a ∈ C ,wehave

b a|CS a b a|CS ,

which implies that any coalition structure that has no objection ac-
cording to the criterion of b-core also has no objection according to
the criterion of the core, meaning that CS is also in the core.

As Theorem 4.4 in Chapter 4, we have a theorem that relates the
accuracy of agents' beliefs to the size of the b-core. Before we present
these theorems, we first define what exactly we mean by belief accuracy
in the context of NTU-Buyer games. The following definition is an
adaptation of Definition 4.8 to the context of NTU-Buyer games.

Definition 5.6 (Is-more-accurate-than Relation over Agent