The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 98

designed for demonstration purposes, has revealed that the b-core is a
more suitable criterion for measuring the stability of coalition forma-
tion game where private beliefs are involved.

The following theorem formally states the relationship between the
b-core and the core for NTU-PB games.

Theorem 4.1 The core of a NTU-PB game is a subset of its b-core.

Proof In Definition 4.6, part of the requirement for a coalitional act
profile S to be in the b-core of an NTU-PB game is that there does
not exist any alternative coalitional action profile S 2 ,sothat S 2 dom α S
for some α , which implies that any coalitional action profile that has
no objection according to the criterion of b-core also has no objection
according to the criterion of the core, meaning that S is also in the
core.

Theorem 4.1 is graphically illustrated in Figure 4.11.

Set of all profiles

b-core

core

General Case

Fig. 4.11 Illustration of Theorem 4.1.

In the next section, we will discuss several interesting properties of
the b-core.