The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 105

coalitional acts. Hence we have the conclusion that the b-core contains
every coalitional act.

Example 4.22 We can restate Example 4.21 in a more formal man-
ner. By Definition 4.6, the b-core of an NTU-PB game g contains all
those coalitional act profile S , such that there does not exist an al-
ternative coalitional act profile S , such that S dom α S and S b-dom α S
for some coalitional act α . However, since here we assume that for any
two coalitional act profiles S 1 and S 2 we always have

( S 1 b-dom α S 2 ) ⇔¬ ( S 1 dom α S 2 )

for some α , therefore, it is just impossible that we have any S and S
such that we have both S dom α S and S b-dom α S at the same time for
any coalitional act α . Hence, it is easy to conclude that any S is in the
b-core.

Therefore, we have several conclusions. First, by Theorem 4.1, the
b-core of a NTU-PB game is always a superset of its core. In the
extreme cases, we prove by Theorems 4.2 and 4.3 that, the b-core
collapses to the core when all the agents' beliefs are accurate, and the
b-core expands to the universal set if for every true preference, there
is always one or more agents that believe the opposite preference, and
any preference believed by agents are wrong. A natural conjecture is
that the b-core expands from the core to the universal set as the agents'
beliefs become more and more inaccurate. This more general result is
given in the following Theorem.

Theorem 4.4 Given two NTU-PB games

g =( N,A, ( i ) ,B ) ,

g =( N,A, ( i ) ,B ) ,

we have

b-core( g ) b-core( g ) ⊆

if B is more accurate than B .