The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 102

are accurate. This is expected because an NTU-PB game where all
agents' beliefs are accurate means that each agent actually knows the
real preferences of all agents. Thus the game is effectively not different
from a classical common-knowledge game in this case and it is not
surprising that its b-core is just the same as a traditional common
knowledge-based core.

Set of all Profiles

b-core and core

All beliefs are accurate

Fig. 4.12 Illustration of Theorem 4.2.

Another extreme case worth mentioning is where the agents have
all the incorrect beliefs. By this we mean that for every true preference
that the agents have, there is always one or more agents that believe
the opposite preference, and any preference believed by agents are
wrong. Here, we have the following theorem, which states that all
coalitional act profiles are in the b-core if this is the case:

Theorem 4.3 GivenanNTU-PBgames

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

and that for any two coalitional act profiles S 1 and S 2 we have

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

for some α .Thenfor any coalitional action profile

S = { ( C 1 ,a 1 ) ,..., ( C k ,a k ) } , C i ∈ N, a i ∈ A, C i ∩ C j = ∅ ,

we have S ∈ b-core( g ).