The Complete Picture: sb-core and wb-core - Belief-based Stability in Coalition Formation with Uncertainty - page 164

Proof By comparing Definitions 6.8 and 6.9, we see that any coali-
tion C that satisfies condition 1) in Definition 6.8 also satisfies the
condition 1) in Definition 6.9. Thus we see that any objections ac-
cording to the criterion of wb-core are also an objection according to
the sb-core. Since the cores are defined as coalition structures without
objections, we see that the sb-core is a subset of the wb-core. Similar
argument also applies to the w-core and s-core. Similarly, by compar-
ing Definitions 6.5 and 6.8, we see that any coalition C that is an
objection according to the criterion of wb-core is also an objection ac-
cording to the of w-core. Hence the w-core is a subset of the wb-core.
Similar argument also applies to the sb-core and the s-core.

6.4.2 Effect of Refinedness

The reason that, in general, the w-core of an NTU-BU differs from the
s-core, and the wb-core differs from the sb-core, is that the environ-
mental beliefs of the agents are often imprecise, meaning that there are
often more than one states that the agents consider to be the possible
current state, i.e., |S i | > 1forsome i ∈ N . To understand the effects
of such imprecision, we have the followings.

Definition 6.10 (Refinedness Relation of Environmental Bel-
iefs) Given two environmental beliefs S 1 ,S 2 ⊆ S ,wesay S 1 is more

refined than S 2 if S 1 ⊆ S 2 .

Definition 6.11 (Refinedness Relation of Environmental Be-
lief Profiles) Given two environmental beliefs profiles

P = { S 1 ,S 2 ,...,S n } ,

S 1 ,S 2 ,...,S n } P = { ,

we say P is more refined than P if there exists i ∈ N such that
S 1 is more refined than S i , and either S j = S j or S j is more refined