The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 95

genuine objection will surely be raised by all agents involved in the
objection. However, in an NTU-PB game, each agent actually has
his own private belief, which, unfortunately, can be either correct or
incorrect. For those coalitional act profiles that are in the core, by
definition there will be no genuine objections. However, there might
still be belief-based, spurious, objections. Fortunately, these belief-
based spurious objections, if they exist, will not be raised by the agents
involved in the objections, because these agents all know their own
genuine preferences. We note that by Definition 4.2, all objections have
to be raised by only those agents involved. Hence, every coalitional act
profile that is in the core remains in the b-core. On the other hand,
for some of the coalitional act profiles that are not in the core, they
might now be in the b-core. To understand this, we first recall that,
for a coalitional act profile not to be in the core, there must be some
objections that can be raised by some agents involved. However, if
it so happens that all agents involved have incorrect beliefs, which
make them believe that such objections do not really exist, then none
of them will really raise the objections, though they genuinely exist.
Consequently, some coalitional act profiles that are not in the core are
now in the b-core. We illustrate this in the following example.

Example 4.16 Consider the situation in Example 4.5. In this exam-
ple, the coalitional act profile }

is in the b-core. Formally, it is in the b-core because we are unable to
find another coalitional act profile that both dominates it, and dom-
inates it based on belief. We know that actually it is dominated by
the coalitional act profile { ( { John , Mary }, Yung Kee ) } , however, un-
fortunately it is not dominated by

{ ( { John , Mary } , Bonheur Restaurant )

based
on belief. Moreover, there is no other coalitional act profile that dom-
inates it. Therefore,

{ ( { John , Mary } , Yung Kee ) }

{ ( { John , Mary } , Bonheur Restaurant ) } is in the

b-core. Intuitively, the coalitional act profile

{ ( { John , Mary } , Bonheur Restaurant ) }

is a stable one based on belief (or more accurate, misbelief) because