The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 109

the alternative (otherwise, if this is not true, no agent will realise the
existence of the objection and it will not be raised). Thus, any inaccu-
racies in the beliefs of the agents will cause less number of objections to
be raised, because some of the agents will wrongly think that an oth-
erwise valid objection is invalid (which causes the objection to fail the
second requirement), which in turn means that there are more num-
ber of stable results as there as less objections. (Another way to think
about this is: the inaccurate beliefs can lead to mutual misunderstand-
ing amongst the agents, making some otherwise suboptimal solution
stable because some better solution cannot be reached because of the
misunderstandings). On the other hand, any invalid objections that
are raised because of incorrect private beliefs would have no effects on
the outcome of the game, because it will be turned down by the fellow
coalition members.
4.5 Discussion
There are two possible interpretations to most game theoretic stability
concepts, which we label the ex-ante interpretation and the ex-post
interpretation, respectively.
In an ex-ante interpretation, we try to predict beforehand, as an
omniscient observer, which outcomes are stable and achievable by the
participants of a game. An example of this type of analysis is the dating
game in Examples 4.4 and 4.18. By an ex-ante interpretation, we can
follow the agents' decision making process, and determine whether a
particular outcome is stable and can be achieved. This interpretation
not only requires the observer to know the preference of each agent, but
each agent's private beliefs also. Thus we see that the core is not really
suitable to this type of interpretation as long as beliefs are involved in
the agents' decision making process, and that the b-core is actually a
better tool for these tasks. This is because b-core can more accurately