The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 110

predict the stable outcomes when private beliefs are involved.
The other type of interpretations is the ex-post analysis, which at-
tempts to answer the question, 'Given this particular coalition struc-
ture and coalitional act profile, what can we say about its stability?'
Thus, while the ex-ante analysis tries to predict which stable out-
comes can be reached, the ex-post analysis tries to decide whether any
arbitrary outcome is indeed stable if all agent preferences are now re-
vealed. By an ex-post interpretation, we can conclude that all those
coalitional act profiles in the core should be stable if all agents' beliefs
are correct; otherwise all those coalitional act profiles in the b-core are
stable. With an ex-post analysis, we can inspect the result obtained by
the randomized mechanisms in Examples 4.6 and 4.19, and determine
whether some achieved solutions are indeed stable according to some
criteria after the games have already completed.
We see that in ex-post analysis, both the core and the b-core can be
used, but they correspond to two different stability concepts. The core
is really asking the question 'assuming that the agents are now told
about the real preferences of the other agents, will any of them now
change their minds and deviate and form new coalitions?' whereas the
b-core is still asking the original question 'will any agents change their
minds and deviate, given what they believe?' Note that in an ex-post
analysis, the beliefs are static in nature, as we do not consider how
those beliefs come into existence during the analysis.
With both tools in hand, we can now provide additional analysis
on coalition formation mechanisms. For example, a mechanism that
can produce a low percentage of results in the b-core and also the core
suggests that the mechanism is not satisfactory, and the mechanism
should be improved. A high percentage of results in b-core but a low
percentage of results in core, however, suggests that there are errors
or mutual misunderstanding in the beliefs of the agents, which pro-
hibit any better results from occurring, and one way to tackle this is
to allow the agents to communicate regarding their beliefs. Finally a
high percentage of results in the core and b-core suggests that there is