The b-Core: Belief-based Coalition Stability - Belief-based Stability in Coalition Formation with Uncertainty - page 85

there are two conditions that the mechanism can terminate: either
termination by time or termination by stagnant criterion. Intuitively,
a mechanism that is frequently terminated by time is less stable than
one that is frequently terminated by stagnant criterion. Now, back to
our mechanism, from the data we can see that, in fact, in the long
run, almost all Random-B games terminates because no agent is able
to make any more proposals, which is a strong suggestion that these
solutions are in fact stable.
To formally verify this observation, we measure the number of
games that terminate by the stagnant criterion for both type of games,
and the result is shown in Figure 4.9. Here, we see that the results actu-
ally suggest that the outcomes of Random-B games should be regarded
as at least as stable as Random-N games in the long run, and even
outperforming Random-N games in the short run.
What these examples suggest is that the traditional stability cri-
teria, which assume all preferences to be common knowledge, are in-
adequate in scenarios where private beliefs are important factors in
determining the behaviour of the agents. In Example 4.4, the core-
based concepts fail to predict a stable outcome that actually exist,
though due to misunderstanding. In Example 4.6, the analytical re-
sult does not reflect the real stability of the solutions achieved by the
mechanisms. The reason is that we are facing a new type of games
where the stability is based on private information instead of common
knowledge. In the next section, we will discuss a new solution concept
that is suitable for this new type of games.
4.3 NTU Games with Private Beliefs
The games depicted in Examples 4.4 and 4.6 are examples of what we
shall call non-transferable utility games with private beliefs (NTU-PB
games), which we define as follows: