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problematic. According to the preference of agent b , he will make such
a switch only if he believes the world is in state s 1 , but this is something
he is uncertain. Therefore, the validity of this objection depends on
how agent b handles his internal uncertainty.
The same can be said about edge 7 to edge 12. We label such cases
potential objections , meaning that it is possible for these objections to
arise, but conditional on how the internal uncertainties are resolved
by at least one agent. In Figure 6.3, potential objections are denoted
by dashed lines.
So far we have not considered the agents' external beliefs. In some
cases, the external beliefs will not affect the validity of the objections.
Consider edge 2 again, for example. Although agent a (incorrectly) be-
lieves that agent b prefers to stay in coalition
{
a, b
}
instead of deviating
and from coalition
, agent b will not be affected as he will raise an
objection to the coalition
{
b
}
regardless of agent a 's (mis-)beliefs.
On the other hand, edge 1 is problematic if the external beliefs
are taken into accounts. Although all three agents prefer the coalition
{
{
a, b
}
in
one of the states (thus making it a potential objection), no agent can
realise this because each of them happens to believe (incorrectly) that
one or more of this potential partners will not agree to switch.
Thus in this case, no agent is likely to raise any objections even
though the objection is a valid one, because raising such an objection
will be in conflict with their beliefs. Similar analysis can be applied to
edge 6. We call this type of objections 'non-raisable objections' and
the rest 'raisable objections'.
The raisable objections for this game are shown in Figure 6.4.
Again, there are two types of raisable objections depending on
whether the objecting agents' preferences are certain or not, and they
are labeled as raisable definite objections if the objecting agents' pref-
erences are certain, and raisable potential objections otherwise.
Regarding the stability of this game, we can analyse as follows.
First, in cases where the agents do not require the objections to be
a, b, c
}
to their original corresponding coalition in
{{
a
}
,
{
b
}
,
{
c
}}
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