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bel
Mary
(
{
(
{
John
,
Mary
}
, Yung Kee
)
}
John
{
(
{
John
,
Mary
}
, Spasso
)
}
)
.
Hence, the particular coalitional act profile
{
(
{
John
,
Mary
}
, Spasso
)
}
is no longer in the b-core, because now there
is
a coalitional act profile,
namely
{
(
{
John
,
Mary
}, Yung Kee
)
},
that both dominates it and dominates it based on belief, through the
coalitional act
(
{
John
,
Mary
}, Yung Kee
)
.
We note that this is still true if we only have
bel
John
(
{
(
{
John
,
Mary
}, Yung Kee
)
}
Mary
{
(
{
John
,
Mary
}, Spasso
)
}
)
,
but we do not have
bel
Mary
(
{
(
{
John
,
Mary
}
, Yung Kee
)
}
John
{
(
{
John
,
Mary
}
, Spasso
)
}
)
.
In this case, the particular coalitional act profile
{
(
{
John
,
Mary
}
, Spasso
)
}
still does not exist in the b-core, because the coalitional act profile
{
still
both dominates it (as a matter of
fact) and dominates it based on belief (of you), through the coalitional
act (
(
{
John
,
Mary
}
, Yung Kee
)
}
{
John
,
Mary
}
, Yung Kee
).
Intuitively, we can understand this as follows. Recall that according
to the definition of b-core, an objection (i.e., an alternative coalitional
action profile) to a coalitional action profile in an NTU-PB game needs
to satisfy two conditions. First, all members of the deviating coalition
must actually prefer the alternative coalitional act to the one that they
are currently in. Second, at least one agent of the coalition correctly
believes, correctly or incorrectly, that all fellow members also prefer
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