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more accurate than another one if at least one of the beliefs in the first
collection is more accurate than the corresponding one in the second
one, and in no case that the opposite is true (i.e., there is not a single
case where it is more accurate in the second collection but not the first
one.)
We are now ready to discuss the relationship between accuracy of
agent beliefs and the b-core of an NTU-PB game. First, we consider
two extreme cases. Regarding the case where all agents' beliefs are
accurate (i.e., all agents beliefs of each other's preferences are the same
as the real preferences of the respective owners), it is not hard to see
that the b-core is the same as the core in this case.
Theorem 4.2
GivenanNTU-PBgames
g
=
N,A,
(
i
)
,B
)
,
we have
b-core(
g
)=core(
g
)
if
bel
i
(
α
1
j
α
2
)
⇒
(
α
1
j
α
2
) for all agents
i
,
j
, and any coalitional
acts
α
1
and
α
2
.
Proof
If all agents' beliefs are accurate, then for any two coalitional
act profiles
S
1
and
S
2
we have
(
S
1
b-dom
α
S
2
)
⇔
(
S
1
dom
α
S
2
)
for any
α
. By comparing Definition 4.2 and Definition 4.6, we see that
the b-core is the same as the core in this case.
Intuitively, Theorem 4.2 means that the core of a NTU-PB is the
same as its b-core if all agents' beliefs are accurate. That is illustrated
in Figure 4.12. This is not di
cult to understand, because if the beliefs
of all the agents are correct, then the situation is effectively the same
as that in which all agents are omniscient, a condition that is always
assumed in conventional game theoretic analysis. In other words, this
theorem says that the b-core reduces to the core when all agents' beliefs
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