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files)
Given two external beliefs profiles
B
=
B
1
,B
2
,...,B
n
}
B
=
{
B
1
,B
2
,...,B
n
}
,
{
,
we say
B
is more accurate than
B
if there exists
i ∈ N
such that
B
i
is more accurate than
B
i
, and either
B
j
=
B
j
or
B
j
is more accurate
than
B
j
for all
j
∈
N
−{
i
}
.
That is, for any two profiles of agent beliefs, one is considered more
accurate than another if at least one of the beliefs in the former is more
accurate than the corresponding one in the latter, while for the other
beliefs, the instances in the former profile are all at least as accurate
as their counterparts in the latter.
Theorem 6.3
Given two games
i
)
,P,B,s
)
,
g
=(
N,S,
(
g
=(
N,S,
(
i
)
,P,B
,s
)
,
the following holds if
B
is more accurate than
B
:
•
wb-core(
g
);
wb-core(
g
)
⊆
sb-core(
g
).
•
sb-core(
g
)
⊆
Proof
Consider an objection in the game
g
against a coalition struc-
ture
CS
1
. By Definitions 6.8 and 6.9, there exists an alternative coali-
tion structure
CS
2
, a coalition
C
∈
CS
1
∩
CS
2
, and an agent
i
∈
C
in
both game such that, for each agent
k ∈ C
,wedonothave
B
i
(coal
k
(
CS
1
)
k
CS
2
)
.
There are only two cases to consider here:
First, if the beliefs of both
B
i
and
B
i
are accurate, then the result
of the two games will be the same, meaning that either there are both
valid objections or both invalid.
Second, if
B
i
is accurate but
B
i
is not, then the latter objection
would be invalid. Thus, we see that any valid objection for the game
g
is also a valid objection for the game
g
, hence b-core(
g
)
b-core(
g
).
⊆
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