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Proof
Consider a coalitional action profile
S
1
in the game
g
.By
Definition 4.4, an alternative coalitional action profile
S
2
is a belief-
based objection to
S
1
in game
g
if there exists a coalitional act
α
=
(
C, a
)
∈ S
2
and an agent
i ∈ C
such that, for each agent
k ∈ C
,wehave
bel
i
(
α
k
α
k
(
S
1
)). There are only two cases to consider here. First,
if the beliefs of both
bel
i
in
B
of
g
and
bel
i
in
B
of
g
are accurate,
then we will conclude that
S
2
is a belief-based objection to
S
1
in
g
if and only if
S
2
is a belief-based objection to
S
1
in
g
. Second, if
bel
i
is accurate but
bel
i
is not, then the latter objection would be invalid.
That is,
S
2
is a belief-based objection to
S
1
in
g
, but if
S
2
is a belief-
based objection to
S
1
in
g
. Hence
S
1
will be in the b-core of
g
but not
theb-coreof
g
. 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
).
⊆
The idea is illustrated in Figure 4.14. Theorem 4.4 suggests that
inaccuracies in the agents' belief actually should lead to more number
of stable results.
b-core(
g
£
)
b-core(
g
)
core
Fig. 4.14
Illustration of Theorem 4.4.
Example 4.23
We continue our discussion in Examples 4.21 and
4.22. Suppose now Mary and you still have all the incorrect beliefs,
except that somehow both believe (correctly this time) that the other
person also prefers eating at
Yung Kee
to
Spasso
. For the sake of
clarity, we assume that this is the only accurate belief the both have.
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