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4.4 The Core, the b-Core and Belief Accuracy
The reason that, in general, the b-core of an NTU-PB game differs from
the core is that the private beliefs of the agents are often inaccurate.
In this section, we shall present a discussion and analysis for better
understanding of the effects of such belief inaccuracy on stability of
coalitions, and the relationship between the core and the b-core in
general.
We note that although it is clear that inaccuracy in agents' beliefs
is the main reason for the existence of the b-core, there are different
levels of inaccuracy in agents' beliefs, as some beliefs are more accurate
than some other. It is worthwhile investigating how the degree of belief
inaccuracy affects the b-core. We first define what we mean by relative
accuracy.
Definition 4.8 (Accuracy Relation of Agents' Private Beliefs)
Given two private beliefs
bel
1
and
bel
2
,wesay
bel
1
is
more accurate
than bel
2
if and only if one or both of the following conditions hold:
1) If
bel
1
(
α
1
j
α
2
) holds but
bel
2
(
α
1
j
α
2
) does not hold, where
α
1
,α
2
∈
2
N
× A
are coalitional acts, and
j ∈ N
,then
α
1
j
α
2
holds.
2) If
bel
1
(
α
1
j
α
2
) does not hold but
bel
2
(
α
1
j
α
2
) holds, where
α
1
,α
2
∈
2
N
×
A
are coalitional acts, and
j
∈
N
,then
α
1
j
α
2
does
not hold.
In other words, a belief
bel
1
is considered to be more accurate than
another belief
bel
2
, if and only if
i)
any preference that is in
bel
1
but
not
bel
2
must be a true preference, and/or
ii)
any preference that is
not in
bel
1
but in
bel
2
must be an false preference. Therefore, for two
particular belief
bel
1
and
bel
2
, it is possible that
bel
1
is considered to
be more accurate than
bel
2
because of a particular preference, but at
the same time
bel
2
is also regarded as more accurate than
bel
1
because
of a different preference. Therefore, the
more-accurate-than
relation is
generally not transitive and not even asymmetric.
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