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Intuitively, Definition 4.3 simply says that a profile
S
1
is dominated
by another profile
S
2
if we can find a coalition and a corresponding
joint action in
S
2
such that every member of that coalition would
real ly
prefer
S
2
to
S
1
. Definition 4.4 says that profile
S
1
is dominated based
on belief by profile
S
2
if at least one member of that coalition believes
that every member of that coalition would prefer
S
2
to
S
1
.
Example 4.13
In Example 4.3, there are many examples of domi-
nation relations. For example, the coalitional act profile
{
(
{
a, b
}
,
movie)
,
(
{
c
}
,
movie)
}
is dominated by
{
(
{a},
movie)
,
(
{b, c},
movie)
}
through the coalitional act (
,
movie) in the latter. Intuitively,
both agents
b
and
c
in the latter coalitional act profile prefer the latter
profile than the former one: for agent
b
, he likes going to see the movie
with
c
more than with
a
; for agent
c
, he likes going to see the movie
with
b
more than going alone. Formally, we have
{
b, c
}
{
(
{
a
}
,
movie)
,
(
{
b, c
}
,
movie)
}
dom
(
{b,c},
movie)
{
(
{
a, b
}
,
movie)
,
(
{
c
}
,
movie)
}
.
Example 4.14
In Example 4.10, we can see that the coalitional act
profile
S
1
=
{
(
{
John
,
Mary
}
, Yung Kee
)
}
dominates any other coalitional act profile
,α
)
S
2
=
{
(
{
John
,
Mary
}
}
where
α
is not
Yung Kee
. This is because if we consider the only
coalitional act
α
=(
{
John
,
Mary
}, Yung Kee
)
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