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1
7
−
1
7
=
−
R
(
6;
{
(
6
,
6
)
,•,•,•,•,•}
)=
6
−
49
×
7
×
2
16
3
−
49
3
×
1
7
−
7
3
×
1
7
=
8
3
R
(
6;
{
(
6
,
5
)
,•,•,•,•,•}
)=
24
5
−
49
5
×
1
7
−
7
5
×
1
7
=
16
5
.
R
(
6;
{
(
6
,
4
)
,•,•,•,•,•}
)=
The expected utility of a type
[
6
,
3
]
male under
{
(
6
,
4
)
,•,•,•,•,•}
is greater than 3.
It follows that
females should not be accepted in the dating subgame.
Since our lower bound (3.2) on the expected utility of a type
[
6
,
0
]
and
[
6
,
6
]
[
,
]
male is less
than the utility obtained from pairing with a female of the same character and the
second highest attractiveness minus the costs of dating,
3
7
6
3
, we now consider strategy
profiles in which type
males solicit dates with females of attractiveness 5 and
6. We only have to consider:
[
6
,
3
]
1. Strategy profiles in which females of type
[
5
,
3
]
are acceptable in the dating
subgame. If this were not the case, then a type
[
6
,
3
]
employer would be incur-
ring unnecessary dating costs.
2. Prospective partners who give a utility higher than the current lower bound on
the expected utility of a type
[
6
,
3
]
male from search.
The ordered preferences of a type
male among the set of females of
attractiveness at least 5 who satisfy criterion 2 above is given by: group 1 is
[
6
,
3
]
{
[
6
,
3
]
}
,
group 2 is
{
[
6
,
2
]
,
[
5
,
3
]
,
[
6
,
4
]
}
and group 3
{
[
6
,
1
]
,
[
6
,
5
]
,
[
5
,
2
]
,
[
5
,
4
]
}
. We only need
to consider strategy profiles of the following two types: (a) type
[
6
,
3
]
males pair with
females from groups 1 and 2 above, i.e. profiles from the set
{
(
5
,
5
)
,•,•,•,•,•}
,(b)
type
[
6
,
3
]
males pair with females from all three groups, i.e. profiles from the set
{
(
5
,
4
)
,•,•,•,•,•}
.Wehave
21
4
−
49
4
×
1
7
−
14
4
×
1
7
=
R
(
6;
{
(
5
,
5
)
,•,•,•,•,•}
)=
3
37
8
−
49
8
×
1
7
−
14
8
×
1
7
=
7
2
.
R
(
6;
{
(
5
,
4
)
,•,•,•,•,•}
)=
We now consider strategy profiles in which type
males solicit dates with
females of attractiveness at least 4. Since the present lower bound on
R
[
6
,
3
]
π
∗
)
(
6;
is
3.5, we only need to consider strategy profiles in which type
[
6
,
3
]
males pair with
the same types of females as in
{
(
5
,
4
)
,•,•,•,•,•}
with the addition of type
[
4
,
3
]
females, i.e. strategy profiles from the set
{
(
4
,
4
)
,•,•,•,•,•}
.Wehave
41
9
−
49
9
×
1
7
−
21
9
×
1
7
=
31
9
<
7
2
.
R
(
6;
{
(
4
,
4
)
,•,•,•,•,•}
)=
[
,
]
It follows that type
males should not solicit dates with females of attractiveness
4. Hence, at a symmetric equilibrium, type
6
3
[
,
]
males solicit dates with females of
attractiveness 5 and 6 and pair with females of type in
M
6
,where
6
3
M
6
=
{
[
6
,
1
]
,
[
6
,
2
]
,
[
6
,
3
]
,
[
6
,
4
]
,
[
6
,
5
]
,
[
5
,
2
]
,
[
5
,
3
]
,
[
5
,
4
]
}.
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