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It should be noted that a type
female given
that they are dating. However, due to the costs of dating, the low probability of find-
ing an acceptable partner of attractiveness 4 and the relatively small gain obtained
from such a partnership compared to the expected utility from future search, a type
[
[
6
,
3
]
male should pair with a type
[
4
,
3
]
6
,
3
]
male should not solicit a date with such a female. Define
M
i
+[
s
,
t
]
to be the set
of
M
i
. From the symmetry of the equilibrium with respect
to character and sex, an individual of type
[
k
+
s
,
j
+
t
]
where
[
k
,
j
]
∈
[
,
+
]
is willing to date prospective
partners of attractiveness 5 and 6 and pair with those in
M
6
6
3
t
+[
,
]
.
Note that searchers are not matched in a block-separated way as in McNamara
and Collins [
28
]. For example, a type
0
t
[
6
,
3
]
male will pair with a type
[
6
,
1
]
female,
who would mate with a type
[
6
,
0
]
male. However, a type
[
6
,
3
]
male will not pair
with a type
female.
Now we consider males of type
[
6
,
0
]
and assume that individuals of maximum
attractiveness follow the strategies derived above and males of attractiveness 5 so-
licit dates with females of attractiveness 5 and 6. Note that from the form of the
equilibrium, a male should always solicit dates with females of the same attrac-
tiveness, together with females of higher attractiveness who solicit dates with him.
From the symmetry of the game with respect to sex and character, since males of
type
[
5
,
3
]
[
6
,
3
]
pair with females of type
[
5
,
2
]
,
[
5
,
3
]
and
[
5
,
4
]
, it follows that males of
type
[
5
,
3
]
will pair with females of type
[
6
,
2
]
,
[
6
,
3
]
and
[
6
,
4
]
. They must also pair
with females of type
[
5
,
2
]
,
[
5
,
3
]
and
[
5
,
4
]
, as such females give a type
[
5
,
3
]
male a
π
∗
)
≥
π
∗
)
utility of 4
≥
R
(
6;
R
(
5;
. The expected utility of a type
[
5
,
3
]
male under
such a strategy profile, i.e. from the set
{
(
5
,
4
)
,
(
5
,
4
)
,•,•,•,•}
,is
10
3
.
This is greater than the expected utility from accepting the next most preferred types
(
29
6
−
49
6
×
1
7
−
14
6
×
1
7
=
R
(
5;
{
(
5
,
4
)
,
(
5
,
4
)
,•,•,•,•}
)=
[
5
,
1
]
and
[
5
,
5
]
). Hence, we can now consider strategy profiles in which males of
type
solicit dates with females of attractiveness at least 4. The only case we
need to consider is extending the set of acceptable females to include those of type
[
[
5
,
3
]
4
,
3
]
,i.e.thesetofstrategyprofiles
{
(
5
,
4
)
,
(
4
,
4
)
,•,•,•,•}
.Wehave
33
7
−
49
7
×
1
7
−
21
7
×
1
7
=
23
7
<
10
3
.
R
(
5;
{
(
5
,
4
)
,
(
4
,
4
)
,•,•,•,•}
)=
It follows that males of type
should solicit dates with females of attractiveness
5 and 6 and pair with females of a type in
[
5
,
3
]
{
[
5
,
2
]
,
[
5
,
3
]
,
[
5
,
4
]
,
[
6
,
2
]
,
[
6
,
3
]
,
[
6
,
4
]
}
.
In these cases acceptance is mutual. It should also be noted that males of type
[
5
,
3
]
should accept females of type
in the dating subgame. However, in
these cases acceptance is not mutual. Females of type
[
6
,
1
]
or
[
6
,
5
]
[
4
,
3
]
would be accepted in
the dating subgame by a type
[
5
,
3
]
male, but such males would not solicit a date
with such a female.
Thus males and females of the top two levels of attractiveness do not date
individuals of any lower level of attractiveness. The problem faced by males of
attractiveness 4 thus reduces to a problem analogous to the one faced by those of
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