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attractiveness 6 (they are the most attractive of the remaining males). It follows that
M
4
=
4. Males of
attractiveness 2 or 4 solicit dates with females of the same attractiveness or of attrac-
tiveness one level lower. Males of attractiveness 1 or 3 solicit dates with females of
the same attractiveness or of attractiveness one level higher. We have
R
M
6
−
[
2
,
0
]
. Arguing iteratively,
M
i
=
M
i
+
2
−
[
2
,
0
]
for
i
=
1
,
2
,
3
,
π
∗
)=
(
4;
3
/
2
4
3
, so in the dating subgame males of attractiveness 3 and 4 accept
any females giving them a utility of at least 2 (in this game the utility from a pairing
is by definition an integer). Also,
R
π
∗
)=
and
R
(
3;
2
3
,sointhe
dating subgame males of attractiveness 1 and 2 accept any female giving them a
utility of at least 0.
It follows that the value of the game to a player of given attractiveness from 1 to
6 must be given by the expected reward of such a player under any strategy profile
from the set
π
∗
)=
−
π
∗
)=
−
(
/
(
2;
1
2and
R
1;
.
Since females of attractiveness 5 and 6 do not solicit dates with males of
attractiveness 4, such males are indifferent between soliciting and not soliciting
dates with such females. We should check the relevant equilibrium condition based
on the concept of a trembling hand equilibrium, i.e. if a female of attractiveness 6
did 'by mistake' accept a date with a male of attractiveness 4, should the male solicit
a date? In the dating subgame, only a female of type
{
(
5
,
4
)
,
(
5
,
4
)
,
(
3
,
2
)
,
(
3
,
2
)
,
(
1
,
0
)
,
(
1
,
0
)
,•}
[
6
,
3
]
would accept a male of
type
[
4
,
3
]
. Hence, the expected utility of a type
[
4
,
3
]
male from dating a female of
attractiveness 6 is
1
7
×
6
7
×
1
7
=
π
∗
)=
π
∗
)
.
(
,
+
.
−
>
(
v
M
4
6;
6
1
5
2
R
4;
It follows that males of attractiveness 4 should solicit dates with females of
attractiveness 6. Arguing similarly, such males should solicit dates with females
of attractiveness 5 and males of attractiveness 2 should solicit dates with females of
attractiveness 3 or 4.
No females of attractiveness 5 or 6 would pair with a male of attractiveness 3 in
the dating subgame. It follows that males of attractiveness 3 should not solicit dates
with females of attractiveness 5 or 6. Arguing similarly, a male of attractiveness 1
should not solicit dates with females of attractiveness above 2.
It remains to determine the strategy used by a type
male. Since no female
of greater attractiveness will date such a male, we only have to consider strategy
profiles where a male pairs with successively less preferred partners of attractive-
ness 0. Note that
R
[
0
,
3
]
π
∗
)
≤
π
∗
)=
−
3, thus in the dating game individuals
of attractiveness 0 must pair with any prospective partner who gives them a utility
of 0 (i.e. with those of the same type). The expected rewards from the game un-
der a strategy profile where individuals of attractiveness at least 1 use the strategies
derived above and males of type
(
0;
R
(
1;
2
/
[
,
]
{
[
,
]
}
0
3
pair with females of types in (a)
0
3
,
(b)
are equal to the ex-
pected rewards from the game under the respective strategy profiles
{
[
0
,
2
]
,
[
0
,
3
]
,
[
0
,
4
]
}
and (c)
{
[
0
,
1
]
,
[
0
,
2
]
,
[
0
,
3
]
,
[
0
,
4
]
,
[
0
,
5
]
}
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