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the searcher from such a date is greater than the expected utility of the prospective
partner. Hence, from Theorem
1
, a searcher of attractiveness
i
should be willing to
date a prospective partner of attractiveness
j
.
Secondly, the proof that searchers of attractiveness
i
should be willing to date
prospective partners of attractiveness
j
,where
j
i
, if and only if
j
is above some
threshold is analogous to the proof of Theorem
2
. Also, if a prospective partner of
type
≤
[
(
+
)
,
]
[
+
,
]
k
1
i
1
l
is mutually acceptable to a searcher of type
i
1
k
in the dating
[
(
+
)
,
]
subgame, then from Theorem
1
a prospective partner of type
k
1
i
1
l
accepts
[
(
+
)
,
]
a searcher of type
(and hence any searcher of the same character and
greater attractiveness) in the dating subgame. Thus, by accepting exactly the same
types of prospective partners of attractiveness
k
1
(
k
1
i
1
k
i
+
1
)
in the dating subgame as
searchers of type
will ensure the same expected
utility from a date with a prospective partner of attractiveness
k
1
(
[
i
+
1
,
k
]
do, a searcher of type
[
i
,
k
]
i
+
1
)
as a searcher
π
∗
)
of type
[
i
+
1
,
k
]
obtains. This expected utility must be at least
R
(
i
+
1;
. It follows
from Theorem
1
that
k
1
(
i
)
≤
k
1
(
i
+
1
)
.
Hence, the general form of the equilibrium of the symmetric game is intuitive.
Individuals date those who are of a similar level of attractiveness. It should also be
noted that at such an equilibrium a type
female.
Due to the assumption that there are no costs associated with soliciting a date
when dating does not follow, at such an equilibrium a searcher of attractiveness
i
is indifferent between soliciting and not soliciting a date with a prospective partner
who is not willing to date. In this case we should check the condition based on the
concept of a trembling hand perfect equilibrium. This states that a searcher should
solicit a date if the expected utility from dating after a 'mistaken' acceptance is
greater than the expected utility from future search. Suppose a prospective partner
of attractiveness
j
would not pair with any searcher of attractiveness
i
in the dating
subgame. A searcher of attractiveness
i
should not solicit a date with a prospec-
tive partner of attractiveness
j
, in order to avoid the dating costs when there is no
prospect of pairing.
It is possible that a prospective partner would wish to pair with a searcher of
lower attractiveness in the dating subgame but, due to the costs of dating and the
risks of obtaining a prospective partner of inappropriate character, would not solicit
a date. This will be considered in the example given in Sect.
17.8.2
.
Note
: At the equilibrium of the classical problem considered by McNamara and
Collins [
28
], the population is partitioned into classes, such that class
i
males only
form pairs with class
i
females. For the game considered here, such a partition only
exists in very specific cases, e.g.:
[
i
,
j
]
male will pair with a type
[
i
,
j
]
1. When the search and dating costs are low enough, type
[
i
,
k
]
males only pair with
females.
2. When the costs of dating are high relative to the importance of character, mate
choice is based entirely on attractiveness.
type
[
i
,
k
]
The difference between the equilibria of these two games is illustrated by the
example in Sect.
17.8.2
.
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