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(b) In the dating subgame, accept any prospective partner who would pair with an
individual of type
[
j
,
k
]
.
Hence, the expected utility of a searcher at such an equilibrium must be
non-decreasing in his/her attractiveness.
Corollary to Theorem 1 .
j . In the dating subgame, if an individual
of attractiveness i accepts one of attractiveness j , then acceptance is mutual. This
follows from the fact that the individual of attractiveness j obtains a greater utility
from the pairing than the individual of attractiveness i , but has a lower expected
utility from future search.
Suppose i
π of the symmetric game, searchers of
maximum attractiveness, i max , are willing to date prospective partners of attractive-
ness above a certain threshold.
Theorem 2. At a symmetric equilibrium
Proof. From Theorem 1 , if a searcher of attractiveness i max accepts a prospective
partner of type
in the dating subgame, then acceptance is mutual. Hence, the
expected utility of such a searcher from the dating subgame is non-decreasing in
the attractiveness of the prospective partner (since character is independent of at-
tractiveness). A searcher should be willing to date a prospective partner if the ex-
pected utility from such a date is at least as great as the expected utility from future
search. If this condition is satisfied for some level of attractiveness i , then it will
be satisfied for all higher attractiveness levels. Note that a searcher of attractiveness
i max is willing to date a prospective partner of attractiveness i max , since such dates
give the highest possible expected utility.
[
i
,
k
]
π of the symmetric game, a searcher
of attractiveness i solicits dates with prospective partners of attractiveness in
[
Theorem 3. At a symmetric equilibrium
(
) ,
(
)]
(
)
is the maximum attractiveness of a prospective partner
willing to date the searcher. In addition, k 1
k 1
i
k 2
i
,wherek 2
i
(
)
(
)
i
and k 2
i
are non-decreasing in i and
k 1
(
i
)
i
k 2
(
i
)
.
Proof. The proof of this theorem is by recursion. Theorem 2 states that for i
i max
the equilibrium strategies are of the appropriate form. Assume that Theorem 3 is
valid for searchers of attractiveness
=
i max .
First, suppose no prospective partner of attractiveness
i
+
1, where i
<
i will date a searcher of
attractiveness i . By ignoring meetings with prospective partners of attractiveness
>
i ,
the game faced by searchers of attractiveness i can be reduced to a game in which
they are the most attractive. From Theorem 2 , it follows that searchers of attractive-
ness i are willing to date prospective partners of attractiveness in
>
[
k 1 (
i
) ,
k 2 (
i
)]
,where
k 2 (
i
)=
i
<
k 2 (
i
+
1
)
and k 1 (
i
)
i
<
k 1 (
i
+
1
)
.
i . Firstly, we show that
searchers are willing to date prospective partners of attractiveness j ,where i
Now assume that k 2 (
i
) >
i . It follows that k 1 (
i
+
1
)
j
k 2 (
. If such a prospective partner finds a searcher of attractiveness i acceptable in
the dating subgame, then acceptance is mutual. It follows that the expected utility of
i
)
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