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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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