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After deriving some of the properties of an equilibrium, we now describe a
procedure for deriving the equilibrium itself. Individuals of maximum attractive-
ness face a one-sided search problem. They should be willing to date a prospective
partner if and only if the expected utility obtained from such a date minus the dating
costs is at least as great as the expected utility from future search. Similarly, in the
dating game a searcher should accept a prospective partner if and only if the utility
gained from such a partnership is at least as great as the expected utility from future
search. By following such a strategy, such individuals will maximise their expected
utility from search, see Whittle [
39
]. Individuals of a lower level of attractiveness
face a similar problem given the strategies followed by those of a higher level of
attractiveness.
Since the solution of the corresponding set of inequalities is difficult to present in
an explicit form, in Sect.
17.8.1
we describe an algorithm which derives a symmetric
equilibrium. The constructive nature of this algorithm leads to the key theorem of
the paper on the uniqueness and existence of a symmetric equilibrium in this game.
17.8.1 The Algorithm
m
2
Define
r
. Since the equilibrium is assumed to be symmetric with respect
to character and sex, it suffices to consider males of character
r
. The advantage of
considering such individuals is that the difference between character
j
and character
r
is simply the standard absolute difference between the two characters.
The game can be solved as follows
=
1. Assume that males of maximum attractiveness are only willing to date females
of maximum attractiveness. Consider strategy profiles
π
t
,
t
=
0
,
1
,
2
,...,
m
/
2
,
[
,
]
where under strategy profile
pair with females whose
characters do not differ by more than
t
(i.e. as
t
increases males accept succes-
sively less preferred females). We calculate
R
π
t
males of type
i
max
r
(
)
,
(
)
,...
i
max
;
π
R
i
max
;
π
in turn
0
1
until
R
(i.e. the expected utility from search is greater
than the utility from mating with any female of maximum attractiveness who is
not acceptable) or
t
(
i
max
;
π
t
)
>
g
(
i
max
,
t
+
1
)
2. This gives us the optimal rule of the form consid-
ered, see Whittle [
39
], which is a lower bound on
R
=
m
/
π
∗
)
(
i
max
;
.
2. If this lower bound is less than the utility obtained by a type
[
i
max
,
r
]
male from
pairing with a type
female minus the dating costs (i.e. the maximum
possible reward from soliciting a date from a female of attractiveness
i
max
−
[
i
max
−
1
,
r
]
1),
it may be optimal for males of maximum attractiveness to solicit dates with
females of the second highest level of attractiveness. We can order females of
the top two levels of attractiveness with regard to the preferences of a type
[
males are
prepared to pair with successively less preferred females as in Point 1, we can
derive the optimal strategy of males given they date females of the two highest
levels of attractiveness.
i
max
,
r
]
male. By considering strategies under which type
[
i
max
,
r
]
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