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denotes the expected values of the dating
subgame to the male and female, respectively, given the strategy profile used by
the population, the measures of attractiveness of the prospective partners and the
fact that a date followed.
From Condition 2, the female should solicit a date if and only if her expected
utility from such a date is at least as great as the expected utility from future
search, i.e.
Here
[
v
M
(
x
a
,
y
a
;
π
)
,
v
F
(
x
a
,
y
a
;
π
)]
(
,
π
)
−
≥
(
π
)
.
v
F
x
a
y
a
;
c
2
R
y
a
;
Similarly, the male should solicit a date if
v
M
(
x
a
,
y
a
;
π
)
−
c
2
≥
R
(
x
a
;
π
)
.
Note that Condition 2 requires the players to use a trembling hand perfect
equilibrium in any soliciting subgame.
In the next section we present an algorithm to find a symmetric equilibrium of the
symmetric game, as an algorithmic game in the tradition of Velupillai [
38
], Nisam
et al. [
29
].
17.8 A Symmetric Equilibrium of the Symmetric Game
Theorems
1
-
3
describe the form of a symmetric equilibrium of the symmetric game.
These results do not fully characterize such an equilibrium. However, they do justify
the logic behind the algorithm presented in Sect.
17.8.1
. The constructive form of
this algorithm allows us to state the key result of this section, Theorem
5
,onthe
existence and uniqueness of a symmetric equilibrium in the symmetric game.
π
∗
of the symmetric game, the utility of an
individual is non-decreasing in attractiveness.
Theorem 1.
At a symmetric equilibrium
π
∗
)
<
π
∗
)
Proof.
Assume that for some
i
>
j
,
R
(
i
;
R
(
j
;
. Consider the dating
subgame. From Condition 1, a type
[
j
,
k
]
(
[
i
,
k
]
) male accepts a female of type
[
i
0
,
j
0
]
π
∗
)
π
∗
)
only if the male's reward is greater than
R
(
j
;
(
R
(
i
;
, respectively). Hence,
males of type
[
i
,
k
]
accept any female that a male of type
[
j
,
k
]
would accept. Simi-
larly, if a type
[
j
,
k
]
male is acceptable to a female of type
[
i
0
,
j
0
]
, then such a female
would accept a type
[
i
,
k
]
male (who gives a greater reward from mating). Hence,
π
∗
)
⊆
π
∗
)
M
2
([
. It follows that a female who is willing to date a male
of attractiveness
j
will also be willing to a date one of attractiveness
i
(who is ex-
pected to be a better partner and at least as likely to be mutually acceptable). Hence,
a searcher of type
j
,
k
]
;
M
2
([
i
,
k
]
;
[
i
,
k
]
can obtain the same expected utility as a searcher of type
[
j
,
k
]
as follows:
(a) In the soliciting subgame, solicit dates with any prospective partner who would
date an individual of attractiveness
j
.
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