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Suppose the male is of type
x
and the female is of type
y
. The payoff matrix is
given by
Female:
a
Female:
r
[
.
Male:
a
Male:
r
g
(
y
a
,|
x
c
−
y
c
|
)
,
g
(
x
a
,|
x
c
−
y
c
|
)]
[
R
(
x
a
;
π
)
,
R
(
y
a
;
π
)]
[
R
(
x
a
;
π
)
,
R
(
y
a
;
π
)]
[
R
(
x
a
;
π
)
,
R
(
y
a
;
π
)]
Note that the payoff matrix depends on the strategy profile used via the ex-
pected utilities of the individuals involved in a subgame. These expected utilities
were derived in Sect.
17.6
.
From Condition 1, at equilibrium an individual accepts 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. Hence, the appropriate Nash equilibrium of this
subgame is for the male to accept the female if and only if
g
(
y
a
,|
x
c
−
y
c
|
)
≥
R
(
x
a
;
π
)
and the female to accept the male if and only if
g
(
x
a
,|
x
c
−
y
c
|
)
≥
R
(
y
a
;
π
)
.
,afemale
always accepts the male (in this case she is indifferent between rejecting and accept-
ing him). Similarly, if
g
For convenience, we assume that when
g
(
x
a
,|
x
c
−
y
c
|
)=
R
(
y
a
;
π
)
, it is assumed that a male always
accepts a female. The implications of this assumption are considered at the end of
Sect.
17.8
.
Note that if a male rejects a female, then the female is indifferent between
accepting or rejecting the male. Under a rule satisfying Condition 1, a female will
make an optimal response whatever action the male takes. Such an equilibrium is a
trembling hand perfect equilibrium (i.e. robust to the other player making a mistake).
Hence, there is a unique trembling hand perfect equilibrium satisfying Condition 1.
Let
v
(
y
a
,|
x
c
−
y
c
|
)=
R
(
x
a
;
π
)
denote the value of the dating subgame cor-
responding to this equilibrium, where
v
M
(
x
,
y
;
π
)=[
v
M
(
x
,
y
;
π
)
,
v
F
(
x
,
y
;
π
)]
(
x
,
y
;
π
)
and
v
F
(
x
,
y
;
π
)
are the values of
the subgame to the male and female, respectively.
We now consider the soliciting subgame.
17.7.2 The Soliciting Subgame
Once the dating subgame has been solved, we may solve the soliciting subgame
and hence the game
G
, played when a male of type
x
meets a female of
type
y
. As before, we assume that the population is following a symmetric strategy
profile
(
x
,
y
;
π
)
.
Both players have two actions:
a
- accept (solicit a date) and
r
- do not solicit
a date. Since the utility an individual expects from a date is independent of his/her
character, the payoff matrix can be expressed as follows:
π
Female:
a
Female:
r
[
.
Male:
a
Male:
r
v
M
(
x
a
,
y
a
;
π
)
−
c
2
,
v
F
(
x
a
,
y
a
;
π
)
−
c
2
]
[
R
(
x
a
;
π
)
,
R
(
y
a
;
π
)]
[
R
(
x
a
;
π
)
,
R
(
y
a
;
π
)]
[
R
(
x
a
;
π
)
,
R
(
y
a
;
π
)]
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