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the assumption of mutual acceptance) under the strategy profile
π
)
to be the set of types of prospective partners that eventually pair with an individual
of type
y
. By definition
M
2
(
π
.Define
M
2
(
y
;
.
The expected length of search of an individual of type
y
,
L
y
;
π
)
⊆
M
1
(
y
;
π
)
, is the recip-
rocal of the probability of finding a mutually acceptable partner at any given stage.
The expected number of dates of such an individual,
D
(
y
;
π
)
, is the expected length
of search times the probability of dating at any given stage. Hence,
(
y
;
π
)
π
)=
∑
x
∈
M
1
(
y
;
π
)
p
(
x
)
1
L
(
y
;
π
)=
;
D
(
y
;
)
.
(17.1)
p
(
x
)
p
(
x
∑
∑
x
∈
M
2
(
y
;
π
)
x
∈
M
2
(
y
;
π
)
Note
: The number of prospective mates seen and the number of individuals dated
by an individual of type
y
have geometric distributions with parameters 1
/
L
(
y
;
π
)
/
(
π
)
and 1
, respectively.
The expected utility of a type
y
individual from forming a pair under the strategy
profile
D
y
;
is the expected utility from pairing given that the type of the prospective
partner is in the set
M
2
(
π
y
;
π
)
. Hence, the individual's expected total utility from
search, denoted
R
since this expected utility depends only on an individual's
attractiveness, is given by
(
y
a
;
π
)
p
(
x
)
g
(
x
a
,|
x
c
−
y
c
|
)
π
)=
∑
x
∈
M
2
(
y
;
π
)
R
(
y
a
;
−
c
1
L
(
y
;
π
)
−
c
2
D
(
y
;
π
)
.
(17.2)
∑
x
∈
M
2
(
y
;π
)
p
(
x
)
Note that it is relatively simple to extend these calculations to non-symmetric
games.
17.7 The Dating and Soliciting Subgames
Since this game is solved by recursion in the manner developed by Spear [
36
,
37
],
we first consider the dating subgame. In this section we consider the general
formulation of the game.
17.7.1 The Dating Subgame
Assume that the population are following a symmetric strategy profile
.Themale
and female both have two possible actions: accept the prospective partner, denoted
a
,
or reject, denoted
r
. Also, we ignore the costs already incurred by either individual,
including the costs of the present date, as they are subtracted from all the payoffs in
the matrix, and hence do not affect the equilibria in this subgame.
π
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