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first class males will only pair with first class females. The problem faced by the
rest of the population then reduces to a problem in which first class individuals are
not present. Define
x
0
=
. Arguing iteratively, it can be shown that
k
classes
of males and females can be defined, such that a male of attractiveness
x
is of class
i
if
x
y
0
=
∞
. Males
of class
i
pair with females of class
i
. There may be a class of males or females who
do not form partnerships.
In the problem considered here, individuals do not always agree on the
desirability of a member of the opposite sex as a partner. It would be natural to
try and reduce the game considered to a sequence of one-sided choice problems.
However, for games within the general framework presented in Sect.
17.2
there are
some technical problems associated with such an approach. For example, consider
a problem in which attractiveness and character are independent, both have a uni-
form distribution over the integers 0
∈
[
x
i
,
x
i
−
1
)
and a female of attractiveness
y
is of class
j
if
y
∈
[
y
j
,
y
j
−
1
)
m
regardless of sex. It is expected that
individuals of maximum attractiveness,
m
, and close to median character will have
a higher expected utility from search (i.e. be choosier) than individuals of attractive-
ness
m
and extreme character, either 0 or
m
(see Alpern and Reyniers [
3
]). In the
problem considered by McNamara and Collins [
28
] it is relatively easy to order in-
dividuals according to how choosy they should be. This ordering is used to derive
the unique equilibrium satisfying the optimality criterion. Such an ordering is not so
easy in the problem considered here. For example, should a male of attractiveness
m
and character 0 be more or less choosy than a male of attractiveness
m
,
1
,
2
,...,
1and
close to median character? Ramsey [
32
] shows that multiple equilibria may exist in
such a problem, i.e. in general there is no unique sequence of one-sided problems
that can be solved to define an equilibrium.
−
17.4 The Symmetric Model with Character Forming a Circle
Due to the problems outlined in the previous section, we present a model which
allows us to adopt a similar (but not identical) approach to the one used by
McNamara and Collins [
28
]. Attractiveness and character are denoted
X
a
and
X
c
,
respectively. The population is assumed to be large. It will be assumed that
(a)
X
a
and
X
c
are independent. The distribution of
X
a
does not depend on sex.
The distribution of
X
c
in both sexes is uniform on the integers 0
1.
(b) The difference between characters is calculated according to modulo
m
,i.e.
character can be thought of as a circle with 0 and
m
,
1
,...,
m
−
−
1 being neighbouring
characters.
(c) Search and dating costs are
c
1
and
c
2
, respectively, independently of sex.
(d) The utility obtained by a type
x
=[
x
a
,
x
c
]
individual from pairing with a
prospective partner of type
y
=[
y
a
,
y
c
]
is given by
g
(
y
a
,|
x
c
−
y
c
|
)
, i.e. the utility
function is independent of sex.
Using such an approach, intuitively an individual's mating prospects do not
depend on his/her character or sex. Such a game will be referred to as symmetric.
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