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attractiveness are more likely to accept them), which in turn might lead to those of
attractiveness
i
2
becoming less choosy at such an equilibrium (as those of slightly
higher attractiveness are less likely to accept them), and so on.
However, suppose the distribution of the types of individuals is fixed and consider
the space of possible cost vectors
R
+
. For nearly all cost vectors
(i.e. apart from on a set of measure 0), it is expected that there will be a unique
equilibrium of the game
R
+
×
[
c
1
,
c
2
]
∈
Γ
.
17.8.2 Example
Suppose that the support of both
X
a
and
X
c
is
and the distributions
of attractiveness and character are uniform. The search costs,
c
1
, and the interview
costs,
c
2
are equal to
{
0
,
1
,
2
,
3
,
4
,
5
,
6
}
1
7
. The utility obtained from a partnership is defined to be the
attractiveness of the partner minus the distance (modulo 7) between the characters
of the pair.
Since the expected payoff of a male does not depend on whether he is willing to
date females who are unwilling to date with him, in order to derive the expected pay-
offs of individuals under any strategy profile it suffices to consider strategy profiles
of the following form: a searcher of attractiveness
i
is willing to date prospective
partners of attractiveness
a
i
and in the dating subgame will pair with a prospec-
tive partner who gives a reward of at least
b
i
,
i
≥
=
0
,
1
,
2
,
3
,
4
,
5
,
6. Denote such a
strategy by
. We derive the equilibrium strategies of
individuals in the order of most attractive to least attractive. Hence, for example, by
{
(
{
(
a
6
,
b
6
)
,
(
a
5
,
b
5
)
,...,
(
a
0
,
b
0
)
}
we denote the set of strategy profiles such that individu-
als of attractiveness levels 5 and 6 use the strategies defined by
a
6
,
b
6
)
,
(
a
5
,
b
5
)
,•,•,•,•}
,
respectively, and the strategies of the remaining individuals are undefined,
but sat-
isfy Conditions 1 and 3
, i.e. a trembling hand equilibrium is always played in the
dating subgame and the strategy profile is stationary. Other similar sets of strategy
profiles will be denoted in an analogous manner.
First we consider males of maximum attractiveness. Suppose they only solicit
dates with females of attractiveness 6. The ordered preferences of a
(
a
5
,
b
5
)
and
(
a
6
,
b
6
)
[
6
,
3
]
male are
as follows: first (group one) -
[
6
,
3
]
, second equal (group two) -
[
6
,
2
]
,
[
6
,
4
]
, fourth
equal (group 3)
. Group 1, 2, 3
and 4 females give a utility from pairing of 6, 5, 4 and 3, respectively. The sets of
strategy profiles in which type three males mate with (a) only those from group 1,
(b) those from groups 1 and 2, (c) those of groups 1, 2 and 3 and (c) those from
all four groups are
[
6
,
1
]
,
[
6
,
5
]
and sixth equal (group 4) -
[
6
,
0
]
,
[
6
,
6
]
{
(
6
,
6
)
,•,•,•,•,•}
,
{
(
6
,
5
)
,•,•,•,•,•}
,
{
(
6
,
4
)
,•,•,•,•,•}
and
{
(
, respectively. We successively include females into the set of
acceptable partners starting from the most preferred until no female of attractiveness
6 outside this set gives a greater utility than the current expected utility of a type
[
6
,
3
)
,•,•,•,•,•}
6
,
3
]
male.
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