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to solicit a date. For example, Härdling and Kokko [
20
] argue that in certain circum-
stances small males should avoid courting attractive females to avoid the possibility
of attacks from larger males.
π
∗
,of
Definition 1.
An equilibrium profile, denoted
is a strategy profile under
which the behaviour of all searchers satisfies Conditions 1-3 in each of the possi-
ble subgames. The value function of
Γ
π
∗
is the set of expected
utilities of each individual according to type under the strategy profile
Γ
corresponding to
π
∗
.
π
∗
must define the appropriate behaviour in all possible dating
subgames, even those that do not occur under
Note that
π
∗
.
In the particular case of the symmetric game, we wish to find an equilibrium
which is symmetric with respect to character and sex. That is to say:
Condition 4: If an individual of type
is willing to date a prospective partner
of attractiveness
y
a
, then any individual of attractiveness
x
a
is willing to date a
prospective partner of attractiveness
y
a
.
Condition 5: If an individual of type
[
x
a
,
x
c
]
[
x
a
,
x
c
]
is willing to pair with a prospective
partner of type
[
y
a
,
y
c
]
in the dating subgame, then an individual of type
[
x
a
,
x
c
+
i
]
is willing to pair with a prospective partner of type
[
y
a
,
y
c
+
i
]
[addition is
carried out mod
(
m
)
].
An equilibrium which satisfies Conditions 4 and 5 will be referred to as a
symmetric equilibrium. Note that under a symmetric strategy an individual's ex-
pected utility from search is independent of sex and character, i.e. only depends on
attractiveness.
Note that at equilibrium, if an individual of type
is willing to date an
individual of attractiveness
j
, then he/she must be willing to pair with some prospec-
tive partners of attractiveness
j
(otherwise unnecessary dating costs are incurred).
Hence, a type
[
i
,
k
]
[
,
]
prospective partner (the most preferred partner of such
attractiveness) must be acceptable in the corresponding dating subgame.
j
k
17.6 Deriving the Expected Utilities Under a Given Strategy
Profile
Consider the symmetric game described above. We will look for a symmetric
equilibrium profile, thus we may assume that the strategy profile used is symmetric
(i.e. satisfies Conditions 4 and 5). Given the strategy profile used by a population,
we can define which pairs of types of individuals proceed to the dating subgame
and which pairs of types of individuals form pairs. From this, it is relatively simple
to calculate the expected length of search and the expected number of dates of an
individual of a given type.
Let
p
to be
the set of types of prospective partners that an individual of type
y
will date (under
(
x
)
be the probability that an individual is of type
x
.Define
M
1
(
y
;
π
)
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