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17.5 Equilibrium Conditions
For a game defined within the general framework, introduced in Sect. 17.2 ,we
require an equilibrium profile to satisfy the following generalisation of the optimality
criterion for the classical two-sided problem. Namely:
Condition 1: In the dating subgame, an individual accepts a prospective partner if
and only if the utility from such a pairing is at least as great as the individual's
expected utility from future search (ignoring previous costs).
Condition 2: An individual is only willing to date if their expected utility from the
resulting dating subgame minus the costs of dating is as least as great as their
expected utility from future search.
Condition 3: The decisions made by an individual do not depend on the moment at
which the decision is made.
It should be noted that the expected future utility of an individual from search,
and thus the exact form of the dating and soliciting subgames, depends on the strat-
egy profile used in the population as a whole. This dependency will be considered
more fully in Sect. 17.7 .
The most preferred prospective partners of a type
individual are those of
maximum attractiveness who have character x c . Condition 1 states that in the dating
subgame an individual will always accept his/her most preferred partner, since an
individual's future expected utility from search must be less than the utility from
obtaining his most preferred partner. Moreover, if in the dating subgame a male
accepts a female who would give him a utility of k , then he must accept any female
who would give him a utility of at least k . It follows that the acceptable difference in
character is non-decreasing in the attractiveness of a prospective partner.
Condition 3 states that the Nash equilibrium strategy should be stationary. This
reflects the following facts:
(a) An individual starting to search at moment i faces the same problem as one
starting at moment 1.
(b) Since the search costs are linear, after searching for i moments and not finding a
partner, an individual maximises his/her expected utility from search simply by
maximising the expected utility from future search (i.e. by ignoring previously
incurred costs).
[
x a ,
x c ]
We might also be interested in profiles that satisfy the following condition.
(i) In the soliciting subgame, an individual of attractiveness y a is willing to date
prospective partners of attractiveness above some threshold, denoted t
(
y a )
,such
that if y 1 >
, i.e. the more attractive an individual, then the
choosier he/she is when choosing a dating partner.
y 2 ,then t
(
y 1 )
t
(
y 2 )
However, it seems reasonable that individuals of low attractiveness may not
solicit dates from highly attractive prospective partners, as by doing so they might
incur dating costs while it is expected that such a date will not lead to pair formation.
Hence, we do not require individuals to use threshold rules when deciding whether
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