A Model of Partnership Formation with Friction and Multiple Criteria - Search Theory: A Game Theoretic Perspective - page 270

3. If required, in a similar way we can derive the optimal strategies of type ]

males given that they date females of the u highest levels of attractiveness,
where u is at least three and not more than the number of attractiveness lev-
els. Hence, we can derive a strategy maximizing the expected utility of a type
[

[ i max , r

male. This strategy defines what attractiveness levels induce solicitation
of a date from a male of maximum attractiveness and what types of females
should be paired with after such dates (i.e. the pattern of dates and partnerships
exhibited by individuals of maximum attractiveness at equilibrium).
4. The strategy defined in Points 1-3 above should be extended to ensure trembling
hand equilibria in all the possible derived dating subgames involving males of
maximum attractiveness. The set of acceptable females in dating subgames can
be easily found using Condition 1: i.e. a male of maximum attractiveness should
accept a prospective partner in the dating subgame if and only if the utility he
obtains from such a partnership is greater than his expected utility from search.
Note that the behaviour of males in dating subgames that do not occur un-
der the equilibrium profile does not affect their expected utility from search
at equilibrium.
Hence, the problem faced by a male of maximum attractiveness can be solved by
solving a sequence of one-sided problems. The strategies used by other individuals
of maximum attractiveness can be found using the symmetry of the profile with
respect to character and sex. Note that it is assumed that if a male is indifferent
between two strategies, then he uses the strategy which maximizes the number of
attractiveness levels inducing willingness to date, together with the number of types
of female that he will eventually pair with.
Suppose we have found the equilibrium strategies of individuals of attractiveness

i max , r ]

> reduces to a one-sided problem in

which the set of females of higher attractiveness who are willing to date and pair
with such males has been derived. The optimal response of such a male can be
calculated in a way analogous to the one described in Points 1-4 above, with the
following adaptations (which take into account the form of the equilibrium):
(a) The initial strategy of a type

i . The problem faced by a male of type [ i , r ]

male is as follows: (A) solicit a date with
(i) any female of a greater attractiveness who would solicit a date from him,
(ii) any female of the same attractiveness and (iii) females of lower attractive-
ness who would be solicited by males of attractiveness i

[ i , r ]

1, (B) in the dating
game always pair with a female of the same type and any female who gives at
least the same utility as the expected utility of an individual of attractiveness
i

+

1.
(b) The set of prospective partners accepted in the dating subgame and solicited
in the soliciting subgame is extended in an analogous way to the case of indi-
viduals of maximum attractiveness. Firstly, we find the optimal strategy which
involves dating individuals of the attractiveness levels derived in (a). This is
done by including successively less preferred mates into the set of those mated
with in the dating subgame. If required, we then find the optimal rules obtained
when an individual solicits dates with successively less attractive prospective
partners.

+