A Model of Partnership Formation with Friction and Multiple Criteria - Search Theory: A Game Theoretic Perspective - page 271

(c) After determining the set of females that a male of type solicits a date with

and the females that he would pair with in the dating subgame, the behaviour of
the male in the dating subgames that would not occur under the partial strategy
profile derived as above is determined using the requirement of a trembling
hand equilibrium in the dating subgame.

[ i , r ]

The full strategy profile can be then defined using the assumption of the
symmetry of the profile with respect to sex and character. It should be noted that
although this algorithm has some similarities to the one presented by McNamara
and Collins [ 28 ], it is clearly different. Their algorithm is purely a one-dimensional
search, which derives the sets of attractiveness levels that define a partition of the at-
tractiveness levels for each sex. The value of the symmetric game described here can
be described by a one-dimensional function. However, in order to derive the equi-
librium, a two-dimensional search over levels of both attractiveness and character is
required.
The following theorem shows that the profile derived in this way is a symmetric
equilibrium profile.

Theorem 4. The strategy profile derived using the algorithm given above defines a
symmetric equilibrium profile.
Proof. From the definition of the algorithm, the form of strategy profile derived
satisfies Theorems 2 - 2 . Also, the behaviour of individuals in dating subgames that
do not occur under such a profile explicitly satisfies the equilibrium conditions. This
behaviour has no effect on a searcher's expected utility from search.
First, consider males of maximum attractiveness. From Whittle [ 39 ], if no female
of a given attractiveness level gives a utility as great as the optimal expected reward
from search, then it cannot be optimal to date such a female. The algorithm considers
soliciting dates with prospective partners of successively decreasing attractiveness,
until the most preferred partner of a given attractiveness, say i , gives a lower utility
than the greatest expected utility from search found so far. From Whittle's condition,
a male of maximum attractiveness should not solicit dates from females of attrac-
tiveness i . Also, in the dating subgame a male should not pair with a female who

gives a utility less than the optimal expected utility from search. Hence, the algo-
rithm considers all the best strategies based on dating females of attractiveness

≤

≥ j ,

for all j i and picks the strategy which maximises the expected utility from search.

Hence, this maximises the expected utility of males of maximum attractiveness from
search.
Now suppose that this algorithm derives the equilibrium strategy of males of
attractiveness

>

≥ [ − , ]

i . From Theorems 1 and 3 a male of type i 1 r should solicit

≥

dates from any female of attractiveness i who solicits dates with him and pair with

any female of attractiveness ≥ i who would pair with him in the dating game. The

strategy of a male of type derived by the algorithm extends the sets of those

females solicited and those paired with starting from a strategy which satisfies this
condition. Given the strategies of the females of attractiveness

[ i − 1 , r ]

≥ i , a male of attrac-

tiveness i 1 faces a one-sided search problem and the optimal strategy in this prob-

lem is derived in an analogous way as for individuals of maximum attractiveness

−