A Model of Partnership Formation with Friction and Multiple Criteria - Search Theory: A Game Theoretic Perspective - page 260

that individuals must always date before forming a pair. The total utility of an
individual is taken to be the utility gained from the partnership minus the sum
of the search costs and dating costs incurred. It is assumed that utility is not
transferable and individuals maximise their expected total utility from search.

This approach implicitly assumes that there is the same number of males as
females. However, the model can be easily adapted to allow the number of males and
females to differ. Suppose that there are r times as many males as females. In this
case, we may assume that at each moment a proportion

( − ) /

r 1 r of males meet a

prospective partner who would give them an expected utility of − ∞ . In reality, such

males do not meet a prospective partner.
The supergame is defined to be the game in which each player observes a

sequence of prospective partners as described above. An encounter between two
prospective partners will be referred to as the encounter game. The encounter game
is split into two subgames, the soliciting subgame, when players decide whether to
date or not and the dating subgame, when they decide whether to form a partnership
or not.

Γ

17.3 Comparison with the Classical Partnership
Formation Game

Two-sided problems are by nature game-theoretic and so we look for a Nash
equilibrium solution at which no individual can improve their expected utility by
changing their strategy. Note that there may be multiple Nash equilibria. For exam-
ple, suppose that mate choice is based only on attractiveness and there are only two
levels of attractiveness: high and low. Suppose individuals of high attractiveness
only accept individuals of low attractiveness as partners. Similarly, individuals of
low attractiveness only accept individuals of high attractiveness as partners. It can
be seen that this is a Nash equilibrium, since e.g. a male of high attractiveness cannot
gain by accepting a female of high attractiveness, since she would not accept him.
Also, he could not gain by rejecting a female of low attractiveness, since he would
not find a partner. However, one would expect that if a male accepts a female of at-
tractiveness x , then he would accept any female of attractiveness

x . McNamara and
Collins [ 28 ] derive an equilibrium for a game in which choice is based only on at-
tractiveness which satisfies such a condition, referred to as the optimality criterion .
This criterion states that any individual accepts a prospective partner if and only if
the utility from such a partnership is as least as great as the expected utility of the in-
dividual given that he/she continues searching. Such an equilibrium can be derived
inductively. Consider a female of maximum attractiveness. She will be acceptable
to any male. Hence, such females face a one-sided problem and their equilibrium
strategy is of the form: accept the first male of attractiveness

>

≥ x 1 . Call such males

first class. It follows that males of attractiveness x 1 are acceptable to any female

(i.e. face a one-sided problem) and their equilibrium strategy is of the form: accept
the first female of attractiveness

≥

≥ y 1 . Call such females first class. It follows that