A Model of Partnership Formation with Friction and Multiple Criteria - Search Theory: A Game Theoretic Perspective - page 272

(using the fact that a male of type [ i − 1 , r ] should pair with any female that a male

of type would pair with). Hence, the algorithm derives the equilibrium strategy

of individuals of attractiveness i

[ i , r ]

− 1 given the strategies used by individuals of at-

tractiveness i . It follows by induction, the symmetry of the strategy profile derived

and the form of the equilibrium strategy from Theorems 1 - 3 that the algorithm gives
a symmetric equilibrium profile.

≥

Due to the form of the equilibrium and the finite number of types, the resulting
strategy profile is well defined and unique. The theorem below follows directly from
the construction of the symmetric equilibrium.

Theorem 5. Assume that if any individual is indifferent between two strategies, then
he/she uses the strategy which maximises the number of attractiveness levels in-
ducing the solicitation of a date, together with the number of types of prospective
partners that he/she will eventually pair with. There exists exactly one symmetric
equilibrium of the symmetric game.

One might ask whether an asymmetric equilibrium exists. Consider a
finite-horizon game where each individual can observe up to n prospective part-
ners. Suppose that in addition to Conditions 1-3, we require that an equilibrium
profile in

Γ is the limit of an equilibrium search profile in the finite-horizon game

when n 1, at the unique equilibrium profile each individual accepts

any prospective mate (i.e. the equilibrium is symmetric). When n steps remain, an
individual (a) should solicit dates from prospective partners of attractiveness i if the
expected utility from such a date is greater than the future expected utility from
search (i.e. when n

→ ∞ .When n =

1 steps remain) and (b) pair with prospective partners in the
dating subgame if the expected utility from pairing is greater than the future ex-
pected utility from search. Given the equilibrium profile in the

−

-step game is
symmetric, all these expected utilities are independent of sex and character. Hence,
the unique equilibrium profile in the n -step game is symmetric. It follows that we
can strengthen Theorem 5 to the following theorem:

( n − 1 )

Theorem 6. If, in addition to the assumptions of Theorem 5 , it is assumed that the
solution to the infinite-horizon game must be the limit of the solution to the appro-
priately defined finite-horizon game, then there is a unique equilibrium profile of the
symmetric game

Γ , which itself is symmetric.

One might consider what equilibria are possible when there is equality between
the future expected reward of a searcher of type [ i , j ] and the reward obtained by

mating with a prospective partner of type [ i 0 , j 0 ] . This will be of importance when

i > i 0 . If searchers of type [ i , j ] do not accept prospective partners of type [ i 0 , j 0 ] ,

then searchers of type will have a lower expected reward from search than at

the equilibrium considered above and thus become less choosy than at the original
equilibrium. This may well have knock on effects on the equilibrium strategy of
individuals of attractiveness below i 0 . Suppose i 0 >

[ i 0 , j 0 ]

i 2 . Those of attractiveness
i 1 may become more choosy at such an equilibrium (as those of slightly higher

i 1 >