Centre for Discrete and Applicable Mathematics |
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CDAM Research Report, LSE-CDAM-2004-05April 2004 |
Steve Alpern and Diane Reyniers
Abstract
We present a two-sided search model in which individuals from two
groups (males and females, employers and workers) would like to form a long
term relationship with a highly ranked individual of the other group, but
are limited to individuals who they randomly encounter and to those who also
accept them.
This article completes the research program, begun in Alpern and Reyniers
(1999), of providing a game theoretic analysis for the Kalick-Hamilton
(1986) mating model in which a cohort of males and females of various
`fitness' or `attractiveness' levels are randomly paired in successive
periods and mate if they accept each other. Their model compared two
acceptance rules chosen to represent homotypic (similarity) preferences and
common (or `type') preferences. Our earlier paper modeled the first kind by
assuming that if a level x male mates with a level y female, both get
utility -|x-y|, whereas this paper models the second kind by giving the male
utility y and the female utility x.
Our model can also be seen as a continuous generalization of the discrete
fitness-level game of Johnstone (1997). We establish the existence of
equilibium strategy pairs, give examples of multiple equilibria, and
conditions guaranteeing uniqueness. In all equilibria individuals become
less choosy over time, with high fitness individuals pairing off with each
other first, leaving the rest to pair off later. This route to assortative
mating was suggested by Parker (1983). If the initial fitness distributions
have atoms, then mixed strategy equilibria may also occur. If these
distributions are unknown, there are equilibria in which only individuals in
the same fitness band are mated, as in the steady state model of MacNamara
and Collins (1990) for the 'job search problem'.
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