Equilibria of dynamic mutual choice mating games

We examine dynamic mutual choice mating games: members of two populations (males and females) are randomly matched in successive periods and form couples only if they mutually accept each other. Players are heterogeneous and their "types" are distributed in an interval. The utility that a player obtains from a mating depends on both his type and the type of his partner. We consider three type of preferences: (i) homotypic (preference for similar types), (ii) common (preference for high types) and (iii) age dependent preferences. In case (i), we explore the equilibrium behaviour when the sex ratio r is 1 : 1. We extend the results of Alpern and Reyniers (1999) two period continuous type game. Next, we develop an algorithm, for reducing the potential equilibrium strategies in the two period discrete type model. Using this algorithm, we are able then to determine the equilibria in some discrete type models; we find multiple equilibria in some cases. Even when we do not assume the sexes adopt identical strategies, we find that this always occurs at equilibrium. We also explore the equilibrium behaviour and the mating patterns when players have mixed preferences (combination of mixed and common preferences) with the help of a discrete type model. In case (ii), we extend the Alpern and Reyniers (2005) common preferences model to the case of a sex ratio r > 1. Males remaining unmated after the end of the game have negative utility -c. We analyse how the equilibria of this mating game are formed, depending on the parameters r and c. It is proved that males are not always choosy at equilibrium and for some (r, c) there are multiple equilibria. In a region of (r, c) space with multiple equilibria, we compare these, and analyse their "efficiency" in several respects (stability and welfare). Finally in (iii), based on an idea of Alpern and Reyniers (1999) and Alpern (2008) we analyse the equilibrium strategies in a steady state model where individuals have age dependent preferences and they seek partners who provide to them the longest possible common fertile life. We determine the equilibrium strategies as the sex ratio of the incoming population changes and comment of the efficiency of equilibria