Multiple equilibria in a Dynamic Mating Game with Discrete Types and Similarity Preferences

We consider the dynamic decentralised non atomic mating game n over n periods, initially presented by Alpern and Reyniers (1999). We are dealing especially with the two period mutual choice game 2(m), where individuals can have m types. In the Alpern and Reyniers game, two populations are randomly matched for n periods. Players have one dimensional types which are uniformly distributed over a continuous or a discrete interval. There exist a continuum of players and no new players can enter the game in any period. In each period, each party of a matched pair (i,j) can either accept or reject the other. If both accept, then they form a mated couple and leave the game, with both paying a cost of ji-jj. Otherwise, they both proceed unmated into the next period. This process is called ‘mutual choice’selection. At the end of the game, all players prefer to be mated than to remain unmated. Players have similarity preferences, searching for a partner whose type is close to their own. Hence, they try to minimise their cost of mating, de…ned above as the absolute distance between their type and the type of their potential partner. In the current paper, we present brie‡y the analysis of the continuoustype n game and focus on the discrete-type n(m) game. Our main result is the existence of multiple equilibria in n(m) which contrasts with the analysis of Alpern and Reyniers and the relevant literature, since in the latter only one equilibrium is described. Moreover, we provide a method for determining all the possible equilibria in the discrete-type game. Finally we comment on the e¤ectiveness and stability of the equilibrium strategies in the game 2(m).