Centre for Discrete and Applicable Mathematics

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 |i-j|. 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, defined above as the absolute distance between their type and the type of their potential partner.

In the current paper, we present briefly the analysis of the continuous-type Γ_n game and focus on the discrete-type Γ_n(m) game. Our main result is the existence of multiple equilibria in Γ_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 effectiveness and stability of the equilibrium strategies in the game Γ₂(m).