Price competition between random and assortive matchmakers

We study price competition between heterogeneous matchmakers in a two-sided matching market. It is well known that when two passive matchmakers who provide random matching services compete with entrance fees, there exists no pure-strategy equilibrium. We show that when a passive matchmaker and an active matchmaker, who provides an assortive matching service, compete with entrance fees, there exists a pure-strategy equilibrium. We also provide welfare comparison between the duopoly and a monopoly operating both the passive and active networks.