Assortative matching by lottery contests

We study two-sided matching contests with two sets, A and B, each of which includes a finite number of heterogeneous agents with commonly known types. The agents in each set compete in a lottery (Tullock) contest, and then are assortatively matched, namely, the winner of set A is matched with the winner of set B and so on until all the agents in the set with the smaller number of agents are matched. Each agent has a match value that depends on their own type and the type of their match. We assume that the agents' efforts do not affect their match values and that they have a positive effect on welfare. Therefore, an interior equilibrium in which at least some of the agents are active is welfare superior to a corner equilibrium in which the agents choose to be non-active. We analyze the conditions under which there exists a (partial) interior equilibrium where at least some of the agents compete against each other and exert positive efforts.