Directed search and optimal production

I consider a model of directed search where strategic sellers advertise general trading mechanisms. A mechanism determines the number of buyers that will get served and the side payments, as a function of ex-post realized demand. Buyers observe these advertisements and visit one seller without being able to coordinate their visiting strategies. Despite the oligopolistic nature of the model, all symmetric equilibria are constrained-efficient. In small markets, multiple equilibria exist that are not payoff equivalent. This indeterminacy vanishes as the market grows large. I provide closed form solutions for equilibrium prices, profits, and the matching function under any parameter values.