Matching markets with middlemen under transferable utility

This paper studies matching markets in the presence of middlemen. In our framework, a buyer-seller pair may either trade directly or use the services of a middleman; and a middleman may serve multiple buyer-seller pairs. For each such market, we examine the associated TU game. We first show that, in our context, an optimal matching can be obtained by considering the two-sided assignment market where each buyer-seller pair is allowed to use the mediation services of any middleman free of charge. Second, we prove that matching markets with middlemen are totally balanced: in particular, we show the existence of a buyer-optimal (seller-optimal) core allocation where each buyer (seller) receives her marginal contribution to the grand coalition. In general, the core does not exhibit a middleman-optimal allocation, not even when there are only two buyers and two sellers. However, we prove that in these small markets the maximum core payoff to each middleman is her marginal contribution. Finally, we establish the coincidence between the core and the set of competitive equilibrium payoff vectors.