Cai, X., Gautier, PieterA. and Wolthoff, RonaldP. (2017). Search frictions, competing mechanisms and optimal market segmentation Journal of Economic Theory, 169:453--473.

In a market in which sellers compete for heterogeneous buyers by posting mechanisms, we analyze how the properties of the meeting technology affect the allocation of buyers to sellers. We show that a separate submarket for each type of buyer is the efficient outcome if and only if meetings are bilateral. In contrast, a single market with all agents is optimal if and only if the meeting technology satisfies a novel condition, which we call “joint concavity.” Both outcomes can be decentralized by sellers posting auctions combined with a fee that is paid by (or to) all buyers with whom the seller meets. Finally, we compare joint concavity to two other properties of meeting technologies, invariance and non-rivalry, and explain the differences.