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We study how matchmakers use prices to sort heterogeneous participants into competing matching markets and how equilibrium outcomes compare with monopoly in terms of prices, matching market structure, and sorting efficiency under the assumption of complementarity in the match value function. The role of prices to facilitate sorting is compromised by the need to survive price competition. We show that price competition leads to a high-quality market that is insufficiently exclusive. As a result, the duopolistic outcome can be less efficient in sorting than the monopoly outcome in terms of total match value in spite of servicing more participants.