We apply the efficiency axioms of [Mortensen, D. (1982). Efficiency of mating, racing and related games. American Economic Review, 72 968–979.] to a set of matching games involving coordination frictions between buyers. These games include markets with finite numbers of buyers and sellers and markets with infinite numbers of heterogeneous buyers and homogenous sellers. We show that the Mortensen rule, but not the Hosios rule, gives constrained efficient allocations. We also show that the Mortensen rule is implemented by a simple auction