Does competition solve the hold-up problem?

In an environment in which heterogeneous buyers and sellers undertake ex ante investments, the presence of market competition for matches provides incentives for investment but may leave inefficiencies, namely hold-up and coordination problems. This paper shows, using an explicitly non-cooperative model, that when matching is assortative and investments precede market competition, buyers' investments are constrained efficient while sellers marginally underinvest with respect to what would be constrained efficient. However, the overall extent of this inefficiency may be large. Multiple equilibria may arise; one equilibrium is characterized by efficient matches, but there can be additional equilibria with coordination failures.