This article considers a two-sided search market where firms and workers are paired to bargain over a unit surplus. The matching market serves as an endogenous outside option for the agents. The market includes inflexible commitment types who demand a constant portion of any match surplus. The frequency of such types is determined in equilibrium.An equilibrium where there are significant delays in reaching an agreement and where negotiations occasionally break down on the equilibrium path is constructed. Such an equilibrium exists and commitment types affect bargaining dynamics even if the equilibrium frequency of such types is negligible. If the inflows of firms and workers into the market are symmetric, then bargaining involves two-sided reputation building and reputation concerns lead to delays and inefficiency. Access to the market exacerbates bargaining inefficiencies caused by inflexible types. If the inflows of workers and firms are sufficiently asymmetric, then bargaining involves one-sided reputation and commitment types determine the terms of trade.

We model a long-run relationship as an infinitely repeated game played by two equally patient agents. In each period, the agents play an extensive-form stage game of perfect information with either locally non-conflicting interests or strictly conflicting interests. There is incomplete information about the type of Player 1, while Player 2's type is commonly known. We show that a sufficiently patient Player 1 can leverage Player 2's uncertainty about his type to secure his highest pay-off, compatible with Player 2's individual rationality, in any perfect Bayesian equilibrium of the repeated game.