  We study dynamic matching in exchange markets with easy- and hard-to-match
agents. A greedy policy, which attempts to match agents upon arrival, ignores
the positive externality that waiting agents generate by facilitating future
matchings. We prove that this trade-off between a ``thicker'' market and faster
matching vanishes in large markets; A greedy policy leads to shorter waiting
times, and more agents matched than any other policy. We empirically confirm
these findings in data from the National Kidney Registry. Greedy matching
achieves as many transplants as commonly-used policies (1.6\% more than
monthly-batching), and shorter patient waiting times.
