Before revision:  A greedy policy, which attempts to match agents upon arrival, does not account for the positive externality  waiting agents generate as they make it easier to match agents that arrive in the future , and may result in more unmatched than other policies. While this trade-off between a "thickermarket" and quick matching is present in small markets we show that it vanishes in large markets and the greedy policy nevertheless dominates any other dynamic matching policy.
After revision:  A greedy policy, which attempts to match agents upon arrival, does not account for the positive externality waiting agents generate by facilitating future matchings. We prove that this trade-off between a "thickermarket" and quick matching is present in small markets we show that it vanishes in large markets and the greedy policy nevertheless dominates any other dynamic matching policy.