Before revision:  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. As arrival rate increases the greedy policy matches a (weakly) higher fraction of agents than any other policy and leads to (weakly) lower average waiting time than any other policy.
After revision:  While this trade-off between a "thickermarket" and quick matching is present in small markets we show that it vanishes in large markets ; A greedy policy leads to shorter waiting times, and more agents matched than any other policy.