  We study a dynamic matching procedure where homogeneous agents arrive at
random according to a Poisson process and form edges at random yielding a
sparse market. Agents leave according to a certain departure distribution and
may leave early by forming a pair with a compatible agent. The primary
objective is to maximize the number of matched agents. Our main result is to
show that a mild condition on the departure distribution suffices to get almost
optimal performance of instantaneous matching, despite operating in a thin
market. We are thus the first to provide a natural condition under which
instantaneous decisions are superior in a market that is both sparse and thin.
This result is surprising because similar results in the previous literature
are based on market thickness. In addition, instantaneous matching performs
well with respect to further objectives such as minimizing waiting times and
avoiding the risk of market congestion. We develop new techniques for proving
our results going beyond commonly adopted methods for Markov processes.
