  We consider an intermediary's problem of dynamically matching demand and
supply of heterogeneous types in a periodic-review fashion. More specifically,
there are two disjoint sets of demand and supply types, and a reward associated
with each possible matching of a demand type and a supply type. In each period,
demand and supply of various types arrive in random quantities. The platform's
problem is to decide on the optimal matching policy to maximize the total
discounted rewards minus costs, given that unmatched demand and supply will
incur waiting or holding costs, and will be carried over to the next period
(with abandonment). For this dynamic matching problem, we provide sufficient
conditions on matching rewards such that the optimal matching policy follows a
priority hierarchy among possible matching pairs. We show those conditions are
satisfied by vertically and unidirectionally horizontally differentiated types,
for which quality and distance determine priority, respectively. As a result of
the priority property, the optimal matching policy boils down to a
match-down-to threshold structure when considering a specific pair of demand
and supply types in the priority hierarchy.
