On the Optimality of Greedy Policies in Dynamic Matching

We study centralized dynamic matching markets with finitely many agent types and heterogeneous match values. A network topology describes the pairs of agent types that can form a match and the value generated from each match. A matching policy is hindsight optimal if the policy can (nearly) maximize the total value simultaneously at all times. We find that suitably designed greedy policies are hindsight optimal in two-way matching networks. This implies that there is essentially no positive externality from having agents waiting to form future matches. We first show that the greedy longest-queue policy with a minor variation is hindsight optimal. Importantly, the policy is greedy relative to a residual network, which includes only non-redundant matches with respect to the static optimal matching rates. Moreover, when the residual network is acyclic (e.g., as in two-sided networks), we prescribe a greedy static priority policy that is also hindsight optimal. The priority order of this policy is robust to arrival rate perturbations that do not alter the residual network. Hindsight optimality is closely related to the lengths of type-specific queues. Queue-lengths cannot be smaller (in expectation) than Ο(1/ε), where ε is the general position gap that quantifies the stability in the network. The greedy longest-queue policy achieves this lower bound.