Optimal Dynamic Matching Under Impatient Demand and Patient Supply

We consider an infinite-horizon dynamic matching problem where the agents on one side, say supply side, are long-lived, and those on the other, say demand side, are impatient and will get lost if unmatched upon arrival. The agents on either side are vertically differentiated with a high- or low-quality level and arrive at the platform sequentially. The matching reward has a supermodular structure, more specifically, it is the multiplication of quality levels of agents in a match. In a centralized setting, we show that the optimal matching priority is different from assortative mating that is optimal for a system with agents on both sides long-lived (see Baccara et al. 2016). In the lost sales setting, it is optimal to still prioritize high-quality supply over low-quality one to satisfy high-quality demand; however, for the low-quality demand, low-quality supply has priority. This is because the centralized planner has an incentive to hold high-quality supply for future arrivals of high-quality demand. In a decentralized setting where the matching reward is split between agents in a match, in equilibrium, a similar protocol prevails, but the expected queues are inefficiently long or short. We show that in some cases the decentralized matching process can be perfectly coordinated as the centralized one by only adjusting the reward allocation to high-quality supply.