Dynamic Matching Under Patience Imbalance
Zhiyuan Chen, Rui, Chen, Ming Hu, Yun Zhou
TL;DR
We study dynamic two-sided matching with unbalanced patience, where long-lived supply incurs per-period waiting costs $h$ while short-lived demand departs if unmatched; arrivals are two-type (H/L) on both sides with probabilities $p$ and $q$, and payoffs $r_{ij}$ are homogeneous and supermodular. The centralized benchmark yields a threshold-based policy with a threshold $k^{ce}$ that governs how many high-type supplies are kept for potential high-quality matches, and we provide a closed-form expression showing $k^{ce}$ rises with the degree of supermodularity $r_{HH}+r_{LL}-r_{HL}-r_{LH}$ and falls with $h$. In the decentralized market, a welfare-maximizing Markov perfect equilibrium exists under FCFS with a priority-threshold matching rule and a supply-queue threshold $k^{de}=ig floor rac{q ext{α} (r_{HH}-r_{HL})}{h}ig floor$, where α splits the match payoff between supply and demand; crucially, the decentralized outcome can be perfectly aligned with the centralized optimum by selecting α appropriately. When arrival rates are identical on both sides, the centralized welfare is weakly increasing with patience (with diminishing returns), while decentralized welfare depends on α and $h$, potentially increasing or decreasing with patience, and the coordination range of α determines when decentralization attains the central benchmark. Overall, one-sided patience captures most welfare gains in centralized settings, while in decentralized settings platform design via payoff allocation can realize the full efficiency potential, offering actionable guidance for designing two-sided platforms such as ride-hailing, freelancing marketplaces, and organ exchange systems.
Abstract
We study a dynamic matching problem on a two-sided platform with unbalanced patience, in which long-lived supply accumulates over time with a unit waiting cost per period, while short-lived demand departs if not matched promptly. High- or low-quality agents arrive sequentially with one supply agent and one demand agent arriving in each period, and matching payoffs are supermodular. In the centralized benchmark, the optimal policy follows a threshold-based rule that rations high-quality supply, preserving it for future high-quality demand. In the decentralized system, where self-interested agents decide whether to match under an exogenously specified payoff allocation proportion, we characterize a welfare-maximizing Markov perfect equilibrium. Unlike outcomes in the centralized benchmark or in full-backlog markets, the equilibrium exhibits distinct matching patterns in which low-type demand may match with high-type supply even when low-type supply is available. Unlike settings in which both sides have long-lived agents and perfect coordination is impossible, the decentralized system can always be perfectly aligned with the centralized optimum by appropriately adjusting the allocation of matching payoffs across agents on both sides. Finally, when the arrival probabilities for H- and L-type arrivals are identical on both sides, we compare social welfare across systems with different patience levels: full backlog on both sides, one-sided backlog, and no backlog. In the centralized setting, social welfare is weakly ordered across systems. However, in the decentralized setting, the social welfare ranking across the three systems depends on the matching payoff allocation rule and the unit waiting cost, and enabling patience can either increase or decrease social welfare.