[Paper Review] Dynamic Matching Under Patience Imbalance

Zhiyuan Chen, Rui+3 more|arXiv (Cornell University)|Feb 3, 2026

Game Theory and Voting Systems0 citations

TL;DR

The paper studies a two-sided dynamic matching platform with one-sided patience (long-lived supply vs. impatient demand), derives centralized optimal policies and welfare-maximizing decentralized equilibria, and compares welfare across backlog configurations. It shows that decentralized outcomes can be perfectly aligned with centralized optimum by payoff sharing, and analyzes how patience affects welfare under different regimes.

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.

Motivation & Objective

• Motivate understanding of dynamic two-sided matching under patience imbalance between supply (long-lived) and demand (impatient).
• Characterize centralized optimal dynamic matching policy and its structural thresholds.
• Characterize welfare-maximizing decentralized equilibrium under FCFS with payoff-splitting.
• Compare centralized and decentralized outcomes and welfare across backlog configurations (full, one-sided, no backlog).
• Provide coordination mechanisms to align decentralized outcomes with the centralized optimum via payoff allocation.

Proposed method

• Model a two-sided platform with one supply and one demand arrival per period; supply are long-lived with waiting cost h, demand are impatient and depart if unmatched.
• Assume two types (H and L) on each side with supermodular payoffs r_ij for pair (i,j).
• Prove centralized optimal policy has a threshold structure with greedy matching for H-demand and threshold-based matching for L-demand; derive optimal threshold k^ce and its properties.
• Develop a decentralized FCFS equilibrium where agents accept/reject matches and payoff allocation is exogenous; show a welfare-maximizing equilibrium with a priority-and-threshold pattern.
• Show that decentralization can be aligned with centralized optimum by tuning payoff shares between supply and demand in matches.
• Provide insights into welfare ranking under three backlog configurations when arrival probabilities are identical on both sides.

Experimental results

Research questions

• RQ1 What is the structure of the centralized optimal dynamic matching policy under one-sided backlog with patience imbalance?
• RQ2 What is the nature of the welfare-maximizing decentralized equilibrium under FCFS when payoff is split across agents?
• RQ3 Can the decentralized outcome be coordinated to coincide with the centralized optimum via payoff allocation?
• RQ4 How does payer/patience level (backlog) affect social welfare under centralized and decentralized settings?
• RQ5 How do arrival probabilities and supermodularity of payoffs influence optimal thresholds and welfare?

Key findings

• Optimal centralized policy is threshold-based: greedily match H-demand with H-supply if available, otherwise with L-supply; match L-demand with L-supply if possible, else with H-supply only if H-supply exceeds threshold k^ce.
• Optimal threshold k^ce increases with the degree of supermodularity and decreases with the unit waiting cost h; explicit forms given for p>q, p=q, p<q.
• Centralized welfare W^ce is decreasing in the waiting cost h and depends on p and q with distinct expressions; assortative matching (HH, LL) dominates cross-matching due to supermodularity.
• In the decentralized FCFS equilibrium, a priority-and-threshold matching pattern emerges, and the equilibrium welfare can be aligned with the centralized optimum by adjusting payoff sharing between sides.
• Patience on at least one side improves centralized welfare, while in the decentralized setting welfare effects depend on payoff allocation; higher waiting costs can, under certain conditions, increase decentralized welfare by reducing backlog.
• When demand and supply arrival probabilities are identical, welfare ranking under centralized systems is: full backlog ≥ one-sided backlog ≥ no backlog; under decentralized systems, ranking depends on payoff allocation and waiting cost, with patience potentially increasing or decreasing welfare.