[Paper Review] Asymptotically Optimal Control of a Centralized Dynamic Matching Market with General Utilities

José Blanchet, Martin I. Reiman+3 more|arXiv (Cornell University)|Feb 8, 2020

Economic Policies and Impacts4 citations

TL;DR

This paper studies a centralized dynamic matching market with Poisson-distributed buyers and sellers who abandon after exponential waiting times, optimizing long-run average utility via threshold-based policies. It proves that under light-tailed matching utilities, a population threshold policy with threshold $ n / \ln n $ is asymptotically optimal; under heavy-tailed utilities, it characterizes optimal thresholds and shows the utility threshold policy outperforms the population threshold policy as tail heaviness increases.

ABSTRACT

We consider a matching market where buyers and sellers arrive according to independent Poisson processes at the same rate and independently abandon the market if not matched after an exponential amount of time with the same mean. In this centralized market, the utility for the system manager from matching any buyer and any seller is a general random variable. We consider a sequence of systems indexed by $n$ where the arrivals in the $n^{\\mathrm{th}}$ system are sped up by a factor of $n$. We analyze two families of one-parameter policies: the population threshold policy immediately matches an arriving agent to its best available mate only if the number of mates in the system is above a threshold, and the utility threshold policy matches an arriving agent to its best available mate only if the corresponding utility is above a threshold. Using a fluid analysis of the two-dimensional Markov process of buyers and sellers, we show that when the matching utility distribution is light-tailed, the population threshold policy with threshold $\\frac{n}{\\ln n}$ is asymptotically optimal among all policies that make matches only at agent arrival epochs. In the heavy-tailed case, we characterize the optimal threshold level for both policies. We also study the utility threshold policy in an unbalanced matching market with heavy-tailed matching utilities and find that the buyers and sellers have the same asymptotically optimal utility threshold. We derive optimal thresholds when the matching utility distribution is exponential, uniform, Pareto, and correlated Pareto. We find that as the right tail of the matching utility distribution gets heavier, the threshold level of each policy (and hence market thickness) increases, as does the magnitude by which the utility threshold policy outperforms the population threshold policy.

Motivation & Objective

• To design and analyze asymptotically optimal matching policies in a centralized dynamic market with general random utilities.
• To address the trade-off between market thickness and agent abandonment in large-scale matching systems.
• To characterize optimal thresholds for population and utility threshold policies under light- and heavy-tailed utility distributions.
• To establish that the population threshold policy with threshold $ n / \ln n $ is asymptotically optimal among arrival-time-only matching policies under light-tailed utilities.
• To compare performance of utility and population threshold policies, showing improved relative performance of utility threshold as utility tail heaviness increases.

Proposed method

• Uses a sequence of systems scaled by $ n $, with arrival rates sped up by $ n $, to analyze asymptotic behavior.
• Applies fluid analysis to a two-dimensional Markov process tracking buyer and seller counts.
• Employs extreme value theory to model the asymptotic distribution of maximum matching utilities from large pools.
• Uses regularly varying functions to characterize tail behavior of utility distributions and derive threshold scaling.
• Derives upper bounds on achievable utility rates and compares them to policy-specific rates.
• Analyzes both population threshold (match only if system size exceeds threshold) and utility threshold (match only if utility exceeds threshold) policies.

Experimental results

Research questions

• RQ1 What is the asymptotically optimal threshold level for the population threshold policy when matching utility distributions are light-tailed?
• RQ2 How do optimal threshold levels scale with system size $ n $ under heavy-tailed utility distributions?
• RQ3 Does the utility threshold policy outperform the population threshold policy, and if so, by how much, as utility tail heaviness increases?
• RQ4 Can the decoupling of queueing dynamics and extremal utility behavior be rigorously justified in a fluid limit?
• RQ5 Are the optimal thresholds for buyers and sellers symmetric in an unbalanced market with heavy-tailed utilities?

Key findings

• For light-tailed matching utility distributions, the population threshold policy with threshold $ n / \ln n $ is asymptotically optimal among all policies that match only at agent arrival epochs.
• Under heavy-tailed utilities, the paper characterizes the optimal threshold levels for both population and utility threshold policies.
• The utility threshold policy consistently outperforms the population threshold policy, with the performance gap increasing as the right tail of the utility distribution becomes heavier.
• For the Gumbel domain of attraction (e.g., exponential, normal), the expected maximum utility scales as $ b_n + a_n \mu $, where $ b_n $ is slowly varying and $ \mu $ is Euler’s constant or a gamma function value.
• For the Frechet domain (e.g., Pareto), the optimal threshold scales with $ \bar{F}^{-1}(1/n) $, and the utility threshold policy’s advantage grows with tail heaviness.
• In an unbalanced market with heavy-tailed utilities, buyers and sellers share the same asymptotically optimal utility threshold, indicating symmetric optimal behavior.