Superiority of Instantaneous Decisions in Thin Dynamic Matching Markets

TL;DR

The paper analyzes dynamic, stochastic matching markets with Poisson arrivals, random compatibility, and finite sojourn times, focusing on two policies: greedy instantaneous matching and patient delayed matching. It shows that a uniformly bounded minimum sojourn time yields exponential loss decay for GDY in thin, sparse markets, challenging the common emphasis on market thickness; GDY also achieves inverse-linear waiting times, indicating strong overall performance. The PAT policy remains compelling, delivering exponential loss guarantees for unit sojourn times and constant waiting times tied to the distribution mass near zero, with simulations corroborating the theoretical findings. Together, the results illuminate how departure behavior can substitute for market thickness in achieving near-optimal matching performance, and they open avenues for extensions to heterogeneous agent populations and hybrid strategies.

Abstract

We study a dynamic matching setting where homogeneous agents arrive at random according to a Poisson process and randomly form edges yielding a sparse market. Agents stay in the market according to a certain sojourn time and wait to be matched with a compatible agent by a matching algorithm. When their maximum sojourn time is reached, they perish unmatched. The primary objective is to maximize the number of matched agents. Our main result is to show that a uniformly guaranteed sojourn time suffices to get almost optimal performance of instantaneous matching. Interestingly, this matching policy essentially keeps as few agents in the market as possible. Hence, in contrast to the common paradigm that market thickness is the crucial property for obtaining strong matching performance, we show that the agents' sojourn behavior can be an equally powerful factor. In addition, instantaneous matching is close to optimal with respect to minimizing waiting time. We develop new techniques for proving our results going beyond commonly adopted methods for Markov processes.

Figures (2)

• Figure 1: Timeline of an agent's sojourn in our stochastic matching market. The greedy and patient matching policies are highlighted in blue.
• Figure 2: Visualization of results of simulations. Each datapoint in the first, third, and fourth figures corresponds to $10$ runs for $m=1000$ and $T=100$. Each box-and-whisker plot in the second figure represents $200$ runs for $m=500$ and $T=100$. Note that the linear behavior in the logarithmic plots of the first and third figure means that the decay is exponential. For reference, we add our theoretically obtained bounds in the large market limit, i.e., the upper bound on the performance of $\text{GDY}$ (cf. \ref{['thm:greedy_loss_upper_bound']}), and our general (cf. \ref{['thm:lossgenlb']}) and $\text{GDY}$-specific (cf. \ref{['cor:greedylbconst']}) lower bounds obtained in \ref{['sec:gdyanalysis']}.

Theorems & Definitions (62)

• Theorem 1.1: Informal version of \ref{['thm:epslowerbound']}
• Theorem 1.2: Informal version of \ref{['thm:lossgenlb']}
• Theorem 1.3: Informal version of \ref{['thm:highloss']}
• Theorem 1.4: Informal version of \ref{['lem:agg_waiting']}
• Theorem 4.1
• Lemma 4.1
• proof
• Theorem 4.2
• Lemma 4.2
• Lemma 4.2