Dynamic Matching Markets with Abandonment

• The paper demonstrates that abandonment mechanisms critically affect matching timing, balancing immediate pairing against market thickening to minimize losses.
• It employs stochastic, fluid, and LP-based analytical frameworks to reveal how patience distributions and compatibility structures drive performance metrics like waiting and congestion.
• The findings indicate that enforcing minimum waiting times or adapting matching policies can optimize outcomes in applications from ride-hailing to organ exchange.

Searching arXiv for recent and foundational papers on dynamic matching markets with abandonment. Dynamic matching markets with abandonment are stochastic matching systems in which agents arrive asynchronously, compatibility is typically incomplete, and unmatched agents may exit before being matched because of impatience, criticality, or finite lifetimes. The literature studies this class through several mathematically distinct but conceptually aligned models: sparse random-graph markets with Poisson arrivals and stochastic departures, double-ended queues, periodic-review bipartite matching, discrete two-sided search with finite horizons, spatial ride-hailing systems, and queueing models with explicit reneging. Across these formulations, the central design question is intertemporal: whether to match immediately, wait to thicken the market, ration admissions, prioritize urgent agents, or reshape the state through pricing, information, or dispatch. The resulting theory links abandonment to throughput, welfare, waiting, congestion, and informational requirements, and it shows that the value of waiting is highly sensitive to the patience distribution, the compatibility structure, and the control objective (Bäumler et al., 2022, Akbarpour et al., 2014, Aveklouris et al., 2021).

1. Core definitions and model classes

A dynamic matching market with abandonment is defined by stochastic arrivals, a compatibility relation, a rule governing when matches can occur, and an exit mechanism for unmatched agents. In the sparse random-graph model of thin markets, time is continuous on $[0,T]$, arrivals follow a Poisson process with rate $\lambda$, each arrival forms independent compatibility edges to agents currently in the pool with probability $p=d/\lambda$, and each agent carries an independent maximum sojourn time drawn from a departure distribution $\mu$; an unmatched agent perishes at arrival time plus that sojourn time, while matched agents leave immediately. The benchmark networked-market model uses Poisson arrivals at rate $m$, independent exponential criticality at rate $\lambda$ normalized to $1$, and persistent acceptable edges that appear independently with probability $d/m$. In both formulations, abandonment is an endogenous outflow hazard from a dynamically evolving compatibility graph rather than a separate service discipline (Bäumler et al., 2022, Akbarpour et al., 2014).

Other formulations compress the state differently. In controlled double-ended queues, buyers and sellers arrive according to independent renewal processes, are matched instantaneously under first-come-first-match, and unmatched agents wait in side-specific buffers subject to generally distributed patience times; the one-dimensional imbalance process

$X^{(n)}(t)=Q_s^{(n)}(t)-Q_b^{(n)}(t)$

is sufficient because at any time only one side can have positive queue length. In heterogeneous bipartite matching with reneging, the state is measure-valued because patience is generally non-exponential: for each type, the model tracks both queue length and the age distribution of potential waiters, and abandonment rates are hazard-weighted integrals over these age measures. In discrete two-sided search with finite lifetimes, agents remain for exactly $T$ periods unless matched, and their remaining lifetime directly enters utility. Spatial ride-hailing models add regional states, repositioning actions, pickup times, and customer abandonment with exponential patience. Periodic-review models with type-dependent survival fractions encode abandonment as carry-over coefficients between periods. Queueing formulations for the $\lambda$0-system represent FCFS matching with reneging through low-dimensional state descriptors and obtain exact stationary distributions (Liu et al., 2021, Aveklouris et al., 2021, Agarwal et al., 2021, Li et al., 3 Apr 2025, Hu et al., 2018, Castro et al., 2020).

The term “abandonment” therefore covers several operational mechanisms: exponential criticality, general patience distributions, finite lifetimes, reneging hazards, and period-to-period survival fractions. A common implication is that waiting is costly not only through explicit holding costs but also through loss of match opportunities. This suggests that the timing of matches is part of the market design problem itself, not a secondary implementation issue.

2. Objectives, state variables, and performance metrics

The dominant objective in sparse random-graph models is to maximize matches, equivalently minimize loss, where loss is the fraction of arrivals that perish. For a horizon $\lambda$1, one formulation defines

$\lambda$2

with asymptotic loss given by a $\lambda$3 as $\lambda$4. The same work defines total waiting time and congestion by

$\lambda$5

and proves the exact Little’s-law identity

$\lambda$6

The benchmark dynamic market design model instead uses discounted welfare

$\lambda$7

with $\lambda$8 recovering loss minimization (Bäumler et al., 2022, Akbarpour et al., 2014).

Queueing-control models replace match counts by discounted linear costs. In admission-controlled double-ended queues, the objective is an infinite-horizon discounted functional combining holding costs and penalties for abandonment and blocking:

$\lambda$9

In heterogeneous matching with general patience, the fluid-scale objective maximizes long-run average match value minus waiting costs, with abandonment entering through invariant queue lengths that depend nonlinearly on patience distributions. In centralized bipartite matching with random utilities, the objective is long-run expected utility rate. In discrete finite-lifetime search, the welfare criterion is collective loss relative to the baseline $p=d/\lambda$0, where utility from a match with agent $p=d/\lambda$1 is

$p=d/\lambda$2

In ride-hailing, the key metrics are request completion, customer abandonment, customer waiting, driver waiting, and pickup time (Liu et al., 2021, Aveklouris et al., 2021, Blanchet et al., 2020, Agarwal et al., 2021, Li et al., 3 Apr 2025).

These metrics are not interchangeable. Loss minimization privileges throughput; welfare objectives trade off throughput against delay or match quality; queueing-control objectives internalize blocking; and utility-rate models may rationally tolerate abandonment to access better matches. A recurring theme is that abandonment changes which state variable is economically relevant: pool size, queue length, age profile, imbalance, or local density can each be sufficient in a different model class.

3. Policy families and control levers

The simplest policies are immediate-matching rules. In sparse random-graph markets, the greedy or instantaneous algorithm matches an arriving agent immediately to a uniformly chosen compatible partner if any exist; otherwise the agent joins the pool. The benchmark “Greedy” policy is analogous. Against this stand waiting policies. “Patient” matches only when an agent becomes critical, choosing a compatible partner uniformly at random if one exists. “Patient($p=d/\lambda$3)” interpolates between the two by adding exponential clocks of rate $p=d/\lambda$4 and attempting a match either at clock ticks or at criticality. These policies isolate the timing decision from the compatibility realization and expose the speed-thickness trade-off directly (Bäumler et al., 2022, Akbarpour et al., 2014).

Threshold and reflection policies are the main alternatives when queue lengths or utilities are the control variables. In centralized bipartite markets with random match values, a population-threshold policy matches an arriving agent to its best available mate only if the opposite-side pool exceeds a threshold, while a utility-threshold policy matches only if the best available utility exceeds a threshold. In heavy traffic double-ended queues, optimal admission control takes the form of zero control, one-sided reflection, or two-sided reflection depending on comparisons between blocking penalties and thresholds

$p=d/\lambda$5

The limiting controlled diffusion satisfies

$p=d/\lambda$6

and the associated HJB equation is

$p=d/\lambda$7

In periodic-review heterogeneous matching, optimal policies often exhibit a priority hierarchy and “match-down-to” thresholds. In vertically differentiated markets, this yields top-down matching; in unidirectional horizontal markets, nearer pairs have priority and lower-priority pairs are used only down to protection levels (Blanchet et al., 2020, Liu et al., 2021, Hu et al., 2018).

Several papers convert abandonment into information or geometry design. In discrete finite-lifetime two-sided search, the “modified reasonable strategy” partitions the value-age grid into strips and accepts proposals only within the same strip, implementing a time-dependent acceptance threshold through geometry. In fluid models with general patience, a discrete-review policy uses optimal steady matching rates, while under increasing hazard rates an asymptotically optimal static priority policy exists. In ride-hailing, the proposed two-matching-radius nearest-neighbor algorithm chooses radii by locally minimizing driver waiting plus pickup time. In recent work on greedy dynamic matching, a family of linear programs $p=d/\lambda$8 selects an acceptable set of matches and induces a greedy policy with provable guarantees in homogeneous-departure settings (Agarwal et al., 2021, Aveklouris et al., 2021, Li et al., 3 Apr 2025, Arnosti et al., 6 Jul 2025).

A plausible implication is that “policy complexity” is model-dependent rather than monotone in realism. Some models admit simple myopic rules with sharp guarantees, whereas others require thresholds, free boundaries, or measure-valued state reduction precisely because abandonment interacts with heterogeneity or spatial frictions.

4. Fundamental theoretical results

A foundational result established that waiting to thicken the market can be substantially more important than increasing the speed of transactions. In the benchmark random-graph market with exponential criticality, Greedy keeps the pool thin and has loss of order $p=d/\lambda$9, while Patient maintains a much thicker steady-state pool and attains loss essentially of order $\mu$0. The same work shows that local timing rules can be close to optimal benchmarks: without criticality information, Greedy is close to the online optimum; with criticality information, Patient is close to the omniscient benchmark. The informational conclusion is equally important: partial knowledge of departure times materially improves performance, and a transfer-free mechanism can elicit truthful criticality reports when agents are not too impatient (Akbarpour et al., 2014).

The most striking counterpoint is the thin-market result for guaranteed minimum patience. In sparse markets where each pair is compatible with probability $\mu$1 and there exists $\mu$2 such that $\mu$3, instantaneous matching becomes nearly optimal even though the market remains thin. For the unit-minimum case $\mu$4 and $\mu$5,

$\mu$6

and after time scaling,

$\mu$7

Under constant unit patience, GDY also satisfies

$\mu$8

while any algorithm obeys

$\mu$9

where $m$0 is the expected maximum sojourn time. By contrast, if patience places substantial mass near zero, then for large $m$1,

$m$2

The same paper proves that under minimum patience both GDY and PAT have exponentially small loss in $m$3, but GDY additionally has low waiting:

$m$4

whereas PAT has large waiting whenever many agents have long patience. This sharply limits any universal claim that waiting is always superior (Bäumler et al., 2022).

Other model classes show equally strong dependence on primitives. In utility-based centralized matching, light-tailed utilities imply that a sublinear population threshold is asymptotically optimal among all arrival-only policies; for Exponential$m$5 utilities, the threshold $m$6 is asymptotically optimal. In heavy-tailed settings, the asymptotically optimal population threshold is linear,

$m$7

and the abandonment fraction converges to $m$8. Utility-threshold policies perform even better in heavy-tailed regimes, and optimal thresholds scale with the extremal growth rate of match utilities. In heterogeneous fluid models with general patience, invariant-state queue lengths depend on excess-life transforms of patience distributions, and when hazard rates are increasing the resulting optimization problem admits an optimal extreme point, yielding a state-independent priority policy that is asymptotically optimal on fluid scale. A concrete robustness result shows that the ranking of edges can change when patience distributions have the same mean but different hazard shapes, so exponential patience is not a robust surrogate (Blanchet et al., 2020, Aveklouris et al., 2021).

Competitive-analysis results have recently tightened the guarantees available for greedy policies. In the homogeneous-departure settings where either all types share the same departure rate or the market is bipartite with side-homogeneous departure rates, the value of $m$9 lower bounds the reward of the induced greedy policy and is at least half of the omniscient reward rate. Accordingly,

$\lambda$0

This improves the earlier $\lambda$1 guarantee and is tight: no online policy can guarantee a factor better than $\lambda$2 in those settings. At the opposite end of the tractability spectrum, exact steady-state analysis is possible for the reneging $\lambda$3-system: the one-sided and two-sided models admit explicit product-form stationary distributions via partial balance, enabling direct computation of abandonment, fill rates, and waiting times (Arnosti et al., 6 Jul 2025, Castro et al., 2020).

Taken together, these results rule out a single monotone relationship between patience and the value of waiting. When abandonment is memoryless or heavily massed near zero, thickening can dominate. When there is a guaranteed minimum waiting window, instantaneous matching can dominate even in sparse thin markets. When utility tails are heavy, deliberate abandonment can be optimal because extreme-value gains outweigh extra losses.

5. Analytical frameworks

The literature uses several distinct analytical reductions to make abandonment tractable. Markovian pool-size analysis is central in the early random-graph models. Under Greedy and Patient, the pool size $\lambda$4 is itself a continuous-time Markov chain with explicit transition rates, unique stationary distribution, and polylogarithmic mixing time. The thin-market minimum-patience analysis goes beyond standard steady-state tools because general patience destroys Markovianity of the pool size. The key device is coupling with an auxiliary process without perishing, for which the pool size is Markovian, combined with stationary tail analysis, random-walk-with-drift bounds, Chernoff bounds, and Mecke’s formula for Poisson processes. The exact loss identity as a time-average of perishing probabilities is a characteristic example of this probabilistic approach (Akbarpour et al., 2014, Bäumler et al., 2022).

Queueing-control papers instead use heavy-traffic and diffusion methods. In admission-controlled double-ended queues, the centered arrival processes satisfy a functional CLT, the scaled imbalance converges to a controlled diffusion, and optimal control reduces to a free-boundary HJB with singular controls interpreted as reflecting barriers. Heterogeneous matching with general patience uses fluid limits and measure-valued age processes: abandonment intensities are hazard integrals up to a state-dependent threshold age, invariant states are computed explicitly, and optimization over steady matching rates yields the fluid benchmark. Under increasing hazard rates, convexity implies that optimal solutions occur at extreme points, which then induce static priority rules (Liu et al., 2021, Aveklouris et al., 2021).

Spatial models introduce mean-field and optimization structures. In multi-region ride-hailing, the long-run state is described by mass rates $\lambda$5, waiting-customer densities, and driver densities. Mean field equilibria coincide with KKT points of an optimization problem, and multiplicity can arise because pickup times are non-monotone in local supply. The two-matching-radius dispatch algorithm restores monotonicity of the effective sojourn time and thereby uniqueness of mean field equilibrium. LP and duality methods appear in the newest greedy-matching analysis: the LP lower bound certifies the chosen greedy policy, while a separate LP upper bounds the omniscient benchmark. Product-form queueing methods furnish a different exact route: for the reneging $\lambda$6-system, partial balance produces explicit stationary measures rather than asymptotic approximations (Li et al., 3 Apr 2025, Arnosti et al., 6 Jul 2025, Castro et al., 2020).

A broader methodological synthesis appears in recent survey work on dynamic market design, which frames dynamic matching through stationary distributions of positive-recurrent regenerative mechanisms. In that perspective, abandonment is compatible with admissions control, priority rules, and information policy, and the dynamic problem can often be transformed into a static program over steady-state distributions. This suggests a common structure beneath otherwise disparate models: one optimizes either transition rates into and out of congestion states or the stationary mass assigned to those states (Che, 1 Jan 2026).

6. Design implications, misconceptions, and research directions

Several design implications recur across domains such as organ exchange, ridesharing, online labor, marriage and job markets, public queues, and ride-hailing platforms. When there is a guaranteed minimum patience window, instantaneous matching can simultaneously achieve near-optimal match rates, low waiting, and congestion control in sparse markets; the explicit suggestion is that fees, commitments, or other mechanisms that enforce minimum participation time can move the market into this regime. When patience is short or approximately memoryless, waiting or batching can dramatically reduce abandonment. In queueing settings, admissions caps, FCFS priority, and coarse information can implement favorable stationary outcomes; when transfers are available, dynamic pricing or reimbursement can replace queue-based screening. In ride-hailing, dynamic control of matching radii can reduce both customer abandonment and driver idle time (Bäumler et al., 2022, Akbarpour et al., 2014, Li et al., 3 Apr 2025, Che, 1 Jan 2026).

One common misconception is that “thickening is always better.” The benchmark exponential-criticality model and the heavy-tailed utility model both show environments in which waiting is valuable, but the minimum-patience thin-market result shows the opposite under a mild guarantee on patience. Another misconception is that only the mean of the patience distribution matters. Fluid analysis with general patience shows that edge rankings can differ for patience distributions with the same mean but different hazard rates, so distributional shape is operationally material. A further misconception is that sophisticated global optimization is always necessary. Several papers find that local or greedy rules, if timed correctly or selected by an auxiliary optimization, can be near-optimal or exactly optimal in relevant asymptotic regimes (Bäumler et al., 2022, Aveklouris et al., 2021, Arnosti et al., 6 Jul 2025).

Open directions are explicit in the literature. Extensions beyond Poisson or renewal arrivals, beyond exponential or monotone-hazard patience, and beyond arrival-only matching remain active. Spatial models point toward learning-based dispatch, time-varying arrivals, richer network topologies, and endogenous pricing. Heterogeneous graph models raise questions about multi-type compatibility, strategic behavior, and information elicitation under partial observability. Periodic-review models suggest broader priority structures under weaker modified Monge conditions. The recent survey view implies that many of these problems may be approached through stationary-distribution design, but the supporting analysis will likely require new martingale, renewal, diffusion, or mean-field tools rather than a single universal template (Liu et al., 2021, Hu et al., 2018, Blanchet et al., 2020, Li et al., 3 Apr 2025, Che, 1 Jan 2026).

The accumulated literature therefore supports a precise but non-universal conclusion: abandonment is not merely a friction appended to dynamic matching; it is the force that determines whether the planner should accelerate matches, hold the market open, block entrants, reprioritize edges, or reshape local search geometry. The correct response depends on patience tails, hazard shape, utility tails, compatibility sparsity, and the informational structure available to the mechanism.