Assortment-Based Matching

• Assortment-based matching is an optimization framework that tailors alternative subsets to agents using choice models such as the MNL.
• It employs techniques like LP relaxations, correlation gap arguments, and dynamic programming to achieve constant-factor approximation guarantees under various constraints.
• Applications span e-commerce and platform design, addressing tradeoffs in congestion control, personalization, scalability, and fairness.

Assortment-based matching refers to a class of optimization problems and algorithmic frameworks in which one or more agents are offered tailored subsets ("assortments") of alternatives (e.g., products, suppliers, or customers), and subsequent matches are realized according to agent choices modeled by discrete-choice systems—most commonly, the multinomial logit (MNL) model. This paradigm underlies a range of settings in operations research, e-commerce, and platform design where participant decisions are both stochastic and endogenous to the displayed menus. The central objectives may vary, including maximizing realized matches, revenue, match quality, or other market efficiency metrics, often under feasibility or inventory constraints.

1. Formal Models of Assortment-Based Matching

In the two-sided assortment-based matching framework, two agent sets (e.g., customers and suppliers, or patients and providers) are represented. The platform selects, possibly adaptively and possibly subject to cardinality or matroid constraints, which assortments $S_i$ of the opposite side to offer to each agent $i$. Agents respond by selecting from their displayed assortment or choosing to opt out, according to a discrete-choice model such as MNL:

$$\text{For agent } i: \quad P\{i \textrm{ chooses } j|S_i\} = \begin{cases} \frac{e^{v_{ij}}}{1 + \sum_{k \in S_i}e^{v_{ik}}} & j \in S_i \ \frac{1}{1 + \sum_{k \in S_i}e^{v_{ik}}} & \textrm{(opt out)} \ \end{cases}$$

The process may be sequential or simultaneous, and can be one-sided (only one agent class chooses assortments), or fully two-sided—with both sides exercising choice:

• One-sided sequential: The platform offers menus to (say) customers, who choose a supplier. Suppliers may then select among their would-be matches, typically via another MNL or submodular choice system (Ashlagi et al., 2019, Torrico et al., 2020, Mohammadreza et al., 5 Jul 2025).
• Fully adaptive: The platform sequentially processes both sides, each time revealing history and possibly adapting assortments as the process unfolds (Housni et al., 2024, Mohammadreza et al., 5 Jul 2025).

The objective varies: maximizing the number of realized matches, total expected revenue, maximum load, or average match quality, depending on the application domain (Ashlagi et al., 2019, Mohammadreza et al., 5 Jul 2025, Housni et al., 2023, Raman et al., 14 Feb 2025).

2. Algorithmic Structures and Approximation Guarantees

Assortment-based matching problems are strongly NP-hard in general, even under the MNL choice model and simple menu constraints (Ashlagi et al., 2019, Housni et al., 2024). Accordingly, the literature has focused on polynomial-time algorithms with proven constant-factor approximation guarantees.

Key algorithmic techniques include:

• LP relaxations and rounding: Linear programming relaxations based on distributions over assortments or marginal choice probabilities, coupled with rounding or sampling schemes that assemble feasible, near-optimal menus (Mohammadreza et al., 5 Jul 2025, Ashlagi et al., 2019, Torrico et al., 2020, Tulabandhula et al., 2020).
• Correlation gap arguments: To quantify the suboptimality incurred by decoupling correlated agent choices, often yielding $1/2$, $1-1/e$, or $1/4$ approximation ratios depending on structural revenue properties and model specifics (Mohammadreza et al., 5 Jul 2025, Housni et al., 2024).
• Continuous greedy/Frank–Wolfe: For submodular or concave extensions of match/revenue objectives, enabling fractional solutions to be efficiently approximated and rounded (Torrico et al., 2020, Mohammadreza et al., 5 Jul 2025).
• Dynamic-prog/DP compression: In maximum load objectives, state-compression techniques allow for quasi-polynomial time (QPTAS) near-optimal adaptive policies (Housni et al., 2023).

A representative sampling of approximation ratios for two-sided assortment-based matching:

Setting	Algorithmic Guarantee	Reference
General revenue, MNL choices	$(1/2 - \epsilon)$-approx	(Mohammadreza et al., 5 Jul 2025)
Uniform per-supplier revenues	$1 - 1/e-\epsilon$	(Mohammadreza et al., 5 Jul 2025)
General submodular supplier demand	$1-1/e$	(Torrico et al., 2020)
Maximum load (static)	PTAS, $1/2$ via WO menus	(Housni et al., 2023)
Maximum load (adaptive)	QPTAS, $1/4$ adaptivity gap	(Housni et al., 2023)
Fully adaptive two-sided (matches)	$1/4$-approx	(Housni et al., 2024)
Static MNL-MNL (matches)	$0.082$-approx	(Housni et al., 2024)

3. Core Structural and Policy Classes

Several canonical policy classes have emerged to differentiate the structural adaptivity achievable by assortment-based matching algorithms (Housni et al., 2024, Housni et al., 2023):

• Static one-side/fully static: Menus are selected in advance for all agents, based only on ex-ante information.
• Adaptive one-side: Menus for one side are adapted sequentially as choices are revealed.
• Fully adaptive: Full alternation between both agent classes, with menu selection exploiting complete past history.

Significant adaptivity gaps separate these classes. Specifically, the gap between static and adaptive one-sided policies is exactly $1-1/e$; the gap between adaptive one-side and fully adaptive is exactly $1/2$ under broad monotonicity and submodularity conditions (Housni et al., 2024). These gaps are tight, and greedy or randomized adaptive algorithms typically achieve the best-possible ratios for their respective classes.

4. Typical Workflow: MNL-Bandit and Dynamic Learning

In the online learning setting—where agent preferences are unknown and must be explored dynamically—the MNL-Bandit framework has provided a template for balancing exploration-exploitation (Agrawal et al., 2017). The workflow:

1. Initialize with prior-attraction estimates for each product/agent.
2. Offer candidate assortments as determined by optimism-under-uncertainty (UCB) bounds.
3. Observe stochastic choices; update counts, parameter estimates, and UCBs.
4. Iterate, adaptively commingling assortment optimization and learning.

Such adaptive UCB-style algorithms provide regret bounds of $O(\sqrt{NT\log NT} + N\log^2(NT))$ matching lower bounds up to logarithmic factors, independent of horizon $T$ or exogenous separability properties (Agrawal et al., 2017).

5. Extensions: Constraints, Arrivals, and Alternative Objectives

Inventory and cardinality constraints

Assortment-based matching formulations readily generalize to cardinality or matroid constraints on server assignments, per-category quotas, and item or menu-level feasibility restrictions. Approximation algorithms and LP relaxations maintain their worst-case guarantees under these constraints (Tulabandhula et al., 2020, Housni et al., 2023, Torrico et al., 2020).

Arrival models: static, fixed-order, fully adaptive

Algorithmic guarantees and adaptivity gaps are robust across static, fixed-order, and adaptive customer/supplier arrival schedules. Static policies retain their factor losses relative to adaptive settings under arbitrary agent orders (Mohammadreza et al., 5 Jul 2025, Housni et al., 2024).

Alternative objectives

While maximizing matches or revenue is standard, assortment-based matching models have been extended to:

• Quality-aware assignment (e.g., patient-provider matching): Maximize total or minimum realized match quality system-wide (Raman et al., 14 Feb 2025).
• Maximum load: Maximize the expected maximal load on any server/resource (Housni et al., 2023).
• Attenuated stochastic matching with patience/timeouts: Algorithms maintain constant-factor performance even in multistage, must-offer, or high-patience scenarios (Fata et al., 2019, Housni et al., 2023).

6. Operational and Practical Insights

• Congestion control vs. coverage: Large assortments increase agent choice but may concentrate demand, resulting in lower realized matches due to collisions. Optimal menu design balances match probability with spread (Ashlagi et al., 2019, Raman et al., 14 Feb 2025).
• Initiation order: Allocating the "first move" to the more selective side (e.g., suppliers with large outside options) may substantially improve welfare—an insight codified through relaxed matching models (Torrico et al., 2020).
• Computational scalability: Near-linear or sublinear-time algorithms (Assort-MNL, LSH-based methods) enable deployment at e-commerce scale ($n \sim 10^5$), with empirical revenue sacrifice <0.1% in constrained regimes and modest loss in general settings (Tulabandhula et al., 2020).
• Personalization and fairness: Menu structure can encode global or group-level fairness, adapt to agent heterogeneity, or optimize for worst-off quality inclusive objectives (Raman et al., 14 Feb 2025).

7. Open Directions and Limitations

• Heterogeneous and learned choice models: While much of the literature assumes known MNL or submodular models, real-world learning of agent preferences, or supporting non-MNL behavioral models, remains challenging and invites online learning adaptations (Agrawal et al., 2017).
• Multi-stage and multi-selection: Classical constructions generally address single-shot, single-choice scenarios. Extending guarantees to richer multi-purchase, sequential, or multi-customer settings continues to be an active area (Fata et al., 2019).
• Strong adaptivity: While tight worst-case bounds for adaptivity have been established, practical settings may offer instance-specific performance well above these worst-case guarantees—a phenomenon borne out empirically (Housni et al., 2024, Raman et al., 14 Feb 2025).
• Interplay of menu size and match quality: Enlarging assortments drives up coverage and match rates but often degrades match quality—an intrinsic tradeoff observed in empirical health-system and synthetic experiments (Raman et al., 14 Feb 2025).

Assortment-based matching thus represents a unifying framework at the intersection of discrete-choice modeling, algorithmic matching, revenue management, and market design, encompassing a spectrum of constrained, sequential, and two-sided optimization paradigms, with rigorous worst-case guarantees and substantial operational significance in real-world platforms.