[Paper Review] Experimental Design for Matching

Chonghuan Wang|arXiv (Cornell University)|Jan 28, 2026

Game Theory and Voting Systems0 citations

TL;DR

The paper introduces the Alternating Path Randomized Design to compare two predetermined matchings on a finite population, addressing interference, and proves unbiased Horvitz–Thompson estimation with a finite-population CLT under a minimax-optimal randomization. It also extends to many-to-one matchings using graph-theoretic decompositions.

ABSTRACT

Matching mechanisms play a central role in operations management across diverse fields including education, healthcare, and online platforms. However, experimentally comparing a new matching algorithm against a status quo presents some fundamental challenges due to matching interference, where assigning a unit in one matching may preclude its assignment in the other. In this work, we take a design-based perspective to study the design of randomized experiments to compare two predetermined matching plans on a finite population, without imposing outcome or behavioral models. We introduce the notation of a disagreement set, which captures the difference between the two matching plans, and show that it admits a unique decomposition into disjoint alternating paths and cycles with useful structural properties. Based on these properties, we propose the Alternating Path Randomized Design, which sequentially randomizes along these paths and cycles to effectively manage interference. Within a minimax framework, we optimize the conditional randomization probability and show that, for long paths, the optimal choice converges to $\sqrt{2}-1$, minimizing worst-case variance. We establish the unbiasedness of the Horvitz-Thompson estimator and derive a finite-population Central Limit Theorem that accommodates complex and unstable path and cycle structures as the population grows. Furthermore, we extend the design to many-to-one matchings, where capacity constraints fundamentally alter the structure of the disagreement set. Using graph-theoretic tools, including finding augmenting paths and Euler-tour decomposition on an auxiliary unbalanced directed graph, we construct feasible alternating path and cycle decompositions that allow the design and inference results to carry over.

Motivation & Objective

• Motivate the need to empirically compare two fixed matching plans without outcome or behavior models.
• Introduce the disagreement set to capture differences between the two matchings and enable feasible randomization.
• Develop the Alternating Path Randomized Design (AP Design) that handles interference via path/cycle decompositions.
• Establish unbiased Horvitz–Thompson inference and a finite-population CLT under AP design.
• Extend the design to many-to-one matchings and provide graph-theoretic conditions for feasible decompositions.

Proposed method

• Define the disagreement set △M(t,c) as the symmetric difference between treatment Mt and control Mc.
• Decompose △M(t,c) into unique disjoint alternating paths and cycles (△P(t,c)).
• Propose the Alternating Path Randomized Design (AP Design) that randomizes along each path/cycle sequentially with conditional probabilities.
• Use Horvitz–Thompson estimator for ATE under AP design and derive unbiasedness independent of potential outcomes.
• Derive variance expressions for path and cycle components under AP, show linear growth with component length, and obtain minimax-optimal p→(√2−1) for long paths.
• Prove a finite-population Central Limit Theorem for the AP estimator under Assumption 1 (bounded outcomes).
• Extend to many-to-one matchings by identifying admissible decompositions via augmenting paths and Euler-tour decompositions in an auxiliary graph.

Experimental results

Research questions

• RQ1 How can two predetermined matchings be compared experimentally on a common finite population when interference prevents simultaneous realization of both matchings?
• RQ2 What is a feasible, principled randomization scheme that accounts for interference in the disagreement between two matchings?
• RQ3 Can unbiased estimation and valid inference be achieved under a design-based framework without outcome or behavioral models?
• RQ4 How does the variance of the estimator behave under the AP design, and what is the optimal randomization probability for long alternating components?
• RQ5 How can the approach be extended to many-to-one matchings while preserving feasibility and inferential guarantees?

Key findings

• The disagreement set admits a unique decomposition into alternating paths and cycles, enabling component-wise randomized experimentation.
• The AP design randomizes along each component sequentially, ensuring feasibility and managing interference.
• Horvitz–Thompson estimator under AP is unbiased for the average treatment effect between the two matchings.
• Under a minimax framework, the optimal long-path probability tends to √2−1 (≈0.4142), reducing worst-case variance.
• A finite-population CLT for the AP estimator is established, accommodating heterogeneous and unstable path/cycle structures as the population grows.
• Extensions to many-to-one matchings are developed using augmenting paths and Euler-tour decompositions to maintain feasibility and inference guarantees.