Experimental Design for Matching

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.

💡 Research Summary

This paper tackles the fundamental problem of experimentally comparing two predetermined matching mechanisms—one representing a new treatment algorithm and the other the status‑quo—on the same finite population without imposing any outcome or behavioral models. The authors adopt a design‑based perspective, treating all potential outcomes as fixed and attributing randomness solely to the randomization scheme.

The central construct is the disagreement set ΔM(t,c), which consists of all pairs that appear in exactly one of the two matchings. By definition, pairs that are common to both matchings cancel out of the average treatment effect, so ΔM contains precisely the information needed for identification. The authors prove that ΔM admits a unique decomposition into disjoint alternating paths and cycles, where edges alternate between the treatment and control matchings. Along any such component, adjacent edges cannot be simultaneously realized because of capacity constraints; this is the essence of “matching interference.”

Leveraging this structural insight, the paper introduces the Alternating Path Randomized Design (AP design). The design proceeds component‑wise: for each path or cycle, edges are randomized sequentially, with the inclusion of an edge conditioned on the realized selection of its predecessor. This sequential conditional randomization guarantees feasibility (no capacity violations) while preserving enough randomness for causal inference.

The authors formulate a minimax optimization problem: choose the conditional inclusion probability p that minimizes the worst‑case variance of the estimator across all possible outcome configurations. Solving the recursion yields a closed‑form optimal p that converges to √2 − 1 ≈ 0.4142 as the length of a path grows without bound. This value reflects a balance between reducing the variance contributed by the current edge and accounting for downstream uncertainty along the same component.

For estimation, the paper adopts the Horvitz–Thompson estimator, which weights each observed outcome by the inverse of its inclusion probability. Under the AP design, the estimator is shown to be unbiased for the average treatment effect τ = (\bar Y_t - \bar Y_c). The authors further derive a finite‑population Central Limit Theorem (CLT) for the estimator. The proof is non‑standard because the number, length, and composition of alternating paths and cycles can fluctuate dramatically with the population size; the authors combine Lindberg–Feller arguments with α‑mixing techniques, invoking the Bolzano–Weierstrass theorem and the subsequence principle to handle this instability. Consequently, the estimator is asymptotically normal even when the decomposition structure does not converge to a stable regime.

Recognizing that many real‑world applications involve many‑to‑one matchings (e.g., hospitals with multiple patients, schools with several students per class), the paper extends the methodology beyond one‑to‑one settings. In many‑to‑one contexts the disagreement set’s decomposition is no longer unique, and many candidate decompositions violate capacity constraints. The authors construct an auxiliary unbalanced directed graph and show that finding a feasible decomposition reduces to two classic graph‑theoretic problems: locating augmenting paths and performing an Euler‑tour decomposition. Sufficient conditions for admissibility are provided, and once a feasible decomposition is obtained, the same AP design and inference results apply unchanged.

A brief simulation study (described in the paper) illustrates that the naive design—randomly selecting the entire treatment or control matching with probability ½—has variance that does not vanish with increasing N, whereas the AP design achieves substantially lower variance, especially when long alternating paths are present. The optimal √2 − 1 probability is shown empirically to approach the theoretical minimax bound.

In summary, the contribution of this work is threefold: (1) a rigorous graph‑theoretic characterization of the difference between two matchings via the disagreement set; (2) a novel, design‑based randomization scheme (AP design) that respects matching interference and is provably minimax‑optimal; and (3) a comprehensive inferential framework—including unbiased estimation and a finite‑population CLT—that remains valid under highly heterogeneous and unstable path/cycle structures, and that extends naturally to many‑to‑one matching environments. This framework provides practitioners with a model‑free, statistically sound method for A/B testing of matching algorithms across a wide range of operational domains.