🤖 AI Summary
This study addresses the challenge of unbiased comparison between two pre-specified matching mechanisms in finite populations, where interference from one mechanism can contaminate the evaluation of the other. The authors propose an alternating-path randomization design based on a decomposition of the disagreement set, which requires no assumptions about outcome or behavioral models. By leveraging unique alternating paths and cycle decompositions from graph theory—such as augmenting paths and Eulerian cycle decompositions—the method constructs a controlled interference structure and sets the optimal randomization probability to √2−1 to minimize worst-case variance. The approach integrates Horvitz–Thompson estimation with minimax optimization, establishing unbiasedness of the estimator, proving a finite-population central limit theorem applicable to complex path structures, and extending successfully to many-to-one matching settings with capacity constraints.
📝 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.