Matching for causal effects via multimarginal optimal transport

Matching on covariates is a well-established framework for estimating causal effects in observational studies. The principal challenge in these settings stems from the often high-dimensional structure of the problem. Many methods have been introduced to deal with this challenge, with different advantages and drawbacks in computational and statistical performance and interpretability. Moreover, the methodological focus has been on matching two samples in binary treatment scenarios, but a dedicated method that can optimally balance samples across multiple treatments has so far been unavailable. This article introduces a natural optimal matching method based on entropy-regularized multimarginal optimal transport that possesses many useful properties to address these challenges. It provides interpretable weights of matched individuals that converge at the parametric rate to the optimal weights in the population, can be efficiently implemented via the classical iterative proportional fitting procedure, and can even match several treatment arms simultaneously. It also possesses demonstrably excellent finite sample properties.