Optimal treatment assignment rules under capacity constraints

We study treatment assignment when treatments are limited in supply, where a planner aims to maximize social welfare by assigning treatments based on observable covariates. Such constraints are common when treatments are scarce and costly, but they complicate the analysis of optimal assignment rules because assignment probabilities must be coordinated across the entire covariate distribution. We develop a new approach that reformulates the planner's problem as an optimal transport problem, which makes the constraints analytically tractable. Using a limits of experiments framework, we establish local asymptotic optimality results for two canonical decision rules-the plug-in rule and the Bayesian rule. We show that the former rule can dominate the latter rule, with simulations demonstrating sizable risk reductions. An empirical illustration using school voucher program data from Angrist et al. (2006) demonstrates how the two rules differ in practice.