Optimal treatment assignment rules under capacity constraints

We study treatment assignment problems under capacity constraints, where a planner aims to maximize social welfare by assigning treatments based on observable covariates. Such constraints, common when treatments are costly or limited in supply, introduce nontrivial challenges for deriving optimal statistical assignment rules because the planner needs to coordinate treatment assignment probabilities across the entire covariate distribution. To address these challenges, we reformulate the planner's constrained maximization problem as an optimal transport problem, which makes the problem effectively unconstrained. We then establish local asymptotic optimality results of assignment rules using a limits of experiments framework. Finally, we illustrate our method with a voucher assignment problem for private secondary school attendance using data from Angrist et al. (2006)