  We consider the problem of learning how to optimally allocate treatments
whose cost is uncertain and can vary with pre-treatment covariates. This
setting may arise in medicine if we need to prioritize access to a scarce
resource that different patients would use for different amounts of time, or in
marketing if we want to target discounts whose cost to the company depends on
how much the discounts are used. Here, we show that the optimal treatment
allocation rule under budget constraints is a thresholding rule based on
priority scores, and we propose a number of practical methods for learning
these priority scores using data from a randomized trial. Our formal results
leverage a statistical connection between our problem and that of learning
heterogeneous treatment effects under endogeneity using an instrumental
variable. We find our method to perform well in a number of empirical
evaluations.

Methodology