Optimal Treatment Allocation under Constraints

Abstract: In optimal policy problems where treatment effects vary at the individual level, optimally allocating treatments to recipients is complex even when potential outcomes are known. We present an algorithm for multi-arm treatment allocation problems that is guaranteed to find the optimal allocation in strongly polynomial time, and which is able to handle arbitrary potential outcomes as well as constraints on treatment requirement and capacity. Further, starting from an arbitrary allocation, we show how to optimally re-allocate treatments in a Pareto-improving manner. To showcase our results, we use data from Danish nurse home visiting for infants. We estimate nurse specific treatment effects for children born 1959-1967 in Copenhagen, comparing nurses against each other. We exploit random assignment of newborn children to nurses within a district to obtain causal estimates of nurse-specific treatment effects using causal machine learning. Using these estimates, and treating the Danish nurse home visiting program as a case of an optimal treatment allocation problem (where a treatment is a nurse), we document room for significant productivity improvements by optimally re-allocating nurses to children. Our estimates suggest that optimal allocation of nurses to children could have improved average yearly earnings by USD 1,815 and length of education by around two months.