Fixed-Confidence Identification of Capacity-Constrained Treatment Allocations with Prognostic Covariates

Randomized experiments are often used to guide treatment allocation under limited capacity. Existing work has studied fixed-confidence identification under structured or constrained decision sets and covariate adjustment for improving treatment effect estimation. However, it remains unclear whether individual-level covariates that predict outcomes, which we call prognostic covariates, can reduce the sample complexity of identifying an optimal capacity-constrained group-level treatment allocation. To address this gap, we study fixed-confidence identification of capacity-constrained group-level treatment allocations with prognostic covariates. We characterize the first-order expected stopping-time benchmark when individual covariates are retained and show that it can be strictly smaller than the benchmark based on group-level information alone. We then develop an adaptive procedure that learns the unknown prognostic functions from accumulating data while controlling the allocation-selection error in finite samples, and prove that its expected stopping time asymptotically attains the same first-order benchmark as when the prognostic functions are known. Experimental results confirm the predicted information gap and show the learned procedure approaching the full-covariate benchmark.