Finite-Sample Certification of Capacity-Constrained Treatment Allocations in Prespecified Cohorts

Capacity-constrained treatment allocation uses experimental evidence to select a fixed number of individuals from a prespecified cohort for treatment. We study a complementary problem: certifying the welfare loss of a selected allocation relative to the best feasible alternative. Under a prespecified treatment effect model linking the randomized trial and target cohort, we characterize which allocation welfare differences are identified. These differences can remain identifiable even when some individual target treatment effects are not. We then construct simultaneous finite-sample upper bounds directly for the welfare differences that determine regret. The resulting regret certificates remain valid when the same trial is used for both allocation selection and certification. The construction applies to a prespecified candidate set and extends to all exact-$k$ allocations through exchange bounds and maximum-weight partial matching. We further bound the excess of the certificate over true cohort-specific regret and show that, under a common confidence region, direct comparison gives no larger uncertainty radius than propagating separate treatment effect or allocation welfare intervals. Experimental results confirm post-selection coverage and show smaller mean certificates from direct comparison in the evaluated three-feature settings.