Optimal A Priori Balance in the Design of Controlled Experiments

We develop a unified theory of designs for controlled experiments that balance baseline covariates a priori (before treatment and before randomization) using the framework of minimax variance. We establish a "no free lunch" theorem that indicates that, without structural information on the dependence of potential outcomes on baseline covariates, complete randomization is optimal. Restricting the structure of dependence, either parametrically or non-parametrically, leads directly to imbalance metrics and optimal designs. Certain choices of this structure recover known imbalance metrics and designs previously developed ad hoc, including randomized block designs, pairwise-matched designs, and re-randomization. New choices of structure based on reproducing kernel Hilbert spaces lead to new methods, both parametric and non-parametric.