On the optimality of randomization in experimental design: How to randomize for minimax variance and design-based inference

I study the minimax-optimal design for a two-arm controlled experiment where conditional mean outcomes vary in a given set and the objective is effect-estimation precision. When this set is permutation symmetric, the optimal design is shown to be complete randomization. Notably, even when the set has structure (i.e., is not permutation symmetric), being minimax-optimal for precision still requires randomization beyond a single partition of units, that is, beyond randomizing the identity of treatment. A single partition is not optimal even when conditional means are linear. Since this only targets precision, it may nonetheless not ensure sufficient uniformity for design-based (i.e., randomization) inference. I therefore propose the inference-constrained mixed-strategy optimal design as the minimax-optimal for precision among designs subject to sufficient-uniformity constraints.

Automated summary

The study reveals that randomization beyond a single partition of units is needed for precision in a two-arm controlled experiment, even when conditional means are linear. Therefore, the inference-constrained mixed-strategy optimal design is proposed as the minimax-optimal for precision among designs under sufficient uniformity constraints.