# On the Optimality of Randomization in Experimental Design: How to Randomize for Minimax Variance and Design-Based Inference

arXiv: https://arxiv.org/abs/2005.03151 · HTML: https://mlsanity.com/paper/2005.03151/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 May 2020.

## Built on by (most cited first, 1 of 1)

- [Synthetic Principal Component Design: Fast Covariate Balancing with Synthetic Controls](https://mlsanity.com/paper/2211.15241/) (2022)

## Abstract (arXiv)

I study the minimax-optimal design for a two-arm controlled experiment where conditional mean outcomes may vary in a given set. When this set is permutation symmetric, the optimal design is complete randomization, and using a single partition (i.e., the design that only randomizes the treatment labels for each side of the partition) has minimax risk larger by a factor of n-1. More generally, the optimal design is shown to be the mixed-strategy optimal design (MSOD) of Kallus (2018). Notably, even when the set of conditional mean outcomes has structure (i.e., is not permutation symmetric), being minimax-optimal for variance still requires randomization beyond a single partition. Nonetheless, since this targets precision, it may still not ensure sufficient uniformity in randomization to enable randomization (i.e., design-based) inference by Fisher's exact test to appropriately detect violations of null. I therefore propose the inference-constrained MSOD, which is minimax-optimal among all designs subject to such uniformity constraints. On the way, I discuss Johansson et al. (2020) who recently compared rerandomization of Morgan and Rubin (2012) and the pure-strategy optimal design (PSOD) of Kallus (2018). I point out some errors therein and set straight that randomization is minimax-optimal and that the "no free lunch" theorem and example in Kallus (2018) are correct.