  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.
