  The design of experiments involves a compromise between covariate balance and
robustness. This paper provides a formalization of this trade-off and describes
an experimental design that allows experimenters to navigate it. The design is
specified by a robustness parameter that bounds the worst-case mean squared
error of an estimator of the average treatment effect. Subject to the
experimenter's desired level of robustness, the design aims to simultaneously
balance all linear functions of potentially many covariates. Less robustness
allows for more balance. We show that the mean squared error of the estimator
is bounded in finite samples by the minimum of the loss function of an implicit
ridge regression of the potential outcomes on the covariates. Asymptotically,
the design perfectly balances all linear functions of a growing number of
covariates with a diminishing reduction in robustness, effectively allowing
experimenters to escape the compromise between balance and robustness in large
samples. Finally, we describe conditions that ensure asymptotic normality and
provide a conservative variance estimator, which facilitate the construction of
asymptotically valid confidence intervals.
