  ML-based predictions are used to inform consequential decisions about
individuals. How should we use predictions (e.g., risk of heart attack) to
inform downstream binary classification decisions (e.g., undergoing a medical
procedure)? When the risk estimates are perfectly calibrated, the answer is
well understood: a classification problem's cost structure induces an optimal
treatment threshold $j^{\star}$. In practice, however, some amount of
miscalibration is unavoidable, raising a fundamental question: how should one
use potentially miscalibrated predictions to inform binary decisions? We
formalize a natural (distribution-free) solution concept: given anticipated
miscalibration of $\alpha$, we propose using the threshold $j$ that minimizes
the worst-case regret over all $\alpha$-miscalibrated predictors, where the
regret is the difference in clinical utility between using the threshold in
question and using the optimal threshold in hindsight. We provide closed form
expressions for $j$ when miscalibration is measured using both expected and
maximum calibration error, which reveal that it indeed differs from $j^{\star}$
(the optimal threshold under perfect calibration). We validate our theoretical
findings on real data, demonstrating that there are natural cases in which
making decisions using $j$ improves the clinical utility.
