  The literature focuses on the mean of welfare regret, which can lead to
undesirable treatment choice due to sensitivity to sampling uncertainty. We
propose to minimize the mean of a nonlinear transformation of regret and show
that singleton rules are not essentially complete for nonlinear regret.
Focusing on mean square regret, we derive closed-form fractions for
finite-sample Bayes and minimax optimal rules. Our approach is grounded in
decision theory and extends to limit experiments. The treatment fractions can
be viewed as the strength of evidence favoring treatment. We apply our
framework to a normal regression model and sample size calculation.
