Treatment Choice with Nonlinear Regret

The literature focuses on minimizing the mean of welfare regret, which can lead to undesirable treatment choice due to sampling uncertainty. We propose to minimize the mean of a nonlinear transformation of regret and show that admissible rules are fractional 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 calculations in randomized experiments.