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.

翻译：文献通常关注最小化福利遗憾的均值，但由于抽样不确定性，这可能导致不理想的治疗选择。我们提出最小化遗憾的非线性变换的均值，并证明对于非线性遗憾，可容许规则是分数形式的。聚焦于均方遗憾，我们推导出有限样本贝叶斯和极小化极大最优规则的闭合形式分数。我们的方法基于决策理论，并可扩展至极限实验。治疗分数可视为支持治疗的证据强度。我们将该框架应用于正态回归模型及随机实验中的样本量计算。