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.

翻译：现有文献主要关注福利遗憾的均值，这可能导致因对抽样不确定性的敏感性而产生不良的治疗选择。我们提出最小化遗憾非线性变换的均值，并证明对于非线性遗憾，单点决策规则并非本质完备。聚焦于均方遗憾，我们推导了有限样本贝叶斯和极小极大最优规则的闭式分配比例。我们的方法基于决策理论，并可推广至极限实验框架。治疗分配比例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样本量计算中。