We derive asymptotically optimal statistical decision rules for discrete choice problems when payoffs depend on a partially-identified parameter $θ$ and the decision maker can use a point-identified parameter $μ$ to deduce restrictions on $θ$. Examples include treatment choice under partial identification and pricing with rich unobserved heterogeneity. Our notion of optimality combines a minimax approach to handle the ambiguity from partial identification of $θ$ given $μ$ with an average risk minimization approach for $μ$. We show how to implement optimal decision rules using the bootstrap and (quasi-)Bayesian methods in both parametric and semiparametric settings. We provide detailed applications to treatment choice and optimal pricing. Our asymptotic approach is well suited for realistic empirical settings in which the derivation of finite-sample optimal rules is intractable.

翻译：本文推导了在离散选择问题中渐近最优的统计决策规则，其中收益依赖于一个部分可识别的参数 $θ$，且决策者可以利用一个点可识别的参数 $μ$ 来推断 $θ$ 的约束条件。示例包括部分识别下的处理选择以及具有丰富未观测异质性的定价问题。我们的最优性概念结合了极小化极大方法（用于处理给定 $μ$ 时 $θ$ 部分识别带来的模糊性）与针对 $μ$ 的平均风险最小化方法。我们展示了如何在参数化和半参数化设定中，使用自助法和（拟）贝叶斯方法实现最优决策规则。我们详细阐述了在处理选择和最优定价中的具体应用。我们的渐近方法非常适用于现实经验设定，其中有限样本最优规则的推导通常是难以处理的。