We derive optimal statistical decision rules for discrete choice problems when payoffs depend on a partially-identified parameter $\theta$ and the decision maker can use a point-identified parameter $P$ to deduce restrictions on $\theta$. Leading examples include optimal treatment choice under partial identification and optimal pricing with rich unobserved heterogeneity. Our optimal decision rules minimize the maximum risk or regret over the identified set of payoffs conditional on $P$ and use the data efficiently to learn about $P$. We discuss implementation of optimal decision rules via the bootstrap and Bayesian methods, in both parametric and semiparametric models. We provide detailed applications to treatment choice and optimal pricing. Using a limits of experiments framework, we show that our optimal decision rules can dominate seemingly natural alternatives. Our asymptotic approach is well suited for realistic empirical settings in which the derivation of finite-sample optimal rules is intractable.

翻译：我们推导了离散选择问题中的最优统计决策规则，其中收益依赖于部分可识别的参数$\theta$，且决策者可利用点可识别的参数$P$推断$\theta$的约束条件。典型应用包括部分识别条件下的最优治疗方案选择，以及存在丰富未观测异质性时的最优定价问题。我们的最优决策规则在给定$P$的条件下，最小化识别集上收益的最大风险或遗憾值，并高效利用数据学习$P$。我们讨论了通过自助法及贝叶斯方法在参数模型与半参数模型中实施最优决策规则的途径，并提供了治疗选择与最优定价的详细应用。基于实验极限框架，我们证明了最优决策规则可超越看似自然的替代方案。这种渐近方法特别适用于有限样本最优规则难以推导的现实实证场景。