Optimal Decision Rules when Payoffs are Partially Identified

We derive asymptotically 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 $\mu$ to deduce restrictions on $\theta$. 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 $\theta$ given $\mu$ with an average risk minimization approach for $\mu$. 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.