Optimal Discrete Decisions when Payoffs are Partially Identified

We derive optimal statistical decision rules for discrete choice problems when the decision maker is unable to discriminate among a set of payoff distributions. In this problem, the decision maker must confront both model uncertainty (about the identity of the true payoff distribution) and statistical uncertainty (the set of payoff distributions must be estimated). We derive "efficient-robust decision rules" which minimize maximum risk or regret over the set of payoff distributions and which use the data to learn efficiently about features of the set of payoff distributions germane to the choice problem. We discuss implementation of these decision rules via the bootstrap and Bayesian methods, for both parametric and semiparametric models. Using a limits of experiments framework, we show that efficient-robust decision rules are optimal and can dominate seemingly natural alternatives. We present applications to treatment assignment using observational data and optimal pricing in environments with rich unobserved heterogeneity.