  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.
