๐ค AI Summary
This paper investigates asymptotically optimal discrete decision-making under partial identification: when the payoff function depends on an incompletely identified parameter ฮธ, while only an auxiliary parameter Pโsubject to known constraints linking it to ฮธโis observable, how can we construct a decision rule conditional on P that minimizes maximum risk or regret? Methodologically, it establishes, for the first time, a systematic statistical theory of optimal decision-making under partial identification, grounded in the limit experiment framework and integrating bootstrap and Bayesian inference techniques, applicable to both parametric and semiparametric models. Theoretically, the proposed rule is proven to strictly dominate conventional plug-in or worst-case substitution rules in asymptotic efficiency. Empirically, it substantially reduces worst-case regret and demonstrates both robustness and practical feasibility in applications such as treatment choice and optimal pricing.
๐ Abstract
We derive optimal statistical decision rules for discrete choice problems when payoffs depend on a partially-identified parameter $ heta$ and the decision maker can use a point-identified parameter $P$ to deduce restrictions on $ heta$. 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.