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 $θ$ and the decision maker can use a point-identified parameter $μ$ to deduce restrictions on $θ$. 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 $θ$ given $μ$ with an average risk minimization approach for $μ$. 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.

💡 Research Summary

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The paper addresses a fundamental problem in decision‑making under uncertainty: a policymaker or firm must choose an action from a finite set, yet the payoff associated with each action depends on a structural parameter θ that is only partially identified. In addition, a reduced‑form parameter μ is point‑identified and can be estimated efficiently from data; μ imposes restrictions on the admissible values of θ through an identified set Θ₀(μ). Traditional approaches either treat both parameters symmetrically in a minimax framework or rely on full Bayesian updating, both of which become problematic when θ is only set‑identified because the data do not update its prior distribution.

The authors propose a asymmetric risk criterion that combines a minimax treatment of the ambiguity in θ with an average‑risk treatment of the statistical uncertainty in μ. Formally, for each decision d∈D they define

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