Decision Theory for Treatment Choice Problems with Partial Identification

TL;DR

This paper applies classical decision theory to treatment choice problems with partial identification, revealing that ignoring data can be optimal and characterizing minimax-regret rules, with implications for policy and estimation.

Contribution

It provides a novel application of decision theory to partial identification, characterizing minimax-regret rules and their properties in treatment choice problems.

Findings

All decision rules are admissible under Gaussian likelihood.

Ignoring data can be maximin-welfare optimal.

Multiple minimax-regret rules may involve randomization.

Abstract

We apply classical statistical decision theory to a large class of treatment choice problems with partial identification. We show that, in a general class of problems with Gaussian likelihood, all decision rules are admissible; it is maximin-welfare optimal to ignore all data; and, for severe enough partial identification, there are infinitely many minimax-regret optimal decision rules, all of which sometimes randomize the policy recommendation. We uniquely characterize the minimax-regret optimal rule that least frequently randomizes, and show that, in some cases, it can outperform other minimax-regret optimal rules in terms of what we term profiled regret. We analyze the implications of our results in the aggregation of experimental estimates for policy adoption, extrapolation of Local Average Treatment Effects, and policy making in the presence of omitted variable bias.