Treatment Choice, Mean Square Regret and Partial Identification

TL;DR

This paper analyzes decision-making under partial welfare identification using mean square regret, revealing that optimal treatment rules are fractional and depend on the width of the identified set, with implications for cautious decision-making.

Contribution

It introduces a finite-sample analysis of treatment choice based on mean square regret, showing that optimal rules are always fractional and depend on a logistic transformation of a t-statistic.

Findings

Optimal treatment rules are always fractional.

The treatment fraction is a logistic transformation of a t-statistic.

Wider identified sets lead to treatment fractions closer to 0.5.

Abstract

We consider a decision maker who faces a binary treatment choice when their welfare is only partially identified from data. We contribute to the literature by anchoring our finite-sample analysis on mean square regret, a decision criterion advocated by Kitagawa, Lee, and Qiu (2022). We find that optimal rules are always fractional, irrespective of the width of the identified set and precision of its estimate. The optimal treatment fraction is a simple logistic transformation of the commonly used t-statistic multiplied by a factor calculated by a simple constrained optimization. This treatment fraction gets closer to 0.5 as the width of the identified set becomes wider, implying the decision maker becomes more cautious against the adversarial Nature.