Statistical Decisions and Partial Identification: With Application to Boundary Discontinuity Design

We are delighted to respond to the excellent surveys by Cattaneo et al. (2026) and Hirano (2026). Our discussion will attempt two things: first, we show how statistical decision theory can be applied to situations with partial identification; second, we connect the surveys' themes by applying these insights to an imagined policy experiment in one of Cattaneo et al.'s (2025) applications. To do so, we lay out a stylized scenario of statistical decision making under partial identification and, drawing on our own and others' earlier work, provide a complete solution for that scenario. We then apply these results to a hypothetical reduction (modelled on actual policies) in eligibility for educational subsidies. We will see that something of interest can be said, but also that bringing the theory to the application involves some leaps of faith and leaves some questions open. This leads to the final section, where we discuss what we see as the main open challenges in statistical decision theory under partial identification.