  We study the question of how best to assign an encouragement in a randomized
encouragement study. In our setting, units arrive with covariates, receive a
nudge toward treatment or control, acquire one of those statuses in a way that
need not align with the nudge, and finally have a response observed. The nudge
can be seen as a binary instrument that affects the response only via the
treatment status. Our goal is to assign the nudge as a function of covariates
in a way that best estimates the local average treatment effect (LATE). We
assume a partially linear model, wherein the baseline model is non-parametric
and the treatment term is linear in the covariates. Under this model, we
outline a two-stage procedure to consistently estimate the LATE. Though the
variance of the LATE is intractable, we derive a finite sample approximation
and thus a design criterion to minimize. This criterion is convex, allowing for
constraints that might arise for budgetary or ethical reasons. We prove
conditions under which our solution asymptotically recovers the lowest true
variance among all possible nudge propensities. We apply our method to a
semi-synthetic example involving triage in an emergency department and find
significant gains relative to a regression discontinuity design.
