  Estimation and evaluation of individualized treatment rules have been studied
extensively, but real-world treatment resource constraints have received
limited attention in existing methods. We investigate a setting in which
treatment is intervened upon based on covariates to optimize the mean
counterfactual outcome under treatment cost constraints when the treatment cost
is random. In a particularly interesting special case, an instrumental variable
corresponding to encouragement to treatment is intervened upon with constraints
on the proportion receiving treatment. For such settings, we first develop a
method to estimate optimal individualized treatment rules. We further construct
an asymptotically efficient plug-in estimator of the corresponding average
treatment effect relative to a given reference rule.
