  There is a fast-growing literature on estimating optimal treatment regimes
based on randomized trials or observational studies under a key identifying
condition of no unmeasured confounding. Because confounding by unmeasured
factors cannot generally be ruled out with certainty in observational studies
or randomized trials subject to noncompliance, we propose a general
instrumental variable approach to learning optimal treatment regimes under
endogeneity. Specifically, we establish identification of both value function
$E[Y_{\mathcal{D}(L)}]$ for a given regime $\mathcal{D}$ and optimal regimes
$\text{argmax}_{\mathcal{D}} E[Y_{\mathcal{D}(L)}]$ with the aid of a binary
instrumental variable, when no unmeasured confounding fails to hold. We also
construct novel multiply robust classification-based estimators. Furthermore,
we propose to identify and estimate optimal treatment regimes among those who
would comply to the assigned treatment under a standard monotonicity
assumption. In this latter case, we establish the somewhat surprising result
that complier optimal regimes can be consistently estimated without directly
collecting compliance information and therefore without the complier average
treatment effect itself being identified. Our approach is illustrated via
extensive simulation studies and a data application on the effect of child
rearing on labor participation.
