On a necessary and sufficient identification condition of optimal treatment regimes with an instrumental variable

Unmeasured confounding is a threat to causal inference and individualized decision making. Similar Cui Tchetgen (2021); Qiu et al. Han (2021), we consider the problem of identification optimal treatment regimes with valid instrumental variable. (2021) provided an alternative identifying condition using conditional Wald estimand when assignment subject endogeneity binary variable available. In this note, provide necessary sufficient for estimand. Our novel necessarily implied by those may continue hold in variety potential settings not covered prior results.