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Unmeasured confounding is a threat to causal inference and individualizeddecision making. Similar to Cui and Tchetgen Tchetgen (2020); Qiu et al.(2020); Han (2020a), we consider the problem of identification of optimalindividualized treatment regimes with a valid instrumental variable. Han(2020a) provided an alternative identifying condition of optimal treatmentregimes using the conditional Wald estimand of Cui and Tchetgen Tchetgen(2020); Qiu et al. (2020) when treatment assignment is subject to endogeneityand a valid binary instrumental variable is available. In this note, we providea necessary and sufficient condition for identification of optimal treatmentregimes using the conditional Wald estimand. Our novel condition is necessarilyimplied by those of Cui and Tchetgen Tchetgen (2020); Qiu et al. (2020); Han(2020a) and may continue to hold in a variety of potential settings not coveredby prior results.

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