a randomized clinical trial, this assumption is satisfied automatically. For an observational study, as clinicians make decisions based only on all past available information, this assumption essentially assumes no unmeasured confounders.

For consistent estimation of the optimal treatment rule, we also need to assume

-   **Positivity Assumption:** $0<\pi({{\bm x}})<1$, $\forall {{\bm x}}\in\mathcal{X}$.

Existing Learning Methods: Q-learning and A-learning
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Define the Q-function $Q({{\bm x}},a)\triangleq{\mbox{E}}(Y|{{\bm x}},a)$. Under assumptions (C1)-(C2), one can show that $g^{\mathrm{opt}}_\mu({{\bm X}})={\mathrm{argmax}}_{a\in\mathcal{A}}Q({{\bm x}},a)={\mathrm{argmax}}_{a\in\mathcal{A}}{\mbox{E}}(Y|{{\bm X}},A=a)$. This suggests that, in order to find $g^{\mathrm{opt}}_\mu$, we only need to estimate the conditional expectation of $Y$ given $({{\bm X}},A)$. This result serves as the foundation of Q- and A-learning framework. We further define the value function $V_\mu(g)={\mbox{E}}_{{{\bm X}}}[Q\{{{\bm X}},g({{\bm X}})\}]$ which is simply the marginal mean outcome under the ITR $g$, and $g^{\mathrm{opt}}_\mu={\mathrm{argmax}}_gV_\mu(g)$.

Define the $\tau$-th conditional quantile of $Y$ given $({{\bm X}},A)$ as $Q_{\tau}({{\bm X}},A)\triangleq\inf\{y: F_{Y|{{\bm X}},A}(y)\geq\tau\}$. Then we define the value function based on the $\tau$-th conditional quantile as $V_{\tau-q}(g)={\mbox{E}}_{{{\bm X}}}[Q_{\tau}\{{{\bm X}},g({{\bm X}})\}]$, which is an analog to the definition of $V_\mu(g)$. The optimal ITR which maximizes the $\tau$-th conditional quantile is then defined as $$g^{\mathrm{opt}}_{\tau}({{\bm x}})=\underset{a\in\mathcal{A}}{\mathrm{argmax}}Q_{\tau}({{\bm x}},a),\;\tau\in[0,1],$$ and $g^{\mathrm{opt}}_\tau={\mathrm{argmax}}_gV_{\tau-q}(g)$.

Consider the general model ${\mbox{E}}(Y|{{\bm X}},A)=h_0({{\bm X}})+AC_0({{\bm X}})$, where $h_0({{\bm 