  With the emergence of precision medicine, estimating optimal individualized
decision rules (IDRs) has attracted tremendous attention in many scientific
areas. Most existing literature has focused on finding optimal IDRs that can
maximize the expected outcome for each individual. Motivated by complex
individualized decision making procedures and the popular conditional value at
risk (CVaR) measure, we propose a new robust criterion to estimate optimal IDRs
in order to control the average lower tail of the individuals' outcomes. In
addition to improving the individualized expected outcome, our proposed
criterion takes risks into consideration, and thus the resulting IDRs can
prevent adverse events. The optimal IDR under our criterion can be interpreted
as the decision rule that maximizes the ``worst-case" scenario of the
individualized outcome when the underlying distribution is perturbed within a
constrained set. An efficient non-convex optimization algorithm is proposed
with convergence guarantees. We investigate theoretical properties for our
estimated optimal IDRs under the proposed criterion such as consistency and
finite sample error bounds. Simulation studies and a real data application are
used to further demonstrate the robust performance of our methods. Several
extensions of the proposed method are also discussed.
