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 popular conditional value at risk
(CVaR) measures, we propose a new robust criterion to estimate optimal IDRs in
order to control the average lower tail of the subjects' 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 method