There is an increasing use of algorithms to inform decisions in many settings, from student evaluations, college admissions, to credit scoring. These decisions are made by applying a decision rule to individual's observed features. Given the impacts of these decisions on individuals, decision makers are increasingly required to be transparent on their decision making to offer the ``right to explanation.'' Meanwhile, being transparent also invites potential manipulations, also known as gaming, that the individuals can utilize the knowledge to strategically alter their features in order to receive a more beneficial decision.

In this work, we study the problem of \emph{robust} decision-making under strategic behavior. Prior works often assume that the decision maker has full knowledge of individuals' cost structure for manipulations. We study the robust variant that relaxes this assumption: The decision maker does not have full knowledge but knows only a subset of the individuals' available actions and associated costs. To approach this non-quantifiable uncertainty, we define robustness based on the worst-case guarantee of a decision, over all possible actions (including actions unknown to the decision maker) individuals might take. A decision rule is called \emph{robust optimal} if its worst case performance is (weakly) better than that of all other decision rules. Our main contributions are two-fold. First, we provide a crisp characterization of the above robust optimality: For any decision rules under mild conditions that are robust optimal, there exists a linear decision rule that is equally robust optimal. Second, we explore the computational problem of searching for the robust optimal decision rule and interestingly, we demonstrate the problem is closely related to distributionally robust optimization. We believe our results promotes the use of simple linear decisions with uncertain individual manipulations.