The quantity of personal data that is collected, stored, and subsequently processed continues to grow rapidly. Given its sensitivity, ensuring privacy protections has become a necessary component of database management. To enhance protection, a number of mechanisms have been developed, such as audit logging and alert triggers, which notify administrators about suspicious activities. However, this approach is limited. First, the volume of alerts is often substantially greater than the auditing capabilities of organizations. Second, strategic attackers can attempt to disguise their actions or carefully choose targets, thus hide illicit activities. In this paper, we introduce an auditing approach that accounts for adversarial behavior by (1) prioritizing the order in which types of alerts are investigated and (2) providing an upper bound on how much resource to allocate for each type. Specifically, we model the interaction between a database auditor and attackers as a Stackelberg game. We show that even a highly constrained version of such problem is NP-Hard. Then we introduce a method that combines linear programming, column generation and heuristic searching to derive an auditing policy. On the synthetic data, we perform an extensive evaluation on the approximation degree of our solution with the optimal one. The two real datasets, (1) 1.5 months of audit logs from Vanderbilt University Medical Center and (2) a publicly available credit card application dataset, are used to test the policy-searching performance. The findings demonstrate the effectiveness of the proposed methods for searching the audit strategies, and our general approach significantly outperforms non-game-theoretic baselines.