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An effective way of preventing attacks in secure areas is to screen for threats (people, objects) before entry, e.g., screening of airport passengers. However, screening every entity at the same level may be both ineffective and undesirable. The challenge then is to find a dynamic approach for randomized screening, allowing for more effective use of limited screening resources, leading to improved security. We address this challenge with the following contributions: (1) a threat screening game (TSG) model for general screening domains; (2) an NP-hardness proof for computing the optimal strategy of TSGs; (3) a scheme for decomposing TSGs into subgames to improve scalability; (4) a novel algorithm that exploits a compact game representation to efficiently solve TSGs, providing the optimal solution under certain conditions; and (5) an empirical comparison of our proposed algorithm against the current state-of-the-art optimal approach for large-scale game-theoretic resource allocation problems.