We consider a zero-sum inspection game, in which a defender positions detectors across a critical system to detect multiple attacks caused by an attacker. We assume that detection is imperfect, and each detector location is associated with a probability of detecting attacks within its set of monitored system components. The objective of the defender (resp. attacker) is to minimize (resp. maximize) the expected number of undetected attacks. To compute Nash equilibria for this large-scale zero-sum game, we formulate a linear program with a small number of constraints that can be solved via column generation. We provide an exact mixed-integer program for the pricing problem and leverage its supermodular structure to design two efficient approaches for computing approximate Nash equilibria with theoretical guarantees: A column generation and a multiplicative weights update algorithm, both with approximate best responses. To address the computational challenges posed by combinatorial attacker strategies, each iteration of our multiplicative weights update algorithm computes a projection onto the polytope of marginal attack probabilities under the unnormalized relative entropy, for which we derive a closed-form expression and a linear-time algorithm. Computational results on real-world gas distribution networks illustrate the performance and scalability of our solution approaches.

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