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.

翻译：我们研究一种零和检查博弈，其中防御者需在关键系统中部署探测器以检测攻击者发起的多次攻击。我们假设检测具有不完美性，每个探测器位置对应一个概率值，表示其能够检测到所监控系统组件集合内攻击的概率。防御者（相应地，攻击者）的目标是最小化（相应地，最大化）未被检测攻击的期望数量。为计算该大规模零和博弈的纳什均衡，我们构建了一个约束数量较少的线性规划，可通过列生成法求解。我们为定价问题提供了精确的混合整数规划，并利用其超模结构设计了两种计算具有理论保证的近似纳什均衡的高效方法：基于近似最优响应的列生成算法和乘性权重更新算法。为应对组合型攻击策略带来的计算挑战，我们乘性权重更新算法的每次迭代需计算在未归一化相对熵下边缘攻击概率多胞形上的投影，为此我们推导出闭式解并提出了线性时间算法。在真实世界燃气配送网络上的计算结果表明了我们解决方案的性能与可扩展性。