Stackelberg Security Games are often used to model strategic interactions in high-stakes security settings. The majority of existing models focus on single-defender settings where a single entity assumes command of all security assets. However, many realistic scenarios feature multiple heterogeneous defenders with their own interests and priorities embedded in a more complex system. Furthermore, defenders rarely choose targets to protect. Instead, they have a multitude of defensive resources or schedules at its disposal, each with different protective capabilities. In this paper, we study security games featuring multiple defenders and schedules simultaneously. We show that unlike prior work on multi-defender security games, the introduction of schedules can cause non-existence of equilibrium even under rather restricted environments. We prove that under the mild restriction that any subset of a schedule is also a schedule, non-existence of equilibrium is not only avoided, but can be computed in polynomial time in games with two defenders. Under additional assumptions, our algorithm can be extended to games with more than two defenders and its computation scaled up in special classes of games with compactly represented schedules such as those used in patrolling applications. Experimental results suggest that our methods scale gracefully with game size, making our algorithms amongst the few that can tackle multiple heterogeneous defenders.

翻译：斯塔克尔伯格安全博弈常被用于建模高风险安全场景中的策略互动。现有模型大多聚焦于单一防御者场景，即由单一主体掌控所有安全资产。然而，许多现实场景涉及多个异质防御者，每个防御者都拥有自身利益与优先级，且嵌于更复杂的系统中。此外，防御者很少直接选择保护目标，而是拥有多种防御资源或时间表可供调配，每种时间表具备不同的防护能力。本文同时研究包含多个防御者及时间表的安全博弈。我们证明，与以往多防御者安全博弈的研究不同，即使在相当受限的环境下，引入时间表也可能导致均衡不存在。我们进一步证明，在"时间表的任何子集仍是时间表"这一温和限制下，不仅可避免均衡不存在的问题，而且对包含两个防御者的博弈，可在多项式时间内计算出均衡。在附加假设下，我们的算法可扩展至超过两个防御者的博弈，并通过紧凑表示时间表（如巡逻应用中使用的类型）的特殊博弈类别实现计算规模的提升。实验结果表明，我们的方法能随博弈规模优雅扩展，使得该算法成为少数能够处理多个异质防御者的方案之一。