In the field of security, multi-objective security games (MOSGs) allow defenders to simultaneously protect targets from multiple heterogeneous attackers. MOSGs aim to simultaneously maximize all the heterogeneous payoffs, e.g., life, money, and crime rate, without merging heterogeneous attackers. In real-world scenarios, the number of heterogeneous attackers and targets to be protected may exceed the capability of most existing state-of-the-art methods, i.e., MOSGs are limited by the issue of scalability. To this end, this paper proposes a general framework called SDES based on many-objective evolutionary search to scale up MOSGs to large-scale targets and heterogeneous attackers. SDES consists of four consecutive key components, i.e., discretization, optimization, evaluation, and refinement. Specifically, SDES first discretizes the originally high-dimensional continuous solution space to the low-dimensional discrete one by the maximal indifference property in game theory. This property helps evolutionary algorithms (EAs) bypass the high-dimensional step function and ensure a well-convergent Pareto front. Then, a many-objective EA is used for optimization in the low-dimensional discrete solution space to obtain a well-spaced Pareto front. To evaluate solutions, SDES restores solutions back to the original space via greedily optimizing a novel divergence measurement. Finally, the refinement in SDES boosts the optimization performance with acceptable cost. Theoretically, we prove the optimization consistency and convergence of SDES. Experiment results show that SDES is the first linear-time MOSG algorithm for both large-scale attackers and targets. SDES is able to solve up to 20 attackers and 100 targets MOSG problems, while the state-of-the-art (SOTA) methods can only solve up to 8 attackers and 25 targets ones. Ablation study verifies the necessity of all components in SDES.

翻译：在安全领域，多目标安全博弈（MOSGs）允许防御者同时保护目标免受多个异构攻击者的威胁。MOSGs 旨在在不合并异构攻击者的情况下，同时最大化所有异构收益（例如生命、金钱和犯罪率）。在现实场景中，异构攻击者和待保护目标的数量可能超出大多数现有最先进方法的能力，即 MOSGs 受限于可扩展性问题。为此，本文提出一个通用框架 SDES，该框架基于多目标进化搜索，将 MOSGs 扩展到大规模目标和异构攻击者。SDES 由四个连续的关键组件组成：离散化、优化、评估和细化。具体而言，SDES 首先通过博弈论中的最大无差异性质，将原始高维连续解空间离散化为低维离散空间。该性质帮助进化算法（EAs）绕过高维阶梯函数，并确保收敛良好的帕累托前沿。然后，使用多目标 EA 在低维离散解空间中进行优化，以获得分布良好的帕累托前沿。为了评估解，SDES 通过贪婪优化一种新颖的散度度量，将解还原到原始空间。最后，SDES 中的细化以可接受的成本提升优化性能。理论上，我们证明了 SDES 的优化一致性和收敛性。实验结果表明，SDES 是首个适用于大规模攻击者和目标的线性时间 MOSG 算法。SDES 能够解决多达 20 个攻击者和 100 个目标的 MOSG 问题，而最先进（SOTA）方法只能解决多达 8 个攻击者和 25 个目标的问题。消融研究验证了 SDES 中所有组件的必要性。