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, restoration and 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 bit-wisely optimizing a novel solution divergence. 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 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，多目标安全博弈使防御者能够同时保护目标免受多个异质攻击者的威胁。该类博弈旨在同时最大化所有异质收益（例如生命、金钱和犯罪率），且无需合并异质攻击者。现实场景中，异质攻击者及待保护目标的数量可能超出多数现有最优方法的能力范围，即多目标安全博弈受限于可扩展性问题。为此，本文提出一种基于多目标进化搜索的通用框架SDES，用于将多目标安全博弈扩展至大规模目标和异质攻击者。SDES由四个连续关键组件构成：离散化、优化、恢复与评估、精炼。具体而言，SDES首先利用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绕过高维阶梯函数并确保获得收敛性良好的帕累托前沿。随后，采用多目标进化算法在低维离散解空间中优化以获取分布均匀的帕累托前沿。为评估解的质量，SDES通过逐位优化新型解散度将解恢复至原始空间。最后，精炼组件以可接受成本提升优化性能。理论上，我们证明了SDES的优化一致性与收敛性。实验结果表明，SDES是首个针对大规模攻击者与目标的线性时间多目标安全博弈算法。SDES可求解包含20个攻击者与100个目标的多目标安全博弈问题，而现有最优方法仅能求解8个攻击者与25个目标的问题。消融实验验证了SDES所有组件的必要性。