[论文解读] The Mysteries of Security Games: Equilibrium Computation Becomes Combinatorial Algorithm Design

Haifeng Xu|arXiv (Cornell University)|Mar 8, 2016

Infrastructure Resilience and Vulnerability Analysis参考文献 29被引用 11

一句话总结

本文通过证明均衡计算（包括极小极大、强Stackelberg及Nash均衡）在多项式时间内等价于在集合系统𝒟上的防御者最佳响应（DBR）优化，为安全博弈建立了一个统一的理论框架。关键贡献在于证明了安全博弈中所有主要均衡概念的计算复杂性，从根本上由防御者纯策略集的组合结构决定，从而将复杂的博弈论问题简化为标准的组合优化任务。

ABSTRACT

The security game is a basic model for resource allocation in adversarial environments. Here there are two players, a defender and an attacker. The defender wants to allocate her limited resources to defend critical targets and the attacker seeks his most favorable target to attack. In the past decade, there has been a surge of research interest in analyzing and solving security games that are motivated by applications from various domains. This paper examines security games from a theoretical perspective and provides a unified view of various security game models. In particular, each security game can be characterized by a set system $E$ which consists of the defender's pure strategies, The defender's best response problem can be viewed as a combinatorial optimization problem over $E$. Our framework captures most of the basic security game models in the literature, including all the deployed systems, The set system $E$ arising from various domains encodes standard combinatorial problems like bipartite matching, maximum coverage, min-cost flow, packing problems, etc. Our main result shows that equilibrium computation in security games is essentially a combinatorial problem. In particular, we prove that, for any set system $E$, the following problems can be reduced to each other in polynomial time: (0) combinatorial optimization over $E$, (1) computing the minimax equilibrium for zero-sum security games over $E$, (2) computing the strong Stackelberg equilibrium for security games over $E$, (3) computing the best or worst (for the defender) Nash equilibrium for security games over $E$. Here, by "games over $E$" we mean the class of security games with arbitrary payoff structures, but a fixed set $E$ of defender pure strategies. This shows that the complexity of a security game is essentially determined by the set system $E$.

研究动机与目标

• 通过在不同模型间识别共同的结构性基础，统一安全博弈的理论分析。
• 阐明安全博弈中关键均衡概念（极小极大、强Stackelberg及Nash均衡）的计算复杂性。
• 证明均衡计算可约化为在固定集合系统𝒟上的防御者最佳响应（DBR）优化。
• 提供一个涵盖所有已部署的安全博弈系统及标准组合优化问题的通用框架。
• 通过将现有文献中的开放复杂性问题与已知的组合优化难题结果关联，解决这些复杂性问题。

提出的方法

• 将安全博弈形式化为双线性博弈，其中防御者策略位于多面体𝒫中，攻击者策略位于单纯形Δₙ中。
• 将防御者纯策略表征为𝒫的顶点，构成定义博弈结构的集合系统𝒟。
• 证明在𝒟上的DBR等价于最大覆盖、最小费用流及二分图匹配等组合优化问题。
• 为任意固定的𝒟，建立DBR、极小极大均衡、强Stackelberg均衡及最优/最差Nash均衡之间的多项式时间归约。
• 利用组合问题的已知复杂性结果（如3-覆盖的NP难，2-覆盖的多项式可解性）推断均衡计算的复杂性。
• 将该框架应用于恢复并强化先前的安全博弈复杂性结果，包括解决零和时空博弈及机场安检博弈中的开放问题。

实验结果

研究问题

• RQ1 安全博弈中均衡计算的计算复杂性能否被归约为单一基础问题？
• RQ2 防御者最佳响应（DBR）计算的复杂性是否与安全博弈中极小极大、强Stackelberg及Nash均衡的计算复杂性等价？
• RQ3 安全博弈的复杂性在多大程度上取决于防御者纯策略集𝒟的结构？
• RQ4 能否利用组合优化中的已知难解性结果来推断安全博弈中均衡计算的复杂性？
• RQ5 该框架是否存在可保持均衡计算与组合优化等价性的推广形式？

主要发现

• 对于任意固定的集合系统𝒟，安全博弈中所有主要均衡概念的计算复杂性均在多项式时间内等价。
• 安全博弈中的均衡计算可约化为在防御者纯策略集𝒟上的组合优化。
• 3-覆盖的DBR为NP难，意味着对应安全博弈中极小极大均衡的计算亦为NP难。
• 2-覆盖的可多项式求解性，意味着对应安全博弈中极小极大均衡计算亦为多项式可解。
• 该框架通过证明极小极大均衡在零和时空安全博弈中为NP难，解决了该领域一个开放问题。
• 结果强化了先前的复杂性结论，例如基于独立集的NP难性，证明了在机场乘客安检博弈中计算极小极大均衡为NP难。