We study a security game over a network played between a defender and k attackers. Every attacker chooses, probabilistically, a node of the network to damage. The defender chooses, probabilistically as well, a connected induced subgraph of the network of \(\lambda \) nodes to scan and clean. Each attacker wishes to maximize the probability of escaping her cleaning by the defender. On the other hand, the goal of the defender is to maximize the expected number of attackers that she catches. This game is a generalization of the model from the seminal paper of Mavronicolas et al. Mavronicolas et al. (in: International symposium on mathematical foundations of computer science, MFCS, pp 717–728, 2006). We are interested in Nash equilibria of this game, as well as in characterizing defense-optimal networks which allow for the best equilibrium defense ratio; this is the ratio of k over the expected number of attackers that the defender catches in equilibrium. We provide a characterization of the Nash equilibria of this game and defense-optimal networks. The equilibrium characterizations allow us to show that even if the attackers are centrally controlled the equilibria of the game remain the same. In addition, we give an algorithm for computing Nash equilibria. Our algorithm requires exponential time in the worst case, but it is polynomial-time for \(\lambda \) constantly close to 1 or n. For the special case of tree-networks, we further refine our characterization which allows us to derive a polynomial-time algorithm for deciding whether a tree is defense-optimal and if this is the case it computes a defense-optimal Nash equilibrium. On the other hand, we prove that it is \({\mathtt {NP}}\)-hard to find a best-defense strategy if the tree is not defense-optimal. We complement this negative result with a polynomial-time constant-approximation algorithm that computes solutions that are close to optimal ones for general graphs. Finally, we provide asymptotically (almost) tight bounds for the Price of Defense for any \(\lambda \); this is the worst equilibrium defense ratio over all graphs.

我们研究了一个防御者和k 个 攻击者之间在网络上进行的安全博弈。每个攻击者都有可能选择要破坏的网络节点。防御者也有概率地选择\(\lambda \)网络的连通诱导子图要扫描和清理的节点。每个攻击者都希望最大化逃脱防御者清理的概率。另一方面，防御者的目标是最大化她抓住的攻击者的预期数量。这个游戏是对 Mavronicolas 等人的开创性论文中模型的概括。马夫罗尼科拉斯等人。（在：计算机科学数学基础国际研讨会，MFCS，第 717-728 页，2006 年）。我们对这个博弈的纳什均衡感兴趣，以及表征允许最佳均衡防御率的防御最优网络；这是k的比率超过防御者在均衡时捕获的攻击者的预期数量。我们提供了该博弈和防御最优网络的纳什均衡的特征。均衡特征使我们能够表明，即使攻击者被集中控制，博弈的均衡也保持不变。此外，我们还给出了计算纳什均衡的算法。我们的算法在最坏的情况下需要指数时间，但对于\(\lambda \)不断接近 1 或n是多项式时间. 对于树网络的特殊情况，我们进一步改进了我们的特征，这使我们能够推导出多项式时间算法来决定树是否是防御最优的，如果是这种情况，它会计算防御最优纳什均衡。另一方面，我们证明了\({\mathtt {NP}}\) - 如果树不是防御最优的，则很难找到最佳防御策略。我们用多项式时间常数近似算法来补充这个负面结果，该算法计算出接近一般图最优解的解。最后，我们为任何\(\lambda \)的防御价格提供渐近（几乎）严格的界限；这是所有图表中最差的均衡防御率。