Inspired by scenarios where the strategic network design and defense or immunisation are of the central importance, Goyal et al. [3] defined a new Network Formation Game with Attack and Immunisation. The authors showed that despite the presence of attacks, the game has high social welfare properties and even though the equilibrium networks can contain cycles, the number of edges is strongly bounded. Subsequently, Friedrich et al. [10] provided a polynomial time algorithm for computing a best response strategy for the maximum carnage adversary which tries to kill as many nodes as possible, and for the random attack adversary, but they left open the problem for the case of maximum disruption adversary. This adversary attacks the vulnerable region that minimises the post-attack social welfare. In this paper we address our efforts to this question. We can show that computing a best response strategy given a player u and the strategies of all players but u, is polynomial time solvable when the initial network resulting from the given strategies is connected. Our algorithm is based on a dynamic programming and has some reminiscence to the knapsack-problem, although is considerably more complex and involved.

翻译：受战略网络设计及防御或免疫具有核心重要性的场景启发，Goyal等人[3]定义了一种包含攻击与免疫的新型网络形成博弈。作者证明，尽管存在攻击，该博弈仍具有较高的社会福利特性，且即便均衡网络可能包含环结构，边的数量仍受到强约束。随后，Friedrich等人[10]针对试图尽可能摧毁节点的最大杀戮对手以及随机攻击对手，提出了计算最佳应对策略的多项式时间算法，但针对最大破坏对手的情况仍留下开放问题。此类对手攻击会使攻击后社会福利最小化的脆弱区域。本文致力于解决此问题。我们证明，给定玩家u及所有其他玩家的策略，当由给定策略生成的初始网络连通时，计算最佳应对策略可在多项式时间内解决。我们的算法基于动态规划，虽与背包问题有相似之处，但更为复杂且涉及更多细节。