Fortification-interdiction games are tri-level adversarial games where two opponents act in succession to protect, disrupt and simply use an infrastructure for a specific purpose. Many such games have been formulated and tackled in the literature through specific algorithmic methods, however very few investigations exist on the completeness of such fortification problems in order to locate them rigorously in the polynomial hierarchy. We clarify the completeness status of several well-known fortification problems, such as the Tri-level Interdiction Knapsack Problem with unit fortification and attack weights, the Max-flow Interdiction Problem and Shortest Path Interdiction Problem with Fortification, the Multi-level Critical Node Problem with unit weights, as well as a well-studied electric grid defence planning problem. For all of these problems, we prove their completeness either for the $\Sigma^p_2$ or the $\Sigma^p_3$ class of the polynomial hierarchy. We also prove that the Multi-level Fortification-Interdiction Knapsack Problem with an arbitrary number of protection and interdiction rounds and unit fortification and attack weights is complete for any level of the polynomial hierarchy, therefore providing a useful basis for further attempts at proving the completeness of protection-interdiction games at any level of said hierarchy.

翻译：防护-阻断博弈是一种三层对抗博弈，其中两个对手依次行动，分别对特定用途的基础设施实施防护、破坏及常规使用。文献中已通过特定算法方法构建并求解了许多此类博弈，然而关于这些防护问题在多项式层次结构中的严格定位及其完备性的研究尚少。本文明确了若干经典防护问题的完备性状态，包括单位防护与攻击权重的三层阻断背包问题、带防护的最大流阻断问题与最短路径阻断问题、单位权重的多层关键节点问题，以及一个被深入研究的电网防御规划问题。对于所有这些问题，我们证明了它们在多项式层次结构中要么属于$\Sigma^p_2$类完备，要么属于$\Sigma^p_3$类完备。同时，我们证明了具有任意防护与阻断轮次且采用单位防护与攻击权重的多层防护-阻断背包问题对多项式层次结构的任意层级都是完备的，这为今后证明防护-阻断博弈在多项式层次结构任意层级的完备性提供了重要基础。