Parallel systems with different types of defence resource expenditure under two sequential attacks

The paper compares the efficiency of single and double attack against a system consisting of identical parallel elements (one-out-of- N system). An attacker tries to maximize the system vulnerability (probability of total destruction) whereas the defender tries to minimize it. The attacker and the defender distribute their constrained resources optimally across two attacks. The attacker chooses the number of elements to attack in the first attack. The defender protects all elements before the first attack and protects all surviving elements before the second attack. Both agents decide how to distribute their resources between the two attacks before the first attack. Two cases are considered. In the first, all the elements that survive the first attack keep their protection in the second attack. In the second, only the elements that are not attacked in the first attack keep their protection in the second attack, while elements that are attacked but survive do not keep their protection. The attacker's resource is expendable and lasts only one attack. Both agents observe which elements are destroyed and not destroyed in the first attack, and apply their remaining resources into attacking and protecting the remaining elements in the second attack. The optimal attack and defence strategy against a system with a fixed number of elements is analysed as a solution of a minmax game.