Asymptotic Nash equilibrium for sequential change detection in adversarial environments

The problem of the quickest detection under adversarial environments is investigated. Initially, the defender aims to design a detection procedure to detect the change of the data stream from a pre-change distribution (Formula presented.) to a post-change distribution (Formula presented.) However, in the adversarial environment, the adversary can modify the actual distributions by picking a distribution p from a set of distributions (Formula presented.) around the designed pre-change distribution (Formula presented.) as the actual pre-change data-generating distribution with the cost (Formula presented.) Assume the adversary knows the true change time and after the change occurs, the adversary picks the actual post-change data-generating distribution q from another distribution set (Formula presented.) around the true post-change distribution with cost (Formula presented.) The defender’s goal is to detect the change as quickly as possible, subject to a false alarm constraint, while the adversary’s goal is to fool the defender, subject to its cost constraint. The problem is formulated as a non-zero-sum game between the adversary and the defender. A pair of strategies for the defender and the attacker is proposed and proved to be a Nash equilibrium pair for the non-zero-sum game asymptotically. Numerical experiments are provided to validate our results.