Strategic Interaction Over Age of Incorrect Information for False Data Injection in Cyber-Physical Systems

Ambient monitoring through remote sensing is the first required step of many control operations in cyber-physical systems to enable accurate decision-making by network intelligence. We consider a controller that sends status updates about a process to a receiver, incurring a cost when doing so. The process is dynamic, implying that the information the receiver has may become outdated due to a natural drift of the process. To determine the correctness of the information at the receiver, we model this interaction using a Markov chain with two states, namely, right (R) and wrong (W). The controller can restore the receiver status to R by performing a new transmission, which comes at a cost. The staleness of information, when the system state is erroneous, is quantified through the average value of the age of incorrect information metric. Moreover, an adversary may inject false data at a price to make the information available at the receiver less fresh, which can only be contrasted by additional measurements by the controller. This results in a game played by strategic agents, namely, the controller and the adversary. The adversary's objective is to maximize the time the receiver is in the W state of the Markov chain, while the controller's objective is to minimize it. We provide a mathematical formulation of this strategic interaction using game-theory, demonstrating the existence of a Nash equilibrium. In our analysis, we discuss the role of different system parameters and the implications on the resulting system performance, providing a quantitative evaluation of the parameter ranges where an adversary can be effectively counteracted is an important guideline to improve the security of cyber-physical systems.