Passive Fault-tolerant Estimation under Strategic Adversarial Bias

This paper is concerned with the problem of fault-tolerant estimation in cyber-physical systems. In cyber-physical systems, such as critical infrastructures, networked embedded sensors are widely used for monitoring and can be exploited by an adversary to deceive the control center by modifying measured values. The deception is modeled as a bias; i.e., there is a misalignment between the objective functions of the control center and the adversarial sensor. Different from previous studies, a Stackelberg equilibrium of a cheap talk setup is adapted to the attacker-defender game setting for the first time. That is, the defender (control center), as a receiver, is the leader, and the attacker (adversarial sensor), as a transmitter, is the follower. The equilibrium strategies and the associated costs are characterized for uniformly distributed variables and quadratic objective functions, and an analysis on the uniqueness of the equilibrium is provided. It is shown that the attacker and defender costs at the equilibrium are increasing with the bias and decreasing with the number of quantization levels. Our results surprisingly show that, under certain conditions, the attacker prefers a public bias rather than a private one.