A Probabilistic Framework for Quantifying Mixed Uncertainties in Cyber Attacker Payoffs

Quantification and propagation of uncertainties in cyber attacker payoffs is a key aspect within multiplayer, stochastic security games. These payoffs may represent penalties or rewards associated with player actions and are subject to various sources of uncertainty, including: (1) cyber-system state, (2) attacker type, (3) choice of player actions, and (4) cyber-system state transitions over time. Past research has primarily focused on representing defender beliefs about attacker payoffs as point utility estimates. More recently, within the physical security domain, attacker payoff uncertainties have been represented as Uniform and Gaussian probability distributions, and mathematical intervals. For cyber-systems, probability distributions may help address statistical (aleatory) uncertainties where the defender may assume inherent variability or randomness in the factors contributing to the attacker payoffs. However, systematic (epistemic) uncertainties may exist, where the defender may not have sufficient knowledge or there is insufficient information about the attacker’s payoff generation mechanism. Such epistemic uncertainties are more suitably represented as generalizations of probability boxes. This paper explores the mathematical treatment of such mixed payoff uncertainties. A conditional probabilistic reasoning approach is adopted to organize the dependencies between a cyber-system’s state, attacker type, player actions, and state transitions. This also enables the application of probabilistic theories to propagate various uncertainties in the attacker payoffs. An example implementation of this probabilistic framework and resulting attacker payoff distributions are discussed. A goal of this paper is also to highlight this uncertainty quantification problem space to the cyber security research community and encourage further advancements in this area.