Robust Stochastic Games and Applications to Counter-Terrorism Strategies

This report presents a new methodology for strategic decision making under uncertainty and presence of adversaries. This investigation is motivated by the need to determine optimal strategies under uncertainty against an adversarial and adaptive opponent. Such problems arise in the context of terrorism threats. To model investment decisions that pertain to homeland security, one should account for both uncertainty and the antagonistic character inherent in the problem. To this end, we propose a novel approach, robust stochastic games. We focus on incomplete information stochastic games and adopt a robust approach to account for uncertainty present in our problem in two dimensions. First, we consider that the adaptive nature of the adversary is uncertain. In other words, we propose a new approach that accounts for the uncertainty in the conversion from one threat category to the other that is based on the alternatives of the adversaries. Second, we consider that payoffs to the opponents are uncertain. We present an interesting new result, existence of equilibrium points in robust stochastic games. A new formulation that uses robust optimization techniques is proposed to solve robust stochastic games. Preliminary results are presented on a simple example with partial unknown data. First, uncertain transition probabilities that belong to convex hull uncertainty sets are considered in this example with exact immediate costs. Next, uncertainty is considered in both transition probabilities and immediate costs. Performance of the nominal solution when parameters attain their worst-case values is compared with the performance of the robust solution when data is certain. It is observed in this small example that the percentage savings resulting from using robust strategies versus the nominal strategies when the parameters attain their worst-case values are higher than the losses caused by using robust strategies when parameters attain their nominal values. It is also observed that compared to the uncertainty in transition probabilities, uncertainty in immediate costs has a greater effect on the robust value of the game. The next phase in this research includes the development of the model for the MANPADS case study, quantification of the model via expert elicitation, and computation of robust optimal strategies.