Playing Stackelberg Security Games in perfect formulations

Protecting critical infrastructure from intentional damage requires foreseeing the strategies of possible attackers. We formulate this problem as a Stackelberg Security Game. A defender must decide which specific targets to protect with limited resources, thus maximizing their expected utility (e.g., minimizing damage value) and considering that a second player (or players), called an attacker, responds in the best possible way. Since Stackelberg Security Games are generally NP-hard, the main challenge in finding optimal strategies in real applications is to develop efficient methodologies for large instances. We propose a general methodology to find a Strong Stackelberg Equilibrium for Stackelberg Security Games, exploiting the structure in the defender's strategy set. This methodology consists of two steps. First, we formulate the problem by using variables representing the probability of defending each target. The formulation must be either a polynomial-size MILP and/or an MILP with an exponential number of constraints that are separable in polynomial time through branch-and-cut. In the second step, we recover the mixed strategies in the original space efficiently (in polynomial time) by using column generation. We apply this methodology to various security applications studied in the last decade. We generalize known examples and propose new examples. Finally, we provide an extensive computational study of the various formulations based on marginal probabilities.