Playing Stackelberg security games in perfect formulations

Protecting critical infrastructure from intentional damage requires foreseeing the strategies of possible attackers. The problem faced by the defender of such infrastructure can be formulated as a Stackelberg security game. A defender must decide what specific targets to protect with limited resources, maximizing their expected utility (e.g., minimizing damage value) and considering that a second player (or players), called 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 developing efficient methodologies for large instances.

We propose a general methodology to find a Strong Stackelberg Equilibrium for Stackelberg se- curity games whose set of defender’s mixed strategies can be represented as a perfect formulation. This methodology consists in two steps. First, we formulate the problem using variables represent- ing the probabilities of each target being defended. The formulation must be either a polynomial-size MILP and/or a MILP with an exponential size of constraints that can be efficiently separated through branch-and-cut. In the second step, we recover the mixed strategies in the original space efficiently (in polynomial time) using column generation. This methodology has been applied in various security applications studied in the last decade. We generalize and propose new examples. Finally, we provide extensive computational study of different formulations based on marginal probabilities.