Computing fortification games in a tri-level Stackelberg Markov chains approach

This paper describes fortification games (FGs), also known as a defender–attacker–defender games model for planning defenses for an infrastructure system, which would increase that system's resistance against attacks by intelligent attackers. FG is usually described as a tri-level Stackelberg game in which, at the top level, the defenders choose strategies for some assets to be protected from prospective harm. The attackers resolve an interdiction game at the middle level by disabling the defenders’ maneuvers (strategies). In response, the defenders again choose strategies based on the surviving or partially disabled maneuvers at the innermost level. Our contribution consists of a method for dealing with defender–attacker–defender issues. The goal is to find the Stackelberg equilibria for such a game. The individual goal of first-level leaders is to achieve one of the Nash equilibria for any fixed strategy of the followers and any fixed strategy of the leaders at the innermost level, in order to satisfy the system of inequalities connected to the Nash condition. We present a solution for computing the equilibrium point for a class of tri-level optimization problem, all of which are represented by nonlinear programs. The solution approach is based on the extraproximal programming method reformulation for fortification variables. We show that the method converges to one of the Stackelberg equilibrium points. The FG problem is restricted to a class of time-homogeneous, finite, ergodic and controllable Markov games. Finally, we present an example of a mall application where one of the most crucial considerations is to provide customers, especially families, a secure space.