Algorithms for a risk-averse Stackelberg game with multiple adversaries

We consider a Stackelberg game that arises in a security domain (SSG), where a defender can simultaneously protect.. out of.. targets from an adversary that observes the defense strategy before deciding on an utility maximizing attack. Given the high stakes in security settings, it is reasonable that the defender in this game is risk averse with respect to the attacker's decisions. Here we focus on developing efficient solution algorithms for a specific SSG, where the defender uses an entropic risk measure to model risk aversion to the attacker's strategies, and where multiple attackers select targets following logit quantal response equilibrium models. This problem can be formulated as a nonconvex nonlinear optimization problem. We propose two solution methods: (1) approximate the problem through convex mixed integer nonlinear programs (MINR) and (2) a general purpose methodology (CELL) to optimize nonconvex and nonseparable fractional problems through mixed integer linear programming approximations. Both methods provide arbitrarily good incumbents and lower bounds on SSG. We present cutting plane methods to solve these problems for large instances. Our computational experiments illustrate the advantages of introducing risk aversion into the defender's behavior and show that MINR dominates CELL, producing in 2 h solutions that are within 2% of optimal on average.