The Deployment-to-Saturation Ratio in Security Games

Stackelberg security games form the backbone of systems like ARMOR, IRIS and PROTECT, which are in regular use by the Los Angeles International Police, US Federal Air Marshal Service and the US Coast Guard respectively. An understanding of the runtime required by algorithms that power such systems is critical to furthering the application of game theory to other real-world domains. This paper identifies the concept of the deployment-to-saturation ratio in random Stackelberg security games, and shows that problem instances for which this ratio is 0.5 are computationally harder than instances with other deployment-to-saturation ratios for a wide range of different equilibrium computation methods, including (i) previously published different MIP algorithms, and (ii) different underlying solvers and solution mechanisms. This finding has at least two important implications. First, it is important for new algorithms to be evaluated on the hardest problem instances. We show that this has often not been done in the past, and introduce a publicly available benchmark suite to facilitate such comparisons. Second, we provide evidence that this computationally hard region is also one where optimization would be of most benefit to security agencies, and thus requires significant attention from researchers in this area. Furthermore, we use the concept of phase transitions to better understand this computationally hard region. We define a decision problem related to security games, and show that the probability that this problem has a solution exhibits a phase transition as the deployment-to-saturation ratio crosses 0.5. We also demonstrate that this phase transition is invariant to changes both in the domain and the domain representation, and that the phase transition point corresponds to the computationally hardest instances.