Non-additive Security Game.

Strategically allocating resources to protect targets against potential threats in an efficient way is a key challenge facing our society. From the classical interdiction game to the recently proposed Stackelberg Security Game, applying computational game models to the security domain has made a real-world impact on numerous fields including military attack and defense, financial system security, political campaign and civil safeguarding. However, existing methods assume additive utility functions, which are unable to capture the inherent dependencies that exist among different targets in current complex networks. In this paper, we introduce a new security game model, called Non-additive Security Game (NASG). It adopts a non-additive set function to assign the utility to every subset of targets, which completely describes the internal linkage structure of the game. However, both the number of utility functions and the strategies exponentially increase in the number of targets, which poses a significant challenge in developing algorithms to determine the equilibrium strategy. To tackle this problem, we first reduce the NASG to an equivalent zero-sum game and construct a low-rank perturbed game via matrix decomposition and random low-dimensional embedding. Then, we incorporate the above low-rank perturbed game into the Augmented Lagrangian and Coordinate Descent method. Using a series of careful constructions, we show that our method has a total complexity that is nearly linear in the number of utility functions and achieves asymptotic zero error. To the best of our knowledge, the NASG is the first computational game model to investigate dependencies among different targets.