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. I. INTRODUCTION Providing security for critical infrastructure, cyber-physical networks and other financial systems is a large and growing area of concern. The key problem in many of these security domains is how to efficiently allocate limited resources to protect targets against potential threats. With the development of computational game theory over the past two decades, such resource allocation problems can be cast in the game-theoretic contexts, which provides a more sound mathematical approach to determine the optimal defense strategy. It allows the analyst to factor differential risks and values into the model, incorporate game-theoretic predictions of how the attacker would respond to the security policy, and finally determine an equilibrium strategy that cannot be exploited by adversaries to obtain a higher payoff. One line of research initiated in the seminal paper of Wollmer (1964) is the interdiction game (IG). It is a two-player normal form game, in which both players move simultaneously. The evader attempts to traverse a path through a network from the source node to the destination node, without being detected by an interdictor, while the defender deploys the interdictor in different nodes or links to halt the possible intrusion. The IG has been successfully applied to United States’ drug interdiction [18], communication network vulnerability [3] and urban network security recently [4]. In some security domains, the defender can build the fortifications before the attack and it is thus in the leader’s position from the point view of the game and able to move first. The branch of research inspired by this phenomenon is called Stackelberg Security Games (SSG), which includes a defender and an attacker, and several targets. The defender moves first by committing to a strategy and then it is observed by the attacker, who plays a best response to the defender’s strategy. The SSG is currently used by many security agencies including Federal Air Marshals Service, US Coast Guard, Transportation System Administration and even in the wildlife protection; see book by Tambe [16] for an overview. A. Motivation A common limitation of existing security game models is that they do not consider the dependency among the different targets. In the SSG, the payoff functions for both players are additive, i.e, the payoff of a group of targets is the sum of the payoffs of each target separately. This assumption means that the security agency measures the importance of several targets without considering the synergy effect among them. In practice, there exists some linkage structure among the targets such that attacking one target will influence the other targets. For instance, an attacker attempts to destroy the connectivity of a network and the defender aims to protect it. The strategy for both players is to choose the nodes of the network. If there are two nodes that constitute a bridge of this network, successfully attacking both of them will split the network into two parts and incur a huge damage, while attacking any one of them will have no significant effect. Hence, traditional models that ignore the inherent synergy effect between the targets are limiting and could lead to catastrophic consequences. Example 1: As shown in Fig. 1, we have a 20−node network. It is clearly that nodes 1, 2, 3 and 4 are the critical battlefields in this network. Suppose attacker’s and defender’s strategies are {1}, {2}, {3}, {1, 2} or {3, 4}, where {v} denotes the index of the nodes. We adopt the network value proposed by [3] as the security measure for different nodes, which calculates the importance of group of nodes via subtracting the value of the network by removing these nodes from the value of the 1Sinong Wang and Fang Liu is with graduate student of Department of Electrical and Computer Engineering, the Ohio State University, 2015 Neil Avenue, Columbus, OH 43210, USA, wang.7691@osu.edu 2Ness Shroff is with the Faculty of Department of Electrical and Computer Engineering, the Ohio State University, 2015 Neil Avenue, Columbus, OH 43210, USA, shroff.11@osu.edu ar X iv :1 60 3. 00 74 9v 1 [ cs . G T ] 2 M ar 2 01 6 Additive Utility Function Strategy 1 2 3 4 1,2 3,4 Benefit 39 39 75 75 78 150 Non-­‐additive Utility Function Strategy 1 2 3 4 1,2 3,4 Benefit 39 39 75 75 238 142 1