The Mysteries of Security Games: Equilibrium Computation Becomes Combinatorial Algorithm Design

The security game is a basic model for resource allocation in adversarial environments. Here there are two players, a defender and an attacker. The defender wants to allocate her limited resources to defend critical targets and the attacker seeks his most favorable target to attack. In the past decade, there has been a surge of research interest in analyzing and solving security games that are motivated by applications from various domains. Remarkably, these models and their game-theoretic solutions have led to real-world deployments in use by major security agencies like the LAX airport, the US Coast Guard and Federal Air Marshal Service, as well as non-governmental organizations. Among all these research and applications, equilibrium computation serves as a foundation. This paper examines security games from a theoretical perspective and provides a unified view of various security game models. In particular, each security game can be characterized by a set system E which consists of the defender's pure strategies; The defender's best response problem can be viewed as a combinatorial optimization problem over E. Our framework captures most of the basic security game models in the literature, including all the deployed systems; The set system E arising from various domains encodes standard combinatorial problems like bipartite matching, maximum coverage, min-cost flow, packing problems, etc. Our main result shows that equilibrium computation in security games is essentially a combinatorial problem. In particular, we prove that, for any set system $E$, the following problems can be reduced to each other in polynomial time: (0) combinatorial optimization over E; (1) computing the minimax equilibrium for zero-sum security games over E; (2) computing the strong Stackelberg equilibrium for security games over E; (3) computing the best or worst (for the defender) Nash equilibrium for security games over E. Therefore, the hardness [polynomial solvability] of any of these problems implies the hardness [polynomial solvability] of all the others. Here, by "games over E" we mean the class of security games with arbitrary payoff structures, but a fixed set E of defender pure strategies. This shows that the complexity of a security game is essentially determined by the set system E. We view drawing these connections as an important conceptual contribution of this paper.

[1]  Rong Yang,et al.  Adaptive resource allocation for wildlife protection against illegal poachers , 2014, AAMAS.

[2]  Lance Fortnow,et al.  On the Complexity of Succinct Zero-Sum Games , 2005, 20th Annual IEEE Conference on Computational Complexity (CCC'05).

[3]  H. Stackelberg,et al.  Marktform und Gleichgewicht , 1935 .

[4]  J. G. Pierce,et al.  Geometric Algorithms and Combinatorial Optimization , 2016 .

[5]  T. Sandler,et al.  Terrorism & Game Theory , 2003 .

[6]  Joan Feigenbaum,et al.  A game-theoretic classification of interactive complexity classes , 1995, Proceedings of Structure in Complexity Theory. Tenth Annual IEEE Conference.

[7]  Vincent Conitzer,et al.  Solving Security Games on Graphs via Marginal Probabilities , 2013, AAAI.

[8]  Milind Tambe,et al.  Protecting Moving Targets with Multiple Mobile Resources , 2013, J. Artif. Intell. Res..

[9]  Vincent Conitzer,et al.  Computing the optimal strategy to commit to , 2006, EC '06.

[10]  Vincent Conitzer,et al.  Approximation Algorithm for Security Games with Costly Resources , 2011, WINE.

[11]  John A. Major Advanced Techniques for Modeling Terrorism Risk , 2002 .

[12]  Adam Tauman Kalai,et al.  Dueling algorithms , 2011, STOC '11.

[13]  Milind Tambe,et al.  One Size Does Not Fit All: A Game-Theoretic Approach for Dynamically and Effectively Screening for Threats , 2016, AAAI.

[14]  Milind Tambe,et al.  Security and Game Theory - Algorithms, Deployed Systems, Lessons Learned , 2011 .

[15]  Paul G. Spirakis,et al.  The Price of Defense , 2006, Algorithmica.

[16]  Vincent Conitzer,et al.  Complexity of Computing Optimal Stackelberg Strategies in Security Resource Allocation Games , 2010, AAAI.

[17]  Robert J. Fowler,et al.  Optimal Packing and Covering in the Plane are NP-Complete , 1981, Inf. Process. Lett..

[18]  Larry Samuelson,et al.  Choosing What to Protect: Strategic Defensive Allocation Against an Unknown Attacker , 2005 .

[19]  Sarit Kraus,et al.  Deployed ARMOR protection: the application of a game theoretic model for security at the Los Angeles International Airport , 2008, AAMAS 2008.

[20]  Yevgeniy Vorobeychik,et al.  Computing Stackelberg Equilibria in Discounted Stochastic Games , 2012, AAAI.

[21]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[22]  Bo An,et al.  PROTECT - A Deployed Game Theoretic System for Strategic Security Allocation for the United States Coast Guard , 2012, AI Mag..

[23]  Milind Tambe,et al.  Urban security: game-theoretic resource allocation in networked physical domains , 2010, AAAI 2010.

[24]  Alan Washburn,et al.  Two-Person Zero-Sum Games for Network Interdiction , 1995, Oper. Res..

[25]  Vincent Conitzer,et al.  Security Games with Multiple Attacker Resources , 2011, IJCAI.

[26]  Eitan Zemel,et al.  Nash and correlated equilibria: Some complexity considerations , 1989 .

[27]  van Leeuwen,et al.  Optimization and approximation on systems of geometric objects , 2009 .

[28]  B. Stengel,et al.  Leadership with commitment to mixed strategies , 2004 .

[29]  Vincent Conitzer,et al.  Solving Zero-Sum Security Games in Discretized Spatio-Temporal Domains , 2014, AAAI.

[30]  Vincent Conitzer,et al.  New complexity results about Nash equilibria , 2008, Games Econ. Behav..

[31]  Milind Tambe,et al.  PAWS: Game Theory Based Protection Assistant for Wildlife Security , 2017 .

[32]  Sarit Kraus,et al.  Deployed ARMOR protection: the application of a game theoretic model for security at the Los Angeles International Airport , 2008, AAMAS.

[33]  Paul G. Spirakis,et al.  A graph-theoretic network security game , 2005, Int. J. Auton. Adapt. Commun. Syst..

[34]  Milind Tambe,et al.  Optimal patrol strategy for protecting moving targets with multiple mobile resources , 2013, AAMAS.

[35]  Todd Sandler,et al.  Counterterrorism , 2005 .

[36]  Bo An,et al.  Security Games with Protection Externalities , 2015, AAAI.

[37]  Bo An,et al.  Deploying PAWS: Field Optimization of the Protection Assistant for Wildlife Security , 2016, AAAI.

[38]  Milind Tambe,et al.  Security and Game Theory: IRIS – A Tool for Strategic Security Allocation in Transportation Networks , 2011, AAMAS 2011.

[39]  Vincent Conitzer,et al.  Stackelberg vs. Nash in Security Games: An Extended Investigation of Interchangeability, Equivalence, and Uniqueness , 2011, J. Artif. Intell. Res..

[40]  Manish Jain,et al.  Security Games with Arbitrary Schedules: A Branch and Price Approach , 2010, AAAI.

[41]  Ruta Mehta,et al.  Bilinear Games: Polynomial Time Algorithms for Rank Based Subclasses , 2011, WINE.

[42]  Milind Tambe,et al.  TRUSTS: Scheduling Randomized Patrols for Fare Inspection in Transit Systems , 2012, IAAI.

[43]  Paul R. Milgrom,et al.  Designing Random Allocation Mechanisms: Theory and Applications , 2013 .

[44]  Paul W. Goldberg,et al.  The complexity of computing a Nash equilibrium , 2006, STOC '06.

[45]  Xiaotie Deng,et al.  Settling the complexity of computing two-player Nash equilibria , 2007, JACM.