There has been significant interest in studying security games for modeling the interplay of attacks and defenses on various systems involving critical infrastructure, financial system security, political campaigns, and civil safeguarding. However, existing security game models typically either assume additive utility functions, or that the attacker can attack only one target. Such assumptions lead to tractable analysis, but miss key inherent dependencies that exist among different targets in current complex networks. In this paper, we generalize the classical security game models to allow for non-additive utility functions. We also allow attackers to be able to attack multiple targets. We examine such a general security game from a theoretical perspective and provide a unified view. In particular, we show that each security game is equivalent to a combinatorial optimization problem over a set system @math , which consists of defender's pure strategy space. The key technique we use is based on the transformation, projection of a polytope, and the elipsoid method. This work settles several open questions in security game domain and significantly extends the state of-the-art of both the polynomial solvable and NP-hard class of the security game.
One line of research focuses on designing an efficient algorithm to solve such a game. @cite_22 proposes a compact representation technique, in which the security game can be equivalently represented by a polynomial-sized mixed-integer linear programming (MILP) problem. The issue in @cite_22 is that they only determine the optimal solution of the compact game instead of the optimal defender's mixed strategy. To solve this problem, @cite_32 introduces the Birkhoff-von Neumann theorem and show that the defender's mixed strategy can be recovered under a specific condition. @cite_20 proposes a double-oracle algorithm to exactly solve the security game with exponential large representation, which can be regarded as a generalization of traditional column generation technique in solving the large-scale linear programming problem. There are also other works such as the Bayesian security game @cite_26 , the security game with quantal response @cite_31 and the security game with uncertain attacker behavior @cite_16 , etc.
@cite_26 @cite_22 @cite_32 @cite_31 @cite_16 @cite_20
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Security games involving the allocation of multiple security resources to defend multiple targets generally have an exponential number of pure strategies for the defender. One method that has been successful in addressing this computational issue is to instead directly compute the marginal probabilities with which the individual resources are assigned (first pursued by (2009)). However, in sufficiently general settings, there exist games where these marginal solutions are not implementable, that is, they do not correspond to any mixed strategy of the defender. In this paper, we examine security games where the defender tries to monitor the vertices of a graph, and we show how the type of graph, the type of schedules, and the type of defender resources affect the applicability of this approach. In some settings, we show the approach is applicable and give a polynomial-time algorithm for computing an optimal defender strategy; in other settings, we give counterexample games that demonstrate that the approach does not work, and prove NP-hardness results for computing an optimal defender strategy. 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 @math , 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. Predictable allocations of security resources such as police officers, canine units, or checkpoints are vulnerable to exploitation by attackers. Recent work has applied game-theoretic methods to find optimal randomized security policies, including a fielded application at the Los Angeles International Airport (LAX). This approach has promising applications in many similar domains, including police patrolling for subway and bus systems, randomized baggage screening, and scheduling for the Federal Air Marshal Service (FAMS) on commercial flights. However, the existing methods scale poorly when the security policy requires coordination of many resources, which is central to many of these potential applications. We develop new models and algorithms that scale to much more complex instances of security games. The key idea is to use a compact model of security games, which allows exponential improvements in both memory and runtime relative to the best known algorithms for solving general Stackelberg games. We develop even faster algorithms for security games under payoff restrictions that are natural in many security domains. Finally, introduce additional realistic scheduling constraints while retaining comparable performance improvements. The empirical evaluation comprises both random data and realistic instances of the FAMS and LAX problems. Our new methods scale to problems several orders of magnitude larger than the fastest known algorithm. Recently, algorithms for computing game-theoretic solutions have been deployed in real-world security applications, such as the placement of checkpoints and canine units at Los Angeles International Airport. These algorithms assume that the defender (security personnel) can commit to a mixed strategy, a so-called Stackelberg model. As pointed out by Kiek- (2009), in these applications, generally, multiple resources need to be assigned to multiple targets, resulting in an exponential number of pure strategies for the defender. In this paper, we study how to compute optimal Stackelberg strategies in such games, showing that this can be done in polynomial time in some cases, and is NP-hard in others.