Algorithms for computing game-theoretic solutions have recently been applied to a number of security domains. However, many of the techniques developed for compact representations of security games do not extend to Bayesian security games, which allow us to model uncertainty about the attacker's type. In this paper, we introduce a general framework of catcher-evader games that can capture Bayesian security games as well as other game families of interest. We show that computing Stackelberg strategies is NP-hard, but give an algorithm for computing a Nash equilibrium that performs well in experiments. We also prove that the Nash equilibria of these games satisfy the interchangeability property, so that equilibrium selection is not an issue.

Gambles in casinos are usually set up so that the casino makes a profit in expectation -- as long as gamblers play honestly. However, some gamblers are able to cheat, reducing the casino’s profit. How should the casino address this? A common strategy is to selectively kick gamblers out, possibly even without being sure that they were cheating. In this paper, we address the following question: Based solely on a gambler’s track record,when is it optimal for the casino to kick the gambler out? Because cheaters will adapt to the casino’s policy, this is a game-theoretic question. Specifically, we model the problem as a Bayesian game in which the casino is a Stackelberg leader that can commit to a (possibly randomized) policy for when to kick gamblers out, and we provide efficient algorithms for computing the optimal policy. Besides being potentially useful to casinos, we imagine that similar techniques could be useful for addressing related problems -- for example, illegal trades in financial markets.

In game theory, it is well known that being able to commit to a strategy before other players move can be beneficial. In this paper, we analyze how much benefit a player can derive from commitment in various types of games, in a quantitative sense that is similar to concepts such as the value of mediation and the price of anarchy. Specifically, we introduce and study the value of pure commitment (the benefit of committing to a pure strategy), the value of mixed commitment (the benefit of committing to a mixed strategy), and the mixed versus pure commitment ratio (how much can be gained by committing to a mixed strategy rather than a pure one). In addition to theoretical results about how large these values are in the extreme case in various classes of games, we also give average-case results based on randomly drawn normal-form games.

Algorithms for finding game-theoretic solutions are now used in several real-world security applications. Many of these applications are based on different but related game-theoretical models collectively known as security games. Much of the research in this area has focused on the two-player setting in which the first player (leader, defender) commits to a strategy, after which the second player (follower, attacker) observes that strategy and responds to it. This is commonly known as the Stackelberg, or leader-follower, model. If none of the players can observe the actions of the others then such a setting is called a simultaneous-move game. A common solution concept in simultaneous-move games is the Nash equilibrium (NE). In the present dissertation, we contribute to this line of research in two ways. First, we consider new ways of modeling commitment. We propose the new model in which the leader can commit to a correlated strategy. We show that this model is equivalent to the Stackelberg model in two-player games and is different from the existing models in games with three or more players. We propose an algorithm for computing a solution to this model in polynomial time. We also consider a leader-follower setting in which the players are uncertain about whether the follower can observe. We describe an iterative algorithm for solving such games. Second, we analyze the computational complexity of computing Stackelberg and NE strategies in security games. We describe algorithms to solve some variants of the security game model in polynomial time and prove NP-hardness of solving other variants of the model. We also extend the family of security games by allowing the attacker have multiple resources. We provide an algorithm for computing an NE of such games in polynomial time, and we show that computing a Stackelberg strategy is NP-hard.

Security Games: Solution Concepts and Algorithms

Firms have ever-increasing amounts of information about possible customers available to them; furthermore, they are increasingly able to push offers to them rather than having to passively wait for a consumer to initiate contact. This opens up enormous new opportunities for intelligent marketing. In this paper, we consider the limit case in which the firm can predict consumers’ preferences and relationships to each other perfectly, and has perfect control over when it makes offers to consumers. We focus on how to optimally introduce a new product into a social network of agents, when that product has significant externalities. We propose a general model to capture this problem, and prove that there is no polynomial-time approximation unless P=NP. However, in the special case where agents’ relationships are symmetric and externalities are positive, we show that the problem can be solved in polynomial time.

Algorithms for finding game-theoretic solutions are now used in several real-world security applications. This work has generally assumed a Stackelberg model where the defender commits to a mixed strategy first. In general two-player normal-form games, Stackelberg strategies are easier to compute than Nash equilibria, though it has recently been shown that in many security games, Stackelberg strategies are also Nash strategies for the defender. However, the work on security games so far assumes that the attacker attacks only a single target. In this paper, we generalize to the case where the attacker attacks multiple targets simultaneously. Here, Stackelberg and Nash strategies for the defender can be truly different. We provide a polynomial-time algorithm for finding a Nash equilibrium. The algorithm gradually increases the number of defender resources and maintains an equilibrium throughout this process. Moreover, we prove that Nash equilibria in security games with multiple attackers satisfy the interchange property, which resolves the problem of equilibrium selection in such games. On the other hand, we show that Stackelberg strategies are actually NP-hard to compute in this context. Finally, we provide experimental results.

The standard approach to computing an optimal mixed strategy to commit to is based on solving a set of linear programs, one for each of the follower's pure strategies. We show that these linear programs can be naturally merged into a single linear program; that this linear program can be interpreted as a formulation for the optimal correlated strategy to commit to, giving an easy proof of a result by von Stengel and Zamir that the leader's utility is at least the utility she gets in any correlated equilibrium of the simultaneous-move game; and that this linear program can be extended to compute optimal correlated strategies to commit to in games of three or more players. (Unlike in two-player games, in games of three or more players, the notions of optimal mixed and correlated strategies to commit to are truly distinct.) We give examples, and provide experimental results that indicate that for 50x50 games, this approach is usually significantly faster than the multiple-LPs approach.

Recent applications of game theory in security domains use algorithms to solve a Stackelberg model, in which one player (the leader) first commits to a mixed strategy and then the other player (the follower) observes that strategy and best-responds to it. However, in real-world applications, it is hard to determine whether the follower is actually able to observe the leader's mixed strategy before acting. In this paper, we model the uncertainty about whether the follower is able to observe the leader's strategy as part of the game (as proposed in the extended version of Yin et al. [17]). We describe an iterative algorithm for solving these games. This algorithm alternates between calling a Nash equilibrium solver and a Stackelberg solver as subroutines. We prove that the algorithm finds a solution in a finite number of steps and show empirically that it runs fast on games of reasonable size. We also discuss other properties of this methodology based on the experiments.

There has been significant recent interest in game-theoretic approaches to security, with much of the recent research focused on utilizing the leader-follower Stackelberg game model. Among the major applications are the ARMOR program deployed at LAX Airport and the IRIS program in use by the US Federal Air Marshals (FAMS). The foundational assumption for using Stackelberg games is that security forces (leaders), acting first, commit to a randomized strategy; while their adversaries (followers) choose their best response after surveillance of this randomized strategy. Yet, in many situations, a leader may face uncertainty about the follower's surveillance capability. Previous work fails to address how a leader should compute her strategy given such uncertainty. We provide five contributions in the context of a general class of security games. First, we show that the Nash equilibria in security games are interchangeable, thus alleviating the equilibrium selection problem. Second, under a natural restriction on security games, any Stackelberg strategy is also a Nash equilibrium strategy; and furthermore, the solution is unique in a class of security games of which ARMOR is a key exemplar. Third, when faced with a follower that can attack multiple targets, many of these properties no longer hold. Fourth, we show experimentally that in most (but not all) games where the restriction does not hold, the Stackelberg strategy is still a Nash equilibrium strategy, but this is no longer true when the attacker can attack multiple targets. Finally, as a possible direction for future research, we propose an extensive-form game model that makes the defender's uncertainty about the attacker's ability to observe explicit.

In response to the Mumbai attacks of 2008, the Mumbai police have started to schedule a limited number of inspection checkpoints on the road network throughout the city. Algorithms for similar security-related scheduling problems have been proposed in recent literature, but security scheduling in networked domains when targets have varying importance remains an open problem at large. In this paper, we cast the network security problem as an attacker-defender zero-sum game. The strategy spaces for both players are exponentially large, so this requires the development of novel, scalable techniques. We first show that existing algorithms for approximate solutions can be arbitrarily bad in general settings. We present Rugged (Randomization in Urban Graphs by Generating strategies for Enemy and Defender), the first scalable optimal solution technique for such network security games. Our technique is based on a double oracle approach and thus does not require the enumeration of the entire strategy space for either of the players. It scales up to realistic problem sizes, as is shown by our evaluation of maps of southern Mumbai obtained from GIS data.

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 Kiekintveld et al. (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.

There has been significant recent interest in game theoretic approaches to security, with much of the recent research focused on utilizing the leader-follower Stackelberg game model; for example, these games are at the heart of major applications such as the ARMOR program deployed for security at the LAX airport since 2007 and the IRIS program in use by the US Federal Air Marshals (FAMS). The foundational assumption for using Stackel-berg games is that security forces (leaders), acting first, commit to a randomized strategy; while their adversaries (followers) choose their best response after surveillance of this randomized strategy. Yet, in many situations, the followers may act without observation of the leader's strategy, essentially converting the game into a simultaneous-move game model. Previous work fails to address how a leader should compute her strategy given this fundamental uncertainty about the type of game faced. Focusing on the complex games that are directly inspired by real-world security applications, the paper provides four contributions in the context of a general class of security games. First, exploiting the structure of these security games, the paper shows that the Nash equilibria in security games are interchangeable, thus alleviating the equilibrium selection problem. Second, resolving the leader's dilemma, it shows that under a natural restriction on security games, any Stackelberg strategy is also a Nash equilibrium strategy; and furthermore, the solution is unique in a class of real-world security games of which ARMOR is a key exemplar. Third, when faced with a follower that can attack multiple targets, many of these properties no longer hold. Fourth, our experimental results emphasize positive properties of games that do not fit our restrictions. Our contributions have major implications for the real-world applications.