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Geometric Security Games

There are many real world scenarios which can be modeled as a security game where there is one player, called the defender, trying to protect a set of potential targets from another player, called the attacker. Game-theoretic security approaches are currently used to set up traffic checkpoints and canine patrols at LAX airport [38, 37], to assign Federal Air Marshals to flights [43], and will soon be used nationally by the TSA for airport protection [40] and by the US Coast Guard for port security [42]. Most work on security games, including the first three applications above, use the following simple model. There are two agents, a defender and an attacker, and a set of targets, which the defender is trying to protect. Some targets may be more valuable than others; also, the attacker and defender may value targets differently. It is assumed that the defender has a limited number of security resources and can be allocated to different schedules, where a schedule can protect one or multiple targets. The outcome, represented by a utility for each agent, is determined by which target the attacker chooses to attack and whether or not there is a security resource protecting it. This model is simple because it does not address possible complexities in the decision space of the attacker and defender. For instance, certain security scenarios may be better modeled as a graph where the attacker chooses a path from a set of possible starting points to a set of possible targets and the defender chooses edges at which to position security resources to try to intercept the attacker [24]. In other scenarios, such as the Coast Guard one above, it may make sense to allow the attacker and defender to update their strategies based upon noisy measurements of their opponents current location. In patrolling security games [6, 4, 5, 2], there is a network of connected cells that the attacker and defender can move between. Both the attacker and defender have some limited ability to detect when the other is close. In these games, the players’ strategies can be modeled as MDPs which make decisions based upon their current location and opponent detection. This is similar to pursuit-evasion games [44, 22, 23], but differs in that the goal of the attacker is to eventually attack a target rather than to simply evade the defender. Most real world scenarios have to consider different types of attackers, e.g. attackers with specific motivations may value targets differently. In this case, it is appropriate to consider Bayesian games [21], where players have some amount of uncertainty about the specifics of the game they are playing. Computing a Bayesian Stackelberg equilibrium is NP-hard [13], although practical algorithms do exist [37]. Much work has also been done on modeling human irrationality in games [10, 46, 47, 33] and using that to calculate better defender strategies for security games [48, 49, 39]. Additionally, people have considered security games with multiple attacker resources [30], a stronger Stackelberg equilibrium refinement for security games [3], and security games with some amount of uncertainty in various ways [50, 36], among other things.