Optimizing periodic patrols against short attacks on the line and other networks

On a given network, a Patroller and Attacker play the following win-lose game: The Patroller adopts a periodic walk on the network while the Attacker chooses a node and two consecutive periods (to attack there). The Patroller wins if he successfully intercepts the attack, that is, if he occupies the attacked node in one of the two periods of the attack. We solve this game in mixed strategies for line graphs, the first class of graphs to be solved for the periodic patrolling game. We also solve the game for arbitrary graphs when the period is even

Continuous Patrolling Games

The continuous patrolling game studied here was first proposed in Alpern et al. (2011), which studied a discrete time game where facilities to be protected were modeled as the nodes of a graph. Here we consider protecting roads or pipelines, modeled as the arcs of a continuous network $Q$. The Attacker chooses a point of $Q$ to attack during a chosen time interval of fixed duration (the attack time, $\alpha$). The Patroller chooses a unit speed path on $Q$ and intercepts the attack (and wins) if she visits the attacked point during the attack time interval. Solutions to the game have previously been given in certain special cases. Here, we analyze the game on arbitrary networks. Our results include the following: (i) a solution to the game for any network $Q$, as long as $\alpha$ is sufficiently short, generalizing the known solutions for circle or Eulerian networks and the network with two nodes joined by three arcs; (ii) a solution to the game for all tree networks that satisfy a condition on their extremities. We present a conjecture on the solution of the game for arbitrary trees and establish it in certain cases

Continuous patrolling games

We study a patrolling game played on a network Q, considered as a metric space. The Attacker chooses a point of Q (not necessarily a node) to attack during a chosen time interval of fixed duration. The Patroller chooses a unit speed path on Q and intercepts the attack (and wins) if she visits the attacked point during the attack time interval. This zero-sum game models the problem of protecting roads or pipelines from an adversarial attack. The payoff to the maximizing Patroller is the probability that the attack is intercepted. Our results include the following: (i) a solution to the game for any network Q, as long as the time required to carry out the attack is sufficintly short, (ii) a solution to the game for all tree networks that satisfy a certain condition on their extremities, and (iii) a solution to the game for any attack duration for stars with one long arc and the remaining arcs equal in length. We present a conjecture on the solution of the game for arbitrary trees and establish it in certain cases

Optimal Patrol of a Perimeter

A defender dispatches patrollers to circumambulate a perimeter to guard against potential attacks. The defender decides on the time points to dispatch patrollers and each patroller's direction and speed, as long as the long-run rate patrollers are dispatched is capped at some constant. An attack at any point on the perimeter requires the same amount of time, during which it will be detected by each passing patroller independently with the same probability. The defender wants to maximize the probability of detecting an attack before it completes, while the attacker wants to minimize it. We study two scenarios, depending on whether the patrollers are undercover or wear a uniform. Conventional wisdom would suggest that the attacker gains advantage if he can see the patrollers going by so as to time his attack, but we show that the defender can achieve the same optimal detection probability by carefully spreading out the patrollers probabilistically against a learning attacker.Comment: 17 pages, 1 figur

The Dormant Second Amendment: Exploring the Rise, Fall, and Potential Resurrection of Independent State Militias

The term “militia” is polarizing, misunderstood, misapplied, and generally difficult for modern Americans to digest. That is not surprising, given the depth and breadth of American militia history and militias’ substantial evolution over four centuries. Historically, militia simply refers to a broad-based civic duty to protect one’s fellow citizens from internal and external dangers and is not limited to activities involving firearms. Reestablishing militia’s true meaning and purpose—and reinvigorating independent state militias in the United States to effect that purpose—has the potential to address states’ emerging financial and security gaps and to produce multiple other significant benefits, including recalibrating federalism. This article suggests a method for how best to reinvigorate independent state militias, addresses the major critique against doing so, and initiates a real discussion about the future of state militias—an issue conspicuously underdeveloped in scholarship today

Designing and Operating Safe and Secure Transit Systems: Assessing Current Practices in the United States and Abroad, MTI Report 04-05

Public transit systems around the world have for decades served as a principal venue for terrorist acts. Today, transit security is widely viewed as an important public policy issue and is a high priority at most large transit systems and at smaller systems operating in large metropolitan areas. Research on transit security in the United States has mushroomed since 9/11; this study is part of that new wave of research. This study contributes to our understanding of transit security by (1) reviewing and synthesizing nearly all previously published research on transit terrorism; (2) conducting detailed case studies of transit systems in London, Madrid, New York, Paris, Tokyo, and Washington, D.C.; (3) interviewing federal officials here in the United States responsible for overseeing transit security and transit industry representatives both here and abroad to learn about efforts to coordinate and finance transit security planning; and (4) surveying 113 of the largest transit operators in the United States. Our major findings include: (1) the threat of transit terrorism is probably not universal—most major attacks in the developed world have been on the largest systems in the largest cities; (2) this asymmetry of risk does not square with fiscal politics that seek to spread security funding among many jurisdictions; (3) transit managers are struggling to balance the costs and (uncertain) benefits of increased security against the costs and (certain) benefits of attracting passengers; (4) coordination and cooperation between security and transit agencies is improving, but far from complete; (5) enlisting passengers in surveillance has benefits, but fearful passengers may stop using public transit; (6) the role of crime prevention through environmental design in security planning is waxing; and (7) given the uncertain effectiveness of antitransit terrorism efforts, the most tangible benefits of increased attention to and spending on transit security may be a reduction in transit-related person and property crimes