Optimal patrol to uncover threats in time when detection is imperfect

Abstract

Consider a patrol problem, where a patroller traverses a graph through edges to detect potential attacks at nodes. An attack takes a random amount of time to complete. The patroller takes one time unit to move to and inspect an adjacent node, and will detect an ongoing attack with some probability. If an attack completes before it is detected, a cost is incurred. The attack time distribution, the cost due to a successful attack, and the detection probability all depend on the attack node. The patroller seeks a patrol policy that minimizes the expected cost incurred when, and if, an attack eventually happens. We consider two cases. A random attacker chooses where to attack according to predetermined probabilities, while a strategic attacker chooses where to attack to incur the maximal expected cost. In each case, computing the optimal solution, although possible, quickly becomes intractable for problems of practical sizes. Our main contribution is to develop efficient index policies-based on Lagrangian relaxation methodology, and also on approximate dynamic programming-which typically achieve within 1% of optimality with computation time orders of magnitude less than what is required to compute the optimal policy for problems of practical sizes.

Continuous patrolling and hiding games

“…Other models of games involving a patroller and an attacker can be found in Basilico et al (2012Basilico et al ( , 2015, in which the authors design algorithms to solve large instances of Stackelberg patrolling security games on graphs. Lin et al (2013Lin et al ( , 2014 use linear programming and heuristics to study a large class of patrolling problems on graphs, with nodes having different values. Zoroa et al (2012) study a patrolling game with a mobile attacker on a perimeter.…”

Timely exposure of a secret project: Which activities to monitor?

“…Atkinson and Wein (2010) examine how a government should allocate its resources over the inspection of terror and criminal networks to exploit the finding of Smith, Damphousse, and Roberts (2006) that, prior to an attack, terrorists frequently participate in crimes such as theft or procuring explosives. Other articles address problems such as predicting the number of undetected terror threats (Kaplan, 2010), estimating the duration of a terrorist plot (Kaplan, 2012a), locating terrorists (Alpern & Lidbetter, 2013;Atkinson, Kress, & Lange, 2016), processing intelligence (Dimitrov, Kress, & Nevo, 2016;Lin, Kress, & Szechtman, 2009), patrolling an area (Lin, Atkinson, & Glazebrook, 2014;Papadaki, Alpern, Lidbetter, & Morton, 2016;Szechtman, Kress, Lin, & Cfir, 2008), and predicting the goal of a suspected terrorist (Tsitsiklis & Xu, 2018). In particular, Atkinson et al (2016) consider a searcher who, based on a stream of unreliable intelligence about a target's location, needs to decide whether to engage or to wait for more information.…”

Games for the optimal deployment of security forces

“…An overview of models for this type of problems is given by Hohzaki [58]. It is possible that the intruder is hidden at a fixed node (e.g., [4,78,95]), or moves through the network (e.g., [55,120]). Neuts [95] introduces a search game in which the intruder hides in one node, while the agent must search in a set of nodes.…”

“…For the security forces this means that patrol strategies or border control strategies at possible entry locations have to be found, in order to intercept the intruder or to prevent the illegal crossing of borders. Several game-theoretic models have been developed to provide solutions for these kinds of problems (e.g., [62,81,78,99]). In this paper we deviate from the traditional stochastic game assumption that both players have full information about the position of the other player.…”

“…However, by adjusting the input parameters, the proposed model can also be used for (barrier) search operation. The game discussed in this chapter is a special variant of a search game (e.g., [11,37,58,78,95,120]). In search games, an intruder is attacking one or multiple cells in a graph while an agent is searching this graph to prevent the attack.…”