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 xed 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 payo 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 suciently 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

Patrolling security games: Definition and algorithms for solving largeinstances with single patroller and single intruder

Security games are gaining significant interest in artificial intelligence. They are characterized by two players (a defender and an attacker) and by a set of targets the defender tries to protect from the attacker\u2bcs intrusions by committing to a strategy. To reach their goals, players use resources such as patrollers and intruders. Security games are Stackelberg games where the appropriate solution concept is the leader\u2013follower equilibrium. Current algorithms for solving these games are applicable when the underlying game is in normal form (i.e., each player has a single decision node). In this paper, we define and study security games with an extensive-form infinite-horizon underlying game, where decision nodes are potentially infinite. We introduce a novel scenario where the attacker can undertake actions during the execution of the defender\u2bcs strategy. We call this new game class patrolling security games (PSGs), since its most prominent application is patrolling environments against intruders. We show that PSGs cannot be reduced to security games studied so far and we highlight their generality in tackling adversarial patrolling on arbitrary graphs. We then design algorithms to solve large instances with single patroller and single intruder

Automated Abstractions for Patrolling Security Games

Recently, there has been a significant interest in studying security games to provide tools for addressing resource allocation problems in security applications. Patrolling security games (PSGs) constitute a special class of security games wherein the resources are mobile. One of the most relevant open problems in security games is the design of scalable algorithms to tackle realistic scenarios. While the literature mainly focuses on heuristics and decomposition techniques (e.g., double oracle), in this paper we provide, to the best of our knowledge, the first study on the use of abstractions in security games (specifically for PSGs) to design scalable algorithms. We define some classes of abstractions and we provide parametric algorithms to automatically generate abstractions. We show that abstractions allow one to relax the constraint of patrolling strategies' Markovianity (customary in PSGs) and to solve large game instances. We additionally pose the problem to search for the optimal abstraction and we develop an anytime algorithm to find it