Continuous Patrolling Games

TL;DR

This paper analyzes a patrolling game on networks, providing solutions for specific network types and attack durations, with conjectures for general trees, modeling protection of infrastructure from adversaries.

Contribution

It offers the first comprehensive solutions for patrolling games on networks, including trees and star networks, with new analytical methods and conjectures for general cases.

Findings

Solution for any network with short attack duration

Solution for all tree networks with certain extremity conditions

Solution for star networks with one long arc

Abstract

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 sufficiently 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.