Game-theoretic approach to search and patrolling problems
Creators: name: Bui, Thuy
Publisher: name: No Publisher Supplied
Year: 2024

description: Search and patrolling problems are of fundamental concern to public safety and security.Whereas patrolling problems address how to continually monitor space to protect itfrom adversarial attacks, search problems focus on the one-time task of finding targets.In the first part of the thesis, we study a patrolling game on a network in continuoustime and space. In this two-person zero-sum game, an Attacker chooses a time and a pointto attack the network for a fixed amount of time. A Patroller picks a patrol of the network.The Patroller wins if he visits the attacked point while the attack is occurring; otherwise, theAttacker wins. The payoff of the game is the probability that the attack is intercepted. ThePatroller is the maximizer and the Attacker is the minimizer. We introduce an ε-optimalstrategy for the Attacker for arbitrary tree networks, thereby settling a conjecture Alpern etal. (2022) on the solution of the game for trees. We provide a complete solution for a starnetwork with one long arc and the remaining arcs equal in length. We also solve the gameon a complete network when the attack duration is smaller than the length of the network.We conclude this part by studying the game on three parallel arc networks and improvingthe bounds on the value of the game.In the second part of the thesis, we consider a discrete search game between a Hiderand a Searcher. The Hider places m targets in n boxes, and the Searcher wants to find kof the targets with the least possible total search time. A search at box i takes ti time unit and finds the target with probability qi if the target is in that box. We study two versionsof this game. First, we consider the classical search game with a single target m = 1. Itis shown in the literature that the Searcher has an optimal mixed strategy that mixes up ton search sequences. In this work, we are interested in investigating the existence of theoptimal pure strategies for the Searcher—a single deterministic sequence that achieves theoptimal expected total search time regardless of the location of the target. We identify theconditions under which the Searcher has an optimal pure strategy and conditions underwhich such optimal pure strategy does not exist. Second, we study the game with multipletargets m &gt; 1. We give the solution to the game for any arbitrary number of locations nand number of targets m, k when the search times and the detection probabilities are allequal. The value of the game is computed explicitly for the case of two targets. In addition,we show that optimal pure strategies for the Searcher exist in some cases.In the third part of the thesis, we introduce a new search game for a moving target. Inthis zero-sum game, the Hider places a target in one of n boxes. The target then movesamong boxes according to a time-homogeneous Markov chain. Each box i has a searchtime ti and a detection probability qi. The Searcher picks a finite search sequence of theboxes of length m corresponding to the order in which the Searcher searches them. Thepayoff is the probability of finding the target in m searches, which the Searcher wishesto maximize and the Hider wishes to minimize. We prove the game has a value and eachPlayer has an optimal strategy. The bounds on the value of the game are given for somecases. We also present the solution of the game for some classes of Markov chains.