Patrolling Adversarial Environments Exploiting an Alarm System

264 Pages·2017·10.31 MB·English

by  Giuseppe De Nittis

POLITECNICO DI MILANO
DIPARTIMENTO DI ELETTRONICA, INFORMAZIONE E BIOINGEGNERIA
DOCTORAL PROGRAM IN INFORMATION TECHNOLOGY
PATROLLING ADVERSARIAL ENVIRONMENTS
EXPLOITING AN ALARM SYSTEM
DoctoralDissertationof:
GiuseppeDeNittis
Supervisor:
Prof. NicolaGatti
Co-supervisor:
Dr. NicolaBasilico
Tutor:
Prof. FrancescoAmigoni
ChairoftheDoctoralProgram:
Prof. AndreaBonarini
2017–CycleXXX

GiuseppeDeNittis
DipartimentodiElettronica,InformazioneeBioingegneria
PolitecnicodiMilano
e-mail: [email protected]
PrintedJanuary2018

Tothosewhostriveeveryday
toleavethisworldalittlebetterthantheyfoundit.

Abstract
P
hysical security is one of the most important challenges of our times.
Due to the terrible events happened in the last decades all around
the world, and especially nowadays in Europe, novel techniques and
methods are being developed to face new threats and dangers. But secu-
rity means also helping people and saving lives, e.g., detecting and rescuing
desperatemigrantsthataremovingacrosstheMediterraneanSea.
Algorithmic Game Theory allows us to scientifically investigate these
phenomena, modeling such interactions as mathematical problems and de-
signingsuitablealgorithmstodealwiththesethreats.
When patrolling large environments or infrastructures, a crucial issue is
to guarantee some level of protection to each area without being able to
constantly surveil them. A common countermeasure is the usage of cheap
butwide-rangedsensors,abletodetectmaliciouseventsthatmayoccur.
Inthisthesis,weproposethefirstSecurityGamemodelwiththepresence
of an alarm system able to trigger alarm signals, carrying the information
abouttargetsthatcanbeunderattack. Specifically,wefocusontheexploita-
tionofsuchinformationtoimprovetheeffectivenessofguards’strategies.
Thedissertationisstructuredinthreeparts,accordingtotheresearchlines
alongwhichthecontributionsaredeveloped.
First, we study the uncertainties that may affect the alarm system. We
start considering spatial uncertainty, i.e., the signal sent to the Defender
communicates that something suspicious is happening in an area, without
specifying the exact location. We show that, without false positives and
misseddetections,thebestpatrollingstrategyreducestostayinaplace,wait
I
for a signal, and respond to it at best. Experimental results show that our
approach could be adopted in real-world applications, e.g., the protection
of Expo 2015. Then, we introduce a positive missed detection rate, i.e., no
alarm signal is generated even though an attack is occurring. We show that
thepreviousapproachisnomoreoptimalandprovidetheDefenderwiththe
best patrolling strategy to move her resource, showing that a deterministic
approachisarbitrarilybetterthanarandomizedone.
The second direction we investigate is the dimension of the problem,
namely, the number of resources both the Defender and the Attacker can
control. We tackle the problem of finding the minimum number of defend-
ingresourcesassuringnon-nullprotectiontoeverytarget,andthenwestudy
howtheDefendershouldmovethem,providingalgorithmstodealwiththree
different levels of coordination among the resources. Then, we investigate
the opportunities the Attacker can take when she can perform multiple at-
tacks, simultaneously or sequentially. Unfortunately, computing the equi-
librium strategies requires the knowledge on the number of Attacker’s re-
sources. Since it is unlikely to have this information, we also provide two
online algorithms to compute the best strategy when the Defender makes a
guessaboutsuchnumberandplaysherbeststrategyaccordingly.
The last line we explore introduces the notion of uncertainty in the type
of the Attacker. We tackle the problem of facing an unknown adversary,
whose profile is just known to be in a list of possible profiles she can as-
sume. Here, a different approach is required: we want to learn the profile
of the Attacker to exploit such information in the future to prevent her from
performing other attacks. To do this, we design two algorithms and provide
aregretanalysisandshowthatourapproachoutperformstheonlinelearning
algorithmsavailableinthestateoftheart.
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Contents
1 Introduction 1
1.1 Problem: SecuringLargeBuildingsandEnvironments . . . . 2
1.2 Solution: AdversarialPatrollingwithanAlarmSystem . . . . 3
1.3 StructureoftheThesis . . . . . . . . . . . . . . . . . . . . . 7
I StartingPoint 9
2 FoundationsofAlgorithmicGameTheoryandOnlineLearning 11
2.1 AlgorithmsandComputationalComplexity . . . . . . . . . . 11
2.1.1 ComputationalComplexityinaNutshell . . . . . . . . 13
2.1.2 ComplexityofProblems . . . . . . . . . . . . . . . . . 14
2.1.3 ApproximabilityofaProblem . . . . . . . . . . . . . . 17
2.1.4 SolvingaProblem . . . . . . . . . . . . . . . . . . . . 19
2.2 GameTheory . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.1 ALittleBitofHistory . . . . . . . . . . . . . . . . . . 22
2.2.2 PlayingaGame . . . . . . . . . . . . . . . . . . . . . 23
2.2.3 NormalFormGames . . . . . . . . . . . . . . . . . . 24
2.2.4 ExtensiveFormGames . . . . . . . . . . . . . . . . . 26
2.2.5 BayesianGames . . . . . . . . . . . . . . . . . . . . . 29
2.3 SolutionConcepts . . . . . . . . . . . . . . . . . . . . . . . 31
2.3.1 NashEquilibrium . . . . . . . . . . . . . . . . . . . . 31
2.3.2 Maxmin/minmaxStrategies . . . . . . . . . . . . . . . 32
2.3.3 Leader-followerEquilibrium . . . . . . . . . . . . . . 34
III
Contents
2.3.4 Zero-sumGames . . . . . . . . . . . . . . . . . . . . . 35
2.4 OnlineLearning . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4.1 PredictionwithExpertAdvice . . . . . . . . . . . . . 37
2.4.2 Multi-ArmedBandit(MAB) . . . . . . . . . . . . . . 39
3 StateoftheArt 43
3.1 SecurityGames . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.2 PatrollingSecurityGames . . . . . . . . . . . . . . . . . . . 47
3.3 LearningSecurityGames . . . . . . . . . . . . . . . . . . . . 49
3.3.1 UncertaintyinSecurityGames . . . . . . . . . . . . . 51
3.3.2 FeaturesofSecurityGamemodels . . . . . . . . . . . 53
II AlarmSystemanditsUncertainties 55
4 AdversarialPatrollingwithSpatiallyUncertainAlarmSignals 57
4.1 OriginalContributions . . . . . . . . . . . . . . . . . . . . . 59
4.1.1 ChapterStructure . . . . . . . . . . . . . . . . . . . . 60
4.2 GameModel . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 BasicPatrollingSecurityGame . . . . . . . . . . . . . 61
4.2.2 IntroducingAlarmSignals. . . . . . . . . . . . . . . . 62
4.2.3 TheGameTreeanditsDecomposition . . . . . . . . . 64
4.2.4 TheComputationalQuestions . . . . . . . . . . . . . . 67
4.3 SignalResponseGame . . . . . . . . . . . . . . . . . . . . . 68
4.3.1 ComplexityResults . . . . . . . . . . . . . . . . . . . 68
4.3.2 Dynamic-programmingApproach . . . . . . . . . . . . 73
4.3.3 Branch-and-boundApproach . . . . . . . . . . . . . . 79
4.3.4 SolvingSRG-v . . . . . . . . . . . . . . . . . . . . . . 84
4.4 PatrollingGame . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.1 StandStill . . . . . . . . . . . . . . . . . . . . . . . . 85
4.4.2 ComputingtheBestPlacement . . . . . . . . . . . . . 88
4.4.3 SummaryofResults . . . . . . . . . . . . . . . . . . . 89
4.5 ExperimentalEvaluation . . . . . . . . . . . . . . . . . . . . 89
4.5.1 Worst-caseInstancesAnalysis . . . . . . . . . . . . . . 89
4.5.2 RealCaseStudy . . . . . . . . . . . . . . . . . . . . . 99
5 IntroducingMissedDetections 105
5.1 OriginalContributions . . . . . . . . . . . . . . . . . . . . . 105
5.1.1 ChapterStructure . . . . . . . . . . . . . . . . . . . . 106
5.2 ProblemFormulation . . . . . . . . . . . . . . . . . . . . . . 107
IV