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Cooperative Control Strategies and Deception in Adversarial Systems

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Title:
Cooperative Control Strategies and Deception in Adversarial Systems
Creator:
Fuchs, Zachariah E
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (171 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Electrical and Computer Engineering
Committee Chair:
Khargonekar, Pramod
Committee Members:
Arroyo, Amauri A
Rao, Anil
Barooah, Prabir
Schwartz, Eric M
Graduation Date:
8/11/2012

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Adjoints ( jstor )
Cost functions ( jstor )
Differential games ( jstor )
Game theory ( jstor )
Linear programming ( jstor )
Mathematical variables ( jstor )
Optimal control ( jstor )
Sensors ( jstor )
Sine function ( jstor )
Trajectories ( jstor )
Electrical and Computer Engineering -- Dissertations, Academic -- UF
control -- cooperative -- deception -- game
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Electrical and Computer Engineering thesis, Ph.D.

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Abstract:
The protection of vulnerable, high-value assets has been a challenge throughout history. These high-value targets may be fixed, mobile, or in the cloud. In any case, it is necessary to deploy and effectively utilize defensive assets in an attempt to neutralize an attack if it occurs, or make the prospect of further engagement so unappealing that the attackers stand down and retreat. My research focused on two aspects of applying game theoretic methods to the development of defensive strategies. The first aspect of my research utilized  tools from differential game theory to develop cooperative, defensive control strategies against a superior, mobile attacker.  It was shown that through cooperation, the defending agents can combine their resources and make engagement more costly to the attacker than if they acted independently. A hierarchical attack-retreat game was then developed that incorporates the concept of player intent. This game demonstrated that it is sometimes optimal for the defending player to cooperate with the attacker in order to encourage retreat. Although cooperation can encourage retreat, cooperating at the wrong time can actually assist attacker in engagement. For this reason, a modified differential game with a value function constraint was developed to prevent the attacker from moving the state of the system to a region where engagement is optimal. The second aspect of my research focused on developing an analytic framework to quantitatively capture and describe concepts of deception. A generic two-player, zero-sum game was developed that incorporated deceptive tactics to corrupt a stochastic sensor network. The stochastic sensor network provides one player an informational advantage over its opponent. Using this framework, an illustrative example was designed and examined to describe a well-known, qualitative principle in the deception field known as the ``Jones' Lemma''. A new sequential variant of the Colonel Blotto game was also developed that included a deterministic sensor network. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Khargonekar, Pramod.
Statement of Responsibility:
by Zachariah E Fuchs.

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Copyright Fuchs, Zachariah E. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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COOPERATIVECONTROLSTRATEGIESANDDECEPTIONINADVERSARIAL SYSTEMS By ZACHARIAHE.FUCHS ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2012

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c 2012ZachariahE.Fuchs 2

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ACKNOWLEDGMENTS IthankmyadvisorDr.PramodKhargonekar.Hisguidance,mentoring,andsupport overthepastfouryearswereextremelyvaluableduringmydoctoratedegree,andIam foreverindebtedtohim. Iacknowledgemycommitteemembers,Dr.AntonioArroyo,Dr.PrabirBarooah, Dr.AnilRao,andDr.EricSchwartz.Theirassitanceandengagingcourseshelped metremendouslythroughoutmygraduatecareer.IwouldespeciallyliketothankDr. AntonioArroyoandDr.EricSchwartzforrecruitingmetotheUniversityofFlorida.Their warmwelcome,opendoor,andhonestprofessionalandpersonalguidancehasmade UFaplacetocallhome. IwouldalsoliketothanktheNationalScienceFoundationforitsnancialsupport throughtheGraduateResearchFellowshipProgram.Additionally,Iwouldliketothank theDepartmentofDefenseforitsnancialsupportthroughtheScience,Mathematics, AndResearchforTransformationSMARTScholarshipprogram. 3

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TABLEOFCONTENTS page ACKNOWLEDGMENTS..................................3 LISTOFTABLES......................................7 LISTOFFIGURES.....................................8 ABSTRACT.........................................10 CHAPTER 1INTRODUCTION...................................12 1.1CooperativeDefensiveControlStrategies..................14 1.1.1ProblemDescription..........................14 1.1.2LiteratureSurvey............................14 1.2DeceptioninAdversarialSystems......................17 1.2.1ProblemDescription..........................17 1.2.2LiteratureSurvey............................18 1.3Contributions..................................20 1.3.1DevelopmentofCohesiveCooperativeDefensiveStrategies....20 1.3.2DevelopmentofaHierarchicalAttack-RetreatGame........22 1.3.3DevelopmentofaGeneralDeceptiveGameFramework......22 1.3.4AnalyticandQuantitativeDescriptionofJones'Lemma......24 1.3.5DevelopmentofaSensorBasedColonelBlottoGame.......24 2COOPERATIVEDEFENSEWITHINASINGLE-PURSUER,TWO-EVADER PURSUITEVASIONDIFFERENTIALGAME...................26 2.1SystemandGameFormulation........................27 2.1.1AgentKinematics............................27 2.1.2InstantaneousCostFunction.....................30 2.1.3GameFormulation...........................32 2.2SolutiontotheGame..............................33 2.2.1OptimalityConditionsfortheGameofAttack............34 2.2.2NumericalAnalysis...........................38 2.2.3IllustrativeCases............................39 2.2.4SingularSurfaces............................40 3ENCOURAGINGATTACKERRETREATTHROUGHDEFENDER COOPERATION...................................45 3.1Introduction...................................45 3.2IntentSelectionStrategies...........................47 3.3SystemandDifferentialGameFormulation.................49 3.3.1AgentKinematics............................50 4

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3.3.2InstantaneousCostFunction.....................52 3.3.3DifferentialGameFormulation.....................52 3.4OptimalityConditionsoftheDifferentialGameofEngagementwith DefenderMaximization............................53 3.4.1HamiltonianandAdjointEquations..................53 3.4.2BoundaryConditions..........................54 3.5OptimalityConditionsoftheDifferentialGameofRetreatwithDefender Minimization..................................56 3.5.1HamiltonianandAdjointEquations..................57 3.5.2BoundaryConditions..........................57 3.6IllustrativeExamples..............................59 3.6.1NumericalAnalysisofGameofEngagement............59 3.6.2NumericalandAnalyticSolutiontotheGameofRetreat......60 3.6.3SingularSurfaces............................61 3.6.4OptimalIntentSelection........................62 3.6.5IllustrativeCases............................62 4GENERALIZEDATTACKRETREATGAMEWITHESCORTREGIONS....67 4.1GeneralGameDescription..........................67 4.2GeneralSolutionTechnique..........................69 4.2.1DifferentialSubgameofEngagement.................69 4.2.2OptimalControl.............................71 4.2.3DifferentialSubgameofRetreat....................71 4.2.4OverallEngageorRetreatGameSolution..............75 4.3AnalyticExample................................89 4.3.1SystemModel..............................90 4.3.2TerminationConditions.........................90 4.3.3PlayerUtilities..............................91 4.3.4GameFormulation...........................92 4.4GameSolution.................................92 4.4.1DifferentialGameofEngagement...................92 4.4.2DifferentialGameofRetreat......................94 4.4.3SolutiontotheEngageorRetreatGame...............100 4.4.4EquilibriumTrajectoriesandRegions.................101 4.4.5DiscussionoftheEscortRegionandTrajectories..........101 5DECEPTIONANDJONES'LEMMAWITHINAZERO-SUMGAME.......106 5.1GameFormulation...............................107 5.1.1PlayerActionsandDeceptionTactics.................107 5.1.2SensorNetwork.............................107 5.1.3PlayerStrategies............................108 5.1.4UtilityFunction.............................108 5.2GeneralGameSolution............................109 5.3Two-ActionGamewithIdenticalInformationChannels...........115 5

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5.3.1MotivatingScenario..........................116 5.3.2GameModelandDescription.....................117 5.3.3GameParameterization........................118 5.3.4SpecialCaseSolutions.........................119 5.3.4.1TheCaseofNoInformants.................119 5.3.4.2TheCaseofZero-CostDeception.............121 5.3.4.3TheCaseofPerfectInformants p id =1 ..........124 5.3.5PiecewiseCharacteristicsoftheGeneralEquilibriumStrategies..128 5.4Jones'LemmaandtheNumberofInformants................138 6ASEQUENTIALCOLONELBLOTTOGAMEWITHASENSORNETWORK..143 6.1GameFormulation...............................144 6.1.1PlayerActions..............................144 6.1.2SensorNetwork.............................145 6.1.3PlayerStrategies............................145 6.1.4UtilityFunctions.............................146 6.1.5GameDenition.............................147 6.2GameSolution.................................148 6.2.1SubgameStructure...........................148 6.2.2ConditionsforPureStrategyEquilibrium...............148 6.2.3First-PriceAllPayAuction.......................150 6.2.3.1Auctiongamedescription..................150 6.2.3.2RelationshiptotheColonelBlottogame..........151 6.2.3.3Auctiongamesolutionnotation...............152 6.2.3.4Auctiongamesolution....................154 6.2.4NumericalSolutionTechnique.....................157 6.3IllustrativeExample..............................158 6.3.1Scenario1: n =5 X A =10 X B =15 =1 .............158 6.3.2Scenario2: n =5 X A =10 X B =15 =9 .............159 6.3.3Scenario3: n =5 X A =10 X B =15 =3 .............159 6.3.4Scenario4: n =20 X A =10 X B =20 =2 ............159 7CONCLUSIONS...................................163 REFERENCES.......................................165 BIOGRAPHICALSKETCH................................171 6

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LISTOFTABLES Table page 6-1Solutionparameterstotheupperboundedrst-priceallpayauctions......160 6-2Solutionparameterstothelowerboundedrst-priceallpayauctions......161 6-3Scenarioresults...................................162 7

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LISTOFFIGURES Figure page 2-1Globalcoordinates..................................42 2-2Relativecoordinates.................................42 2-3Optimaltrajectoriesfor d 2 f =1.5 f =2.2 ,and v p =1.5 .............43 2-4Optimaltrajectoriesfor d 2 f =7 f =.8 ,and v p =2.5 ...............43 2-5EnlargedviewofnearcapturetrajectoriesofFigure2-3.............44 2-6Optimaltrajectoriesfor d 2 f =7 f =2.8 ,and v p =1.1 ..............44 3-1IntentSelectionTree.................................64 3-2Globalcoordinates..................................64 3-3Relativecoordinates.................................65 3-4AttacktrajectoriesScenario1............................65 3-5AttacktrajectoriesScenario2............................66 3-6AttacktrajectoriesScenario3............................66 4-1GameRegions....................................103 4-2Equilibriumtrajectoriesfor R 1 and R 2 ........................103 4-3Equilibriumtrajectoriesfor R 4 ............................104 4-4Equilibriumtrajectoriesfor R 3 ............................104 4-5Equilibriumtrajectoriesfor R A ............................105 5-1OverallGameStructure...............................108 5-2Overallgamestructure................................140 5-3Informationnetworkfortwo-actiongame......................140 5-4Equilibriumutilityvs p id withnodeception.....................140 5-5Equilibriumutilityvaluevs p id when c d =2 .....................141 5-6ProbabilityofPlayerAcorruptinginformantswhen c d =2 ............141 5-7Equilibriumutilityvaluevs p id when c d =.2 ....................141 5-8ProbabilityofPlayerAcorruptinginformantswhen c d =.2 ............142 8

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5-9Equilibriumutilityvaluevs p id when c d =.01 ....................142 5-10ProbabilityofPlayerAcorruptinginformantswhen c d =.01 ...........142 9

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy COOPERATIVECONTROLSTRATEGIESANDDECEPTIONINADVERSARIAL SYSTEMS By ZachariahE.Fuchs August2012 Chair:PramodP.Khargonekar Major:ElectricalandComputerEngineering Theprotectionofvulnerable,high-valueassetshasbeenachallengethroughout history.Thesehigh-valuetargetsmaybexed,mobile,orinthecloud.Inanycase,itis necessarytodeployandeffectivelyutilizedefensiveassetsinanattempttoneutralize anattackifitoccurs,ormaketheprospectoffurtherengagementsounappealingthat theattackersstanddownandretreat. Myresearchfocusedontwoaspectsofapplyinggametheoreticmethodstothe developmentofdefensivestrategies.Therstaspectofmyresearchutilizedtoolsfrom differentialgametheorytodevelopcooperative,defensivecontrolstrategiesagainsta superior,mobileattacker.Itwasshownthatthroughcooperation,thedefendingagents cancombinetheirresourcesandmakeengagementmorecostlytotheattackerthanif theyactedindependently.Ahierarchicalattack-retreatgamewasthendevelopedthat incorporatestheconceptofplayerintent.Thisgamedemonstratedthatitissometimes optimalforthedefendingplayertocooperatewiththeattackerinordertoencourage retreat.Althoughcooperationcanencourageretreat,cooperatingatthewrongtimecan actuallyassistanattackerinengagement.Forthisreason,amodieddifferentialgame withavaluefunctionconstraintwasdevelopedtopreventtheattackerfrommovingthe stateofthesystemtoaregionwhereengagementisoptimal. Thesecondaspectofmyresearchfocusedondevelopingananalyticframework toquantitativelycaptureanddescribeconceptsofdeception.Agenerictwo-player, 10

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zero-sumgamewasdevelopedthatincorporateddeceptivetacticstocorrupta stochasticsensornetwork.Thestochasticsensornetworkprovidesoneplayeran informationaladvantageoveritsopponent.Usingthisframework,anillustrativeexample wasdesignedandexaminedtodescribeawell-known,qualitativeprincipleinthe deceptioneldknownastheJones'Lemma.AnewsequentialvariantoftheColonel Blottogamewasalsodevelopedthatincludedadeterministicsensornetwork. 11

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CHAPTER1 INTRODUCTION Theprotectionofvulnerable,high-valueassetshasbeenachallengethroughout history.Thesehigh-valuetargetsmaybexedstockpiles,factories,powerplants, orpopulationcenters,mobiletransports,supplyconvoys,orVIPs,orinthecloud servers,databases,orterminals.Inanycase,itisnecessarytodeployandeffectively utilizedefensiveassetsinanattempttoneutralizeanattackifitoccurs,orbetteryet, maketheprospectoffurtherengagementsounappealingthattheattackersstanddown andretreat. Whilepreparingagainstpossibleattacks,thedefendingsiderarelypossesses enoughresourcestosafeguardagainsteverypossiblescenario.Instead,astrategy mustbedevelopedthatbalancestheriskthattheattackerswillemployaparticular tacticagainstthecosttodefendagainstit.Inthissituation,knowledgeofanattacking force'sstrengthanddistributioncanprovecriticalwhenallocatingdefensiveassets.If defendingforcesknowthatparticularareaswillcomeunderheavyattackwhileother areaswillremainrelativelyuncontested,theycaneffectivelyshiftresourcesfromthe safeareastotheareasofhigherthreatinordertoimprovetheirchancesofsuccessfor alloftheareasonthewhole.Giventhatthedefendingforceshavetheabilitytoestimate therelativestrengthandlocationofattackingforces,theattackersmustimplement anattackstrategythatprovidesthedefenderswithaslittleforeknowledgeaspossible whilestillremainingeffectiveinanattack.Itisalsopossiblethattheattackstrategy couldincorporatedeceptivetacticsthatintentionallyprovidefalse,misleading,orcorrupt informationtothedefenderinordertoneutralizethedefenders'informationaladvantage. Evenifthedefenderknowsofthepossibilitythatdeceptivetacticsarebeingdeployed, itcannotsimplythrowawaytheinformationonthechancethatthesourcehasbeen compromised.Instead,thedefendermustbalancetheriskthattheinformationis corruptedagainstthepotentialadvantagethattheinformationprovides. 12

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Aftertheresourceshavebeendistributedbybothsides,itisstillnecessaryfor eachsidetodevelopstrategiesthatutilizetheavailableresourcestotheirmaximum potential.Fromtheattackersperspective,ithopesthatthedefenderhasmadeanerror initspreparationsandtheattackcanstartfromaninitialconditionofattackersuperiority. Ontheotherhand,thedefendermayhaveeffectivelydeployeditsresourcessothat itiscapableofinictingseverecoststotheattacker.Inthissituation,itmaybemore productivefortheattackertoabandonitsplansforengagementanddecidetoretreat inordertocutitslosses.Additionally,itmaybebenecialforthedefendertocooperate withtheattackerinretreatbyloweringtheattacker'scostinordertomakeretreatamore attractiveoption. Gametheoryprovidesapowerfulframeworktoanalyzetheconictinginterests ofthedefenders'desiretopreventanattackandtheattackers'desiretosuccessfully engagethehigh-valuetargetwithminimumcost.Gametheoreticmethodscanbe appliedtotheresourceallocationstageoftheprobleminordertodeterminethebest high-levelstrategyfortheattackingforcesaswellasthedefendingforces.Additionally, gametheoreticmethodscanbeusedtodevelopcontrolstrategiesforeachsidethat optimallyutilizetheresourcesaftertheyaredeployed.Becausetheplayersinthese gameshaveconictinggoals,theconceptofoptimalityinvolvestheideaofequilibrium, particularlyNashEquilibrium.ThestrategiesofthetwosidesareinNashEquilibriumif neitherplayercanimprovethevaluefunctionthroughaunilateraldeviation.Therefore, thestrategiesdevelopedusinggametheoryrepresentthebestresponsetotheworst casescenariofromtheopposingside. Mystudyfocusedontwoaspectsofapplyinggametheoreticmethodstothe developmentofdefensivestrategies.Inthersttopic,differentialgametheorywasused todevelopcooperativestrategiesfordefenseagainstasuperiormobileattacker.An introductiontothistopicandrelatedworksarediscussedinSection1.1.Furtherdetails regardingcompletedworkaregiveninChapter2throughChapter4.Theremaining 13

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focusisgiventothedevelopmentofhigh-levelstrategiesthatincorporatedifferenttypes ofdeceptionwithinadversarialsystems.Anintroductiontothistopicandrelatedworks arediscussedinSection1.2.Furtherdetailsregardingcompletedworkonthistopicare giveninChapters5andChapter6. 1.1CooperativeDefensiveControlStrategies 1.1.1ProblemDescription Theuseofunmannedmobilesystemsisrapidlyincreasingduetoavarietyof reasons,suchastheirrelativelowcostandabilitytooperateinhazardousenvironments withminimalrisktohumanlife.Averyimportantapplicationofunmannedsystemsisin themodernbattleeldtoperformtasksrangingfromsurveillancetodirectengagement. Duetotheirlowcost,multipleinexpensiveunmannedsystems,oragents,canbe deployedsimultaneouslytoaccomplishataskormission.Inthesescenarios,thegroup ofagentsareoftenindirectcompetitionwithanopposingforce.Itisthereforeimportant tondalgorithmsorstrategiesthatcansystematicallymaximizethevalueofferedby suchgroupsofagentsinthepresenceofanuncooperativeadversaryorenvironment. Asmentionedbefore,gametheoryisanaturalsettingtostudysuchadversarial situations.Inparticular,differentialgametheoryprovidesatoolsettodevelopoptimal controlstrategiesinthepresenceofanopponentwithconictinginterests.Withina differentialgame,eachplayerdrivesacommondynamicsystemthroughtheuseof theirown,independentsetofcontrolofvariables.Generally,thedynamicsystemcan representanything,butinthissetting,thedynamicsystemwouldrepresentthemotionof themobileagentsthroughtheirenvironment.Eachplayerwithinthegamepossessesa functional,whichtheystrivetomaximize.Thisfunctionalcanbebasedonthetrajectory ofthesystem,controlhistory,elapsedtime,oracombinationofallthree. 1.1.2LiteratureSurvey FormallyintroducedbyRufusIsaacs[1],pursuit-evasiongamesandtheirvariants havebeenusedtosolveawiderangeofproblems.Oneearlyprototypicalexample 14

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wouldbetheHomicidalChauffeurgame.OriginallyposedbyIsaacs,thecomplete solutionwasdevelopedseveralyearslaterbyMerz[2].Thetwoplayersinthisgame representapursuerwhostrivestocapturetheevaderinminimumtime,whiletheevader strivestomaximizethetimeuntilthegameends.Thepursuermoveswithgreaterspeed butpossessesanonzeroturningradius,whichisasimplieddynamicmodelofacar. Theevaderontheotherhandmoveswithsimplemotionandaturningradiusofzero butpossessesaslowerspeed.Thissimple,two-dimensionalgameproducesacomplex solutioncontainingfourtypesofsingularsurfaces;dispersal,equivocal,universal,and focal.Alongthesesurfaces,thestandardoptimalityconditionsbreakdownandsingular characteristicsmustbeusedtoderivetheequilibriumsolution.Thisgamedemonstrates boththepromiseofdifferentialgametheorybydevelopingrobustcontrolstrategies againstanadversaryattemptingtomaximizinghisowninterest,butitalsohighlightsthe possiblecomplexitiesthateventhemoststraightforwardsystemdynamicsgenerate. OtheravorfullynamedprototypicalexamplesincludetheLadyintheLake[3]andMan intheLionCage[4]games.Bothofwhichcontaincomplicatedsingularitiesthatare generatedbyotherwisesimplesystems. Sinceitsoriginalintroductionandsolution,severalvariantsoftheHomicidal Chauffeurgamehavebeenposed.In[5],theauthorsmodifythestandardgameby allowingmultiplepursuers.Thepaperposesadaisy-chainformationthatenables quickercaptureofthesingleevaderforawiderrangeparameters.Additionally,the solutiondoesnotcontainthesingularitiespresentwithintheoriginalproblem.There wasasimilarvariationin[6],whichaddedanadditionalpursuerandposedalgorithms thatcouldbeusedtodevelopasolution.Astochasticvariantwasposedin[7],which examinestheeffectsofnoisewhenintroducedtothedynamicsofthestandardgame. Interestingly,itisshownthatthenoiseeliminatessomeofthediscontinuitiesthatexistin theoriginalgame.Theauthorsgoontosuggestthatthisisthereasonsimplerpursuit 15

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strategiesareoftenimplementedinbothmanmadeandnaturalscenarioswherenoise isalmostalwayspresent. Researchershavemovedbeyondtheseillustrativeproblemstoattempttosolve problemswithmorerealisticdynamicsorreal-worldapplications.Therehavebeen severalpapersthathaveusednumericaltechniquesinordertosolvecombatgames withrealisticdynamics.In[8],aghterplanegamewithrealisticdynamicswas addressedforparticularinitialconditionsthatdidnotposethepotentialforsingular surfaces.Asimilartechniquewasusedtosolveathree-dimensionalorbitalmissile pursuit-evasiongame[9,10].Ineachofthesepapers,asemi-directoptimizationroutine wasusedinwhichoneplayer'soptimalcontrolstrategywassolvedforintermsof thestateandadjointvariables.Thesecontrolstrategieswerethensubstitutedinto thedynamicsandthecorrespondingadjointequationswereappendedontothestate dynamics.Theremainingcontrolvariablesfortheopposingplayerwerethensolvedfor directlyusingnumericalcollocationtechniques. Recently,theauthorsin[11,12]usedtheanalysistechniquesdevelopedbyIsaacs toexamineacontinuoustime,visibilitybased,single-pursuer,single-evadergamein anenvironmentcontainingpolygonalobstacles.Inthisgame,eachagentmoveswith simplemotion.Theevaderstrivestobreakthelineofsightbetweenthepursuerand itselfbyrunningaroundthecornerofapolygonalobstacle.Thepursuerstrivesto maximizethetimeperiodforwhichthelineofsightismaintained.Intheirrstpaper [11],theauthorsfocusonthegamenearasinglecornertherebyavoidingthepossibility ofsingularitieswithinthesolution.Butinthesecondpaper[12],theyhavebegunto investigatetheeffectsthatmultipleobstacleshaveontheglobalsolution.Inparticular, theyarefocusingonthenumericalcalculationofsingulardispersalsurfacesthatdivide thestatespaceintolocalsolutionsaroundthecornersoftheobstacles.Additionalwork onthistopiccanbefoundin[1315]. 16

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Singularsurfacesposeasignicantchallengewhensolvingdifferentialgames. Oftentheexistenceofthesesurfacesisnotknownbeforehand,andmanytypesof surfacesarenotreadilyapparentevenaftersolvingfortheoptimalityconditions.To addtothecomplexity,thereareseveraldifferenttypesofsingularsurfaces[16],and eachtypeischaracterizedbydifferentformsofdiscontinuitiesofthevaluefunctionor itsgradient.Themostcommontypeofsingularsurfaceisreferredtoasadispersal surface,whichwasrstdescribedbyIsaacs[1].Ifagameisinitiatedalongthissurface, oneplayeriscapableofforcingthestateoffofthesurfaceintwodifferentdirections. Eitherdirectionwillyieldanequivalentvaluewhenthegameterminates.Another commonsingularsurfaceisknownasasingularfocalsurface[17,18],andcanbe thoughtofasthedualofthedispersalsurface.Inaneighborhoodaroundthissurface, oneplayeriscapableofforcingthestatetothissurfaceandholdingittheir.Eventually theoptimaltrajectoriesmaypassontoadispersalsurfaceandbreakoff,orthestate mayremainonthefocalsurfaceuntilthegameterminates.Inmostoftheliterature, singularsurfacesareaddressedindividuallywhenevertheyariseinaparticularproblem. Forasurveyofpossiblesingularsurfacesandtheircharacteristicssee[19]. 1.2DeceptioninAdversarialSystems 1.2.1ProblemDescription Deceptionplaysanimportantroleinavarietyofadversarialsituationsranging fromcompetitioninnature[2023]towarfare[24,25].In TheArtofWar ,thelegendary militarystrategistSunTzuoftenpreachedtheimportanceofdeceptioninmilitary strategy.Theextremelyhighvalueheplacedondeceptioncanbesummedupina singlequote, Allwarfareisbasedondeception [26].Perhapsthemostwell-known useofdeceptioninmodernmilitaryhistorywouldbetheAlliesuseofdeceptionand secrecyleadinguptotheD-Dayinvasion[27].DuringthebuilduptoD-Day,theAllies implementedanelaborateruse,code-namedFortitudeSouth,inwhichtheycreated theillusionofalargeinvasionforcebeingmassedinKent,Englandusingdummy 17

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landingcraft,inatabletanks,andanarrayofotherdeceptivetactics.Thisresultedinthe GermansshiftingdefensiveresourcestoPas-de-Calais,whichleftNormandyvulnerable andenabledtheAlliestosuccessfullyestablishabeachheadandsecondfronttothe war.Morerecently,theroledeceptionplayedduringOperationDesertStormhasbeen examined[28]. Althoughitiscommonknowledgethatdeceptionisanintegralcomponentofmilitary planning,systematicanalysisofdeceptionremainsachallengingeldthatcontains manyunansweredquestions.ThissentimentwasrecentlyemphasizedbyMcEneaney [29]whereitisstatedthat, Deceptionisacriticalcomponentofreal-worldgamesincomplexand imperfectlyobservedenvironments.However,eventhebasicmathematical denitionsofissuesindeceptionarenotcomplete.Thisisanimportant practicalproblem,whichisnaturaltohumans,butpresentsdeepdifculties. Itisagoalofmyresearchtosystematicallyaddressparticularaspectsofdeception andtheroleitplaysinmakingstrategiclevelchoiceswithinadversarialsystems.In myresearch,deceptionisanytacticoractionthatisdesignedtomisleadanopponent bymanipulating,distorting,orfalsifyinginformationinordertoreduceanopponents capabilityofperformingactionsthatareundesirable.Thesedeceptivetacticscantake manyformssuchasactivelycorruptingsensormeasurementsorsimplyadapting strategiesthatprovideaminimumamountofinformation.Althoughmuchofthe discussionofdeceptioninthisdissertationisinthecontextofdefensivestrategies, theresultingframeworksandconceptscreatedcanbeappliedtomanyotherscenarios withslightmodicationsorsimplyreinterpretations. 1.2.2LiteratureSurvey Gametheoreticmethodshavebeenleveragedtoincorporatesecrecyanddeception intodefensivestrategiesinmanystudiesofdeception.In[30],Brownetal.present atwo-sidedoptimizationmodelforplanningtheplacementofdefensivemissile 18

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interceptorstocounteranattackthreat.Thepaperexaminesthebenecialrolesecrecy anddeceptioncanplayforeitherside.Intheirmodel,adefenderpre-positionsballistic missiledefenseplatformstominimizetheworst-casedamageanattackercanachieve. Itisassumedthattheattackerwillhaveknowledgeofthelocationofdefensiveplatforms aswellasthetargetvalues.Usingthismodel,theydemonstratetwohypotheticalNorth Koreanattackscenariosandexaminethevalueofsecrecyanddeceptiontoeitherside. Inasimilarscenarioinvolvingpotentialterroristattacks,ZhuangandBier[31] explorewhethertherstmoverinatwo-stepgameshoulddisclosetheallocationof defensiveresourcesorattempttoprovidefalseinformationinthehopesofmisleading theattackerinthesecondstageofthegame.Theyshowthatsecrecyanddeception cansometimesbestrictlypreferredtotruthfuldisclosureinagamewithdefenderprivate information.In[32],ageneralasymmetriczero-sum,two-playergameisanalyzed,and deceptionisusedbyoneplayertotrickthesecondplayerintoselectinganon-optimal action. Thecompetitiveallocationofresourcesacrossmultipleregionshasbeenstudied sincethe1920's.ItisknownastheColonelBlottogame,andalongwiththePrisoner's Dilemma,itservedasaverystrongsourceofmotivationintheearlydevelopmentof gametheory.OriginallyposedbyBorel[33],anumberofresearchershavemadevery importantcontributionstomanyvariantsofthisgame[3440]andmanyothers.Beyond theobviousdefenseapplications,thisgameanditsvariantshavebeenusedtostudy advertising,politicalcampaignstrategies,researchanddevelopmentfunding,and lobbyingstrategies. Inarecentwork[41],motivatedbycriticalinfrastructureprotectionscenarios, PowellposedasequentialdefensiveColonelBlottogameinwhichthedefensiveplayer allocatesitsassetstoprotectmultipletargets.Theattackingplayerthenreceivesperfect informationregardingtheexactdistributionofthoseresources.Itisshownthatapure strategyequilibriumalwaysexistsinthisgame.BhattacharyaandBasarhaverecently 19

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investigatedadversarialgamesinwhichoneplayerattemptstojamorbreakinformation channelsbetweenmobileagents[42,43].Krichmannetal.posedatheaterlevel militaryallocationgame[44]inwhichtheyanalyzeandmodelthetemporalaspectofa mutli-stageallocationproblemusingdynamicprogrammingtechniques.In[45],Cruzet al.constructamodelrepresentingawarofattritionbetweenattackinganddefending forcesanddemonstrateazero-sumNashequilibriumforaparticularexample. 1.3Contributions 1.3.1DevelopmentofCohesiveCooperativeDefensiveStrategies Thisareaofmystudyaimedtoutilizethetoolsprovidedbydifferentialgametheory todevelopcooperative,defensivecontrolstrategiesagainstasuperiormobileattacker. Throughcooperation,thedefendingagentscancombinetheirdefensiveresourcesand makeengagementmorecostlytothatattackerthaniftheyactedindependently.By maximizingtheattacker'scost,thedefendershopetoencouragetheattackertoretreat becausethepotentialforinjuryorhighcostoutweighsthebenetoftheattackmission. Therehasbeenworkdoneondifferentialgameswithmultiplepursuerscompeting againstasingleevader,butgamescontainingmultiplecooperativeevadersremained arelativelyunexploredtopic.Amulti-evaderpursuitevasiongamewasposedin[46], butthecostfunctionalwasbasedsolelyonelapsedtime.Becausetherewasnodirect costgeneratedbytheevaders,theresultingevaderbehaviorsexhibitascattered,eeing patterninsteadofacohesive,defensivestrategy.Thesituationinwhichtheevadercan potentiallycapturethepursuerispresentedin[47]andalsoin[48].Inthesepapers,the agentsareallowedtoswitchrolesbetweenpursuerorevaderbasedontheparticular initialconditions.Thisrepresentsmoreofadogghtoraduelingscenario,inwhicheach playeractuallyrepresentsanaggressortryingtogettheupperhandasopposedtoa defensivestrategy. Fromtheseexamples,itcanbeseenthatthesetupofthegameandthedesign oftheplayercostfunctionalsarecriticalinthedevelopmentofthedesiredcooperative 20

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behaviors.Asarststeptowardsthedevelopmentofcooperativedefensivestrategies, asingle-pursuer,two-evadergamewithanovelintegralcostfunctionalisintroduced inChapter2.Thecostfunctionalisintendedtorepresenttheriskofdamageorinjury tothepursuerortheadditionalenergyorcomputationalexpenseneededtomonitor multipleevaders.Theproposedcostfunctionalisacombinationofaconstanttime penaltyandevadergeneratedcost.Theevadergeneratedcostcomponentisbasedon therelativecongurationofthethreeagentsandpossessesparticularcharacteristics thatencouragetheevaderstoattemptankingmaneuverstosurroundthepursuer.As adirectresultoftheevadergeneratedcostcomponent,theoptimalevaderstrategies exhibitcooperativedefensivebehaviors.Itshouldbenotedthatcooperationisnot directlyimposedasarequirementofthesolution.Instead,cooperationemergesasthe optimalstrategyduetothedesignofinstantaneouscostfunction. Thecooperativebehaviorsexhibitedinthesolutiontothisgamearequalitatively similartonumerousexamplesofpreystrategiesusedinresponsetoattacking predators.Someexamplesincludered-wingblackbirdnestdefense[49],meerkat predatormobbing[50],andpredatoridenticationinguppyschools[51].Suchanimal behaviorshavebeenstudiedextensivelywithinthebiologicalcommunity,andtheories thatexplaintheirevolutionarystabilityandadvantageshavebeenproposed[22].Often, thesetheoriesutilizeprinciplesfromgametheory.Inparticulartheconceptofrepeated gamesiscommonlydeployedforthispurpose[21,52].Intheseapproaches,the potentialbehaviorsarerepresentedasstrategieswithassignedutilitiesthatareinferred fromempiricaldataorbasedonthegeneticsimilaritybetweenindividuals.Thedifferent strategiesarethenshowntoincreasethesurvivabilityortnessofthegenesthat describethesebehaviorsovertimeormultiplegenerations.Althoughtheseapproaches explainhowcooperationisoptimalattheevolutionarylevel,theydonotdirectlyaddress howcooperationisbenecialattheday-to-day,systemlevel.Thedevelopmentand resultingsolutionofthegamepresentedinChapter2showshowcooperationcanarise 21

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astheoptimalstrategyatthesystemlevelgivenparticularsystemdynamicsandcost functional.Theresultingutilityvaluesforparticularinitialconditionscanbeusedata higherlevelbytheattackertomakethedecisiontoengageorretreat. 1.3.2DevelopmentofaHierarchicalAttack-RetreatGame Theadditionofastrategichierarchyisincorporatedintoanewgamepresentedin Chapter3.Inthistwo-playergame,oneplayerrepresentsanattacker,andtheother playerrepresentsadefensiveteamthatconsistsofamobile,high-valuetargetand N protectiveagents.Itisassumedthattheprotectiveagentsgenerateacosttothe attacker,whichcanrepresentcasualtiesincurred,resourcesused,ortheriskofinjury ordamage.Atthestartofthegame,theattackermustchoosebetweenengagementor retreat.Aftertheattackerhasmadeitsdecision,thedefendingteammustthendecide whethertomaximizeorminimizetheattacker'scostinresponse.Theattackingand defendingteams'choicesarereferredtoastheirintent.Onceeachsidehasselectedan intent,adifferentialpursuit-evasiongameisplayedinwhichtheterminalconditionsand theplayers'optimalcontrolstrategiesaredictatedbytheintentselections.Inallvariants ofthedifferentialgame,thevalueofthedifferentialgamerepresentstheintegralcostto theattacker.Forcertainconditions,itisshownthatitisoptimalforthedefensiveteam tocooperatewiththeattackerinretreatsothatretreatbecomesamoreattractiveoption thanengagementfromtheattackersperspective. ThehierarchicalgameposedinChapter3isparticularlynovelduetheincorporation ofintentanditsresultingeffectontheprocedingdifferentialgame.Althoughthisgame onlyprovidestwooptionstoeachplayer,theintentselectionstrategycouldbefurther generalizedtoincorporatetheselectionofinitialpositions.Additionally,thisalsoleadsto thepotentialincorporationofdeceptivetactics. 1.3.3DevelopmentofaGeneralDeceptiveGameFramework Inmostofthepreviousworks,theplayerthatfallsvictimtothedeceptionis assumedtobeignorantofthepossibilityofdeception.However,inmanycases,itis 22

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commonknowledgethatone'sopponentmaybetryingtoimplementsomeformof deception.Thisdoesnotmeanthatallinformationshouldbeignored,butinstead,the riskthataparticularpieceofinformationmaybecompromisedneedstobebalanced withthepotentialadvantagethattheinformationprovides.Anexampleofsucha scenariocanbefoundin[53]wheretheauthorsexamineaparticulartwo-playergamein whichoneplayerutilizescost-free,passivedeceptionthroughconcealmentordisclosure ofdefensiveresourceallocationsinordertoneutralizetheopponent'sinformational advantage.Therehavealsobeensomeresultsonthedetectionofdeceptionwithin repeatedgames[54]. InChapter5,agenerictwo-player,zero-sumgameisdevelopedthatincorporates deceptivetactics.Inthisgameastochasticsensornetworkprovidesoneplayer,Player B,aninformationaladvantageoveritsopponent,PlayerA.Simultaneously,PlayerA possessestheabilitytocorruptthesensornetworkoutput,atacost,inanattempt tomanipulatePlayerB'sactions.ThepossibleuseofdeceptionallowsPlayerAto neutralizetheinformationaladvantageofPlayerBandshiftthegame'sequilibrium valueclosertothesolutionofthegamewheretheinformationnetworkisremoved.Itis assumedthatPlayerBknowsofthepossibilityofdeception,butiftheriskofdeception issmallenough,PlayerBwillstillutilizetheinformationprovidedbyitssensornetwork. Autilityfunctionforthegameisdesignedwhichtakesintoaccounttheeffectsthat thedeceptivetacticshaveonthesensornetworkanditscorrespondingcost.The solutiontothisgameconsistsoftheoptimalstrategiesforeachoftheplayersand thecorrespondingvalueoftheutilityfunction.WithrespecttoPlayerA,theoptimal strategyrepresentsthebestmixofactionsalongwiththecomplimentarydeceptive tactic.TheoptimalstrategyofPlayerBrepresentsthebeststochasticcontrollawbased onthemeasuredsensornetworkoutput.Utilizingtherelationshipbetweentheminimax theoremandthestrongdualitytheoremoflinearprograming,Ishowthatthesolution ofthezero-sumgamecanbecomputedbysolvingapairofduallinearprogramming 23

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problems.Thisframeworkisgeneralenoughtoincorporateawiderangeofbothactive andpassiveformsofdeceptionaslongastheireffectsonthesensornetworkare known. 1.3.4AnalyticandQuantitativeDescriptionofJones'Lemma Usingthisframework,anillustrativeexampleisdesignedandexamined,which canbemodiedtorepresentalargerangeofscenarios.Closedformsolutionsare analyticallyexaminedfortwospecialcasesofthisgame.Theresultingequilibrium playerstrategiesandutilityvaluesnicelycaptureawell-known,qualitativeprinciple inthedeceptioneldknownastheJones'Lemma.ThismaximisattributedtoR.V. Joneswhoisconsideredtobethefatherofscienticintelligence.Itstates, Deceptionbecomesmoredifcultasthenumberofchannelsavailableto thetargetincreases.Nevertheless,withinlimits,thegreaterthenumberof channelsthatarecontrolledbythedeceiver,thegreaterthelikelihoodthat thedeceptionwillbebelieved.[25] IfurtherexploreJones'Lemmaforarangeofparametervaluesbynumericallysolving thegenerallinearprogramsolution. 1.3.5DevelopmentofaSensorBasedColonelBlottoGame InChapter6,anewsequentialvariantoftheColonelBlottogameisdeveloped. Thisgameconsistsoftwoplayers,PlayerAandPlayerB,whomustallocatenite resourcesamong N regionsofabattleeld.Akeyfeatureofthisproblemisthe introductionofasensornetworkemployedbyPlayerBtogainaninformational advantageoverPlayerA.Thesensornetworkconsistsofsensorsthatproduce binaryoutputtriggeredwhenevertheresourceallocationinitsregionexceedsa certainthreshold.Thissensormodelcouldbeadaptedtorepresentseveralrealworld applications.Inthegame,PlayerAallocatesitsresourcesrst.PlayerBthenreceivesa sensorvalueforeachregionindicatingwhetherPlayerA'sallocationisaboveorbelowa 24

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threshold.Usingthisearlywarninginformation,PlayerBcanthenallocateitsresources moreeffectively. AfterposingtheabovescenarioasasequentialColonelBlottogame,necessary conditionsforNashequilibriumoptimalmixedstrategiesforthetwoplayersareobtained intermsoftheeachplayer'savailableresourcesandthesensornetworkcharacteristics. Theseconditionsapplytothemarginalprobabilitydistributionfunctionsofeachplayer's strategy.Themarginaldistributionsrepresenttheresourceallocationstrategyrelating toaspecicregionofthebattleeld.Todeveloptheseconditions,Iuseanapproach thatissimilartothehighlyinuentialrecentpaperbyRoberson[40].Thecontestfor eachregionisrepresentedasarst-priceallpayauctionwherethevalueawarded toeachplayerintheauctionincorporatesanadjointvariablethatcorrespondstothat player'sresourceconstraint.Theexistenceofthesensornetworkinthegameleads toupperandlowerboundsonPlayerA'sallocationstrategiesforeachregion.Froma technicalviewpoint,thisisthekeynewfeatureofourformulationandthecorresponding results.Asolutionisobtainedforeachoftheseindividualauctiongamesintermsof thesensornetworkcharacteristicsandadjointvariables.Theequilibriumsolutions haveapiecewisebehaviorthatmakesndingageneralclosed-formanalyticsolution particularlychallenging.Efcientnumericaltechniquesaredevelopedtosolveforthe adjointvariablesforparticularvaluesoftheplayers'resourcesandsensornetwork characteristics.Theseanalyticalcharacterizationsareusedtodevelopaneffective computationalsolutiontothegame.Resultsareillustratedviaanumericalexample. 25

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CHAPTER2 COOPERATIVEDEFENSEWITHINASINGLE-PURSUER,TWO-EVADERPURSUIT EVASIONDIFFERENTIALGAME Thischapterisbasedontheworkpresentedatthe2010CDC[55].Iintroduced asingle-pursuer,two-evadergamewithanovelintegralcostfunctional.Thecost functionalisintendedtorepresenttheriskofdamageorinjurytothepursuerorthe additionalenergyorcomputationalexpenseneededtomonitormultipleevaders.During thegame,thepursuerstrivestominimizethiscostwhileattemptingtocaptureoneof theevaders.Simultaneously,theevadersattempttomaximizethepursuer'scostinthe hopesofmakingpursuitunattractivefromcertaininitialconditions,therebyprotecting themselvesandtheirfellowevader.Theproposedcostfunctionalisacombination ofaconstanttimepenaltyandevadergeneratedcost.Theevadergeneratedcost componentisbasedontherelativecongurationofthethreeagentsandpossesses particularcharacteristicsthatencouragetheevaderstoattemptankingmaneuvers tosurroundthepursuer.Asadirectresultoftheevadergeneratedcostcomponent, theoptimalevaderstrategiesexhibitcooperativedefensivebehaviors.Itshouldbe notedthatcooperationisnotdirectlyimposedasarequirementofthesolution.Instead, cooperationemergesastheoptimalstrategy. Althoughbiologicallyinspired,mymainmotivationforthescenariopresentedin thispaperandtheresultingcooperativedefensivebehaviorscomesfromtheidea ofcooperativedefenseofhighvalueassets.Justasinnature,therearerarelyany defenselesstargets,andattackingforcesusuallyelectnottoattackatargetifthe potentialforinjuryorhighcostoutweighsthebenetoftheattackmission.Thus, bycooperatingtocombinetheirdefensiveresources,agroupofevaderscanmake engagementmorecostlytotheattackerthaniftheyactedindependently.Thisincreased costmaythensurpassatolerancelevelforthepotentialattackerandpreventanattack beforeiteveroccurs.Forexample,throughcooperationagroupofunmanneddrones couldbeusedtoprotectvulnerablehigh-valuetargets,suchasslowmovingcargo 26

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planes,supplyships,oraveryimportantperson.Ifthehighvaluetargetwasattacked, thedronescouldthenengageinacooperativedefensivemaneuver.Thiscooperative defensivemaneuvercouldbesufcienttoprotecttheintendedasset. InSection2.1,Idescribethesystemunderconsideration.Ialsodevelopan alternativecoordinatesystem,whichwillsimplifylateranalysis.Anovelevader generatedcostfunctionisthendevelopedthatcapturesthesynergybetweenthetwo evadersandservesastheprimarymotivationforcooperation.Usingthedeveloped instantaneouscost,Ithendescribethepursuitevasiongameunderanalysis.In Section2.2,Ideveloptheoptimalityconditionsandperformthenecessaryintegrationto generatetheoptimalagenttrajectories. 2.1SystemandGameFormulation Inthissection,Iwilldescribethethreeagentsystemunderanalysisanddenethe kinematicequationsthatcontroltheirmotion.Iwillalsointroducearelativecoordinate systemandcorrespondingkinematicequations,whichwillprovetobemorecompact andintuitiveinlateranalysis.Afterthesystemkinematicsaredened,Idevelopan integralcostfunctionwhichisbasedontherelativecongurationofthethreeagents. Inthethirdsection,Ilayoutthemotivationsforatwo-teamdifferentialgameusingthe denedsystemkinematicsandpursuercostfunction. 2.1.1AgentKinematics Consideradynamicsystemwiththreeagents:twoevadersandapursuer.For brevity,IwilloftenrefertothetwoevadersasE 1 andE 2 andthepursuerasP.The threeagentsaremodeledasmasslessparticlesmovingwithsimplemotionaboutan obstacle-free,inniteplane.Withinthispaper,twodifferentbutequivalentcoordinate systemsareused.Therstcoordinatesystemisreferredtoasthe global coordinate systemandwillbeusedtoplotagenttrajectoriesandothervisualizations.Inthis coordinatesystem,thepositionofeachagentisdenedbyitsownpairofstandard Cartesiancoordinates x y .ThevelocitiesofE i i =,2 andParedenedas v i ^ i 27

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and v p ^ respectively.Here,thecomponents v i and v p representthemagnitude ofvelocitiesand ^ i and ^ representtheheading.Theheadinganglesaremeasured counter-clockwisefromthepositivex-direction.Theheadingangleisthecontrolvariable foreachagent,andIassume v i and v p areconstant.Thestateofsystemiscompletely denedbythe6-tuple, x G = x 1 y 1 x 2 y 2 x p y p .Theglobalcoordinatesystemis depictedgraphicallyinFigure2-1.Theglobalkinematicequationsofthesystemare thus x p = v p cos ^ y p = v p sin ^ x 1 = v 1 cos ^ 1 y 1 = v 1 sin ^ 1 x 2 = v 2 cos ^ 2 y 2 = v 2 sin ^ 2 Iwillnowintroduceasecondcoordinatesystem,whichwillrepresentthelocations ofeachoftheevadersrelativetothepositionofthepursuer.Thisrepresentationwill allowustoreducethenumberofdimensionsinlateranalysisandwillbereferredto asthe relative coordinatesystem.Inthiscoordinatesystem,thestateofthesystemis representedbythefollowing6-tuple, x R = d 1 d 2 x y .Thersttwocoordinates, d 1 and d 2 ,representthedistancebetweenE 1 andPandthedistancebetweenE 2 andPrespectively.Theangle ismeasuredcounter-clockwisefrom )430()430(! PE 1 to )430()430(! PE 2 .The angle representstheglobalrotationofthethreeagentsystemandismeasured counter-clockwisefromthepositivex-directionto )430()430(! PE 1 .Thexandycoordinates representglobalpositionofthepursuer.Thesixcoordinatescanbeseparatedinto twogroups.Therstgroup, d 1 d 2 ,containsallnecessaryinformationtodescribe therelativecongurationofthethreeagents.Thesecondgroup, x y ,contains theglobalrotationalandtranslationinformation.Intherelativecoordinatesystem, theevaderheadingangle, i ,ismeasuredcounter-clockwisefrom )229(! PE i inorderto simplifythekinematicequations.Similarly,thepursuerheadingangle, ,ismeasured 28

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counter-clockwisefrom )430()430(! PE 1 .Therelativecoordinatesystemisgraphicallydepictedin Figure2-2. Theglobalandrelativerepresentationsarerelatedthroughthefollowingequations. x p = x y p = y x 1 = d 1 cos + x y 1 = d 1 sin + y x 2 = d 2 cos + + x y 2 = d 2 sin + + y Thecontrolvariablesarerelatedasfollows. ^ 1 = 1 + ^ 2 = 2 + + ^ = + UsingthevariablesintherelativemodelwiththedynamicsinEquation2,the relativekinematicequationsareshownbelow. d 1 = v 1 cos 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v p cos d 2 = v 2 cos 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v p cos )]TJ/F25 11.9552 Tf 11.955 0 Td [( = v 2 d 2 sin 2 )]TJ/F40 7.9701 Tf 13.397 4.884 Td [(v 1 d 1 sin 1 + v p 1 d 1 sin )]TJ/F23 7.9701 Tf 15.032 4.707 Td [(1 d 2 sin )]TJ/F25 11.9552 Tf 11.955 0 Td [( = 1 d 1 sin 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v p sin x = v p cos + y = v p sin + 29

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Wefurtherconditiontheseequationswiththefollowingrestrictions. d 1 d c d 2 d c v 1 < v p v 2 < v p Therstpairofrestrictions,Equation2andEquation2,requiresthat bothdistancesaregreaterthanorequaltothecapturedistance, d c .Thesecondpair ofrestrictions,Equation2andEquation2,requiresthatthepursuerisfaster thanbothoftheevaders,whichensuresthatthepursueriscapableofcapturingan evaderinnitetime. 2.1.2InstantaneousCostFunction Inthissection,Idevelopaninstantaneouscostfunctiondependentontherelative positionsofthetwoevadersandpursuer.Thedevelopedcostfunctioncapturesthe synergybetweentheevadersandservesastheprimaryincentiveforcooperationwithin theevadingteam.Withrespecttothebiologicalinspiration,thiscostcouldmodelthe riskofinjurytoapredatorcausedbytheprey.Intermsofaman-madeexample,the evader-generatedcostcouldrepresenttheriskofdamagetoanattackingaircraftfrom thetargets'defensivecapabilities. Eachevadergeneratesanindividualcost,whichisafunctionofdistancebetween theevaderandpursuer.Inthisgame,exponentialcostfunctionsareusedforE 1 andE 2 : C 1 d 1 = k 1 e k 2 d c )]TJ/F40 7.9701 Tf 6.586 0 Td [(d 1 C 2 d 2 = k 1 e k 2 d c )]TJ/F40 7.9701 Tf 6.587 0 Td [(d 2 wheretheconstant k 1 denesthemaximumvalueofthecostand k 2 controlshow quicklythecostdecaysasafunctionofdistance.Thesefunctionswerechosen 30

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becauseoftheirsimplicity,butmorecomplexfunctionscouldbeusedtomodelparticular predator-preyorattacker-targetinteractions. Weprovidethepursuertheabilitytocounteractorreducetheseindividualcosts. Returningtotheaircraftattackexample,theaircraftmaybeabletoperformevasive maneuversordeploycountermeasuresifathreatisdetected.Thedetectionofthe threatmayberelativelystraightforwardifonlyasingletargetexists,butinthecaseof multipletargets,itmaybenecessarytoallocatenitesensoryorprocessingcapabilities betweenmultiplethreats.Thedecreasedvigilanceofthetargetsattheindividuallevel increasestheoverallriskofdamage. Wemodelthiseffectbydeningadirectionofsensoryfocus, ,forthepursuer.The directionoffocusisindependentofthemotionofPandismeasuredcounter-clockwise from )405()404(! PE 1 .Bysteeringthedirectionoffocustowardanevader,thepursuerreducesthe costgeneratedbythatevader.Theresultingreducedcostsareaproductofthecost reductionfunctionandtheoriginalevadercost: C E 1 x = S C 1 d 1 a C E 2 x = S )]TJ/F25 11.9552 Tf 11.956 0 Td [( C 2 d 2 b where S representsthecostreductionasafunctionofthedifferencebetweenthe sensoryfocusangleandtheangletowardstheevader.Inthispaper,Iusethefollowing denitionfor S S = 1 2 [ 1 )]TJ/F22 11.9552 Tf 11.955 0 Td [(cos ] Thetotalevader-generatedcostforthepursueristhesumoftheindividualevader costs: C E x = C E 1 x + C E 2 x 31

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Thepursuermustthenselect suchthatthetotalcostisminimizedatanymomentin time.Theminimizing satisesthefollowingconditions cos = C 1 +cos C 2 sin = sin C 2 where = q C 2 1 +2 C 1 C 2 cos + C 2 2 Substitutingtheoptimal -strategyEquation2-Equation2intoEquation 2providestheminimumcost: C E x = 1 2 C 1 + C 2 )]TJ/F30 11.9552 Tf 11.955 14.564 Td [(q C 2 1 +2 C 1 C 2 cos + C 2 2 Itshouldbenotedthatthisfunctionevaluatestozerowhen =0 .Thissituation allowsthepursuertomonitorbothevaderssimultaneously.Theevadercostfunction ismaximizedwhen = ,whichrepresentsthescenarioinwhichtheevadershave ankedthepursueranditcanonlydirectitsbeamoffocusatthemostcostlypursuer. Because doesnotaffectthesystemdynamicsandthepursuercaninstantaneously chooseanyvaluefor ,Iwillassumethepursueralwayschooses .Asaresult,Iwill considertheinstantaneousevader-generatedcostasafunctionofstatealoneandno longerconsider inthedevelopmentofthegame.Anadditionalconstantcostterm, c t isaddedtotheevader-generatedcostinordertorepresentatimeorenergypenaltyfor thepursuer.Thetotalpursuerinstantaneouscostisthen C T x = C E x + c t 2.1.3GameFormulation TheinstantaneouscostfunctionEquation2isintegratedovertimetocalculate thetotalcosttothepursueroverasingleplayofthegame.Inthisgame,termination 32

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occurswhenthepursuercapturesoneoftheevaders,whichhappenswhenthestate passesthroughtheterminalsurface: \050 x = d 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c d 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c =0. Thecosttothepursuerforagamestartingatinitialtime t 0 andreachingtheterminal surfaceattime t f isthendenedas: J = Z t f t 0 C E x + c t dt Wecannowposeadifferentialgameinwhichthegoalofthetwoevadersisto maximizetheintegralcosttothepursuerEquation2.Byinspection,itcanbeseen thatingeneraltheevadersshouldstrivetodelayterminationofthegameinorderto continuetheintegrationofcost.Simultaneously,thepursuerstrivestominimizeitscost byterminatingthegameassoonaspossiblewhileattemptingtoavoidpotentialanking maneuversoftheevaders. Althoughtherearethreeagentsinthissystem,thetwoevadersshareacommon goal,maximizethepursuer'scost.Therefore,theevaderscanbethoughtofasasingle playerwithtwocontrolvariables.Thisperspectiveresultsinatwo-playerzero-sum game;oneplayeristhepursuer,whiletheotherplayerrepresentstheevadingteam. Wecanthendeneafunction V x ,whichrepresentsthevalueofagamethatstartsat point x andinwhichbothplayersimplementtheiroptimalstrategies. Inthispaper,Iassumethatallagentspossesscompleteknowledgeofallstate variables.Thepursuerdoesnotpossessknowledgeofeitherevader'scontrolwhilethe evadersareignorantofthepursuer'scontrolaswell. 2.2SolutiontotheGame InthissectionIwilldevelopthesolutiontothegame.Forthispaper,Iwillexamine thecasewhere v 1 = v 2 =1 k 1 = k 2 =1 ,and c t > 0 .Itisassumedthat t 0 =0 .Iwillrst calculatetheoptimalityconditionsthatdescribetheoptimalcontrolstrategies.Usingthe 33

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calculatedoptimalityconditions,Inumericallyintegratebackwardsintimetogenerate theoptimaltrajectories.Allofthefollowingcalculationsareperformedusingtherelative coordinatesystem. 2.2.1OptimalityConditionsfortheGameofAttack Theoptimalityconditionsofdifferentialgamesarerstcalculatedasdescribedby RufusIsaacs[1].UsingthedenedkinematicequationsEquation2-Equation 2andthecostfunctionalEquation2,theHamiltonian,H,isintroducedas H = T f x u p u e + C T = 1 d 1 + 2 d 2 + + + x x + y y + C T Thevector = 1 2 x y T containstheadjointvariablesconjugatetothe kinematicequations.Theadjointequationsarefoundbytakingthepartialderivativeof theHamiltonianwithrespecttotheirrespectivestatecomponent: 1 = )]TJ/F26 7.9701 Tf 13.957 4.707 Td [(@ H @ d 1 = )]TJ/F25 11.9552 Tf 9.299 0 Td [( @ @ d 1 )]TJ/F26 7.9701 Tf 13.151 5.112 Td [(@ C T @ d 1 2 = )]TJ/F26 7.9701 Tf 13.957 4.707 Td [(@ H @ d 2 = )]TJ/F25 11.9552 Tf 9.299 0 Td [( @ @ d 2 )]TJ/F25 11.9552 Tf 11.955 0 Td [( @ @ d 2 )]TJ/F26 7.9701 Tf 13.151 5.112 Td [(@ C T @ d 2 = )]TJ/F26 7.9701 Tf 13.151 4.707 Td [(@ H @ = )]TJ/F25 11.9552 Tf 9.299 0 Td [( 2 @ d 2 @ )]TJ/F25 11.9552 Tf 11.955 0 Td [( @ @ )]TJ/F26 7.9701 Tf 13.15 5.112 Td [(@ C T @ = )]TJ/F26 7.9701 Tf 13.151 4.707 Td [(@ H @ = )]TJ/F25 11.9552 Tf 9.299 0 Td [( x @ x p @ )]TJ/F25 11.9552 Tf 11.955 0 Td [( y @ y p @ x = )]TJ/F26 7.9701 Tf 13.151 4.708 Td [(@ H @ x =0 y = )]TJ/F26 7.9701 Tf 13.151 4.707 Td [(@ H @ y =0 Theboundaryconditions, ,forthegameare = 0 B B B B B B B B B B B B B B B @ d 1 t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d 10 d 2 t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d 20 t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 t 0 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 0 x t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x 0 y t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 0 d 1 t f )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c d 2 t f )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c 1 C C C C C C C C C C C C C C C A =0 34

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where d 10 d 20 0 0 x 0 ,and y 0 aretheinitialvaluesoftheirrespectivestatecomponents atthestartofthegame.Inordertodeterminetheboundaryconstraintsontheadjoint variables,IusetheboundaryconditionsEquation2tocreateafunctionofterminal conditions, : = T where = 1 2 3 4 5 6 7 T containstheadjointvariablesconjugatetothe boundaryconstraintsofthestate.TakingthepartialderivativesofEquation2 withrespecttothestatecomponentsprovidestheterminalconditionsfortheadjoint variables: 1 t f = @ @ d 1 t f = 8 d 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c 2 t f = @ @ d 2 t f = 8 d 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c t f = @ @ t f =0 t f = @ @ t f =0 x t f = @ @ x t f =0 y t f = @ @ y t f =0 UsingtheadjointderivativesEquation2-Equation2,andtheterminal constraintsEquaiton2-Equation2,itisfoundthat t =0 x t =0 y t =0. SubstitutingEquation2-Equation2intoEquation2,resultsina simpliedHamiltonian,whichisdependentonlyonthecomponentsofthestatethat 35

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describetherelativecongurationofthethreeagents: H = 1 d 1 + 2 d 2 + + C T Thenextstepinsolvingthegameistodeterminetheoptimalstrategiesforthe threeagents,whichIwilldenoteas 1 2 ,and .Forregionsinwhichthegradient ofthevaluefunctioniscontinuous,theoptimalstrategiesmustsatisfytwoconditions, whichareoftenreferredtoasIsaacsConditions.Theregionsinwhichthevaluefunction oritsgradientisdiscontinuousarecalledsingularsurfacesandwillbediscussedina latersection. Theorem2.1. Supposethatthevaluefunctionandthevaluefunctiongradientare continuous.Thecontrolstrategiesforthethreeagentsarethengivenby cos 1 = 1 1 sin 1 = )]TJ/F25 11.9552 Tf 15.15 8.088 Td [( d 1 1 1 = q 2 1 + d 1 2 cos 2 = 2 2 sin 2 = d 2 2 2 = q 2 2 + d 2 2 cos = )]TJ/F39 11.9552 Tf 10.829 8.087 Td [(c 1 p sin = )]TJ/F39 11.9552 Tf 10.829 8.087 Td [(c 2 p p = q c 2 1 + c 2 2 where c 1 = d 2 sin )]TJ/F25 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 cos c 2 = d 1 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 2 sin )]TJ/F26 7.9701 Tf 13.15 4.707 Td [( d 2 cos 36

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Proof. Alongtheoptimaltrajectories,theHamiltonianmustsatisfythefollowing conditions[1]: H x H x H x H x =0 where = 1 2 .FromEquation2Indthat =argmin H 1 =argmax 1 H 2 =argmax 2 H Becausethecontrolvariablesareunbounded,theoptimalstrategiesEquation 2-Equation2mustsatisfythefollowingconditions: @ H @ =0 @ H @ 1 =0 @ H @ 2 =0 and @ 2 H @ 2 0 @ 2 H @ 2 1 0 @ 2 H @ 2 2 0 Therstsetofconditions,Equation2-Equation2,gauranteetheHamiltonian isstationarywithrespecttothecontrolvariables.ThesecondsetofequationsEquation 2-Equation2,representthenecessarysecond-orderconditionssothat 1 and 2 maximizeand minimizes.SolvingEquation2-Equation2,intermsof 1 2 ,and provideouroptimalcontrolstrategiesEquation2-Equation2. 37

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2.2.2NumericalAnalysis Findingananalyticsolutiontotheoptimaltrajectoriesisnotpracticaldueto thenonlinearandcouplednatureofthestateandadjointequations.Inorderto numericallygeneratetheoptimaltrajectories,Irstsubstitutetheoptimalcontrol strategiesEquation2-Equation2intothethekinematicequationsEquation 2-Equation2andtheadjointequationsEquation2-Equation2. Theresultingsystemoftwelveordinarydifferentialequationsdescribetheoptimal trajectoriesofthethreeagentsandthecorrespondingcostatesforthisgame.Wecan thennumericallyintegratebackwardsintimefromtheterminalsurfacetogeneratethe optimaltrajectories. TondtheinitialconditionsforintegrationIconsiderapointontheterminalsurface: x f = d 1 f d 2 f f f x f y f T where d 1 f = d c and d 2 f > d c .FromEquation2-Equation3,Indtheterminal adjointvector: f = 8 d 2 )]TJ/F39 11.9552 Tf 11.956 0 Td [(d c ,0,0,0,0,0 T AftersubstitutingtheoptimalcontrolstrategiesintotheHamiltonianandevaluatingatthe terminalstate,wecansolvedirectlyfor 1 f : j 1 f j = C E x f + ct v p )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 KnowingthatE 1 attemptstodelaycapturebyincreasing d 1 ,Iusethepositivevalue for 1 .Itshouldbenotedthatontheportionoftheterminalsurfacethatrepresents thecaptureofE 1 ,theterminalcontrolforE 2 isundenedatthemomentofcapture. ConceptuallythismakessensebecauseE 2 candonothingtofurtherdelaycaptureof E 1 ,andanychangeitcanproducein C E willhavenoeffectontheintegralcostbecause thegamehasended.Butinordertoperformthenumericalintegration,itisnecessaryto knowthecontrolforE 2 tostartthenumericalintegration.Forthispurpose,wecanuse 38

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thecontroljustbeforecapture,whichcanbefoundbytakingthelimit: lim t t f tan 2 =lim t t f d 2 2 =lim t t f d 2 2 + d 2 2 = t f d 2 f 2 t f Wecannowusethegiventerminalstate x f Equation2,terminalvalues foundfor f Equation2,andtheterminalcontrolforE 2 Equation2asinitial conditionsforourbackwardsintimenumericalintegration.Thestateequationsare thenintegratedoverthetimeperiodofinterestoruntilthetrajectoryreachesadispersal surface. 2.2.3IllustrativeCases Aftertheintegrationisperformed,theresultingtrajectoriesinthereduced coordinatesystemcanthenbemappedtotrajectoriesintheglobalcoordinatesystem usingEquation2-Equation2.SeveralillustrativecasesaredisplayedinFigure 2-3throughFigure2-6.Ineachofthesegures,thesolidlinerepresentsthetrajectory ofthepursuer;thedashedlinerepresentsthetrajectoryofE 1 ;andthedottedline representsthetrajectoryofE 2 .Inallthreecases,thegameisterminatedwhenE 1 is captured.Theterminalpositionofthethreeagentsaremarkedbyanx.Themarkers alongthecurvesrepresenttheagentlocationsintwosecondincrements. InFigure2-3,E 2 rushestomeetnearthepointofcaptureinordertoperformalast ditchankingmaneuverandcreatealargeaccumulationofcostjustbeforecapture. Thisresultsinacounterankingmaneuverbythepursuerjustbeforecapture.An enlargedviewofthetrajectoriesjustbeforecapturecanbeseeninFigure2-5. InFigure3-5,thepursuerutilizesitsspeedadvantageandperformsacounter ankingmaneuveragainstthetwoevadersinordertominimizetheevadergenerated cost.Inthisscenario,E 1 canincreasethecosttothepursuermorebyattemptingto remaincloseandankingasopposedtoastrategyofmaximizingthetimeofthegame byeeing.Thisissimilartoaghtorightdecisioninnature.E 1 knowsthatitpresents moreofacosttothepursuerbymakingastand,andtheevadershopethatthiscost 39

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maybemorethanthepursueriswillingtoacceptandthereforeabortstheattack. AlthoughtheinitialconditionsofthisscenarioshowninFigure2-6aresimilartoFigure 3-5,thepursuerdoesnotpossessthesamespeedadvantage.Therefore,itdoesnot trytooutanktheevadersandinsteadtakesamoredirectapproachtowardsE 1 .Also, E 1 canaccumulatemorecostbyrunningawayanddraggingthegameoutforalonger periodoftime.Again,thisisaghtorightsituation,butitismoreadvantageousfor E 1 toee.Throughoutthegame,E 2 continuestoharassthepursuerfrombehindand accumulatecost. 2.2.4SingularSurfaces Thevaluefunctiongeneratedbytheoptimalcontrolstrategiesdividesthestate spaceintomutuallydisjointregions.Withintheseregions,thevaluefunctioniswell denedbytheoptimalityconditions.Themanifoldsthatdividetheseregionsare calledsingularsurfacesandarecharacterizedbyatleastoneofthefollowingthree characteristics:theoptimalcontrolstrategiesarenotuniquelydeterminedbyoptimality conditionspreviouslydescribed,thevaluefunctionisnotcontinuouslydifferentiable, orthevaluefunctionisdiscontinuous[16].Mostsingularsurfacesarenotidentiedby backwardintegrationoftheoptimaltrajectoriesandrequirefurtheranalysisinorderto describethesystembehavioronornearthesesurfaces. Withinthisgame,symmetryinthekinematicequationsandcostfunctionhintat theexistenceofparticularsingularsurfaces.Iwillbegintheanalysisofthesingular surfacesbylookingatthe =0 plane.Onthisplane,thethreeagentsareinacollinear congurationwithbothevadersononesideofthepursuer.Thepursuercanthendirect itsbeamoffocusatbothevaderssimultaneously,therebycompletelynegatingthe evader-generatedcost.Asaresult,thepursuerwouldliketokeepthestatenearthe =0 planewhiletheevadersattempttoforcethestateawayfromthisplane.For thecasewhere c t > 0 ,the =0 planerepresentsadispersalsurface.Adispersal surfaceisasurfacewithinstatespaceinwhichoneorbothoftheplayerscanselect 40

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frommultipleoptimalcontrolstrategies.Eachofthesestrategiesmovesthestateoff ofthesurfaceindifferentdirections,butwillresultinthesamevalueforthegame.For anygamethatbeginswiththeinitialstateonthe =0 plane,theevadersmakean initialchoicetoforcethe -componentofthestateawayfromzeroineitherthepositive ornegativedirection.Eitherdirectionresultsinthesamevalueofthegamebecauseof thesymmetryofthestateequations,costfunction,andtheresultingadjointequations. Althoughthepursuercouldattempttoholdthestatenearthe =0 plane,theslight reductionofevader-generatedcostwouldbeoutweighedbytheincreasedtimepenalty. Thisdispersalsurfaceappearsasadiscontinuityofthegradientofthevaluefunctionin thealpha-direction. The d 1 = d 2 planeisalsosingularsurface.Theportionofthisplanewhere > 2 is clearlyadispersalsurfacewherethepursuerchoosesanevadertocaptureandforces thestateoffoftheplaneinthatdirection.Undercertainconditions,theregionofthe d 1 = d 2 planeneartheintersectionwiththe =0 planehasthepotentialforasingular focalsurface.Inmyresearch,Ionlyconsiderinitialstartingpositionsabove = 2 onthe d 1 = d 2 plane. 41

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Figure2-1.Globalcoordinates Figure2-2.Relativecoordinates 42

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Figure2-3.Optimaltrajectoriesfor d 2 f =1.5 f =2.2 ,and v p =1.5 Figure2-4.Optimaltrajectoriesfor d 2 f =7 f =.8 ,and v p =2.5 43

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Figure2-5.EnlargedviewofnearcapturetrajectoriesofFigure2-3 Figure2-6.Optimaltrajectoriesfor d 2 f =7 f =2.8 ,and v p =1.1 44

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CHAPTER3 ENCOURAGINGATTACKERRETREATTHROUGHDEFENDERCOOPERATION 3.1Introduction Thischapterisbasedonworkpresentedatthe2011CDC-ECC[56].Inthis chapter,Iexamineatwo-playergameinwhichoneplayerrepresentsan attacker ,and theotherplayerrepresentsa defensiveteam thatconsistsofamobile,high-value target and N protectiveagents .Itisassumedthattheprotectiveagentsgenerateacostto theattacker,whichcanrepresentcasualtiesincurred,resourcesused,ortheriskof injuryordamage.Itisalsoassumedthattheattackerpossessessuperiorperformance capabilities,allowingittosuccessfullycapturethetargetfromallinitialconditions.At thestartofthegame,theattackermustchoosebetweenengagementorretreat.After theattackerhasmadeitsdecision,thedefendingteammustthendecidewhether tomaximizeorminimizetheattacker'scostinresponse.Iwillrefertotheattacking anddefendingteams'choicesastheirintent.Ialsodiscussthescenarioinwhichthe attackeranddefensiveteamareallowedtoupdatetheirintentsthroughoutthegame. Onceeachsidehasselectedanintent,adifferentialpursuit-evasiongameisplayed inwhichtheterminalconditionsandtheplayers'optimalcontrolstrategiesaredictated bytheintentselections.Therearefourvariantsofthedifferentialgamebasedonthe fourpossiblecombinationsofintent.Iftheattackerchoosestoengage,thedifferential gameterminateswhenthedistancebetweentheattackerandhigh-valuetargetisequal toapredenedcapturedistance.Ifretreatischosenbytheattacker,thedifferential gameterminateswhenalldistancesbetweentheattackerandtheprotectiveagentsare greaterthanorequaltoadenedretreatdistance.Inallvariantsofthedifferentialgame, thevalueofthedifferentialgamerepresentstheintegralcosttotheattacker. Itisassumedthatboththeattackeranddefensiveteamcancalculatetheresulting integralcostofthefourpossibledifferentialgamesfromanyinitialcondition.Additionally, whentheattackerchoosestoengagethedefensiveteam,theattackerisawardeda 45

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bonusandthedefensiveteamassessedapenalty.Usingthevaluesfromthepossible differentialgamesinconjunctionwiththegivencapturebonusandpenaltyvalues,the optimalintentstrategiesforeachplayercanbecalculated.Forcertainconditions,it willbeshownthatitisoptimalforthedefensiveteamtocooperatewiththeattacker inretreatsothatretreatbecomesamoreattractiveoptionthanengagementfromthe attackersperspective. Differentialgametheoryhasbeenusedforseveraldecadestoanalyzepursuit-evasion gamessinceitsformalintroductionbyIsaacs[1].Inparticular,therehavebeenseveral papersthataddresscombatusingrealisticdynamics[10,57].Inthesepapers,pursuit andcaptureistheonlyconditioninwhichthegameterminates.Therehasbeensome workthatallowstheplayerstoswitchrolesbetweenevaderorpursuerdependingon theinitialconditions[48].In[42],agameisanalyzedinwhichateamofUAVsattempts topostponeanattackbyanaerialjammeronthecommunicationchannel.There hasalsobeenworkdoneondefensivestrategieswithinsequentialgames.In[41], theauthorexaminesasequentialgameinwhichthedefensiveplayermustdistribute limitedresourcesinpreparationfortheopposingplayer'sattack.Itisassumedthat theattackerwillalwaysengage,buthigher-valueassetscanbeprotectedthroughthe properallocationofresources.Inthecontextofthisliterature,theprimarydistinguishing featureofmyworkisthattheattackeriscapableofsuccessfulcapturefromeveryinitial position,butthroughtheselectionofappropriatecontrolstrategies,thedefensiveteam makesretreatamoreattractivestrategyfortheattacker. Inthepreviouschapter,Ianalyzedasingle-pursuer,two-evadergameusingsimple motionwherethepursuerwasallowedtocaptureeitherevaderandthecostfunction wasdependentonbothevaderdistancesandtheanglebetweenthem.Byrestricting capturetoaspecicevaderthatgeneratesnocostandintroducingacostfunctionwith particularconvergenceproperties,Iintroducethepossibilityofretreatinthispaper.This resultsinnotonlypursuitbehaviorsbutalsoretreatbehaviors.Also,thisworkdiffers 46

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fromtheresultsofChapter2inthatIdeveloptheoptimalityconditionsintermsofan arbitrarynumber, N ,ofprotectiveagents. Ideveloptheoptimalintentselectionstrategiesoftheattackeranddefensiveteam inSection3.2.Next,Idescribethecoordinatesystems,systemkinematics,andcost functionoftheresultingpursuit-evasiongames.Followingthat,Idenetheoptimality conditionsofthegameofattackandgameofretreatinSections3.4andSection3.5 respectively.Usingtheintentselectionstrategiesandoptimalityconditions,Iexamine theresultingbehaviorsforthreeillustrativeexamplesinSection3.6. 3.2IntentSelectionStrategies Atthestartofthegame,eachplayermustdeterminetheirintentfortheentirety ofthegame.Theattackeranddefensiveteam'sintentarerepresentedbythediscrete controlvariables I A and I D respectively.Oncetheplayershavemadetheirselection, theycannotswitch.TherelaxationofthisrestrictionisdiscussedlaterinChapter4. Theselectionofintentisperformedinatwo-stepsequence.Theattackermustrst determinewhethertoengage, I A = i E ,orretreat, I A = i R .Aftertheattackermakesits selection,itisassumedthedefensiveteamknowstheattacker'sintentandmustthen choosetomaximize, I D = i + ,orminimize, I D = i )]TJ/F20 11.9552 Tf 7.085 -4.339 Td [(,theattacker'scostinresponse. Oncetheattackeranddefensiveteamhavemadetheirintentselection,their respectiveutilityfunctions, U A I A I D and U D I A I D ,areevaluated.Theutilityvalue pairs, U A U D ,arelistednexttotherightmostnodesinFig.3-1.Thesevaluesare basedontheintegralattackercostinthefourpossibledifferentialgamesaswellasany relevantbonusesorpenalties.Thegoalofeachplayeristomaximizetheirrespective utilities.Thevalues C E + and C E )]TJ/F20 11.9552 Tf 10.408 1.793 Td [(representthecostofengagementtotheattackerwhen thedefensiveteammaximizesorminimizesthecostrespectively.Thevalues C R + and C R )]TJ/F20 11.9552 Tf 10.409 1.794 Td [(representthecostofretreattotheattackerwhenthedefensiveteammaximize orminimizethecostrespectively.Itisassumedthat C E + C E )]TJ/F20 11.9552 Tf 10.408 1.793 Td [(and C R + C R )]TJ/F20 11.9552 Tf 7.085 1.793 Td [(. Theseassumptionswillbeveried,throughtheanalysisofthedifferentialgamesinthe 47

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followingsections.Thequantity B c 0 representsthebonustheattackerreceivesfor capturingthetarget,and B d 0 representsthepenaltythedefensiveteamreceives whenthetargetiscaptured.Itisassumedthatallofthesevaluesareknownorcanbe calculatedbyboththeattackerandthedefensiveteam. Inthisgame,thedefendingteam'ssolegoalistopreventcaptureofthehigh-value targetinthisone-shotgame.Idonotconsideranyfutureconfrontationswiththe attacker.Sinceitisassumedthattheattackerpossessessuperiorcapabilitiesthat guaranteesuccessfulcaptureifengagementisselected,thedefendingteam'sonly optionistomakeengagementsocostlythatitoutweighsanybonustheattacker gainsfromcapture.Thedefensiveteamgainsnodirectutilityfrominictingcoston theattacker.Instead,theattacker'scostisusedasatoolbythedefensiveteamto discourageengagement.Also,itisassumedthatanydifferenceinresourceusage bythedefensiveteambetweenmaximizingandminimizingattackercostisnegligible comparedtothepenaltyincurredwhenthehigh-valuetargetiscaptured.Therefore, defensiveteamutilityisdependentonlyontheattacker'sintent. Thefollowingtheoremdelineatestheoptimalintentstrategiesforboththeattacker andthedefensiveteamasafunctionofthedenedutilities. Theorem3.1. Let C E + representthecostofengagementtotheattackerwhenthedefensiveteammaximizescost.Let C R )]TJ/F48 11.9552 Tf 10.076 -0.299 Td [(representthecostofretreatwhenthedefensive teamminimizescost,and B c denotetheterminalbonusawardedtotheattackerfor engagement.Theoptimalintentstrategiesfortheattacker, I A ,anddefensiveteam, I D are I P = 8 > < > : i E if B c )]TJ/F39 11.9552 Tf 11.956 0 Td [(C E + )]TJ/F39 11.9552 Tf 21.918 0 Td [(C R )]TJ/F39 11.9552 Tf -116.538 -24.206 Td [(i R if B c )]TJ/F39 11.9552 Tf 11.956 0 Td [(C E + < )]TJ/F39 11.9552 Tf 9.299 0 Td [(C R )]TJ/F20 11.9552 Tf 133.762 13.448 Td [( I D = 8 > < > : i + if I A = i E i )]TJ/F48 11.9552 Tf 17.047 -4.339 Td [(if I A = i R 48

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Proof. Inordertocalculatetheoptimalintentstrategies,Ibeginbyrepresentingthe intentselectionprocessasthedirectedtreeinFig.3-1.Theupperbranchofthis treerepresentsthescenarioinwhichtheattackerhaselectedtoengageandforms asubgameforthedefensiveteam.Inthissubgame,choosingeithertomaximize orminimizeyieldsthesameutilityforthedefendingteambecausetheattackeris guaranteedsuccessfulcaptureduetoitssuperiorcapabilities.Similarly,inthelower subgameformedwhentheattackerchoosestoretreat,maximizingorminimizingyields equalutilitiesforthedefendingteam.However,thedefensiveteamdoespreferthat theattackerretreatssince 0 )]TJ/F39 11.9552 Tf 22.896 0 Td [(B d .Inordertoencourageretreat,thedefensiveteam mustminimizeattackerutilityifengagedandmaximizeattackerutilityinretreat,which isachievedbyEquation3.Theattackerassumesthatthedefendingteamposesa crediblethreatandwillimplementEquation3.Themaximumattackerutilityisthen achievedbyEquation3. FromTheorem3.1,itcanbeseenthatthevalues C E + C R )]TJ/F20 11.9552 Tf 7.084 1.793 Td [(,and B c playacritical roleinthecalculationoftheoptimalplayerintent.Thevalue C E + representsthevalue ofthedifferentialgameofengagementwhenthedefensiveteamismaximizingattacker cost,andthevalue C R )]TJ/F20 11.9552 Tf 10.408 1.793 Td [(representsthedifferentialgameofretreatwhenthedefensive teamisminimizingattackercost.Fortheremainderofthepaper,Iwillformulateand denetheoptimalityconditionsofthesedifferentialgames.Theresultingsolutionsto thesegameswillthenbeusedtocalculatetheoptimalintentstrategiesforgiveninitial conditionsandvaluesof B c 3.3SystemandDifferentialGameFormulation Inthesystemunderconsideration,theattackerisrepresentedbyapursuer,and thedefendingteamconsistsofthemobile,high-valuetargetand N protectiveagents. Theattacker,mobiletarget,and N protectiveagentswillbedenotedby P E 0 ,and E i for i =1,..., N ,respectively.Forthesakeofbrevity,Iwilloftenomittheclarication 49

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that i =1,..., N .Thereforewheneveravariableusesthesubscript i ,itisassumedthat i =1,..., N unlessexplicitlystatedotherwise. 3.3.1AgentKinematics Eachagentmoveswithsimplemotionandconstantspeedonanobstaclefree plane.Iwillusetwocoordinatesystems.Therstcoordinatesystemrepresentsthe locationofeachagentusingapairofCartesiancoordinates.Theattackerlocationis representedbythepair x p y p while x 0 y 0 and x i y i representthepositionsofthe mobiletargetandprotectiveagents.Thestateofthesystemiscompletelydenedby theN+4-tuple x G = x p y p x 0 y 0 ,..., x N y N .Iwillrefertothisrepresentationasthe GlobalCoordinates ,andtheresultingdynamicequationsareasfollows: x p = v p cos ^ x 0 = v 0 cos ^ 0 x i = v i cos ^ i y p = v p sin ^ y 0 = v 0 sin ^ 0 y i = v i sin ^ i Theconstants v p v 0 ,and v i ,representthespeedsoftheattacker,thetarget, andtheprotectiveagents.Theangles ^ ^ 0 ,and ^ i arethedirectionoftraveland representthecontrolvariablesforeachoftheirrespectiveagents.Allcontrolvariables aremeasuredcounterclockwisefromthex-axis.Thiscoordinatesystemisdepicted graphicallyinFig.3-2. Inordertosimplifylateroptimalitycalculations,Iwillnowintroducethe Relative Coordinates .Intherelativecoordinatesystem,thestateiscompletelydenedbythe N+4-tuple x R = d 0 d 1 ,..., d N 1 ,..., N x y .ThedistancebetweenthePandE 0 willberepresentedby d 0 .Similarly,thecomponent d i representsthedistancebetween PandE i .Theanglemeasuredcounterclockwisefrom )430()430(! PE 0 to )229(! PE i isrepresentedby thestatecomponent i .Theangle ismeasuredcounterclockwisefromthex-axis to )430()430(! PE 0 andrepresentstheglobalrotationoftherelativecongurationoftheagents.In ordertosimplifythedynamicequations,theheadingangles and 0 aremeasured counterclockwisefrom )430()430(! PE 0 ,andtheangles i aremeasuredcounterclockwisefrom 50

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)229(! PE i .Theremainingstatecomponents, x and y ,aretheglobal x and y positionofthe attackerandrepresenttheglobaltranslationalpositionoftheN+2-agentsystem. ThiscoordinatesystemisdepictedinFig.3-3.Thedynamicequationsfortherelative coordinatesystemareasfollows: d 0 = v 0 cos 0 )]TJ/F39 11.9552 Tf 11.956 0 Td [(v p cos d i = v i cos i )]TJ/F39 11.9552 Tf 11.955 0 Td [(v p cos )]TJ/F25 11.9552 Tf 11.955 0 Td [( i i = v i d i sin i )]TJ/F40 7.9701 Tf 13.397 4.884 Td [(v 0 d 0 sin 0 + v p 1 d 0 sin )]TJ/F23 7.9701 Tf 14.373 4.708 Td [(1 d i sin )]TJ/F25 11.9552 Tf 11.955 0 Td [( i = 1 d 0 sin 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v p sin x = v p cos + y = v p sin + where d i > 0 and v i < v p for i =0,1,..., N .Theseinequalitiesrequirethatall distancesremainpositiveandthatPisfasterthan E 0 and E i .Theglobalandrelative representationsarerelatedusing, x p = x y p = y x p = d 0 cos + x y p = d 0 sin + y x i = d i cos + i + x y i = d i sin + i + y Thecontrolvariablesarerelatedthoughthefollowingequations. ^ 0 = 0 + ^ i = i + + i ^ = + 51

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3.3.2InstantaneousCostFunction Inowdenetheinstantaneouscosttotheattackerwhichisintegratedoverthe totalcourseofthegame.Thisfunctioncanrepresenttheriskofinjuryortheamountof damagethattheattackerincursatanyinstantintime.Forthisgame,Ihavechosenthe followingcostfunction: C T = c 0 + N X i =1 c i 1 d 2 i where c i and c 0 areweightingparameters.Asany d i 0 ,theinstantaneouscost explodestoinnity.Asaresult,itisimpossiblefortheattackertopassdirectlythrough anyoftheprotectiveagentswithnitecost.Therefore,theattackermustgoaroundthe protectiveagentsinordertocapturethetarget.Sinceitisassumedthattheattacker hasaspeedadvantageoverallagentswithinthedefensiveteam,theattackerwillthen guaranteethat d i t > 0 inordertomaintainnitecost.Also,itwillbeshowninalater sectionthatthegameofretreattoaninnitedistancehasnitecostwhen c 0 =0 Althoughthisparticularcostfunctionwaschosenbecauseofitssimplicity,morerealistic andcomplexfunctionscouldbeusedinordertomodelparticularattacker-defender interactions.Ifthesemorerealisticcostfunctionspossesscharacteristicssimilarto thosejustdescribed,theresultingoptimalagentbehaviorswillbesimilartothose developedinthispaper. 3.3.3DifferentialGameFormulation Dependingontheplayers'intentselection,variousdifferentialgamescanbe formulated.Ineverygame,theinstantaneouscostfunctionEquation3isintegrated untilthegameterminatesandrepresentsthetotalcosttotheattacker.Theterminal conditionswillbediscussedinSection3.4andSection3.5.Thecosttotheattackerfora gamestartingat t 0 andterminatingat t f isthendenedas V := Z t f t 0 C T dt 52

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Iftheintentofthedefendingteamistomaximizetheattacker'scost,Icanuse thevaluefunctionEquation3toposeatwo-player,zero-sumdifferentialgame. Althoughthereare N +1 agentswithinthedefendingteam,theyallsharethesame goalofmaximizingtheattacker'scostandcanthereforeberepresentedasoneplayer withmultiplecontrolvariables.Iftheintentofthedefendingteamistocooperatewiththe attackerandminimizecost,thedifferentialgamenowreducestoastandardoptimization problemwith N +2 controlvariables. 3.4OptimalityConditionsoftheDifferentialGameofEngagementwithDefender Maximization Inthissection,Idevelopthesolutionforthegameofengagement.Inthisgame, thedefensiveteamstrivestomaximizetheattacker'sintegralcostoverthecourseofthe game.Simultaneously,theattackerattemptstominimizethiscost.Thegameterminates whenthedistancebetweentheattackerandhigh-valuetarget, d 0 ,isequaltothecapture distanceof d c 3.4.1HamiltonianandAdjointEquations IbegincalculatingthesolutiontothegameofattackbyconstructingtheHamiltonian: H := T f x + C T =0 = N X i =0 d i d i + N X i =1 i i + + x x + y y + C T Thevector := d 0 ,..., d N 1 ,..., N x y T containstheadjointvariables conjugatetothekinematicequations.Theadjointequationsarefoundbytakingthe partialderivativeoftheHamiltonianwithrespecttoeachofthestatecomponents: d 0 = )]TJ/F26 7.9701 Tf 13.957 4.707 Td [(@ H @ d 0 = )]TJ/F40 7.9701 Tf 16.634 14.944 Td [(N X i =1 @ i @ d 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( @ @ d 0 d i = )]TJ/F26 7.9701 Tf 13.299 4.707 Td [(@ H @ d i = )]TJ/F25 11.9552 Tf 9.299 0 Td [( i @ i @ d i )]TJ/F26 7.9701 Tf 13.151 5.112 Td [(@ C T @ d i i = )]TJ/F26 7.9701 Tf 13.832 4.707 Td [(@ H @ i = )]TJ/F25 11.9552 Tf 9.299 0 Td [( d i @ d i @ i )]TJ/F25 11.9552 Tf 11.955 0 Td [( i @ i @ i = )]TJ/F26 7.9701 Tf 13.151 4.707 Td [(@ H @ = )]TJ/F25 11.9552 Tf 9.299 0 Td [( x @ x p @ )]TJ/F25 11.9552 Tf 11.955 0 Td [( y @ y p @ 53

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_ x = )]TJ/F26 7.9701 Tf 14.379 4.707 Td [(@ H @ x P =0 y = )]TJ/F26 7.9701 Tf 14.379 4.707 Td [(@ H @ y P =0. 3.4.2BoundaryConditions Usingthedenitionofcapture, d 0 = d c ,theboundaryconditions, A ,forthegame ofattackare A := d 0 t f )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c d 0 t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d 00 ,..., d N t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d N 0 1 t 0 )]TJ/F25 11.9552 Tf 11.956 0 Td [( 10 ,..., N t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( N 0 t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 x t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x 0 y t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 0 T where d i 0 i 0 0 x 0 ,and y 0 aretheinitialvaluesoftheirrespectivestatecomponents atthestartofthegame.Wecanthenconstructafunctionofterminalconditions, A = T A where isavectorofLagrangemultiplierscorrespondingtotheboundary conditions. Theterminalvaluesoftheadjointvariablesarefoundbytakingthepartialderivative of A withrespecttoeachofthestatecomponents: d 0 t f = @ @ d 0 t f = 1 t f = @ @ t f =0 d i t f = @ @ d i t f =0 x t f = @ @ x t f =0 i t f = @ @ i t f =0 y t f = @ @ y t f =0 InordertosimplifytheHamiltonian,wecananalyticallysolvefor x ,and y Combiningtheterminalvaluesof t f x t f ,and y t f withtheirrespectiveadjoint 54

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equationsEquation3-Equation3itcanbeseenthat t =0 x t =0 y t =0. UsingEquation3,wecansimplifytheHamiltonianEquation3: H = N X i =0 d i d i + N X i =1 i i + C T =0. UsingthereducedHamiltonianEquation3,theoptimalcontrolstrategiesforeach oftheagentsarecalculatedinthefollowingtheorem. Theorem3.2. Supposethatthevaluefunctionandthevaluefunctiongradientare continuous.Thecontrolstrategiesfortheagentsarethengivenby OptimalMaximizingControlStrategyofE 0 : cos 0 = d 0 0 sin 0 = )]TJ/F31 7.9701 Tf 7.998 5.978 Td [(P N i =1 i d i 0 0 = r 2 d 0 + )]TJ 7.472 1.674 Td [(X N i =1 i d i 2 OptimalMaximizingControlStrategyofE i i =1,..., N : cos i = d i i sin i = i d i i i = q 2 d i + i d i 2 OptimalMinimizingControlStrategyofP : cos = )]TJ/F39 11.9552 Tf 10.53 8.088 Td [(b 1 p sin = )]TJ/F39 11.9552 Tf 10.53 8.088 Td [(b 2 p p = q b 2 1 + b 2 2 55

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where b 1 = N X i =1 i d i sin i )]TJ/F25 11.9552 Tf 11.955 0 Td [( d i cos i )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 b 2 = N X i =1 i d 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( d i sin i + i d i cos i Proof. Alongtheoptimaltrajectories,theHamiltonianmustsatisfythefollowing conditions[16]: H x H x H x H x =0 where = 0 ,..., N .FromEquation3,wendthat =argmin H 0 ,..., N =argmax 0 ,..., N H Becausethecontrolvariablesareunbounded,theoptimalstrategiesofEquation 3mustsatisfythefollowingconditions: @ H @ =0 @ H @ i =0 for i =0,..., N @ 2 H @ 2 0 @ 2 H @ 2 i 0 for i =0,..., N Therstsetofconditions,Equation3,gauranteestheHamiltonianisstationary withrespecttothecontrolvariables.ThesecondsetofequationsEquation3, representthenecessarysecond-orderconditionssothat maximizesand minimizes. SolvingEquation3andEquation3,intermsof 0 ,..., N ,and provideour optimalcontrolstrategiesEquation3andEquation3. 3.5OptimalityConditionsoftheDifferentialGameofRetreatwithDefender Minimization Inthisgame,theattackerisattemptingtoreachtheretreatconditionwithminimal integralcost.DenitionoftheretreatconditionrequirestheuseoftheminimumofN quantities.Inprinciple,itispossibletodothis,butsincetheminimumfunctionisnot 56

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differentiableeverywhere,thereisalargenumberofsingularsurfaceswhichmake analysisofthegamecomplicated.Instead,Idenetheretreatcondition, d m t f )]TJ/F39 11.9552 Tf 11.678 0 Td [(d r =0 usingthe p -normwithrespecttothe 1 d i 'scorrespondingtotheprotectiveagentswhere d m := X 1 d k i )]TJ/F23 7.9701 Tf 7.979 4.707 Td [(1 k and k > 1. SinceIrestrict d i > 0 ,thefunctionisdifferentiableeverywherewithintheadmissible statespace.As k !1 d m convergesto )]TJ/F22 11.9552 Tf 5.48 -9.684 Td [(max 1 d 1 ,..., 1 d N )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 =min d 1 ,..., d N [58]. Theattackeranddefendingteamarebothminimizingthecostfunction.Therefore, thedifferentialgamereducestoastandardoptimalcontrolproblemwithrespecttoall agents. 3.5.1HamiltonianandAdjointEquations Sincethedynamicsandcostfunctionarethesameasinthegameofattack,the gameofretreathasanidenticalHamiltonianEquation3andresultingadjoint equationsEquation3-Equation3. 3.5.2BoundaryConditions Usingtheconditionofretreat, d m t f )]TJ/F39 11.9552 Tf 12.222 0 Td [(d r =0 ,Icanformtheboundaryconditions, R ,forthegameofretreat: R := d m t f )]TJ/F39 11.9552 Tf 11.955 0 Td [(d r d 0 t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d 00 ,..., d N t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d N 0 1 t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 10 ,..., N t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( N 0 t 0 )]TJ/F25 11.9552 Tf 11.955 0 Td [( 0 x t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(x 0 y t 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y 0 T where d 00 ,..., d N 0 1 ,..., N 0 x 0 ,and y 0 aredenedthesameasintheprevious section.Wewillnowcalculatetheterminalboundaryconditionsoftheadjointvariables. Afterconstructingafunctionofboundaryconditions, R := T R ,andtakingpartials withrespecttoeachofthestatecomponents,wehavetheterminalconstraintsonthe 57

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adjointvariables: x t f = @ @ x t f =0 t f = @ @ t f =0 i t f = @ @ t f =0 y t f = @ @ y t f =0 d 0 t f = @ @ d 0 t f =0 d i t f = @ @ d i t f = 1 1 d i t f k +1 X 1 d i t f k )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F40 7.9701 Tf 6.586 0 Td [(k k Asinthegameofattack,theadjointvariablescorrespondingtothe x ,and y componentsofthestatearealwayszeroEquation3andwecanfurtherreduce theHamiltonianasbefore.UsingthereducedHamiltonianEquation3,wecannow calculatetheoptimalcontrolstrategiesforeachoftheagentsintermsofthestateand adjointvariables. Theorem3.3. Supposethatthevaluefunctionandthevaluefunctiongradientare continuous.Thecontrolstrategiesfortheagentsarethengivenby OptimalMinimizingControlStrategyofE 0 andE i : cos 0 = )]TJ/F25 11.9552 Tf 10.494 8.088 Td [( d 0 0 sin 0 = P N i =1 i d i 0 0 = r 2 d 0 + )]TJ 7.472 1.674 Td [(X N i =1 i d i 2 OptimalMinimizingControlStrategyofE i i =1,..., N : cos i = )]TJ/F25 11.9552 Tf 10.494 8.088 Td [( d i i sin i = )]TJ/F25 11.9552 Tf 12.141 8.088 Td [( i d i i i = q 2 d i + i d i 2 58

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OptimalMinimizingControlStrategyofP : cos = )]TJ/F39 11.9552 Tf 10.53 8.088 Td [(b 1 p sin = )]TJ/F39 11.9552 Tf 10.53 8.088 Td [(b 2 p p = q b 2 1 + b 2 2 wheretheterms b 1 and b 2 aredenedthesameasinTheorem3.2. 3.6IllustrativeExamples 3.6.1NumericalAnalysisofGameofEngagement Inmostcases,ndingananalyticsolutiontotheoptimaltrajectoriesforthe differentialsubgamesisnotpracticalduetothenonlinearandcouplednatureofthe stateandadjointequations.Inordertonumericallygeneratetheoptimaltrajectories thatresultfromthepreviouslydevelopedoptimalityconditions,Irstsubstitutethe optimalcontrolstrategiesintothekinematicequationsEquation3-Equation3 andtheadjointequationsEquation3-Equation3.Thisresultsinasystem of 4 N +6 ordinarydifferentialequationsinadditiontotheintegralcostfunction.These equationscanbenumericallyintegratedbackwardsintimefromanypermissiblepoint ontheterminalsurfaceforadenedtimespanoruntilthetrajectorycrossesasingular ordispersalsurface,whichwillbediscussedinalatersection. Inthegameofengagement,Icancompletelydenetheterminalconditions.After substitutingtheoptimalcontrolstrategiesintotheHamiltonianEquation3and evaluatingatthepointofcapture,Icansolvedirectlyfor d 0 t f = C T t f v p )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 .Itcanalsobe seenthattheprotectiveagents'terminalcontrolanglesareundenedatthemoment ofcaptureduetothefactthat d i t f = i t f =0 .Conceptuallythismakessense becauseatthemomentofcapture,theprotectiveagentscannotpreventthecaptureof E 0 .Also,anyincreaseinthecostfunction C T willnotbeintegratedbecausethegame willterminate.ItisstillnecessarytodeneaterminalcontrolforE i inordertotakethe rststepofintegration.Forthisvalue,IwillusethelimitofE i 'scontrolas t approaches 59

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t f .Takingthelimitof tan i yields lim t t f tan i t =lim t t f i d i d i =lim t t f i d i d i + d i d i = 0 2 ThecombinationofEquation3andthefactthat d i t f = 2 d i t f 3 > 0 impliesthat lim t t f i t f = .Inowhaveacompletesetofterminalvaluesforthestate,theadjoint variables,andcontrol,whichallowustoinitializethenumericalintegration.Icanthen useshootingtechniquestosolveforparticularinitialconditions. 3.6.2NumericalandAnalyticSolutiontotheGameofRetreat Foranarbitrarynumberofdefendingagents N > 1 andaniteretreatdistance d r thesamenumericalshootingmethodsasintheprevioussectionareusedtosolvefor theoptimalagenttrajectories.Inthiscase,theminimizingdefendercontrolissubstituted intothedynamicandadjointequationsinordertogeneratedthesystemof 4 N +6 differentialequations.Additionally,theterminalretreatsurfaceandcorrespondingadjoint conditionsareusedfortheterminalconstraints. When N =1 ,ananalyticsolutiontothegameofretreatcanbecalculated.First,the terminalconditionofretreatreducesto d 1 t f )]TJ/F39 11.9552 Tf 12.431 0 Td [(d r =0 .Aftersubstitutingtheterminal constraintsofthestateandadjointvariablesintotheHamiltonian,Icansolvedirectlyfor d 1 t f = )]TJ/F40 7.9701 Tf 12.326 5.699 Td [(c 0 + c 1 d 2 r d 2 r + v p 0 .Usingtheterminalvaluesoftheadjointvariablesandstate,I canalsondtheterminalcontrolof E 1 andtheattacker: 1 t f =0 and t f = + Substitutingtheterminalcontrolintotheadjointderivativesevaluatedattheterminal surfaces, d 2 t f = d r ,yields 1 t f =0 2 t f = 2 c 0 d 3 r ,and t f =0 .Afterintegrating backwardsintime,wendthat 1 t =0 2 t < 0 t =0. 60

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Fortheentiregameofretreat,theoptimalcontrolofE 0 isundenedbecauseE 0 hasnoeffectonthecostfunctionorwhenthegameterminates.Thereforeanycontrol strategyistriviallyoptimal,andIwillassumethat 0 t =0 .Wendtheoptimalcontrol strategiesofE 1 andP: cos 1 t =0 and cos t = + .Wecanthencalculatethe optimaltrajectoryofthe d 1 -componentfromanyinitialcondition: d 1 t = d 10 ++ v p t Assumingthattheinitialdistance, d 10 ,islessthantheretreatdistance,theterminal timeiscalculatedusingEquation3: t f = d r )]TJ/F40 7.9701 Tf 6.587 0 Td [(d 10 1+ v p .Icanthencalculatethevalueof thegame: V d 10 = Z t f t 0 c 1 d 1 t 2 + c 0 = Z t f 0 c 1 d 10 ++ v p t 2 + c 0 = t f c 0 + c 1 d 10 d 10 ++ v p t f Forthespecialcasewere c 0 =0 and d r !1 ,thevalueofthegameofinniteretreat converges: lim d r !1 V d 10 j c 0 =0=lim t f !1 t f c 1 d 10 d 10 ++ v p t f = c 1 d 10 + v p 3.6.3SingularSurfaces Withinthisgame,therearecertaincongurationsinwhicheithertheattackeror defendingagents'optimalcontrolisnotuniquelydened.Thisistypicallyaresultof symmetrywithinthedynamics.Forexample,whentheattacker,target,anddefending agentsareinacollineararrangement,theattackercanchoosetorotateclockwiseor counterclockwise.Eitheroptionisoptimal.Theseregionsofthestate-spacearereferred toassingularsurfacesandarecharacterizedbydiscontinuitiesofthevaluefunctionor 61

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itsgradient.Inthispaper,Ionlyconsiderinitialconditionsonsingularsurfacesknownas dispersalsurfacesinwhichthestateinstantaneouslymovesoffandtheoptimalcontrol isthenuniquelydened.Fullanalysisofsingularsurfacesisbeyondthescopeofthis researchandisatopicleftforfuturework. 3.6.4OptimalIntentSelection Afterthedifferentialgamesaresolvedforparticularinitialconditions,wemayuse thevaluesofgameofengagementandgameofretreatfor C E + and C R )]TJ/F20 11.9552 Tf 10.408 1.794 Td [(inEquation 3andEquation3.Also,weknowthattheassumptions C E + C E )]TJ/F20 11.9552 Tf 10.408 1.794 Td [(and C R + C R )]TJ/F20 11.9552 Tf 10.409 1.793 Td [(holdfromTheorem3.2andTheorem3.3.Therefore,foranygiven B c ,Ican calculatetheoptimalintentstrategiesusingTheorem3.1.Foraxedvalueof B c thestatespacecanbedividedintotworegions.Inoneregion,thecapturebonus, B c ,offsetsthecostofattack, C E + ,andtheattackerelectstoengagedespitethe maximizationofcostbythedefensiveteam.Intheremainderofthestatespace,the maximizedcostofattacknegatesanybenetthatthecapturebonusgrantswhen comparedtotheminimizedcostofretreat.Therefore,theattackerelectstoretreatand thedefensiveteamcooperatestominimizethecost. 3.6.5IllustrativeCases Inthefollowingscenarios,Isetthesystemparameters d c =1 v 0 = v i =1 c 0 =0 and c i =1 .Ashootingmethodisusedinordertosolvefortheinitialvalueoftheadjoint variablesinthegameofengagementaswellasthegameofniteretreatfor N > 1 InFig.3-4throughFig.3-6thetrajectoryoftheattackeristhesolidline,thetrajectory ofthemobiletargetisthedashedline,andthetrajectoryoftheprotectiveagentis thedottedline.AlltrajectoriesareplottedintheglobalcoordinatesusingEquation 3-Equation3toconvertfromtherelativecoordinatesystem. InScenario1,thereisonedefendingagent, N =1 ,andtheretreatdistanceis takentoinnity.Theattackerhasamoderatespeedadvantage, v p =2 .Thisscenario isshowninFigure3-4.Sincetheattackerstartsfarawayfromtheprotectiveagent,very 62

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littlecostisgeneratedearlyinthepursuit.Theprotectiveagentforcestheattackerto comeclosemid-pursuitinordertooutankitandcapturethetarget.Theresultingcost ofengagementandretreatforthisscenarioare1.4and.02respectively.Engagementis optimalwhen B c 1.38 .InScenarios2and3,therearefourdefendingagents, N =4 andtheretreatdistanceistakentobe d r =20 .AscanbeseeninFigure3-5andFigure 3-6,themultipleprotectiveagentsconvergeonthepursuerfrommultipledirections forcingtheattackertoweavethroughthem.InScenario2,thecostsofengagementand retreatare10.4and1.7respectively.Engagementisoptimalwhen B c 8.7 .InScenario 3,thecostsofengagementandretreatare16.2and2.3respectively.Engagementis optimalwhen B c 13.9 63

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Figure3-1.IntentSelectionTree Figure3-2.Globalcoordinates 64

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Figure3-3.Relativecoordinates Figure3-4.AttacktrajectoriesScenario1 65

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Figure3-5.AttacktrajectoriesScenario2 Figure3-6.AttacktrajectoriesScenario3 66

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CHAPTER4 GENERALIZEDATTACKRETREATGAMEWITHESCORTREGIONS InworkpresentedinChapter2andChapter3,Iexploredtheconceptofdefensive strategiesinadversarialgames.Inthegamepresentedin[56],oneplayerrepresenteda mobileattackerandtheotherplayerrepresentedadefendingteamconsistingofseveral protectiveagentsandamobile-highvaluetarget.Similartothegamepresentedchapter, theattackerselectedwhethertoengageorretreatandthedefendingteamchose whethertomaximizeorminimizetheattacker'sutility.Intheformulationconsideredin [56],eachsidemadeitsselectionatthestartofthegameandwasnotallowedtoswitch strategiesaftertheinitialselection.Thiswasasignicantrestrictionandsometimes generatedcounterintuitiveretreattrajectorieswherethedefensiveteamhelpedthe attackermoveclosertothehigh-valuetarget. Theformulationpresentedinthischapterremovesthisrestrictionandallowseither sidetoswitchgoalsifitbecomesoptimal.Throughtheincorporationofavaluefunction constraint,theattackerispreventedfrommovingintoregionsofoptimalengagement. Ishowthatitisneveroptimalforeithersidetoswitchfromtheirinitiallyselectedgoals, whichdependontheinitialstatesofeachplayerandthevalueoftherelateddifferential games. 4.1GeneralGameDescription ThegeneralEngageorRetreatGameconsistsoftwoplayers,PlayerAandPlayer B,thatmanipulatethestateofthesystemthroughtheuseoftheirrespectivecontrol vectorsinordertomaximizetheirrespectiveutilityfunctionals.PlayerA'scontrolvector isdenedas u A 2 R j ,andPlayerB'scontrolvectorisdenedas u B 2 R k .Thestate ofthesystemisrepresentedbyann-dimensionalvector x 2 R n .Thebehaviorofthe systemisdescribedbyasystemof n ordinarydifferentialequations: x = f x u A u B 67

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PlayerA'sutilityfunctional U A u A t u B t consistsofaterminalvaluefunctionin additiontoanintegralcostfunction.PlayerB'sutilityfunctional U B u A t u B t consists ofonlyaterminalvaluefunction.Theutilityfunctionalsaredenedas: U A u A t u B t := Z t f t 0 C A x t u A t u B t dt + A x f U B u A t u B t := B x f InEquation4,thefunction A x f istheterminalvaluefunctionforPlayerA,and C A x t u A t u B t istheinstantaneouscostfunctionthatisintegratedoverthe courseofthegame.Thefunction B x f istheterminalvaluefunctionforPlayerB. Thevectors x 0 and x f representthestateatinitialtime, t 0 ,andthestateatterminal time, t f ,respectively.Theterminaltime, t f ,isdenedasthemomentthestateofthe systemsatiseseithertheretreatcondition )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(R x = 0 ortheengagementcondition )]TJ/F40 7.9701 Tf 6.941 -1.793 Td [(E x = 0 .Thefunctions )]TJ/F40 7.9701 Tf 6.941 -1.793 Td [(R and )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(E arevectorvalued,andthedimensionoftheirrange mustbelessthanorequalto n )]TJ/F22 11.9552 Tf 10.745 0 Td [(1 .Thenotation 0 representsavectoroftheappropriate dimensioninwhicheachelementiszero.Vectorequalityisappliedelement-wise.We willrefertothesetsofstatevaluesthatsatisfytheseconditionsastheretreatsurface, X R := f x 2 R n j )]TJ/F40 7.9701 Tf 6.941 -1.794 Td [(R x = 0 g andtheengagementsurface, X E := f x 2 R n j )]TJ/F40 7.9701 Tf 6.941 -1.794 Td [(E x = 0 g Eachplayerpossessesapreferenceastowhetherthegameterminatesinretreator engagement.Iftheintegralcostisignored,itisassumedthatPlayerAprefersthatthe gameterminatesinengagementoverretreat: A x E > A x R 8 x E 2 X E x R 2 X R 68

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Conversely,itisassumedthatPlayerBprefersthatthegameterminatesbyreachingthe retreatsurface: B x E < B x R 8 x E 2 X E x R 2 X R ItisalsoassumedthatPlayerApossessessuperiorcontrolanddynamiccharacteristics suchthatitiscapableofforcingthestateto x E 2 X E and x R 2 X R fromanyinitial state x 0 .Therefore,PlayerAisfreetochoosetoterminatethegameinretreator engagement,andPlayerBisunabletodirectlypreventPlayerAfromengaginginattack. PlayerB'sonlyoptionistoattempttomanipulatetheintegralportionof U A insucha waythatthereexistsatrajectoryfrom x 0 to x R thatprovidesabetterutilityvalueforthe attackerthanalltrajectoriesfrom x 0 toany x E 2 X E .IfPlayerBcancreatethissituation, PlayerAwillthenelecttoretreatratherthanengageinattack,whichwillmaximize PlayerB'sutilityfunctionsince x R < x E .Inthefollowingsection,wepropose asolutionmethodtodetermineifthisconditioncanbemetandwhattheresulting equilibriumstrategiesareforeachoftheplayers. 4.2GeneralSolutionTechnique InordertothesolvetheEngageorRetreatGame,werstaddresstwodifferential subgames,whichwewillrefertoastheDifferentialSubgameofEngagementDSEand theDifferentialSubgameofRetreatDSR.Wewillthenshowthatforanygiveninitial state x 0 theequilibriumsolutiontothegeneralEngageorRetreatGameisprovidedby thesolutiontoeithertheDSEorDSR. 4.2.1DifferentialSubgameofEngagement WewillrstexaminetheDSE.Inthisgame,PlayerAattemptstoterminatethe gameinengagementwhilemaximizingitsrespectiveutilityfunction.Simultaneously, PlayerBattemptstominimizePlayerA'sutilityfunction.Usingtheseplayergoalsalong withthesystemdynamicsEquation4,theresultingdifferentialgameisdenedas V E x 0 :=max u A t min u B t U A u A t u B t 69

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=min u B t max u A t U A u A t u B t withtheconstraintthat )]TJ/F40 7.9701 Tf 6.941 -1.793 Td [(E x f =0 Thefunction V E x 0 representstheequilibriumvalueofthegamestartingat x 0 whenPlayerAandPlayerBimplementtheirrespectiveoptimalcontrolstrategies u A t and u B t ,where u A t u B t =argmax u A t min u B t U A u A t u B t =argmin u B t max u A t U A u A t u B t Webegincalculatingthesolutiontothegameofengagementbyconstructingthe Hamiltonian: H := T E f x u A u B + C A x u A u B =0. Thevector E containstheadjointvariablesconjugatetothekinematicequationsand alsorepresentsthegradientofthevaluefunction: E := 1 2 ,..., n T = @ V E @ x 1 @ V E @ x 2 ,..., @ V E @ x n T Thetimederivativeoftheadjointvariablesarefoundbytakingthepartialderivative oftheHamiltonianwithrespecttoeachofthestatecomponents: E = )]TJ/F25 11.9552 Tf 10.494 8.088 Td [(@ H @ x T = )]TJ/F26 7.9701 Tf 11.054 4.707 Td [(@ H @ x 1 )]TJ/F26 7.9701 Tf 11.054 4.707 Td [(@ H @ x 2 ,..., )]TJ/F26 7.9701 Tf 11.104 4.707 Td [(@ H @ x n T Usingtheterminalconstraintforengagement, )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(E x f =0 ,wecanconstructanadjoined terminalvaluefunction: E x f := A x f + T E )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(E x f where E isavectorofLagrangemultipliersoftheappropriatedimensioncorresponding totheterminalconstraints.Theterminalvaluesoftheadjointvariablesarefoundby 70

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takingthepartialderivativeof E x f withrespectto x f : E t f = @ E @ x f T = @ E @ x 1 @ E @ x 2 ,..., @ E @ x n T 4.2.2OptimalControl UsingtheHamiltonianEquation4,theoptimalcontrolstrategiesforeachof theagentsarestatedinthefollowingtheorem. Theorem4.1. Supposethatthevaluefunctionandthevaluefunctiongradientsare continuous.Theequilibriumcontrolstrategiesfortheagentsarethengivenby u E A u E B =argmax u A min u B H =argmin u B max u A H Thesolutionsfor u E A and u E B thatresultfromTheorem4.1arefunctionsof x and E .Althoughthevaluesfor x t and E t areunknownforarbitraryvaluesof t ,we cansubstitutethefunctionsfor u E A and u E B intothesystemdynamicsEquation4 andadjointequationsEquation4.Theresultingsystemof 2 n ordinarydifferential equationsalongwiththeinitialstate x 0 ,terminalstateconstraints )]TJ/F40 7.9701 Tf 6.941 -1.794 Td [(E x =0 ,and terminaladjointconstraintsEquation4createaboundaryvalueproblemthat canbesolvedanalyticallyornumerically.Thissolutionconsistsoftheequilibrium statetrajectory x E t x 0 ,adjointvalues E t x 0 ,valuefunction V E x 0 ,andcontrol strategies u E A t x 0 and u E B t x 0 forthegameofengagementthatbeginswithinitial state x 0 .Iftheequilibriumstrategiesareimplemented,theresultingutilityfunctionvalue forPlayerAandPlayerBare U E A x 0 := U A u E A t x 0 u E B t x 0 U E B x 0 := U B u E A t x 0 u E B t x 0 4.2.3DifferentialSubgameofRetreat Inthisgame,PlayerAterminatesthegameinretreatwhileattemptingtomaximize itsutilityfunction U A .Simultaneously,PlayerBalsoattemptstomaximizePlayerA's 71

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utilityfunctionwhilepreventingPlayerAfrommovingthestatetoaregionwhereit wouldbeadvantageoustoengage.Usingtheseplayerobjectivesalongwiththesystem dynamicsEquation4,theresultingdifferentialgameisdenedas V R x 0 :=max u A t max u B t U A u A t u B t =max u A t ,u B t U A u A t u B t withtheterminalconstraintthat )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(R x f = 0 .Sincebothplayersarestrivingto maximizingPlayerA'sutilityfunction,thedifferentialgamecanbeviewedasatraditional continousoptimalcontrolproblem. Inordertoensurethatthestatedoesnotmoveintoaregionwhereengagementis anoptimaloptionforPlayerA,weimposeaconstraintonthevaluefunctionofretreat: V R x t )]TJ/F39 11.9552 Tf 11.955 0 Td [(V E x t 0 8 t 2 [ t 0 t f ]. Thisvaluefunctionconstraintcanbeconvertedintoastateinequalityconstraintby addinganadditionalstatecomponent, c t ,withatimederivativedenedas c := )]TJ/F39 11.9552 Tf 9.299 0 Td [(C A x t u A t u B t Thisstatecomponentrepresentstheremainingintegralcostforrestofthegame,andit hasaterminalvalueconstraintof c t f =0. Thevaluefunction V R x cannowbecalculatedusingthisstatecomponent: V R x = A x t f + Z t f t 0 C A x t u A t u B t = A x t f )]TJ/F39 11.9552 Tf 11.956 0 Td [(c t f + c t 0 = A x t f + c t 0 72

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UsingEquation4,wecannowtransformEquation4intoastateinequality constraint: g x := A x t f + c t )]TJ/F39 11.9552 Tf 11.955 0 Td [(V E x t 0. InordertoincorporatethisconstraintintotheHamiltonian,acontrolvariable mustbepresent.Ingeneral g x isnotanexplicitfunctionofcontrol.Therefore, g x issuccessivelydifferentiatedwithrespecttotimeandthedynamicequations4 aresubstitutedinto x untilanexpressionthatisexplicitlydependenton u A or u B appears.Thisresultsinacontrolconstraint h x thatwillbeactivatedonlywhenthe stateconstraint, g x isactive: h x := d m dt m g x Eachtime g x isdifferentiatedaconstanttermislostinthedifferentiation.Therefore, wemustalsoimposetheadditionalinternaltangencyconditionsatthemoment t 1 that thestateconstraintbecomesactive[ ? ]: N x t 1 := 2 6 6 6 6 6 6 6 4 g x t 1 g 0 x t 1 g m )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 x t 1 3 7 7 7 7 7 7 7 5 =0 WecannowdenetheHamiltonianofthedifferentialgameofretreat: H R := R x + h x + C x t u A t u B t where R istheadjointvariablevectorthatcontainsthegradientofthevaluefunctionfor theGoR: R := 1 2 ,..., n T = @ V R @ x 1 @ V R @ x 2 ,..., @ V R @ x n T 73

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Theadditionaladjointvariable isascalarandmustsatisfy t =0 when g x t > 0 t < 0 when g x t =0. Theadjointequationsareagainfoundbytakingthepartialderivativeofthe Hamiltonianwithrespecttoeachstatecomponent,buttheresultingderivativesnow possessapiece-wisebehaviorcorrespondingtoconstrainedarcsandunconstrained arcs: R := @ H @ x T = 8 > < > : @ _x @ x + @ h @ x + @ C @ x g x =0 @ _x @ x + @ C @ x g x > 0 Thestateconstraintalsocreatesinternalboundaryconstraintson R : T R t 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(= T R t 1 ++ T @ h @ x where t 1 isthetimethatthestateenterstheconstrainedarcand isavectorof appropriatedimensionofadditionaladjointvariables. Usingtheterminalconstraintforretreat, )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(R x f =0 ,wecanconstructanadjoined terminalvaluefunction: R x f := A x f + T R )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(R x f where R isavectorofLagrangemultipliersoftheappropriatedimensioncorresponding totheterminalconstraints.Theterminalvaluesoftheadjointvariablesarefoundby takingthepartialderivativeof R x f withrespectto x f : R t f = @ R @ x f T = @ R @ x 1 @ R @ x 2 ,..., @ R @ x n T UsingtheHamiltonian,theoptimalcontrolstrategiesforeachoftheagentsare statedinthefollowingtheorem. 74

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Theorem4.2. Supposethatthevaluefunctionandthevaluefunctiongradientsare continuous.Theequilibriumcontrolstrategiesfortheagentsarethengivenby u R A u R B =argmax u A max u B H =argmax u A ,u B H UsingtheoptimalcontrolfunctionsdescribedinTheorem4.2,wecansolve thegameusingthesamemethodsdescribedintheprevioussection.Theresulting solutionconsistsoftheequilibriumstatetrajectory x R t x 0 ,adjointvalues R t x 0 valuefunction V R x 0 ,andcontrolstrategies u R A t x 0 and u R B t x 0 forthegame ofengagementthatbeginswithinitialstate x 0 .Iftheequilibriumstrategiesare implemented,theresultingutilityfunctionvalueforPlayerAandPlayerBare U E A x 0 := U A u E A t x 0 u E B t x 0 U E B x 0 := U B u E A t x 0 u E B t x 0 Itshouldbenotedthatitispossiblethatthereexistvaluesfor x 0 inwhichthere doesnotexistanycontrolstrategies u A t and u B t thatsatisfythestateconstraint g x t 0 andtheterminalconstraints x t 0 = x 0 and )]TJ/F40 7.9701 Tf 6.94 -1.794 Td [(R x t f =0 .Thiscondition dividestheadmissiblestatespaceintotwodisjointregions.Oneregionwhereasolution totheGoRexists,andtheotherregioninwhichasolutiontotheGoRdoesnotexist. Theseregionswillplayacriticalroleinthesolutiontheoverallretreatorengagement gameandwillbeformallydenedandanalyzedinthefollowingsection. 4.2.4OverallEngageorRetreatGameSolution Theoverallsolutiontechniquefortheengageorretreatgameconsistsoftwoparts. Intherstpart,onesolvesthegameofengagement.Usingthevaluefunctionfromthat solution,onecanthenposetheGoR.Asmentionedintheprevioussection,itispossible thatasolutionmaynotexisttotheGoRforsomevaluesof x 0 .Therefore,wecandivide theadmissiblestatespaceintotworegions:aregionthatcontainsstatesthatpossess 75

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asolutiontotheGoRandaregionwhosestatesdonotpossesasolutiontotheGoR. Theyareformallydenedasfollows. R R := f x 2 R A j9 u R A t x u R B t x g R E := R A n R R Theexistenceof u R A t x and u R B t x isdependentonanumberconditionsthat arerelatedtothedynamics,terminalconditions,and V E x fortheparticularscenario underconsideration.Therefore,wewillnotaddresstherequirementsforexistencein thegeneralcase.Instead,wewillassumethattheexistenceofasolutiontotheGoRis knownorcanbedeterminedfortheparticularsystemorscenariounderconsideration. Theequilibriumtrajectories x R t x and x E t x possessusefulpropertiesthat willbeusedintheproofofTheorem4.3.Weaddressthesepropertiesinthefollowing lemmas. Lemma1. Supposethat x 0 2 R R andtheplayersimplementtheirrespectiveopenloop equilibriumcontrolstrategiesfortheGoR, u R A t 0 x 0 and u R B t 0 x 0 .Thestateofthe systemwillremainin R R alongtheentireequilibriumtrajectory x R t x 0 i.e. x R t 2 R R 8 t 2 [ t 0 t R f x 0 ]. Proof. Bydenition,asolutiontotheGoRexistsforany x 0 2 R R .Pickanarbitrary pointalongtheequilibriumGoRtrajectory x 1 := x R t 1 x 0 where t 1 2 [ t 0 t R f x 0 ] .Ifa newGoRisstartedatthispoint,thesolutionofthisgameissimplytheremainderofthe solutionoftheoriginalgamethatwasinitiatedat x 0 .Inparticular, u R A t x 1 = u R A t x 0 8 t 2 [ t 1 t R f x 0 ] u R B t x 1 = u R B t x 0 8 t 2 [ t 1 t R f x 0 ] x R t x 1 = x R t x 0 8 t 2 [ t 1 t R f x 0 ]. 76

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SinceasolutionexiststotheGoRinitiatedat x 1 ,wecanconcludethat x 1 2 R R and therefore x R t 2 R R 8 t 2 [ t 0 t R f x 0 ]. AstatementsimilartoLemma1canbemadefortheequilibriumtrajectoriesofany GoEinitiatedin R E Lemma2. Supposethat x 0 2 R E andtheplayersimplementtheirrespectiveopen-loop equilibriumcontrolstrategiesfortheGoE, u E A t 0 x 0 and u E B t 0 x 0 .Thestateofthe systemwillremainin R E alongtheentireequilibriumtrajectory x E t x 0 i.e. x E t x 0 2 R E 8 t 2 [ t 0 t E f ]. Proof. Forthesakeofcontradiction,supposethatforsome x 0 2 R E theequilibrium trajectory x E t x 0 passesthrough R R .Thatis, 9 t 1 2 [ t 0 t E f ] s t x E t 1 x 0 2 R R Denethestateat t = t 1 as x 1 := x E t 1 x 0 NowconsiderthefollowingpairofcontrolstrategiesforPlayerAandPlayerB respectively. ^ u A t := 8 > < > : u E A t x 0 t 2 [ t 0 t 1 u R A t x 1 t 2 [ t 1 t R f ] ^ u B t := 8 > < > : u E B t x 0 t 2 [ t 0 t 1 u R B t x 1 t 2 [ t 1 t R f ] Withthesecontrolstrategies,eachplayerimplementstheiroriginalequilibriumcontrol fortheGoEinitiatedat x 0 untiltime t 1 .Atthistime,eachplayerthenswitchestotheir equilibriumcontrolfortheGoRinitiatedat x 1 .Weknowthattheequilibriumcontrol strategiesexistfortheGoRbecausewehaveassumedthat x 1 2 R R .Wedene ^ x t 77

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totherepresentthetrajectorythatresultswhenPlayerAandPlayerBimplement ^ u A t and ^ u B t respectively. SimilartothemethodusedforthedevelopmentoftheGoR,wewilldenean additionalstatecomponent ^ c t representingthecost-to-gofortheremainderofthe trajectory ^ x t where ^ c = )]TJ/F39 11.9552 Tf 9.299 0 Td [(C ^ x t ,^ u A t ,^ u B t ^ c t R f =0. SincePlayerAandPlayerBplayoutaGoRinitiatedat x 1 for t 2 [ t 1 t R f ] weknowthat ^ c t = Z t t R f C ^ x ,^ u A ,^ u B dt = Z t t R f C x R u R A u R B dt = c R t x 1 8 t 2 [ t 1 t R f ] where c R t x 1 representsthecosttogostatecomponentfortheGoRinitiatedat x 1 Inparticular, ^ c t 1 = c R t 1 x 1 .Thefollowinginequalityconstraintmustbesatised because x 1 2 R R A x R f + c R t 1 x 1 V E x 1 UsingtheequalitycostconstraintEquation4,itcanbeseenthat A x R f +^ c t 1 V E x 1 WecanrepresentthevaluefunctionfortheGoEatanypointalong x E t x 0 in termsof V E x 1 andtheintegralcost: V E x E t x 0 = V E x 1 + Z t t 1 C x E u E A u E B dt 78

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Itcanbeshownthat ^ x t = x E t x 0 for t 2 [ t 0 t 1 becauseeachplayeremploystheir GoEcontrolstrategyoverthatperiod.Therefore, V E ^ x t = V E x 1 + Z t t 1 C x E u E A u E B dt 8 t 2 [ t 0 t 1 Additionally,wecanrepresentthecosttrajectory ^ c t asfunctionof ^ c t 1 andthe integralcost: ^ c t =^ c t 1 + Z t t 1 C ^ x ,^ u A ,^ u B dt Thecosttrajectoryfunctioncanbewrittenas ^ c t =^ c t 1 + Z t t 1 C x E u E A u E B dt 8 t 2 [ t 0 t 1 Addingtheintegralcostovertheinterval t 2 [ t 0 t 1 tobothsidesoftheinequality provides A x R f +^ c t 1 + Z t t 1 C x E u E A u E B dt V E x 1 + Z t t 1 C x E u E A u E B dt for t 2 [ t 0 t 1 .Aftersubstituting,wecanseethat A x R f +^ c t V E ^ x t 8 t 2 [ t 0 t 1 Since x 1 2 R R ,itisknownbythedenitionof R R andtheGoRthat A x R f +^ c t V E ^ x t 8 t 2 [ t 1 t R f ]. CombiningEquations4andEquations4yields A x R f +^ c t V E ^ x t 8 t 2 [ t 0 t R f ]. Thisimpliesthattheopen-loopcontrolstrategies ^ u A t and ^ u B t areadmissiblecontrol strategiesfortheGoRinitiatedat x 0 .Sincethereexistsanadmissiblecontrol,therealso existsanadmissiblemaximizingcontrolduetoFillipov'stheoremandtheassumptions onthesystemcharacteristics.Therefore,thereexistsa u R A t x 0 and u R B t x 0 for 79

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x 0 2 R E .Thisisacontradictionbythedenitionof R R and R E .Therefore, 69 t 1 2 [ t 0 t R f ] s t x E t 1 x 0 2 R R Thisimpliesthat x E t x 0 2 R E 8 t 2 [ t 0 t R f ]. Lemma3. Theengagementsurface, X E ,isasubsetof R E .Thatis X E R E Proof. Let x 0 2 X E .AGoEinitiatedat x 0 terminatesimmediatelyresultinginavalueof V E x 0 = x 0 Since x E > x R forall x E 2 X E and x R 2 X R ,thevaluefortheGoEprovidesa greaterutilitythananyalternativestrategythatterminatesinretreat: V E x 0 = x 0 x R + Z t f t 0 C u A t x B t dt Therefore,itisimpossiblethata u A t and u B t existthatsatisfythevalueconstraintfor theGoRstatedinEquation4,whichimpliesthatall x 0 2 X E arein R E TheoverallsolutiontothegeneralGameofAttackorRetreatcanbefound foraparticular x 0 byidentifyingwhichregiontheinitialstatebelongstoandthen implementingthecontrolstrategiesdenedinthefollowingtheorem. Theorem4.3. Supposethatregions R R and R E areknownorcanbecalculated. Additionally,supposethat V R x = V E x arecontinuesalongtheboundaryof R R .The followingcontrolstrategiesareinequilibrium. u A t x 0 = 8 > < > : u R A t x 0 x 0 2 R R u E A t x 0 x 0 2 R E 80

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and u B t x 0 = 8 > < > : u R B t x 0 x 0 2 R R u E B t x 0 x 0 2 R E Theequilibriumutilities,statetrajectories,andadjointvaluesaregivenbythesolutions ofeithertheGoEorGoRthatcorrespondtoequilibriumcontrolstrategies. Proof. InordertoprovethatthecontrolstrategiesgivenbyEquation4and Equation4areinequilibrium,itmustbeshownthatneitherplayerbenetsfroma unilateraldeviation. TheequilibriumcontrolstrategiesprovidedbythesolutionsoftheGoEandGoRare open-loopstrategiesforaparticularinitialstate x 0 .Foruseinthisproof,weintroducea feedbackcontrollawforeachplayer.Thiscontrollawimplementstheinitialequilibrium controlstrategyforeitheraGoEorGoRusingthecurrentstateastheinitialposition. TheselectionofwhethertoinitiateaGoEorGoRisdependentontheregionthatthe stateisin.Thetimevalue t 0 indicatesthatweareimplementingtheinitialcontrolvalue foragameinitiatedat x u F A x := 8 > < > : u E A t 0 x x 2 R E u R A t 0 x x 2 R R u F B x := 8 > < > : u E B t 0 x x 2 R E u R B t 0 x x 2 R R Let x F t x 0 representtheresultingtrajectoryifbothplayersimplementtheir respectivefeedbackcontrolsstartingfromstate x 0 .ThefollowingLemmastatesthat thefeedbackcontrollawsgeneratetheequilibriumsolutiontoeithertheGoRorGoE dependingtheregioninwhichtheinitialstateresides. 81

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Lemma4. Theimplementationofthefeedbackcontrollaws, u F A x and u F B x ,byboth playerswillgeneratethefollowingcontrolstrategies,trajectories,andutilityfunctions. u A t = 8 > < > : u R A t x 0 2 R R u E A t x 0 2 R E u B t = 8 > < > : u R B t x 0 2 R R u E B t x 0 2 R E x t x 0 = 8 > < > : x R t x 0 x 0 2 R R x E t x 0 x 0 2 R E U A = 8 > < > : V R x 0 x 0 2 R R V E x 0 x 0 2 R E U B = 8 > < > : B x R x 0 2 R R B x E x 0 2 R E Wewillnowdemonstratethatneitherplayerbenetsfromaunilateraldeviation fromthecontrolstrategiesstatedinEquation4andEquation4.Weexamine thefourcombinationsofplayerdeviationandinitialstartingregioninLemma5through Lemma8. Lemma5. Supposethat x 0 2 R E .PlayerAdoesnotbenetfromaunilateraldeviation fromtheequilibriumGoEcontrolstrategy x E A t x 0 Proof. Consideranalternativecontrolstrategy ^ u A t .Dene x t tobetheresulting trajectory, u B t tobePlayerB'sresultingcontrolhistory,and t f tobetheresulting terminaltimewhenPlayerAimplements ^ u A t andPlayerBimplements u F B x .Suppose thatthealternativecontrolstrategy ^ u A t terminatesthegameinengagement, i.e. x t f 2 R E .Iftheresultingtrajectoryremainsin R E x t 2 R E forall t 2 [ t 0 t f ] ,then theresultingutilitycanbeupperboundedbythevalueoftheGoEinitiatedat x 0 dueto 82

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Theorem3.2: U A u E A t x 0 u E B t x 0 U A ^ u A t u B t Thereisthepossibilitythat x t couldleave R E andeventuallyreturntoterminate inengagement.Ifthisisthecase,thereexiststimes t 1 < t 2 < < t n where n iseven suchthat x t possessthefollowingstructure, x t 2 8 > > > > > > > > > > < > > > > > > > > > > : R E t 2 [ t 0 t 1 R R t 2 [ t 1 t 2 ] R E t 2 t 1 t 3 R E t 2 t n t f ] Dene x i := x t i .Fromtheassumptionsonthevaluefunctions,thevaluefunctions fortheGoRandGoEareequalateachcrossing: V E x i = V R x i Dene c i for i =1,..., n tobetheaccumulatedcostsovertheperiod [ t i )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 t i ] : c i := Z t i t i )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 C x t ,^ u A t u B t dt Foreachcrossingfrom R E to R R ,wecanimposethefollowinginequalityusingTheorem 3.2. V E x i V E x i +1 + c i +1 i =0,2,4,... Similarly,wecanimposeaninequalityforeachcrossingfrom R R to R E using Theorem3.3: V R x i V R x i +1 + c i +1 i =1,3,5,... 83

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CombiningtheinequalitiesEquation4andEquation4withtheboundary conditionsEquation4provides V E x 0 V E x 1 + c 1 ... V E x i + n X i =1 c i Thisimpliesthat U A u E A t x 0 u E B t x 0 U A ^ u A t u B t Therefore,PlayerAcannotgenerateabetterutilityvalueforanalternativestrategy ^ u A t thatterminatesinengagement. Nowsupposethatthealternativestrategyterminatesinretreat.Thisrequires thatthealternativetrajectoryenters R R since X R R R .Thealternativetrajectorywill possessthebehaviordescribedinEquation4exceptthat n isodd.Wewillagain dene c i asinEquation4andenforcetheinequalitiesdenedinEquation496 andEquation4.Combiningtheinequalitiesweget V E x 0 V R x 1 + c 1 ... V R x i + n X i =1 c i Thisimpliesthat U A u E A t x 0 u E B t x 0 U A ^ u A t u B t Therefore,PlayerAcannotgenerateabetterutilityvalueusinganalternativestrategy ^ u A t thatterminatesinretreat. Thus,PlayerAcannotgenerateabetterutilityvalueusinganyalternativestrategy whetheritterminatesinretreatorengagement. Lemma6. Supposethat x 0 2 R E .PlayerBdoesnotbenetfromaunilateraldeviation fromtheequilibriumGoEcontrolstrategy u E B t x 0 Proof. Considerthealternativecontrolstrategy, ^ u B t .Dene x t tobetheresulting trajectory, u A t tobePlayerA'sresultingcontrolhistory,and t f tobetheresulting terminaltimewhenPlayerAimplements u F A x andPlayerBimplements ^ u B t 84

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Supposethatthealternativestrategyterminatesthegameinengagement.This situationwillprovidethesameutilitytoPlayerBasdoestheequilibriumstrategy u E B t x 0 : U B u A t ,^ u B t = U B u E A t x 0 u E B t x 0 = B x E Therefore,thereisnoimprovementinutilityforPlayerBforthistypeofdeviation. PlayerBcanonlyimproveitsutilityifitiscapableofterminatingthegameinretreat. Forthistooccurthealternativetrajectorymustpassinto R R atwhichpointPlayerA wouldswitchtoitsretreatstrategy.Wewillshowbyproofofcontradictionthatthisisnot possible.Supposeforthesakeofcontradictionthatthealternativestrategy ^ u B t moves thestateinto R R .Let t 1 representthetimethatthestatepassesthroughtheboundaryof R R .Dene x 1 := x t 1 .Fromtheassumptionsontheboundaryof R R ,weknowthat V E x 1 = V R x 1 WewillnowintroducetwootheralternativecontrolstrategiesforPlayerB.The alternativecontrolstrategy ^ u E B t implements ^ u t until t = t 1 ,atwhichpointPlayer Bimplements u E B t x 1 untilthestateterminatesontheengagementsurface: ^ u 1 B t = 8 > < > : ^ u B t t 2 [ t 0 t 1 u E B t x 1 t 2 [ t 1 t f 1 ]. Usingthecontrolstrategy ^ u E B t ,PlayerBimplements ^ u t until t = t 1 ,atwhichpoint PlayerBimplements u R B t x 1 untilthestateterminatesontheretreatsurface: ^ u 2 B t = 8 > < > : ^ u B t t 2 [ t 0 t 1 u R B t x 1 t 2 [ t 1 t f 2 ]. 85

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SincethevaluefortheGoEandGoRareequalattheboundarystrategies ^ u E B t and ^ u E B t resultinthesameutilityforPlayerA: U A u E A t ,^ u E B t = E A + Z t E f t 0 C u E A t ,^ u E B t x E t dt = E A + Z t E f t 1 C u E A t ,^ u E B t x E t dt + Z t 1 t 0 C u A t ,^ u B t x t dt = V E x 1 + Z t 1 t 0 C u A t ,^ u B t x t dt = V R x 1 + Z t 1 t 0 C u A t ,^ u B t x t dt = R A + Z t R f t 1 C u R A t ,^ u R B t x R t dt + Z t 1 t 0 C u A t ,^ u B t x t dt = R A + Z t R f t 0 C u R A t ,^ u R B t x R t dt = U A u R A t ,^ u R B t Theorem4.4providesPlayerB'sequilibriumminimizingcontrolforanygame thatterminatesinengagement.Therefore,anyalternativecontrolthatPlayerBwould implementwouldprovideagreaterutilitytoPlayerA.Inparticular,thealternative strategy ^ u E B providesautilityofatleast V E x 0 whenPlayerArespondswithitsoptimal engagementcontrolstrategydenedinTheorem4.4: V E x 0 = U A u E A t x 0 u E B t x 0 U A u E A t ,^ u E 1 B t Since U A u E A t ,^ u E B t = U A u R A t ,^ u R B t ,wecanalsolowerbound U A u R A t ,^ u R B t : V E x 0 U A u R A t ,^ u R B t Thislowerboundappliesalongtheentiretrajectory x t for t 2 [ t 0 t 1 : V E x t U A u A t ,^ u B t 8 t 2 [ t 0 t 1 86

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Therefore, ^ u R B t isanadmissiblecontrolstrategyfortheGoR,whichimpliesthat x 0 2 R R .However,thisisacontradictionsinceitisassumedthat x 0 2 R E .Therefore, therecannotexistacontrolstrategyforPlayerBthatmovesthestatefrom R E to R R becausethatcontrolstrategywouldrepresentanadmissiblecontrolfortheGoR.Thus, PlayerBcannotterminatethegameinretreatsincePlayerAwillforcethegameto terminateinengagementforany x 0 2 R E Sinceeverygamewillterminateinengagement,anyalternativestrategypresented byPlayerBwillyieldthesameequilibriumutility.Therefore,PlayerBdoesnotbenet fromaunilateraldeviationfrom u E B t x 0 when x 0 2 R E Lemma7. Supposethat x 0 2 R R .PlayerBdoesnotbenetfromaunilateraldeviation fromtheequilibriumGoRstrategy u R B t x 0 Proof. IfPlayerBimplementstheequilibriumcontrolstrategy u R t x 0 ,thegamewill terminateinretreat.FromPlayerB'sperspective,thisisthebestpossibleoutcome sinceitisassumedthat B x R > B x E where x R 2 X R and x E 2 X E .Therefore,it isimpossibleforPlayerBtoimproveitsutilitythroughaunilateraldeviationinstrategy sinceitisalreadyreceivingitsmaximumpossibleutility. Lemma8. Supposethat x 0 2 R R .PlayerAdoesnotbenetfromaunilateraldeviation fromtheequilibriumGoRstrategy u R A t x 0 Proof. Consideranalternativecontrolstrategy ^ u A t .Dene x t tobetheresulting trajectory, u B t tobePlayerB'sresultingcontrolhistory,and t f tobetheresulting terminaltimewhenPlayerAimplements ^ u A t andPlayerBimplements u F B x Supposethatthealternativecontrolstrategy ^ u A t terminatesthegameinretreat, i.e. x t f 2 R R .Iftheresultingtrajectoryremainsin R R x t 2 R R forall t 2 [ t 0 t f ] ,then theresultingutilitycanbeupperboundedbythevalueoftheGoRinitiatedat x 0 dueto Theorem4.6: U A u R A t x 0 u R B t x 0 U A ^ u A t u B t 87

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Thereisthepossibilitythat x t couldleave R R andeventuallyreturntoterminate inretreat.Infact,thestateofthesystemcouldpassbackandforthfrom R R to R E any numberoftimesaslongasiteventuallyterminatesin R R .Ifthisisthecase,therewill existtimes t 1 < t 2 < < t n where n isevensuchthat x t possessthefollowing structure, x t 2 8 > > > > > > > > > > < > > > > > > > > > > : R R t 2 [ t 0 t 1 R E t 2 [ t 1 t 2 R R t 2 [ t 1 t 3 R R t 2 [ t n t f ] Dene x i := x t i .Fromtheassumptionsonthevaluefunctions,thevaluefunctionsfor theGoRandGoEareequalateachcrossing: V E x i = V R x i Dene c i for i =1,..., n tobetheaccumulatedcostsovertheperiod [ t i )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 t i ] : c i := Z t i t i )]TJ/F24 5.9776 Tf 5.756 0 Td [(1 C x t ,^ u A t u B t dt Foreachcrossingfrom R E to R R ,wecanimposethefollowinginequalityusing Theorem4.4. V E x i V E x i +1 + c i +1 i =0,2,4,... Similarly,wecanimposeaninequalityforeachcrossingfrom R R to R E usingTheorem 4.6: V R x i V R x i +1 + c i +1 i =1,3,5,... CombiningtheinequaltiesEquation4andEquation4withtheboundary conditionsEquation4provides V R x 0 V R x 1 + c 1 ... V R x i + n X i =1 c i 88

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Thisimpliesthat U A u R A t x 0 u R B t x 0 U A ^ u A t u B t Therefore,PlayerAcannotgenerateabetterutilityvalueforanalternativestrategy ^ u A t thatterminatesinretreat. Nowsupposethatthealternativestrategyterminatesinengagement.Thisrequires thatthealternativetrajectoryenters R E .Thealternativetrajectorywillpossessthe behaviordescribedinEquation4exceptthat n isodd.Wewillagaindene c i asin Equation4andenforcetheinequalitiesdenedinEquation4andEquation 4.Combiningtheinequalitiesweget V R x 0 V E x 1 + c 1 ... V E x i + n X i =1 c i Thisimpliesthat U A u R A t x 0 u R B t x 0 U A ^ u A t u B t Therefore,PlayerAcannotgenerateabetterutilityvalueusinganalternativestrategy ^ u A t thatterminatesinengagement. Thus,PlayerAcannotgenerateabetterutilityvalueusinganyalternativestrategy whetheritterminatesinretreatorengagement. Lemma5throughLemma8showthatneitherplayerbenetsfromadeviation fromtheirrespectiveequilibriumcontrolstrategiesdenedbyEquation48and Equation4when x 2 R E norwhen x 2 R R .Therefore,thesecontrolstrategiesare inequilibrium. 4.3AnalyticExample Inthisexamplegame,PlayerArepresentsamobileattackermovingwithsimple motionaboutanobstacle-free,inniteplane.PlayerAiscapableofterminatingthe differentialgamebyeitherengagingahigh-value,stationarytargetorretreatingtoa predenedretreatboundary.PlayerBrepresentsthedefender.PlayerBdoesnothave 89

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directinuenceoverthemotionofPlayerA,butitiscapableofinictingacostonPlayer Ainanefforttomaketheprospectofengagingthehigh-valuetargetunattractive. Asaconcreteexample,consideranoutpostdefensescenario.Inthisscenario, PlayerAwouldrepresentanattackingforcesuchasagroupofinsurgentsoranarmed vehicle.Ithastheoptionofattackingthebaseorretreatingacrossanearbyborder tosafety.PlayerBwouldrepresentthecommanderoftheoutpostwhoiscapableof directingrepowerontheattackingagent.Thecommander'sprimaryobjectiveisto protecttheoutpostfromattack.Therefore,hereceivesnorewardfrominictingcostor damageontheattacker.Instead,therepowerisonlyusedasadeterrent. 4.3.1SystemModel ThepositionofPlayerAisdescribedbytheCartesiancoordinates x := x y .The statichigh-valuetargetislocatedattheorigin, x T :=,0 .Thecontrolvariablesfor PlayerAareitsspeed v p andheadingangle .Thekinematicequationsofthesystem are 2 6 4 x y 3 7 5 = 2 6 4 v p cos v p sin 3 7 5 =: f u A u B x PlayerBdoesnotpossessdirectinuenceonthestateofthesystem.Instead, PlayerB'scontrolvariableisincorporatedintoPlayerA'sutilityfunctional.PlayerA's control, u A ,andPlayerB'scontrol, u B ,aredenedas u A := v p where 0 v p v p u B := where 0 c 1 4.3.2TerminationConditions Theterminaltime, t f ,isdenedasthemomentthestateofthesystemsatises eithertheretreatcondition, )]TJ/F40 7.9701 Tf 6.941 -1.793 Td [(R x := y )]TJ/F39 11.9552 Tf 11.955 0 Td [(y r =0, 90

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ortheengagementcondition, )]TJ/F40 7.9701 Tf 6.941 -1.793 Td [(E x := p x 2 + y 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c =0. Thevalue d c > 0 representsthecapturedistance,andthevalue y r 0 representsthe retreatboundary. Wedenetheretreatsurface, X R ,andengagementsurface, X E ,via X R := x 2 R 2 j )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(R x = 0 and X E := x 2 R 2 j )]TJ/F40 7.9701 Tf 6.941 -1.793 Td [(E x = 0 Theretreatboundaryrestrictstheregionofadmissibleinitialconditionsforthe game.Wedenetheregionofadmissibleinitialconditionsas R A := f x 2 R 2 j y y r g 4.3.3PlayerUtilities PlayerAandPlayerB'sutilityfunctionalsaredenedas U A u A t u B t := A x f )]TJ/F30 11.9552 Tf 11.955 16.273 Td [(Z t f t 0 C u B t dt U B u A t u A t := B x f wheretheinstantaneouscostfunction, C u B t ,isdenedas C u B t := + c 2 Theterminalvaluefunctionsaredenedas A x f := 8 > < > : a 1 x f 2 R E 0 x f 2 R R B x f := 8 > < > : )]TJ/F39 11.9552 Tf 9.299 0 Td [(b 1 x f 2 R E 0 x f 2 R R Itisassumedthat a 1 > 0 and b 1 > 0 91

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4.3.4GameFormulation Adifferentialgamecanbeconstructedinwhicheachoftheplayerssimultaneously attemptstomaximizeitsrespectiveutilityfunctionsdescribedinEquation4and Equation4: U A =max u A t U A u A t u B t U B =max u B t U B u A t u B t 4.4GameSolution InordertosolvetheEngageorRetreatGame,Irstsolvethetwodifferential subgamesGoEandGoR. 4.4.1DifferentialGameofEngagement WewillrstexaminetheGoE.Inthisgame,PlayerAattemptstoterminatethe gameinengagementwhilemaximizingitsrespectiveutilityfunctional.Simultaneously, PlayerBattemptstominimizePlayerA'sutilityfunction.Usingtheplayers'goalsalong withthesystemdynamics,theresultingdifferentialgameisdenedas V E x 0 :=max u A t min u B t U A u A t u B t x 0 =min u B t max u A t U A u A t u B t x 0 withtheconstraintthat )]TJ/F40 7.9701 Tf 6.941 -1.793 Td [(E x f =0 Webegincalculatingthesolutiontothegameofengagementbyconstructingthe Hamiltonian: H E := T E f x u A u B )]TJ/F39 11.9552 Tf 11.955 0 Td [(C A x u A u B = x E v p cos + y E v p sin )]TJ/F22 11.9552 Tf 11.955 0 Td [( + c 2 =0. TheequilibriumcontrolforPlayerAandPlayerBarefoundbymaximizingand minimizingtheHamiltonian. 92

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Theorem4.4. Supposethatthevaluefunctionandthevaluefunctiongradientare dened.TheequilibriumcontrolofPlayerAandPlayerBareasfollows. cos E = x E p 2 x E + 2 y E sin E = y E p 2 x E + 2 y E v E p = v p and E = c 1 Thetimederivativeoftheadjointvariablesarefoundbytakingthepartialderivative oftheHamiltonianwithrespecttoeachofthestatecomponents: x E = @ H @ x =0 y E = @ H @ y =0. Usingthecapturecondition,aterminalvaluefunctionisconstructed: E x f := a 1 + E q x 2 f + y 2 f )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c Takingtheappropriatepartialderivatives,providesterminalconditionsthattheadjoint variablemustsatisfy: x E t f = @ E @ x = E x f p x 2 f + y 2 f y E t f = @ E @ y = E y f p x 2 f + y 2 f UsingthefactthattheHamiltonianisalwayszero,wecansolvefor E bysubstituting theoptimalcontrolEquation4-Equation4intotheHamiltonianand evaluatingitat t = t f .SubstitutingthesolutionintoEquation4andEquation 4providesterminaladjointvaluesasafunctionoftheterminalstate: x E t f = x f c 1 + c 2 v p d c y E t f = y f c 1 + c 2 v p d c 93

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Bycombiningtheoptimalcontrolstrategies,systemdynamics,adjointequations, andterminaladjointconditionswecancalculatethecompletesolutiontotheGoEfor anyinitialcondition x 0 = d 0 0 .ThesolutionisgiveninTheorem4.5. Theorem4.5. Supposethedifferentialgameofengagementisinitiatedat x 0 = x 0 y 0 Theequilibriumcontrolstrategies, u E A t x 0 = E t x 0 v E p t x 0 u E B t x 0 = E t x 0 andresultingstatetrajectoriesareasfollows. E t x 0 = E v E p t x 0 = v p E t x 0 = c 1 x E t x 0 = x 0 + v p cos E t y E t x 0 = y 0 + v p sin E t t E f x 0 = p x 2 0 + y 2 0 )]TJ/F40 7.9701 Tf 6.586 0 Td [(d c v p V E x 0 = a 1 )]TJ/F40 7.9701 Tf 13.151 4.884 Td [(c 1 + c 2 v p q x 2 0 + y 2 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c where cos E = )]TJ/F40 7.9701 Tf 23.47 4.883 Td [(x 0 p x 2 0 + y 2 0 and sin E = )]TJ/F40 7.9701 Tf 23.469 5.034 Td [(y 0 p x 2 0 + y 2 0 4.4.2DifferentialGameofRetreat WewillnowaddresstheGoR.Inthisgame,PlayerAterminatesthegameinretreat whileattemptingtomaximizeitsutilityfunctional.Simultaneously,PlayerBalsoattempts tomaximizePlayerA'sutilityfunctionalwhilepreventingPlayerAfrommovingthestate toaregionwhereitwouldbeadvantageoustoengage.Usingtheseplayerobjectives, 94

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theresultingdifferentialgameisdenedas V R x 0 :=max u A t max u B t U A u A t u B t =max u A t ,u B t U A u A t u B t withtheterminalconstraintthat )]TJ/F40 7.9701 Tf 6.94 -1.793 Td [(R x f =0 Inordertoensurethatthestatedoesnotmoveintoaregionwhereengagementis anoptimaloptionforPlayerA,weimposeaconstraintonthevaluefunctionofretreat: V R x t )]TJ/F39 11.9552 Tf 11.955 0 Td [(V E x t 0 8 t 2 [ t 0 t f ]. Weconvertthisstateinequalityconstraintintoacontrolequalityconstraintbydifferentiating withrespecttotime: g 0 x u A u B := d dt g x = + c 2 + c 1 + c 2 x cos + y sin p x 2 + y 2 =0. Thecontrolconstraintisimposedonlywhenthestateinequalityconstraintisactive. TheHamiltonianfortheGoRisconstructedusingthesystemdynamics,PlayerA's utilityfunctional,andthecontrolequalityconstraint: H R = x R v p cos + y R v p sin )]TJ/F22 11.9552 Tf 11.956 0 Td [( + c 2 + + c 2 )]TJ/F23 7.9701 Tf 13.15 5.478 Td [( c 1 + c 2 x cos + y sin p x 2 + y 2 =0. Theadditionaladjointvariable actstoimposethecontrolconstraint,Equation4, whenthestateinequalityconstraintbecomesactiveandhasthefollowingbehavior. =0 when g x > 0 0 when g x =0 95

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WecalculatetheoptimalcontrolforeachplayerbymaximizingtheHamiltonian: Theorem4.6. Supposethatthevaluefunctionanditsgradientsarecontinuous.The equilibriumcontrolstrategiesforPlayerAandPlayerBare cos R = k 1 sin R = k 2 = q k 2 1 + k 2 2 R = 8 > < > : 0,0 +1 c 1 ,0 > +1 v R p = v p where k 1 = v p x R )]TJ/F25 11.9552 Tf 11.955 0 Td [( c 1 + c 2 x p x 2 + y 2 k 2 = v p y R )]TJ/F25 11.9552 Tf 11.955 0 Td [( c 1 + c 2 y p x 2 + y 2 k 1 = v p x R )]TJ/F25 11.9552 Tf 11.955 0 Td [( c 1 + c 2 x p x 2 + y 2 k 2 = v p y R )]TJ/F25 11.9552 Tf 11.955 0 Td [( c 1 + c 2 y p x 2 + y 2 AsintheGoE,wemustnowdevelopoptimalityconditionsfortheadjointvariables. TheadjointequationsarefoundbytakingthepartialderivativeoftheHamiltonian: x R = )]TJ/F26 7.9701 Tf 10.494 4.707 Td [(@ H @ x = c 1 + c 2 y 2 cos )]TJ/F39 11.9552 Tf 11.955 0 Td [(xy sin x 2 + y 2 3 2 y R = )]TJ/F26 7.9701 Tf 10.494 4.707 Td [(@ H @ y = c 1 + c 2 x 2 sin )]TJ/F39 11.9552 Tf 11.955 0 Td [(xy cos x 2 + y 2 3 2 Tocalculatetheterminalvaluesfortheadjointvariables,theterminalvaluefunction isconstructed, R x f =0+ R y )]TJ/F39 11.9552 Tf 11.955 0 Td [(y r 96

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andtheappropriatepartialderivativesaretaken: x R t f = @ R @ x =0 y R t f = @ R @ y = R Wecansolvefor y R t f bysubstitutingtheoptimalcontrolstrategiesintotheHamiltonian andevaluatingat t f : y R t f = )]TJ/F39 11.9552 Tf 10.55 8.088 Td [(c 2 v p Wecannowconstructtheoptimaltrajectoriesusingtheoptimalityconditions.We willonlyformallyaddressthecaseswhere x 0 0 inthispaper.Thesolutionswhen x 0 0 aresymmetricaboutthe y -axis.Denetheoptimalretreatcontrolstrategiesfor PlayerAandPlayerBas u R A t x 0 = R t x 0 v R p t x 0 u R B t x 0 = R t x 0 Wewillgivetheexplicitformulafor u R A t x 0 and u R B t x 0 inTheorem4.7through Theorem4.9.ThestateinequalityconstraintimposedbyEquation4complicates thestructureoftheoptimaltrajectoriesforsomeinitialconditions.Inthesecases, theoptimaltrajectoriespossesapiecewisestructureconsistingofconstrainedand unconstrainedsegments.Additionally,forotherinitialconditions,asolutiondoesnot existthatsatisesthevalueconstraintimposedbyEquation4.Theoptimal controlandresultingtrajectoriesforeachofthesesituationsarestatedinTheorem4.7 throughTheorem4.9.Ineachofthesetheorems,theinitialstate x 0 = x 0 y 0 belongsto oneoffourdifferentregions.Thesefourregionsaredenedas R 1 := f x 0 2 R A j x 2 x 0 g R 2 := f x 0 2 R A j x 0 < x 2 ,0 m 1 x 0 g R 3 := f x 0 2 R A j x 0 < x 2 m 1 x 0 < 0,0 < m 2 x 0 g 97

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R 4 := f x 0 2 R A j x 0 < x 2 m 2 x 0 0 g where m 1 x := c 1 + c 2 p x 2 + y 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d c )]TJ/F39 11.9552 Tf 11.955 0 Td [(v p a 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c 2 y )]TJ/F39 11.9552 Tf 11.955 0 Td [(y r m 2 x := p x 2 + y 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d 2 e cot ^ R arctan y = x )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 ^ R =arccos c 2 c 1 + c 2 x 2 = c 1 + c 2 d c + a 1 v p )]TJ/F40 7.9701 Tf 6.586 0 Td [(c 2 y r p c 1 + c 2 2 )]TJ/F40 7.9701 Tf 6.586 0 Td [(c 2 2 y 2 = c 2 c 1 + c 2 d c + a 1 v p )]TJ/F40 7.9701 Tf 6.586 0 Td [(c 2 y r c 1 + c 2 2 )]TJ/F40 7.9701 Tf 6.586 0 Td [(c 2 2 d 2 = q x 2 2 + y 2 2 2 =arctan y 2 x 2 TheregionsaredepictedgraphicallyinFigure4-1. Theorem4.7. Supposethat x 0 2 R 1 [ R 2 .Theoptimalcontrolstrategiesandresulting trajectoriesaregivenby R t x 0 = )]TJ/F26 7.9701 Tf 10.494 4.708 Td [( 2 v R p t x 0 = v p R t x 0 =0 x R t x 0 = x 0 y R t x 0 = y 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(v p t V R x 0 = )]TJ/F40 7.9701 Tf 10.543 4.884 Td [(c 2 v p y 0 )]TJ/F39 11.9552 Tf 11.955 0 Td [(y r t R f x 0 = y )]TJ/F40 7.9701 Tf 6.586 0 Td [(y r v p Theorem4.8. Suppose x 0 2 R 4 .Theoptimalcontrolstrategiesandresultingtrajectories haveapiecewisebehaviorcomposedofunconstrainedandconstrainedsegments. For t 0 t t 1 R t x 0 = 1 + ^ R 98

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R t x 0 =0 x R t x 0 = x 0 + v p cos 1 + ^ R t y R t x 0 = y 0 + v p sin 1 + ^ R t For t 1 t t 2 R t x 0 = ^ R + R x 0 t R t x 0 =0 x R t x 0 = d R t x 0 cos R x 0 t y R t x 0 = d R t x 0 sin R x 0 t d R t x 0 = d 2 + v p cos ^ R t 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(t R t x 0 =tan ^ log d R t ,x 0 d 2 + 2 For t 2 t t f R t x 0 = )]TJ/F25 11.9552 Tf 9.298 0 Td [(= 2 R t x 0 =0 x R t x 0 = x 2 y R t x 0 = y T )]TJ/F22 11.9552 Tf 12.243 0 Td [( v p t )]TJ/F39 11.9552 Tf 11.955 0 Td [(t 2 Thevaluefunctionforthisregionis V x 0 = t R f c 2 .Thevaluesof t 1 t 2 ,and t R f fora particular x 0 arefoundbysolvingthefollowingequationsfor 1 d 1 t 1 t 2 ,and t R f x 0 = d cos 1 )]TJ/F22 11.9552 Tf 12.244 0 Td [( v p cos 1 + ^ R t 1 y 0 = d sin 1 )]TJ/F22 11.9552 Tf 12.244 0 Td [( v p sin 1 + ^ R t 1 d = d 2 e cot R 1 )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 0= d 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(d 1 + v p cos ^ R t 2 )]TJ/F39 11.9552 Tf 11.955 0 Td [(t 1 0= y r )]TJ/F39 11.9552 Tf 11.956 0 Td [(y 2 + v p t R f )]TJ/F39 11.9552 Tf 11.955 0 Td [(t 2 99

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Theorem4.9. Supposethat x 0 2 R 3 .TheredoesnotexistanequilibriumcontrolstrategyforPlayerAandPlayerBthatsatisesthevalueconstraintimposedbyEquation 4: 8 u A t u B t U A u A t u B t x 0 < V E x 0 when \050 x f R =0 4.4.3SolutiontotheEngageorRetreatGame Wecandividetheadmissiblestatespaceintotworegions:anengageregion, R E andaretreatregion, R R .Wedene R R and R E intermsof R 1 R 2 R 3 ,and R 4 : R R = R 1 [ R 2 [ R 4 R E = R 3 TheoverallsolutiontotheEngageorRetreatGamecanbefoundforaparticular x 0 byidentifyingwhichregiontheinitialstatebelongstoandthenimplementingthecontrol strategiesdenedinthefollowingtheorem. Theorem4.10. Let R R and R E bedenedasinEquation4andEquation4. Supposethat V R x = V E x alongtheboundaryof R R .Thefollowingcontrolstrategies areinequilibrium. u A t x 0 := 8 > < > : u R A t x 0 x 0 2 R R u E A t x 0 x 0 2 R E u B t x 0 := 8 > < > : u R B t x 0 x 0 2 R R u E B t x 0 x 0 2 R E Theequilibriumutilities,statetrajectories,andadjointvaluesaregivenbythesolutions ofeithertheGoEorGoRthatcorrespondtoequilibriumcontrolstrategies. 100

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4.4.4EquilibriumTrajectoriesandRegions Inthissection,wewillexaminetheequilibriumtrajectoriesandresultingregionsfor thefollowingparametervalues. v p =1 a 1 =1 y r = )]TJ/F22 11.9552 Tf 9.299 0 Td [(5 c 1 =1 c 2 =1 d c =1 Inthefollowingplots,theretreatsurfaceisdepictedasadashedlineat y = )]TJ/F22 11.9552 Tf 9.298 0 Td [(5 ,and thecapturesurfaceisdepictedasadashedlinealong p x 2 + y 2 =1 unitcircle.First, theoptimaltrajectoriesforregions R 1 and R 2 aredisplayedinFigure4-2.Thebluelines representthetrajectoriesforseveralinitialconditions.Thesolidblacklinerepresents m 1 x =0 andformsanupperboundfortheunconstrainedtrajectoriesof R 2 .Thesolid markersthatterminatethesurfaceindicatethetangencypoints x 2 y 2 and )]TJ/F39 11.9552 Tf 9.299 0 Td [(x 2 y 2 Figure4-3displaystheequilibriumtrajectoriesforregion R 4 .Inthisplot,thesolidblack linerepresents m 2 x =0 .Inthisregion,theoptimaltrajectoriesarriveat m 2 x =0 tangentiallyandmovealongthissurfaceuntilreachingthetangencypoints.Atthispoint, thestatemovesawayfromthestateconstraintandeventuallyterminatesontheretreat surface.Thesurface x =0 isadispersalsurfacewithinthisregion.Figure4-4displays theequilibriumengagementtrajectorieswithin R 3 InFigure4-5,theequilibriumtrajectoriesfromeachregionarecombinedtodisplay thecompletesolutiontotheengageorretreatdifferentialgame.Theengagement region, R E ,isthearealledwiththeredengagementtrajectoriesandisenclosedby m 1 x =0 and m 2 x =0 .Theretreatregion, R R ,isthearealledwithblueretreat trajectories. 4.4.5DiscussionoftheEscortRegionandTrajectories Region R 4 representsan escort region.Inthisregion,PlayerBhasagreedto cooperatewithPlayerAinordertomaximizeitsutilityifPlayerAretreats.But,PlayerA musttakeaconstrainedretreattrajectory,whichavoidstheregionwhereengagement becomestheoptimalstrategy.IfPlayerAwouldbreakfromthistrajectoryandviolate 101

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thevaluefunctionconstraint,PlayerBwouldswitchtoaminimizingstrategythatwould provideaworseutilityvalueforPlayerA.Therefore,PlayerBeffectivelyescortsPlayer Aoutofregion R 4 untilstatereachesthetangencypoint.Afterthispoint,theoptimal retreattrajectorymovesawayfrom R E ,andPlayerBnolongerneedstoconcernitself withPlayerA. Thisescortbehaviorbecomesevenmorepronouncedinmorecomplexsystems, particularlywithsystemsinwhichPlayerBpossessessomeinuenceonthesystem dynamics.Somepreliminaryresultsofapplyingthevaluefunctionconstrainttoasystem thatcontainsmobiledefensiveagentsgeneratesdefensiveagenttrajectoriesthatfollow themobileattackerthroughouttheescortregion.Thesetrajectoriesmaintainthevalue constraintandneutralizethepotentialoftheattackerswitchingtoengagement. 102

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Figure4-1.GameRegions Figure4-2.Equilibriumtrajectoriesfor R 1 and R 2 103

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Figure4-3.Equilibriumtrajectoriesfor R 4 Figure4-4.Equilibriumtrajectoriesfor R 3 104

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Figure4-5.Equilibriumtrajectoriesfor R A 105

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CHAPTER5 DECEPTIONANDJONES'LEMMAWITHINAZERO-SUMGAME Thischapterisbasedonworkpresentedatthe2011ACC[59].Inthischapter, Iposeagenerictwo-player,zero-sumgameinwhichastochasticsensornetwork providesoneplayer,PlayerB,aninformationaladvantageoveritsopponent,PlayerA. Simultaneously,PlayerApossessestheabilitytocorruptthesensornetworkoutput,at acost,inanattempttomanipulatePlayerB'sactions.Thepossibleuseofdeception allowsPlayerAtoneutralizetheinformationaladvantageofPlayerBandshiftthe game'sequilibriumvalueclosertothesolutionofthegamewheretheinformation networkisremoved.ItisassumedthatPlayerBknowsofthepossibilityofdeception, butiftheriskofdeceptionissmallenough,PlayerBwillstillutilizetheinformation providedbyitssensornetwork.Iproposeautilityfunctionforthegamewhichtakes intoaccounttheeffectsthatthedeceptivetacticshaveonthesensornetworkandits correspondingcost.Thesolutiontothisgameconsistsoftheoptimalstrategiesforeach oftheplayersandthecorrespondingvalueoftheutilityfunction.WithrespecttoPlayer A,theoptimalstrategyrepresentsthebestmixofactionsalongwiththecomplimentary deceptivetactic.TheoptimalstrategyofPlayerBrepresentsthebeststochasticcontrol lawbasedonthemeasuredsensornetworkoutput.Utilizingtherelationshipbetween theminimaxtheoremandthestrongdualitytheoremoflinearprograming,Ishowthat thesolutionofthezero-sumgamecanbecomputedbysolvingapairofduallinear programmingproblems. Usingthisframework,Iexamineanillustrativeexample,whichcanbemodied torepresentalargerangeofscenarios.Inthisexample,PlayerAmustselectoneof thetwolocationsinordertostoreorhideahighvalueitem.Thereareanumberof informationchannelsthatprovidePlayerBanoisyestimateofthelocationoftheitem, whichitcanthenusetodeterminewhichlocationtoattackorsearch.Itiscommon 106

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knowledgethatPlayerAcancorruptthisinformationnetwork,butthecorruptionisnot costfreeandthecostofcorruptionisincorporatedintothevaluefunction. Ithenanalyticallydevelopclosedformsolutionstoseveralspecialcasesofthis game.Theresultingequilibriumplayerstrategiesandutilityvaluesverynicelycapture awell-known,qualitativeprincipleinthedeceptioneldknownastheJones'Lemma. ThismaximisattributedtoR.V.Joneswhoisconsideredtobethefatherofscientic intelligence.Itstates,Deceptionbecomesmoredifcultasthenumberofchannels availabletothetargetincreases.Nevertheless,withinlimits,thegreaterthenumberof channelsthatarecontrolledbythedeceiver,thegreaterthelikelihoodthatthedeception willbebelieved.[25]IfurtherexploreJone'sLemmaforarangeofparametervaluesby numericallysolvingthegenerallinearprogramsolution. 5.1GameFormulation Inthissection,Idevelopazero-sumgamewithtwoplayers,PlayerAandPlayerB, attemptingtominimaximizeavonNeumann-Morgensternutilityfunction. 5.1.1PlayerActionsandDeceptionTactics Thesets A := f a 1 a 2 ,..., a l g and B := f b 1 b 2 ,..., b m g representthesetsoffeasible actionsforPlayerAandPlayerBrespectively.Eachplayermustselectoneaction fromtheirrespectivesets.PlayerAselectsanaction a 2 A ,andPlayerBselectsan action b 2 B .PlayerAmustalsoselectadeceptiontactic, d 2 D ,fromthesetof possibledeceptiontactics D := f d 1 d 2 ,..., d p g .PlayerA'sselectedactionanddeception tacticalongwithPlayerB'sselectedactionarepassedintothegamesvaluefunction V a d b ,whichgeneratesthevaluethatbothplayersstrivetominimaximize. 5.1.2SensorNetwork PlayerAmustselectitsactionanddeceptiontacticbeforePlayerB.Theselected actionanddeceptiontacticarethenpassedintothestochasticsensornetwork.The stochasticsensornetworkproducesasensorvalue s 2 S accordingtotheprobability distribution P S j A D s j a d ,where S := f s 1 s 2 ,..., s n g isthesetof n possiblesensor 107

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Figure5-1.OverallGameStructure values.Thesensorvalueswithin S canrepresentavarietyofinformationsources, suchasstrategypredictions,objectclassications,orrawsensormeasurements.The conditionalprobabilitydistributionfullydenesthecharacteristicsofthesensornetwork andiscommonknowledgewithinthegame,i.e.,bothplayersknow P S j A D s j a d 5.1.3PlayerStrategies Aplayer'sstrategyisdenedastheprobabilitydistributionrepresentingthe likelihoodofselectingaparticularactionfromitsactionset.PlayerA'sstrategyis denedasthejointprobabilitydistribution P A D a d becauseitmustselectboth anactionanddeceptiontactic.PlayerB'sstrategyisrepresentedbytheconditional probabilitymassfunction P B j S b j s duetothepossibledependenceonthemeasured sensorvalue. 5.1.4UtilityFunction IconstructavonNeumann-Morgensternutilityfunction[60]usingtheplayer strategiesandvaluefunction: U P A D a d P B b := X A D S B P A D a d P S j A D s j a d P B j S b j s V a d b Theutilityfunctionrepresentstheexpectedvalueproducedfromthevaluefunction wheneachplayerimplementstheirrespectivestrategies.Thedirectlyconictinggoals ofeachplayerleadtothefollowingzero-sumgameinwhichPlayerAstrivestominimize theutilityfunctionwhilePlayerBsimultaneouslyattemptstomaximize. U :=min P A D a d max P B j S b j s U P A D a d P B b 108

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=max P B j S b j s min P A D a d U P A D a d P B b Although,thegamepossessesasequentialstructure,PlayerBonlyhasinformation generatedbythesensornetworkanddoesnotpossesdirectknowledgeofPlayerA's selectedaction.Therefore,PlayerB'soptimalstrategyintermsof s canbegeneratedat thesametimePlayerAdevelopsitsoptimalstrategy. 5.2GeneralGameSolution Inordertodevelopasolutiontothegame,Iwillrstexpressthegamedescribedby Equation5anditscomponentsinmatrixform.PlayerA'sstrategyisparameterized usingthe l p matrix =[ i j ] ,where i j = P A D a i d j .Similarly,PlayerB'sstrategy isparameterizedusingthe n m matrix =[ k l ] ,where k l = P B j S b l j s k .The valuefunction V a d b isalsoparameterizedinmatrixformusingthe lp m matrix V =[ V l j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1+ i k ] ,where V l j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1+ i k = V a i d j b k .Theconditionalprobabilityfunctionthat describethesensornetworkisparameterizedinthe lp n matrix =[ l j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1+ i k ] such that l j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1+ i k = p s k j a i d j .Usingthesematrixparameterizations,Irewritetheutility functionEquation5inmatrixform: W := vec T V s vec = U P A D a d P B b wherevec M representsacolumnvectorformedbystackingthecolumnsofmatrix M belowoneanother.The lp mn matrix V s isdenedastherow-wiseKroneckerproduct of and V : V s := 2 6 6 6 6 6 6 6 4 1 V 1 2 V 2 lp V lp 3 7 7 7 7 7 7 7 5 where i and V i representthei th rowof and V respectivelyandthesymbol indicatestheKroneckerproduct.FromEquation5,theoriginalgameEquation5 109

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canberewritteninmatrixform: W :=min max vec T V s vec =min P A D a d max P B j S b j s U P A D a d P B b = U withtheconstraints ij 0 and X i j ij =1 ij 0 and X i ij =1 8 j =1,2,..., n TheconstraintsofEquation5andEquation5areemposedtogaraunteethat thesolutionsfor and representvalidprobabilitymassfunctions. TheNashEquilibriumsolutiontothisgameisthepairofequilibriumstrategies and andthecorrespondingequilibriumutility U thatsatisfytheNashEquilibrium condition W W = U W Usingthestrongdualitytheoremoflinearprogramming,thesolutioncanbefound bysolvingasetofduallinearprogrammingproblems[60,61].Iwilldenotecolumn vectorscontainingallonesandallzerosby e and 0 respectively.Vectorinequalitiesare evaluatedelement-wise.Inthefollowingtheorem,theoptimizationvectors x A and x B are denedas x A := 2 6 4 vec u 3 7 5 and x B := 2 6 4 vec v 3 7 5 where u isan n 1 matrixand v isscalar. Theorem5.1. Thesolutiontothefollowingzero-sumgame U =min max vec T V s vec =max min vec T V s vec canbefoundbysolvingthefollowingduallinearprogrammingproblems: 110

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LinearProgramALPA min x A 0 e T x A subjecttotheconstraints 2 6 4 V T s )]TJ/F44 11.9552 Tf 9.299 0 Td [(I 2 6 4 1 1 3 7 5 3 7 5 x A 0 I 0 x A 0 e 0 x A =1 LinearProgramBLPB max v 0 1 x B subjecttotheconstraints V s )]TJ/F44 11.9552 Tf 9.299 0 Td [(e x B 0 I 0 x B 0 I [1,1], 0 x B = e EquilibriumSolutionofGame Theequilibriumstrategiesare vec = I 0 x A andvec = I ,0 x B Theequilibriumvalueofthegameis U = 0 e T x A = 0 ,1 x B Proof. IwillrstverifythatLPAandLPBareinfactduals.Tobegin,LPAwillbeputinto standardformbyreplacingthesingleequalityconstraintEquation5byapairof inequalityconstraints: X i j ij 1 and )]TJ/F30 11.9552 Tf 11.955 11.357 Td [(X i j ij )]TJ/F22 11.9552 Tf 21.917 0 Td [(1. 111

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Iwillalsoreplace u by u 0 )]TJ/F44 11.9552 Tf 11.956 0 Td [(u 00 where u 0 0 and u 00 0 Aftermakingthesesubstitutions,LPAcannowberepresentedinstandardform: StandardFormLinearProgramASLPA min ,u 0 ,u 00 0 e T )]TJ/F44 11.9552 Tf 9.299 0 Td [(e T 2 6 6 6 6 4 vec u 0 u 00 3 7 7 7 7 5 subjecttotheconstraints 2 6 6 6 6 6 6 6 4 V T s )]TJ/F44 11.9552 Tf 9.299 0 Td [(I 2 6 4 1 1 3 7 5 I 2 6 4 1 1 3 7 5 e T 00 )]TJ/F44 11.9552 Tf 9.299 0 Td [(e T 00 3 7 7 7 7 7 7 7 5 2 6 6 6 6 4 vec u 0 u 00 3 7 7 7 7 5 2 6 6 6 6 4 0 1 )]TJ/F22 11.9552 Tf 9.298 0 Td [(1 3 7 7 7 7 5 vec 0 u 0 0 u 00 0. Icannowusethegeneralstandardformofthestandarddualtocalculatethedual ofSLPA[62]: StandardFormofLinearProgramADual max v 0 v 00 0 1 )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 2 6 6 6 6 4 vec v 0 v 00 3 7 7 7 7 5 subjecttotheconstraints 2 6 6 6 6 4 V s e )]TJ/F44 11.9552 Tf 9.298 0 Td [(e )]TJ/F44 11.9552 Tf 9.299 0 Td [(I [1,1] 00 I [1,1] 00 3 7 7 7 7 5 2 6 6 6 6 4 vec v 0 v 00 3 7 7 7 7 5 2 6 6 6 6 4 0 e )]TJ/F44 11.9552 Tf 9.298 0 Td [(e 3 7 7 7 7 5 112

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vec 0 v 0 0 v 00 0. Wecannowsubstituteanunbounded v for v 0 )]TJ/F39 11.9552 Tf 13.243 0 Td [(v 00 .Additionally,thebottom 2 n inequalityconstraintscanbereplacedwith n equalityconstraints.Afterthese substitutions,itcanbeseenthatthedualofLPAisinfactLPB: LinearProgramB max v 0 1 2 6 4 vec v 3 7 5 subjecttotheconstraints V s )]TJ/F44 11.9552 Tf 9.299 0 Td [(e 2 6 4 vec v 3 7 5 0 vec 0 I [1,1], 0 x B = e Next,wewillshowthatthesolutionstotheduallinearprograms,LPAandLPB,provide theequilibriumstrategiesandvaluetothezero-sumgameposedinSection5.1.For notationalclarity,Idene i asthe i thelementofvec j asthe j thelementof vec ,and v ij astheelementinthe i throwand j thcolumnof V s .Recalltheutility functionforthegame: U = vec V s vec = X i =1 X k =1 i v ij j Fromthecomplementaryslacknesspropertyofduallinearprograms[63],the optimalplayerstrategies, i and j ,satisfy i =0 if X j v ij j > v 113

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and i > 0 onlyif X j v ij j = v Icannowsubstitutetheseconditionsintotheutilityfunction: X i =1 X j =1 i v ij j = X i =1 i v = v Thelastequalityresultsfromthefactthat P i =1 duetotheequalityconstraint Equation5ofLPA.ThesameprocedurecanbeappliedtoLPB.Againusingthe complementaryslacknessproperty,theoptimalsolutionssatisfy j =0 if X i v ij i > u k and j > 0 onlyif X i v ij i = u k where u k referstothe k thelementof u where k = oor j l +1 .Putsimply, u k isthe elementof u thatappearsinthe j thinequalityconstraintinEquation5ofLPA. UsingpropertiesEquation5andEquation5withtheutilityfunctionprovides X i =1 X j =1 i v ij j = X j =1 j u k = X k =1 u k Since v and P k =1 u k aretheoptimalvaluesofduallinearprograms,Iknowthatthey mustbeequalduetothestrongdualitytheoremoflinearprogramming[64]: v = X k =1 u k Iwillnowlookattheutilityvaluesthatresultfromanarbitraryadmissiblestrategy andoptimal .Since isoptimal,itmustsatisfyeachinequalityconstraintfrom Equation5: X j =1 v ij j v 8 i =1,2,..., lp 114

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IfImultiplythe i thconstraintofLPBby i andsumover i ,itisseenthat X i =1 i X j =1 v ij j X i =1 i v v Usingthesameprocedure,Icanexaminetheresultingutilityforanadmissible andoptimal .Since isoptimal,itmustsatisfyeachinequalityconstraint: X i =1 i v ij u k Imultiplyby j andsumover j tondtheupperbound: X j =1 j X i =1 i v ij X j =1 u k j X k =1 u k CombiningthelowerboundEquation5,upperboundEquation5,and equalityconditionEquation5yieldstheNashequilibriumconditionfor and : X i =1 X j =1 i v ij j X i =1 X j =1 i v ij j = v = X k =1 u k X i =1 X j =1 i v ij j Therefore,thestrategiesrepresentedbytheoptimalsolutionstoLPAandLPBare inequilibrium.Additionally,theoptimalvalue v = P k =1 u k istheequilibriumutilitytothe game. UsingTheorem5.1,Icannumericallysolveawiderangeofpotentialscenariosas longasthesensornetworkcharacteristicsandvaluefunctionareknown.InSection 5.3,Iaddressaspecicexamplewithstructuralcharacteristicsthatallowusetodevelop analyticsolutionstotheselinearprogramsforspecialcases. 5.3Two-ActionGamewithIdenticalInformationChannels Inthissection,Iwillpresentaspecicillustrativeexampleofthedeceptivegame denedinSection5.1.Iwillrefertothisparticularexampleasthe intelligencegame .I analyticallysolvespecialcasesofthisgame,andanalyzeinterestingcharacteristicsof thesolutionthatarequalitativelysimilartoconceptsfromtheintelligenceanddeception 115

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community.Inparticular,IaddresstheconceptknownasJones'Lemmaandits relationshiptothisparticulargame. 5.3.1MotivatingScenario Considerascenarioinwhichtheleaderofanillegaldrugdistributionnetwork, representedbyPlayerA,knowsthatthereisanimpendingraidononeoftwopossible locationswherethedrugscanbehidden.PlayerAalsoknowsthatithasnotyetbeen decidedwhichsitewillbetargetedbythelawenforcementorganization,whoseleader isrepresentedbyPlayerB.PlayerAmustchoosewhichlocationtohidethedrugs.The rstlocationisalocalcommunitycenter.Hidingthedrugsatthecommunitycenterwill beinitiallymorecostly;however,ifthecommunitycenterisraidedwhilethedrugsare present,therewillbelessevidencelinkingPlayerAtothedrugs,anditwillbeeasierto avoidprosecution.Ifthecommunitycenterisraidedandthedrugsarenotthere,there willbesignicantcommunityoutcryagainstPlayerB,whichwillhinderfutureoperations againstPlayerA.FromPlayerA'sperspective,thisisthebestpossibleoutcomewhile itistheworstoutcomefromPlayerB'sperspective.TheotherlocationisPlayerA's warehouse.Itisrelativelyeasytohidethedrugsatthewarehouse,butifthedrugsare discoveredthere,itwillbedifculttodenyinvolvementandPlayerAwillmostlikelybe convicted.ThisisconsideredtheworstpossibleoutcomefromPlayerA'sperspective; however,thisisthebestpossibleoutcomefromthelawenforcement'sperspective. Withinthecommunity,thereareseveralinformantswhoarewillingtoprovide informationtothelawenforcementaboutthelocationofthedrugs.However,these informantsarenotparticularlyaccurateandcanonlycorrectlyidentifythelocation withaprobabilityof p id .ItisalsocommonknowledgethatPlayerAiscapableof persuadingasubsetoftheseinformantstoprovidespecicinformationifheiswilling topayaparticularprice, c d ,foreachinformantcorrupted.PlayerAdoesnotpickwhich particularinformantstopayoff,butinsteaddirectsalower-levelagenttopayoffa specicnumberofinformants.Itisassumedthatanyresourcesusedtopaythese 116

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informantsareresourcesnotusedtoproducemoredrugs,whichisbenecialfromthe lawenforcement'sview. Thedrugnetworkleadermustdecideonthelocationtohidethedrugsandhow manyinformantsshouldbepaidoffinanattempttomisleadPlayerB.Ontheother hand,PlayerBmustdecidewhichlocationtoraidbasedontheinformationprovidedby theinformantswhiletakingintoaccountthattheinformantscouldpossiblybecorrupt. 5.3.2GameModelandDescription PlayerAandPlayerBeachhavetwopossibleactionstochoosefrom.Actions a 1 and a 2 representPlayerA'schoicestohidethedrugsatthecommunitycenterorhis warehouserespectively.PlayerB'schoicestoraidthecommunitycenterofPlayerA's warehousearerepresentedby b 1 and b 2 respectively.Wecanthendenetheaction sets A = f a 1 a 2 g and B = f b 1 b 2 g forthisgame. Thesensornetworkconsistsof N identicalinformantsandisdepictedgraphically inFigure5-3.Eachinformantproducesascalarvalue ^ s .Whenaninformantisnot corrupted,itcorrectlyidentiesPlayerA'sactionwithalikelihoodof p id byoutputtinga zerofor a 1 andaonefor a 2 Althougheachinformantcanbemanipulatedthroughitscorrespondingdeception signal ^ d ,PlayerAisonlyallowedtoselectthetotalnumberofinformantstocorrupt throughitsdeceptionstrategy d .ItisassumedthatPlayerAmustforceallcorrupted informantstothesamevalue.Forexample,PlayerAmaychoosetosetveinformants tozero,butPlayerAisnotallowedtosimultaneouslysettwoinformantstozeroand threeotherinformantstoone.Therefore,PlayerAhastheoptionofforcingupto N informantstozero,forcingupto N informantstoone,ornotmanipulatinganyinformants foratotalof 2 N +1 differentdeceptiontactics.Aparticulardeceptiontactic d j 2 D where D := f d 1 ,..., d 2 N +1 g ,denesthenumberofcorruptedinformantsandtheir correspondingvalueaccordingtothefollowingrule. 117

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Force N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j channelstozero j < N +1 NoCorruption j = N +1 Force j )]TJ/F22 11.9552 Tf 11.955 0 Td [( N +1 channelstoone j > N +1 Theoutputsoftheindividualinformantsareaddedtogether,andthesumisusedas thesensornetworkoutput s 2 S := f s 1 ,..., s n g ,where s i = i )]TJ/F22 11.9552 Tf 12.387 0 Td [(1 and n = N +1 .The valuefunction V a i d j b k = V ik + c d j N +1 )]TJ/F39 11.9552 Tf 12.131 0 Td [(j j ,where c d isthecosttocorruptasingle informant.BasedontheplayerpreferencesdescribedinSection5.3.1,itisassumed that V 21 V 12 V 11 V 22 5.3.3GameParameterization InordertosolvethegameusingthemethoddescribeinTheorem5.1,wemustrst parameterizethegame.WerepresentPlayerAandPlayerB'sstrategiesusing =[ i j ] and =[ i j ] ,respectively. Theelementsofthesensornetworkcharacteristicmatrix arecalculatedas 2 j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1+ i k = p s k j a i d j = 8 > > > > > > > < > > > > > > > : B j p a i k k j j N 0 k > j j < N 0 k < j )]TJ/F39 11.9552 Tf 11.955 0 Td [(N j > N B N )]TJ/F39 11.9552 Tf 11.955 0 Td [(j p a i k )]TJ/F39 11.9552 Tf 11.955 0 Td [(j + N k j )]TJ/F39 11.9552 Tf 11.955 0 Td [(N j > N where B n p k = n k p k )]TJ/F39 11.9552 Tf 11.955 0 Td [(p n )]TJ/F40 7.9701 Tf 6.586 0 Td [(k and p a = 8 > < > : 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(p id a = a 1 p id a = a 2 Theelementsofvaluematrix V are V 2 j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1+ i k = V a i d j b k = V a i b k + c d j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j 118

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5.3.4SpecialCaseSolutions Inthissection,Iwilldiscussthesolutionoftheintelligencegameforparticular specialcases. 5.3.4.1TheCaseofNoInformants Sincetherearenoinformants,thesensornetworkcanonlygenerateonvalue s 1 ,whichrepresentsnodata.PlayerB'sstrategymatrix issimplyacolumnvector representingtheprobabilityofplayingeither b 0 or b 1 .PlayerAhasnochoicebutto deploythedegenerativenodeceptiontacticsincetherearenoinformantstocorrupt. Therefore,PlayerA'sstrategymatrix alsoreducestoacolumnvectorrepresentingthe probabilityofeachaction.Thesensorcharacteristicmatrixreducestoascalar: =1 Therefore,thegamesvaluefunctionmatrixisdenedas V s = V .Thesesimplications convertthisgameintoastandardtwo-actionzero-sumgamewhosesolutionfollows easilyfromstandardresults[60].Theproofofthistheoremisomittedforthesakeof brevity. Theorem5.2. Assumingthat n =0 andthevaluefunction V a d b possessesthe structuredenedbyEquation5andEquation5,theequilibriumstrategiesand resultingequilibriumvaluearegivenasfollows. EquilibriumStrategies = s := V 22 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 + V 22 T = s := V 22 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 + V 22 T EquilibriumUtility U = W = V s := V 11 V 22 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 ThesolutionpresentedinTheorem5.2representsthebaselinescenarioforPlayer A.Asmoreinformantsareaddedtothegame,PlayerBwillgainmoreinformationabout PlayerA'sselectedaction.Usingthisinformation,PlayerBcanthenpotentiallyincrease theexpectedutilityvalue.PlayerAcanattempttocorrupttheinformantsinorderto 119

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reducetheinformationPlayerBreceivesfromsensornetworkoutputandtherebyhold theequilibriumvalueclosertothestandardgameequilibrium.However,PlayerAcannot reducetheequilibriumvaluebelow V s bycorruptingsensorsbecausePlayerBcan alwaysguaranteeavalueofatleast V s byplaying s foreachsensorvalue.Thisis proveninthethefollowingTheorem. Theorem5.3. Foranyinformantgamewithparametervalues n 0 0 p id 1 ,and c d 0 ,theequilibriumvalueoftheresultinggame, U ,isgreaterthanorequaltothe baselinevalue V s : U V s Proof. First,Iwilldeneabaselinestrategy, ~ =[ ~ k l ] ,forPlayerB: ~ k l := 8 > < > : V 22 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 l =1 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 l =2 TheexpectedutilitywhenPlayerAplaysanarbitraryadmissiblestrategy andPlayerB implements ~ is W ~ = 2 X i =1 2 N +1 X j =1 N +1 X k =1 2 X l =1 p a i d j p s k j a i d j p b l j s k V il + j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j = 2 X i =1 2 N +1 X j =1 N +1 X k =1 2 X l =1 p a i d j p s k j a i d j p b l j s k V il + 2 X i =1 2 N +1 X j =1 p a i d j j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j = 2 X i =1 2 N +1 X j =1 N +1 X k =1 p a i d j p s k j a i d j V s + 2 X i =1 2 N +1 X j =1 p a i d j j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j = 2 X i =1 2 N +1 X j =1 p a i d j V s + 2 X i =1 2 N +1 X j =1 p a i d j j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j = V s + 2 X i =1 2 N +1 X j =1 p a i d j j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j V s 120

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FromEquation5,itcanbeseenthat ~ providesautilityofatleast V s nomatter whatstrategyPlayerAemploys.FromtheNashEquilibriumconditionEquation5, weknowthattheequilibriumstrategiessatisfy W W Inparticular, V s W ~ W .Therefore, V s formsalowerboundforthe equilibriumutility, U ,foranyinformantgame: V s W = U 5.3.4.2TheCaseofZero-CostDeception InTheorem5.3,Ishowedthat V s formsalower-boundontheequilibriumutility forallinformantgames.IwillnowshowthatPlayerAiscapableofmaintainingthis equilibriumvalueforany p id and N whendeceptioniscostfree, c d =0 Theorem5.4. Supposethat n > 0 1 p id 0 c d =0 andthevaluefunction V a d b possessesthestructuredenedbyEquation5andEquation5.Let V s beas inEquation5.Then,theequilibriumstrategiesandresultingequilibriumvalueare givenasfollows. PlayerStrategies i j = P A D a i d j = 8 > > > > > > > < > > > > > > > : V 22 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 11 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 i =1, j = n +1 0 i =1, j 6 = n +1 ^ j V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 i =2, j n +1 0 i =2, j > n +1 k l = P B j S b l j s k = 8 > < > : V 22 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 + V 22 l =1 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 + V 22 l =2 121

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where ^ j isthe j thelementofthe N +1 1 vector ^ whichsatises ^ ^ =^ Theelementsofthematrices ^ =[ ^ k j ] and ^ =[^ k ] aredenedas ^ k j := p s k j a 2 d j ^ k := p s k j a 1 d n +1 EquilibriumUtilities U = V s Proof. Tobegintheproof,Iwillshowthataunique ^ existsthatsatisesEquation 5.FromEquation5,Iknowthat p s k j a 2 d j = 8 > < > : )]TJ/F40 7.9701 Tf 5.479 -4.379 Td [(j )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 k p k id )]TJ/F39 11.9552 Tf 11.955 0 Td [(p id j )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 )]TJ/F40 7.9701 Tf 6.586 0 Td [(k k j 0 k > j Sinceallelementsbelowthemaindiagonalof ^ arezeroandallelementsalongthe maindiagonalarenon-zero,thematrix ^ isupper-triangularandinvertable.Icanthen solvedirectlyfor ^ : ^ = ^ )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 ^ Next,IshowthatthestrategiesdenedinEquation5andEquation5 areinequilibriumandproducetheequalibriumutilitystatedinEquation5.Idene ^ u =[^ u k ] where ^ u k = V s p s k j a 1 d n +1 = V 11 V 22 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 + V 22 p s k j a 1 d n +1 SubstitutingEquation5intothevaluefunctionofLPAEquation5producesa valueof V s : n X k =1 u k = n X k =1 V s p s k j a 1 d n +1 = V s 122

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SubstitutingEquation5andEquation5intothe j thconstraintofLPAprovides: 2 X i =1 2 n +1 X j =1 p s k j a i d j p a i d j V il + c d j n +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j )]TJ/F39 11.9552 Tf 11.955 0 Td [(u k = 2 X i =1 2 n +1 X j =1 p s k j a i d j p a i d j V il )]TJ/F39 11.9552 Tf 11.956 0 Td [(u k = p s k j a 1 d n +1 p a 1 d n +1 V 1 l + n +1 X j =1 p s k j a 2 d j p a 2 d j V 2 l )]TJ/F39 11.9552 Tf 11.955 0 Td [(u k SincePlayerA'sstrategysatisesEquation5,weknowthat n +1 X j =1 p s k j a 2 d j p a 2 d j = p s k j a 1 d n +1 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 22 SubstitutingEquation5intoEquation5yields p s k j a 1 d n +1 V 22 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 11 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 22 V 1 l + V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 22 V 2 l )]TJ/F40 7.9701 Tf 19.737 4.884 Td [(V 11 V 22 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 V 12 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 + V 22 Whether l =1 or l =2 ,thetermswithintheparenthesisreducetozero.Thisimplies thateachinequalityconstraintofLPAissatised.Thereforethestrategydenedby Equation5andthevector ^ u k denedbyEquation5formanadmissibleinput forLPAthatyieldsavalueof V s WewillnowexaminePlayerB'sstrategydenedbyEquation5.Idene ^ v := V s .SubstitutingEquation5and ^ v intothe i thconstraintofLPBprovides: n X k =1 2 X l p s k j a i d j p b l j s k V il + c d j n +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = n X k =1 p s k j a i d j V 22 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 + V 22 V i 1 + V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 + V 22 V i 2 )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = n X k =1 p s k j a i d j V s )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v =0. FromEquation5,wecanseethatEquation5and ^ v satisfyeachinequality constraintofLPBandproduceavalueof V s 123

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SinceEquation5andEquation5areadmissibleinputstoduallinear programsandtheirrespectivevaluesareequal,theymustbeoptimalsolutionsfrom thedualitytheoremoflinearprogramming.ThereforefromTheorem5.1,theyarealso equilibriumstrategiesthatproduceanequilibriumutilityof V s 5.3.4.3TheCaseofPerfectInformants p id =1 Whentheinformantscanperfectlyidentifythedruglocation, p id =1 ,PlayerAcan selectwhichsensorvaluetogeneratesincethesensornetworkbehavedeterministically. Althougheachcombinationofactionanddeceptiontacticgeneratesasinglesensor value,theresultingvaluesarenotnecessarilyunique.Forexample,thecombinations a 1 d 2 and a 2 d n +2 bothresultinthesamesensorvalue s n )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 .Therefore,PlayerBis incapableofuniquelydeterminePlayerA'simplementedactionordeceptiontacticbased onthemeasuredsensorvalue.Thisspecialcasecanberepresentedasasequential gamewithimperfectinformationfromPlayerB'sperspective.Theresultingclosedform solutiontothisgameisdescribedinTheorem5.5. Theorem5.5. Supposethat n > 0 p id =1 ,andthevaluefunction V a d b possesses thestructuredenedbyEquation5andEquation5.Let V s beasinEquation 5anddenetheJones'Cost J c andtheJones'Threshold k respectivelyas J c := V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 nc d V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 + V 22 and k := V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 N +1 )]TJ/F40 7.9701 Tf 6.587 0 Td [(k c d Then,theequilibriumstrategiesandresultingequilibriumvaluearegivenasfollows. EquilibriumStrategies ij = P A D a i d j = 8 > > > > < > > > > : ^ i =1, j = n +1 1 )]TJ/F22 11.9552 Tf 12.903 0 Td [(^ i =2, j =1 0 otherwise k l = P B j S b l j s k = 8 > < > : ^ k i =1 1 )]TJ/F22 11.9552 Tf 13.428 2.657 Td [(^ k i =2 124

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where ^ = 8 > < > : V 22 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 + V 22 1 1 1 1 < 1 ^ k = 8 > < > : V 22 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 + N )]TJ/F23 7.9701 Tf 6.587 0 Td [(2 k c d V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 + V 22 k 1 1 k < 1 EquilibriumUtility U = U = U J := 8 > < > : V s + J c 1 1 V 11 1 < 1 Proof. WewillbegintheproofbyverifyingthatPlayerA'sstrategystatedinEquation 5isanadmissibleinputforLPA.Wedeneacorresponding n 1 vector ^ u =[^ u k ] where ^ u k := 8 > < > : U J k =1 0 k > 1 Ifitisassumedthat 1 1 ,substitutingthestrategydenedinEquation5for PlayerAand ^ u intothe 1 stconstraintofEquation5resultsin 2 X i =1 2 N +1 X j =1 i j p s 1 j a i d j V i 1 + j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(U J = 1, N +1 V 11 + 2,1 V 21 + Nc d )]TJ/F39 11.9552 Tf 11.955 0 Td [(U J =0. Substitutionintothe 2 ndconstraintofEquation5provides 2 X i =1 2 N +1 X j =1 i j p s 1 j a i d j V i 2 + j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(U J = 1, N +1 V 12 + 2,1 V 22 + Nc d )]TJ/F39 11.9552 Tf 11.955 0 Td [(U J =0. 125

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AllotherconstraintsinEquation5arealsosatisedbecause p a i d j p s k j a i d j =0 forall k > 1 : 2 X i =1 2 N +1 X j =1 i j p s k j a i d j V i 1 + j N +1 )]TJ/F39 11.9552 Tf 11.956 0 Td [(j j c d =0 8 k > 1. Therefore, [ T J ,^ u T ] T satisesallconstraintsinLPAandproducesavalueof U J when 1 1 Ifitisassumedthat 1 < 1 ,substituting [ J ,^ u T ] intotherstconstraintofEquation 5resultsin 2 X i =1 2 N +1 X j =1 i j p s 1 j a i d j V i 1 + j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(U J = 1, N +1 V 11 )]TJ/F39 11.9552 Tf 11.956 0 Td [(U J = V 11 )]TJ/F39 11.9552 Tf 11.955 0 Td [(V 11 =0. Substitutionintothesecondconstraintresultsin 2 X i =1 2 N +1 X j =1 i j p s 1 j a i d j V i 2 + j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j c d )]TJ/F39 11.9552 Tf 11.955 0 Td [(U J = 1, N +1 V 12 )]TJ/F39 11.9552 Tf 11.956 0 Td [(U J = V 12 )]TJ/F39 11.9552 Tf 11.955 0 Td [(V 11 0. Again,allotherconstraintsinEquation5aresatisedbecausethecorresponding sensorvectorwillneverbeproduced.Therefore, [ T J ,^ u T ] T satisesallconstraintsin LPAandproducesavalueof U J when < 1 IwillnowexaminePlayerB'sstrategydenedinEquation5.Idenea corresponding ^ v where ^ v = U J Ifweassumethat 1 1 ,substituting [ vec J T ,^ v ] intothe N +1 constraintof Equation5yields X k X l p s k j a 1 d n +1 k l V 1 l + N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(N )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 c d )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = 1,1 V 11 + 1,2 V 12 )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v =0. Similarly,substituting [ vec J T ,^ v ] intothe 2 ndconstraintofEquation5yields X k X l p s k j a 2 d 1 k l V 2 l + N +1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 c d )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = 1,1 V 21 + 1,2 V 22 + Nc d )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v =0. 126

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Becauseitisassumedthat 1 1 ,itisimpliedthat k 1 forall k .Therefore,when examiningallotherconstraintsinEquation5,Iseethat X k X l p s k j a i d j k l V 1 l + j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j c d )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = k ,1 V i 1 + k ,2 V i 2 + j N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(j j c d )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 N )]TJ/F40 7.9701 Tf 6.587 0 Td [(k c d + V 22 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 kc d V 11 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 22 )]TJ/F23 7.9701 Tf 22.031 5.477 Td [( V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 Nc d V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 22 V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 Nc d V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 22 )]TJ/F23 7.9701 Tf 22.031 5.478 Td [( V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 Nc d V 11 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 21 )]TJ/F40 7.9701 Tf 6.586 0 Td [(V 12 )]TJ/F40 7.9701 Tf 6.587 0 Td [(V 22 =0 Wehavenowshownthat [ vec J T ,^ v ] satisesallconstraintsofLPBandproducesa valueof U J when 1 1 IfIassumethat < 1 ,substituting [ vec J T ,^ v ] intothe N +1 constraintof Equation5yields X k X l p s k j a 1 d n +1 k l V 1 l + N +1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(N )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 c d )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = V 11 )]TJ/F22 11.9552 Tf 12.243 0 Td [(^ v =0. Substituting [ vec J T ,^ v ] intothe 2 ndconstraintofEquation5yields X k X l p s k j a 2 d 1 k l V 2 l + N +1 )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 c d )]TJ/F22 11.9552 Tf 12.244 0 Td [(^ v = V 21 + Nc d )]TJ/F39 11.9552 Tf 11.955 0 Td [(V 11 0. Theotherremainingconstraintsarealsosatisedwhether k 1 or k < 1 .Therefore, thevector [ vec J T ,^ v ] isanadmissibleinputtoLPBandproducesavalueof U J when 1 < 1 WehaveshownthatthestrategiesstatedinEquation5andEquation5 areadmissibleinputstoLPAandLPB.Additionally,theyproducethesamevalue U J whichmustthenbeoptimalduetothedualitytheoremoflinearprogramming.Therefore, thesestrategiesmustalsobeinequilibriumfromTheorem5.1. Inthisspecialcase,theequilibriumvalueforthegameEquation5issplit intotwocases.Thevalue J c ,whichIwillrefertoasthe Jones'Cost ,capturesthe additionalcostforPlayerAtomixinthedeceptivetactics.TheJonesCostiscritically 127

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dependentonthenumberofinformantsandthecostofcorruption.Aseitherthenumber ofinformationchannelsorthecostofcorruptionincreases,theexpectedutilityincreases aswell.Oncethenumberofinformantsorthecostofcorruptionhaveincreasedsuch that 1 < 1 ,thecostofthedeceptivetactic d 1 outweighsanybenetPlayerAwould gainfromemployingdeception.Therefore,PlayerAfallsbacktoanon-deceptivesafety strategy a 1 d n +1 .Iwillreferto 1 asthe Jones'Threshold .ThevalueoftheJones' Thresholdindicateswhetherornottoengageindeceptivetactics.Thisthresholding behaviorleadsustothenextsectionwhereIexaminetheeffectsofthenumberof informantsandtheircorrespondingcorruptioncostsforgeneralvaluesof p id and c d 5.3.5PiecewiseCharacteristicsoftheGeneralEquilibriumStrategies Ingeneral,itisnotpossibletoderiveaclosedformsolutionforarbitraryvaluesof p id and c d .However,itispossibletoshowforaxed p id ,theequilibriumstrategy ispiecewiseconstantwithrespecttochangesin c d .Similarly,theequilibriumstrategy ispiecewiselinearwithrespecttochangesin c d when p id isheldconstant.These characteristicsarecapturedinTheorem5.8andTheorem5.9respectively. BeforeweaddressTheorem5.8andTheorem5.9,wewillformalizeafewnew concepts.Denetheconstraintmatrix A asthecollectionofinequalityconstraintsin LPA: A := 2 6 6 6 6 4 V T s )]TJ/F44 11.9552 Tf 9.299 0 Td [(I 2 6 4 1 1 3 7 5 )]TJ/F44 11.9552 Tf 9.298 0 Td [(I 0 3 7 7 7 7 5 where Ax A 0 .Similarly,denetheconstraintmatrix B asthecollectionofinequality constraintsinLPB: B := 2 6 4 V s )]TJ/F44 11.9552 Tf 9.298 0 Td [(e I 0 3 7 5 where Bx B 0 128

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Foranylinearprogramwithdimension N ,ifthereare N linearlyindependent constraintsthatareactiveatvector x ,thenwesaythat x isabasicfeasiblesolution. Thisterminologyismotivatedbythefactthattheactiveconstraintsformabasisfor R N Considerabasicfeasiblesolution x A forLPA.Since x A isabasicsolutionandLPAhasa dimensionof 3 n +1 ,weknowthatthereare 3 n +1 activeconstraintsinLPAat x A .From thedenitionofLPA,thereisalwaysoneactiveequalityconstraintEquation5. Therefore,theremaining 3 n linearlyindependentactiveconstraintsat x A correspond toactiveinequalityconstraints.Wenowdeneabasismatrix A ,whichiscomposed oftherowsof A thatcorrespondtoactiveinequalityconstraintsat x A andtheequality constraintEquation5: A := 2 6 6 6 6 6 6 6 6 6 6 4 A i 1 A i 2 A i k e T 0 T 3 7 7 7 7 7 7 7 7 7 7 5 where A i correspondstothe i throwofconstraintmatrix A .Itisassumedthatthe orderingoftherowsof A isthesameastheorderof A .Becausetheactiveconstraints ofthebasicsolutionformabasis,thesquarematrix A hasfullrankandistherefore invertible. SimilarlyforLPB,thebasismatrix B correspondingtoabasicsolution x B is composedoflinearlyindependentrowsof B andtheequalityconstraintsEquation 5: B := 2 6 6 6 6 6 6 6 6 6 6 4 B i 1 B i 2 B i k [ I [1,1], 0 ] 3 7 7 7 7 7 7 7 7 7 7 5 129

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where B i correspondstothe i throwoftheconstraintmatrix B .Everybasicsolution anditscorrespondingbasismatrixsatisfythefollowingconditions A x A = 2 6 4 0 1 3 7 5 and B x B = 2 6 4 0 e 3 7 5 Fromthefundamentaltheoremoflinearprogramming,weknowthatifanoptimal solutionexistsforalinearprogrammingproblemthenthereexistsanoptimalbasic feasiblesolution.ThisimpliesthatifanoptimalsolutionforLPAisunique,itmustmust alsobebasic.Althoughthesolutionmaynotnecessarilybeunique,Theorem ?? shows thatasolutionexistsforeveryLPAandLPBthatcorrespondtoanintelligencegame.Ifa solution x A foraLPAisuniqueforaparticularsetofparametervalues,wecanshowthat anyLPAwithidentical N and p id butwithaslightlyperturbedvalueof c d willalsohavea uniquesolution;moreover,bothsolutionswillhavethesamecorrespondingbasismatrix. Theorem5.6. Supposethatthesolution x 0 A isuniqueforLPAwithparametervalues c d = c 0 d > 0 N > 0 ,and 0 < p id < 1 .Dene A 0 tobethecorrespondingbasismatrix forthissolution.Thereexistsan > 0 ,suchthateachLPAwithparametervalues c d 2 c 0 d )]TJ/F25 11.9552 Tf 12.109 0 Td [( c 0 d + N ,and p id willalsohaveauniquesolution x A withacorresponding basismatrix A 0 Proof. Denethenitesetofallfeasiblebasicsolutionsas F A := f x 1 x 2 ,..., x z g andthe setoftheircorrespondingbasicmatricesas B A := f A 1 A 2 ,..., A z g .Wecancalculate theresultingvalueofLPA U i foreachofthebasicfeasiblesolutions x i A 2 F A : U i :=[ 0 e T ] x i =[ 0 e T ] A )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 6 4 0 e 3 7 5 130

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= a 1 i + c d a 2 i Supposethat x j isauniqueoptimalbasicsolution.Sincethissolutionisunique,we knowthat U k > U i 8 i 6 = k a 1 k + c d a 2 k > a 1 i + c d a 2 i 8 i 6 = k Itcanthenbeseenthatthereexistsan > 0 suchthat a 1 k + c d + a 2 k > a 1 i + + c d a 2 i 8 i 6 = k Therefore,theuniquefeasibleoptimalsolution x k willremaintheuniquefeasibleoptimal solutionwithinaneighborofvaluesof c d Theorem5.7. Supposethatthesolution x 0 B isuniqueforLPBwithparametervalues c d = c 0 d > 0 N > 0 ,and 0 < p id < 1 .Dene B 0 tobethecorrespondingbasismatrix forthissolution.Thereexistsan > 0 ,suchthateachLPBwithparametervalues c d 2 c 0 d )]TJ/F25 11.9552 Tf 12.109 0 Td [( c 0 d + N ,and p id willalsohaveauniquesolution x B withacorresponding basismatrix B 0 Fortheinformantgame,eachbasismatrixofLPAandLPBpossessparticular structuralcharacteristicswhicharedescribedinLemma9andLemma10. Lemma9. Foreverybasicfeasiblesolution x A ofLPA,thecorrespondingactivebasis matrix, A ,canbeexpressedas A = 2 6 4 A 1 e T 0 T 3 7 5 0 B @ I + c d 2 6 4 00 ^ A 0 3 7 5 1 C A wherethesizeof A is n +1 n +1 ,thesizeof A 1 is n n +1 ,and ^ A isan n +1 n +1 matrix. Proof. Thisproofisomittedforthesakeofbrevity.Theproofissimilartothefollowing proofofLemma10 131

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Lemma10. Foreverybasicfeasiblesolution x B ofLPB,thecorrespondingbasismatrix, B ,canbeexpressedas B = 0 B @ I + c d 2 6 4 0 ^ B 00 3 7 5 1 C A 2 6 4 B 1 [ I [1,1], 0 ] 3 7 5 wherethesizeof B is n +1 n +1 ,thesizeof B 1 is n +1 n +1 ,and ^ is an n n matrix. Proof. InLPB,theoptimizationvector x B contains 2 n +1 elements.Forany x B ,the correspondingbasismatrix B mustbeafullrank n +1 x n +1 matrix,whereeach rowcorrespondstoanactiveconstraint.FromEquation5,weknowthatthere arealways n activeequalityconstraintsthatensurethatthemarginalconditionalpdfs sumtoone.Therefore,theremaining n +1 rowsof B correspondtoactiveinequality constraints.Sincetheparticularorderingoftherowsisnotimportant,letusarrange therowsof B suchthattheactiveinequalityconstraintsareplacedonthetopandthe equalityconstraintsareplacedonthebottom: B = 2 6 4 B 3 I [1,1], 0 3 7 5 where B 3 isan n +1 n +1 matrixwithrowscorrespondingtotheactiveinequality constraints.Let B m representthe m throwof B 3 .Eachrowof B 3 correspondstoan activeconstraintineitherEquation5orEquation5.If B m correspondstoan activeconstraintinEquation5,thenitpossessesthefollowingform B m = [ j m V j m )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 ] where k m and V k m representthe k m throwof and V respectively.Usingthedenition of V S ,row B m canbedecomposedasasumofatwocolumnvectors: B m = B m 1 + c d B m 2 132

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where B m 1 = [ k m [ V i 1 V i 2 ], )]TJ/F22 11.9552 Tf 9.299 0 Td [(1 ] B m 2 = [ ^ m I [1,1],0 ] Therowvector ^ m isdenedas ^ m = j n +1 )]TJ/F39 11.9552 Tf 11.956 0 Td [(j j k m Thevaluesfor i and j aredeterminedbythe k m throwof V If B m correspondstoanactiveconstraintinEquation5,thenitpossessesthe sameform B m = B m 1 + c d B m 2 butthevaluesfor B m 1 and B m 2 aredenedas B m 1 = [ 0,0,...,0,1,0,...,0 ] B m 2 = [ ^ m I [1,1],0 ] where ^ m =[0,...,0]. Thelocationofthesingle1in B m 1 isdeterminedbywhichconstraint B m representsin Equation5. UsingEquation5andEquation5,matrix B 2 canberepresentedasa sumoftwomatrices: B 3 = B 1 + c d B 2 133

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where B 1 and B 2 arecreatedbystackingvectors B m 1 and B m 2 respectively: B 1 = 2 6 6 6 6 6 6 6 4 B 11 B 21 B n 1 3 7 7 7 7 7 7 7 5 and B 2 = 2 6 6 6 6 6 6 6 4 B 12 B 22 B n 2 3 7 7 7 7 7 7 7 5 Matrix B 2 canberepresentedasthematrixproduct B 2 = ^ [ I [1,1], 0 ] wherethematrix ^ iscomposedofstackingtherowvectors ^ m : ^ := 2 6 6 6 6 6 6 6 4 ^ 1 ^ 2 ^ n 3 7 7 7 7 7 7 7 5 UsingEquation5,Equation5,andEquation5,wecannowexpress B as aproductofblockmatrices: B = 0 B @ I + c d 2 6 4 0 ^ 00 3 7 5 1 C A 2 6 4 B 1 I [1,1], 0 3 7 5 Theorem5.6showsthatthereexistsaneighborhoodofvaluesfor c d around c 0 d inwhichLPApossessesauniquesolution.Thisneighborhoodmayinfactcontain arelativelywiderangeofvalues c 1 c 2 .Becauseeachvalueof c d 2 c 1 c 2 producesauniquesolutioninLPA,wecanconstructafunctionthatmapseach c d toitscorrespondinguniquesolutiontoLPA.Duetothestructureofthebasismatrix describedLemma9,wecanshowthatthisfunctionislinear.Moreover,theequilibrium strategy containedwithinthesolutionofLPAisconstantwithrespecttochanges 134

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in c d withintherange c 1 c 2 .ThislocalbehaviorisdescribedinTheorem5.8.Using thesameargument,PlayerB'sequilibriumstrategy isshowntobelinearintermsof changesin c d inTheorem5.9.InTheorem5.10,theequilibriumutility U isshowntobe linearintermsofchangesin c d Theorem5.8. SupposethatLPApossessesauniquesolutionforeach c d 2 c 1 c 2 for axed 0 < p id < 1 and N > 0 withacorrespondingbasismatrix A .Theequilibrium strategy isconstantwithrespectto c d andcanbeexpressedas = 1 where 1 =[ I 0 ] 2 6 4 A 1 e T 3 7 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 6 4 0 1 3 7 5 Theorem5.9. SupposethatLPBpossessesauniquesolutionforeach c d 2 c 1 c 2 for axed 0 < p id < 1 withacorrespondingbasismatrix B .Theequilibriumstrategy is linearwithrespectto c d andcanbeexpressedas = 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d 2 where 1 = [ I 0 ] 2 6 4 B 1 [ I [1,1], 0 ] 3 7 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 4 0 e 3 7 5 2 = [ I 0 ] 2 6 4 B 1 [ I [1,1], 0 ] 3 7 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 4 ^ B e 0 3 7 5 2 6 4 0 e 3 7 5 Proof. Fromthedenitionofthebasismatrix,wecanexpresstheoptimalsolutionto LPBintermsofthebasis B : x B = B )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 b 135

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FromLemma10,thebasismatrix B canbeexpressedas B = 0 B @ I + c d 2 6 4 0 ^ 00 3 7 5 1 C A 2 6 4 B 1 I [1,1], 0 3 7 5 UsingEquation5,wecancalculate B )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 : B )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 = 2 6 4 B 1 I [1,1], 0 3 7 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 0 B @ I )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d 2 6 4 0 ^ 00 3 7 5 1 C A Fromthedenitionofthebasismatrix,weexpresstheoptimalsolutiontoLPBinterms ofthebasismatrix B : x B = B )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 b = 2 6 4 B 1 I [1,1], 0 3 7 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 0 B @ I )]TJ/F39 11.9552 Tf 11.956 0 Td [(c d 2 6 4 0 ^ 00 3 7 5 1 C A 2 6 4 0 e 3 7 5 Inordertoextract, from x B ,wemustmultiplyby [ I 0 ] : =[ I 0 ] 2 6 4 B 1 I [1,1], 0 3 7 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 0 B @ I )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d 2 6 4 0 ^ 00 3 7 5 1 C A 2 6 4 0 e 3 7 5 =[ I 0 ] 2 6 4 B 1 I [1,1], 0 3 7 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 4 0 e 3 7 5 + c d [ I 0 ] 2 6 4 B 1 I [1,1], 0 3 7 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 4 ^ e 0 3 7 5 2 6 4 0 e 3 7 5 = 1 + c d 2 Proof. Denethenitesetofallfeasiblebasicsolutionsas F A := f x 1 x 2 ,..., x z g andthe setoftheircorrespondingbasicmatricesas B A := f A 1 A 2 ,..., A z g .Wecancalculate theresultingvalueofLPA U i foreachofthebasicfeasiblesolutions x i A 2 F A : U i :=[ 0 e T ] x i 136

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=[ 0 e T ] A )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 4 0 e 3 7 5 = a 1 i + c d a 2 i Supposethat x j isauniqueoptimalbasicsolution.Sincethissolutionisunique,we knowthat U k > U i 8 i 6 = k a 1 k + c d a 2 k > a 1 i + c d a 2 i 8 i 6 = k Itcanthenbeseenthatthereexistsan > 0 suchthat a 1 k + c d + a 2 k > a 1 i + + c d a 2 i 8 i 6 = k Therefore,theuniquefeasibleoptimalsolution x k willremaintheuniquefeasibleoptimal solutionwithinaneighborofvaluesof c d Theorem5.10. SupposethatLPBpossessesauniquesolutionforeach c d 2 c 1 c 2 foraxed 0 < p id < 1 withacorrespondingbasismatrix B .Theequilibriumutility v is linearwithrespectto c d andcanbeexpressedas U = U 1 )]TJ/F39 11.9552 Tf 11.955 0 Td [(c d U 2 where U 1 = [ 0 ,1 ] 2 6 4 B 1 [ I [1,1], 0 ] 3 7 5 )]TJ/F23 7.9701 Tf 6.586 0 Td [(1 2 6 4 0 e 3 7 5 U 2 = [ 0 ,1 ] 2 6 4 B 1 [ I [1,1], 0 ] 3 7 5 )]TJ/F23 7.9701 Tf 6.587 0 Td [(1 2 6 4 ^ B e 0 3 7 5 2 6 4 0 e 3 7 5 FromTheorem5.8throughTheorem5.10,wecanseethattheequilibriumsolution totheintelligencegameislinearlydependenton c d inaneighborhoodaroundunique solutions.Theseneighborhoodscanberelativelylarge,andleadtoapiecewiselinear 137

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structureglobally.Thelinearregionsofthepiecewisesolutionareseparatedbypoints orregionsinwhichthesolutiontoLPAdoesnothaveauniquesolution.Analytically calculatingtheseregionsaredifcultandareasubjectorcurrentresearch. 5.4Jones'LemmaandtheNumberofInformants AlthoughPlayerApossessestheabilitytocorruptasmanyinformantsas desired,thecostrequiredfordeceptiontobeeffectivecanpotentiallyoutweighany benetreceived.Thisisparticularlytrueiftherearemanyaccurateinformantswithin community.IfPlayerAonlymanipulatesafewandleavesthemajoritytoreportfreely, theinformationprovidedbytheuncorruptedinformantswillovershadowanddrownout thedeceptionintroducedbyPlayerA.Therefore,PlayerAmustpayoffalargeportion oftheavailableinformantsinordertobeeffective.Iftherearemanyinformants,this canbecomeaverycostlystrategy.Theimpactofthenumberofinformationchannels andthecostofcorruptiononthesuccessofdeceptioniscapturedinamaximfromthe intelligenceliteratureknownasJones'Lemma: Deceptionbecomesmoredifcultasthenumberofchannelsavailableto thetargetincreases.Nevertheless,withinlimits,thegreaterthenumberof channelsthatarecontrolledbythedeceiver,thegreaterthelikelihoodthat thedeceptionwillbebelieved[25]. ThisconceptappliedtotheintelligencegameisquantitativelycapturedbyTheorem 5.5.ItsshowsthatwhenPlayerAdecidestoemploydeceptivetactics,allavailable informantsmustbecontrolledinordertobeeffective.However,thisdeceptivestrategy isonlyemployedforarangeof c d denedbytheJones'Threshold.WhentheJones' Thresholdislessthanone,thecostsofdeceptionaretoohighandPlayerAdecidesnot toemployanydeceptivetactics.WhentheJones'Thresholdexceedsone,deception becomesaneffectivetactic. AlthoughtheanalyticsolutioninTheorem5.5onlyappliesforthespecialcase p id =1 ,Icanexaminethebehaviorforthegeneralcasewhere 0 p id 1 n > 0 ,and 138

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c d > 0 bycalculatingnumericalsolutionsusingTheorem5.1.Iwillnowexamineseveral scenariosfordifferentvaluesof c d n ,and p id .Inthefollowingsimulations,thevalues of V ij are V 11 =4 V 12 =3 V 21 =1 ,and V 22 =5 .Usingthesevalues,thestandard equilibriumvalue V s equals3.4. Figure5-4showsfourcurvesdepictingtheequilibriumutilityas p id sweepsfrom .5to1forthegamewherePlayerAisnotallowedtocorruptanyinformants.Thiswill beusedasthebaselinecase.Astheinformantsbecomemoreaccurate,PlayerAis eventuallyforcedtoplayitssafetystrategyandtheutilityvaluetopsoutatvalueof4. Asthenumberofinformantsincreases,thegamereachesthesafetyvalueforsmaller valuesof p id Ifthecostofcorruptionisrelativelyhigh,PlayerAwilldecidetonotemploy deceptivetacticsandtheresultingequilibriumvaluesarethesameasthebaseline case.ThiscanbeseeninFigure5-5.Figure5-6showstheprobabilityofPlayerA implementingsomeformofdeception.Decidingtoemploydeceptivetacticsisawaste ofresourcesforeverycaseexcept n =1 .Figure5-7showsthatbyloweringthecostof deceptionto c d =.2 ,PlayerAcanholdthevalueclosetotheoriginalNashequilibrium for n =1 andcanslowtherateofincreasefor n =5 and n =15 .Foralargenumber ofinformants,thePlayerAwillstillfallbackonitssafetystrategy,whichresultsinan equilibriumvalueof 4 .InFigure5-8,itcanbeseenthatPlayerAstartsusingdeception whentheinformantsarerelativelyinaccurate,butdecidestoabandondeceptivetactics forlargenumbersofinformantsasinformantaccuracyincreases. Wheninformantsareverycheaptocorrupt,PlayerAcanslowtheincreasein equilibriumutilitysubstantiallyandcaneffectivelyholdtheequilibriumvalueveryclose totheoriginalNashequilibriumvalueevenwithmanyhighlyaccurateinformants.This canbeseeninFigure5-9.Themoreinformantsthereareintheinformationnetwork,the soonerPlayerAdecidestoemploymoredeceptivetacticsasseeninFigure5-10. 139

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Figure5-2.Overallgamestructure Figure5-3.Informationnetworkfortwo-actiongame Figure5-4.Equilibriumutilityvs p id withnodeception 140

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Figure5-5.Equilibriumutilityvaluevs p id when c d =2 Figure5-6.ProbabilityofPlayerAcorruptinginformantswhen c d =2 Figure5-7.Equilibriumutilityvaluevs p id when c d =.2 141

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Figure5-8.ProbabilityofPlayerAcorruptinginformantswhen c d =.2 Figure5-9.Equilibriumutilityvaluevs p id when c d =.01 Figure5-10.ProbabilityofPlayerAcorruptinginformantswhen c d =.01 142

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CHAPTER6 ASEQUENTIALCOLONELBLOTTOGAMEWITHASENSORNETWORK Thischapterisbasedonworkthatwaspresentedatthe2012ACC[65].Inthis chapter,IposeanewsequentialvariantoftheColonelBlottogame,whichconsistsof twoplayers,PlayerAandPlayerB,whomustallocateniteresourcesamong N regions ofabattleeld.Akeyfeatureofmyproblemistheintroductionofasensornetwork employedbyPlayerBtogainaninformationaladvantageoverPlayerA.Thesensor networkconsistsofsensorsthatproducebinaryoutputtriggeredwhenevertheresource allocationinitsregionexceedsacertainthreshold.Thissensormodelcouldbeadapted torepresentseveralrealworldapplications.Oneparticularlyapplicableexampleisthe UnattendedGroundSensorUGS.Thesesmallgroundbasedsensorsprovideanearly warningsystemtonotifydefendingforcesofaperimeterbreach[66]. Thus,inthisgamePlayerAallocatesitsresourcesrst.PlayerBthenreceivesa sensorvalueforeachregionindicatingwhetherPlayerA'sallocationisaboveorbelowa threshold.Usingthisearlywarninginformation,PlayerBcanthenallocateitsresources moreeffectively.Aplayerwinsaregionifitallocatesmoreresourcesthanitsopponent. Intheeventoftie,PlayerBisawardedthewin.Thistiebreakingpolicywaschosento representPlayerB'sgoalofneutralizingPlayerA'sattack.Ifeachplayerallocatesequal amountsofforcetothesameregion,theseforcesmutuallydestroyeachother,butthis isconsideredavictoryfromPlayerB'sperspectivesincetherearenomoreattacking forces.[Historically,manydifferenttiebreakingpolicieshavebeenusedinstudiesofthe ColonelBlottogame,andthisformulationcouldbemodiedifothertiebreakingpolicies aredeemedtobemorecompelling.] AfterposingtheabovescenarioasasequentialColonelBlottogame,Iobtain necessaryconditionsforNashequilibriumoptimalmixedstrategiesforthetwoplayers intermsoftheeachplayer'savailableresourcesandthesensornetworkcharacteristics. Theseconditionsapplytothemarginalprobabilitydistributionfunctionsofeachplayer's 143

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strategy.Themarginaldistributionsrepresenttheresourceallocationstrategyrelating toaspecicregionofthebattleeld.Todeveloptheseconditions,Iuseanapproach thatissimilartothehighlyinuentialrecentpaperbyRoberson[40].Thecontestfor eachregionisrepresentedasarst-priceallpayauctionwherethevalueawarded toeachplayerintheauctionincorporatesanadjointvariablethatcorrespondstothat player'sresourceconstraint.Theexistenceofthesensornetworkinthegameleads toupperandlowerboundsonPlayerA'sallocationstrategiesforeachregion.Froma technicalviewpoint,thisisthekeynewfeatureofthisformulationandthecorresponding results.Iobtainanalyticsolutionsforeachoftheseindividualauctiongamesinterms ofthesensornetworkcharacteristicsandadjointvariables.Theequilibriumsolutions haveapiecewisebehaviorthatmakesndingageneralclosed-formanalyticsolution particularlychallenging.Idevelopefcientnumericaltechniquestosolvefortheadjoint variablesforparticularvaluesofplayers'resourcesandsensornetworkcharacteristics. Theseanalyticalcharacterizationsareusedtodevelopaneffectivecomputational solutiontothegame.Resultsareillustratedviaanumericalexample. 6.1GameFormulation Inthissection,IwillformulateanovelsequentialvariantofthefamousColonel Blottogame.Iwillformallyintroducethedifferentcomponentsofthegameincluding playeractions,sensornetwork,playerstrategies,andutilityfunctions. 6.1.1PlayerActions Thebattleeldisdividedinto N regions, R j j =1,2,..., N .Thetotalamountof resourcesavailabletoPlayerAandPlayerBare X A 2 R + and X B 2 R + respectively. AparticularallocationbyPlayerAisdenedbythenonnegative N -dimensionalvector x A := x A 1 x A 2 ,..., x A N ,where x A i 0 istheamountresourcesallocatedtothe i -th regionofthebattleeldbyPlayerA.AnallocationbyPlayerBisdenedsimilarly by x B := x B 1 x B 2 ,..., x B N .Itisrequiredthateachplayermustallocateallavailable resources.Therefore,thesetoffeasibleallocationsforPlayerA, A ,andPlayerB, B 144

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are A := x A 2 R N + X A = N X i =1 x A i and B := x B 2 R N + X B = N X i =1 x B i 6.1.2SensorNetwork Adeterministicthresholdsensor, s i ,monitorsregion, R i ,alongthebattleeld.The outputofthesensorisgovernedbythefollowingrule: s i x A i = 8 > < > : 0 x A i < 1 x A i where representsthesensorthreshold.PlayerBreceivesthemeasurement, s := f s 1 s 2 ,..., s N g ,whichcontainstheoutputofeachoftheindividualsensoroutputs.The behaviorofthesensornetworkiscommonknowledgewithinthegame.Ihavechosen allsensorthresholdstobeidenticalforthesakeofsimplicity.Theseresultscanbe generalizedeasilytothecaseofdifferentsensorthresholds. 6.1.3PlayerStrategies ItiswellknownthatColonelBlottogamesdonotadmitpurestrategyNash equilibriumsolutionsforcertainconditionsonplayers'availableresources[67].This turnsouttobethecaseforthisproblemaswell.Therefore,Iwillalsofocusmostlyon mixedstrategies.AmixedstrategyforPlayer i 2f A B g isdenedasthe N -variate jointprobabilitydensityfunction P i : R N + [0,1] withsupportcontainedin i .The one-dimensionalmarginalcumulativedistributionfunctioncdf F i j y := P x i j y representstheprobabilityofPlayer i allocatinglessthanorequalto y resourcesto R j .It iscommoninsolutionstoColonelBlottogamestosolveforthemarginalcdf'sandthen 145

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constructthejointdistributionfunctionwiththerequisiteproperties.Thisisgenerally adifcultproblem.Roberson[40]hasshownhowSklar'stheoremalongwithCopula functioncanbeusedtoconstructthejointmultivariatedistributionfunctionfromthe marginalcdf's.Inmyresearch,Iwillnotaddressthisaspectoftheproblemandleaveit forfutureresearch. 6.1.4UtilityFunctions Inthisgame,eachplayerreceivesarewardof 1 N forwinningaparticularregion R j .Thisischosenforsimplicityinordertomaketheresultseasiertointerpret,but thesolutionobtainedinthispapercanbegeneralizedtohandledifferentvaluesfor eachregion.Aplayerwinsaparticularregioniftheyallocatemoreresourcesthanits opponentatthatparticularlocation.Intheeventofatie,itisassumedthatPlayerBwins theregion.TheprobabilityofPlayerAwinning R j whenallocating x A j forcesisthen P x B j x A j = F B j x A j IcanthendenePlayerA'sexpectedpayofffrom R j whenimplementingastrategywith themarginalcdf F A j as u A j F A j F B j := Z 1 0 1 N F B j x A j dF A j Bysummingtheexpectedpayoffsforeach R j Icancalculatethetotalexpectedpayoff, U A ,forastrategypair P A P B : U A P A P B = N X j =1 Z 1 0 1 N F B j x A j dF A j ThesamecanbedoneforPlayerB: U B P A P B = N X j =1 Z 1 0 1 N F A j x B j dF B j TheexpectedpayoffsinEquation6andEquation6representtheplayerutilities. 146

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6.1.5GameDenition Eachplayerstrivestomaximizetheexpectednumberofregionswonandtherefore attemptstomaximizetheirrespectiveutilityfunction U i .Ihavealsorequiredthateach playermustallocateallavailableresources.Itisthennecessarytoincorporatethe constraint N X j =1 x i j )]TJ/F39 11.9552 Tf 11.955 0 Td [(X i =0 foreachplayer.Itisshownin[40],thatthisisequivalenttoimposingtheconstraintthat N X j =1 E [ x i j ] )]TJ/F39 11.9552 Tf 11.955 0 Td [(X i =0 where E [ x i j ] representstheexpectedvalueofPlayer i 'sallocationofforcewithinthe j th region.IcanthenconstructPlayer i 'sLagrangianbyappendingthisconstraintonto U i withtheadjointvariable i > 0 : L A := N X j =1 Z 1 0 1 n F i B x dF i A )]TJ/F25 11.9552 Tf 11.955 0 Td [( A N X j =1 E [ x A j ] = A N X j =1 Z 1 0 1 n A F i B x )]TJ/F39 11.9552 Tf 11.956 0 Td [(x dF i A and L B := B N X j =1 Z 1 0 1 n B F i A x )]TJ/F39 11.9552 Tf 11.956 0 Td [(x dF i B Agamecannowbedenedinwhicheachplayermaximizesitsrespective Lagrangian: max F A j 2f 1,..., n g A N X j =1 Z 1 0 1 n A F B j x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x dF A j x max F B j 2f 1,..., n g B N X j =1 Z 1 0 1 n A F A j x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x dF B j x Lookingattheintegralwithinthesummation,Icanseethattheyaredependent onlyonthemarginalcdf'scorrespondingtoaparticularregion R j .Therefore,Icansolve 147

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forthenecessaryconditionswithrespecttothemarginalcdf'sindependentlyforeach region.Thisisthesubjectofthenextsection. 6.2GameSolution IwillnowdescribethesolutiontotheColonelBlottogameposedinSection6.1. 6.2.1SubgameStructure Theinclusionofthedeterministicsensornetworkcreatesahierarchicalstructure totheoverallgame.Intherststage,PlayerAimplementsanallocationofforce x A whichgeneratessensorvector s .Let I 0 := f j j s j =0 g and I 1 := f j j s j =1 g representthe indexsetscorrespondingtosensorvaluesofzeroandonerespectively.Afterreceiving informationfrom s ,PlayerBknowsthat x A j < for j 2 I 0 and x A j for j 2 I 1 because s isdeterministicallygeneratedaccordingtoEquation6,butPlayerAisstillfreeto mixstrategieswithinthesubsetof A suchthat x A j < for j 2 I 0 and x A j for j 2 I 1 Foraparticularsensorvector s ,theincorporationofthisinformationimposesinequality constraintsonPlayerA'sallocationforeachregionwithintheoriginalColonelBlotto game. ThisresultsinavariantofthestandardColonelBlottoGameinwhichPlayerA hascontrolconstraints.InthefollowingsectionsIwilldevelopthenecessaryconditions forthisboundedcontrolvariantoftheColonelBlottogame.Sincethesensornetworkis deterministic,PlayerAknowsaprioriwhichsensorvectorPlayerBwillreceive.PlayerA canthenchoosetheboundedsubgamecorrespondingtosensorvector s k thatyieldsthe bestexpectedutilitythroughselectionofappropriatecontrolstrategies. 6.2.2ConditionsforPureStrategyEquilibrium Iwillnowstatetheresultforthepurestrategyequilibriumtothegameandthe conditionsunderwhichitexists. Theorem6.1. Let k = j I 1 j .ThereexistsapurestrategyforPlayerBwhichguarantees victoryforeachregioniff X B k X A )]TJ/F22 11.9552 Tf 11.955 0 Td [( k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 + N )]TJ/F39 11.9552 Tf 11.956 0 Td [(k min X A )]TJ/F39 11.9552 Tf 11.955 0 Td [(k 148

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Ifthisconditionissatised,anydeployment x B thatsatises x B j 8 > < > : min X A )]TJ/F39 11.9552 Tf 11.955 0 Td [(k j 2 I 0 X A )]TJ/F22 11.9552 Tf 11.956 0 Td [( k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 j 2 I 1 willguaranteethatPlayerBwillwinallregions. Furthermore,theredoesnotexistapurestrategyequilibriumif X B < k X A )]TJ/F22 11.9552 Tf 11.955 0 Td [( k )]TJ/F22 11.9552 Tf 11.956 0 Td [(1 + N )]TJ/F39 11.9552 Tf 11.955 0 Td [(k min X A )]TJ/F39 11.9552 Tf 11.955 0 Td [(k Proof. Duetotherequirementsthat X A = P N j x A j and x A j forall j 2 I 1 ,Iknowthat max x A j = X A )]TJ/F22 11.9552 Tf 11.955 0 Td [( k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 8 j 2 I 1 evenif x A j =0 forall j 2 I 0 .SincePlayerisrequiredtoallocateatleast resourcesto x A j forall j 2 I 1 ,thisleavesatmost X A )]TJ/F39 11.9552 Tf 12.47 0 Td [(k resourcestodistributeto x A j forall j 2 I 0 Therefore,Iknowthat max x A j =min X A )]TJ/F39 11.9552 Tf 11.956 0 Td [(k 8 j 2 I 0 IfPlayerBimplementsadeploymentthatsatisesEquation6itwillbeguaranteed towineachregionsince x B j min X A )]TJ/F39 11.9552 Tf 11.955 0 Td [(k x A j 8 j 2 I 0 x B j X A )]TJ/F22 11.9552 Tf 11.955 0 Td [( k )]TJ/F22 11.9552 Tf 11.955 0 Td [(1 x A j 8 j 2 I 1 InorderforPlayerBtoimplementanallocationthatsatisesEquation6,PlayerB's availableresourcesmustsatisfyEquation6. IfPlayerB'savailableresourcesdonotsatisfyEquation6,itcanbeshownthat thereexistsabenecialdeviationforatleastoneplayerforanyallocationpair x A x B ThisconditionisidenticaltothatofthestandardColonelBlottogame[40]. 149

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6.2.3First-PriceAllPayAuction InordertocalculatethemixedstrategieswhenEquation6isnotmet,I developthenecessaryconditionsfortheoptimalstrategiesintermsofthemarginal distributionfunctions F i j .Considertheintegralswithinthesummationoftheplayer LagrangiansEquation6andEquation6: Z 1 0 1 n A F B j x )]TJ/F39 11.9552 Tf 11.956 0 Td [(x dF A j x Z 1 0 1 n A F A j x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x dF B j x TheseformulaarecriticalinconnectingtheColonelBlottogametoauctiontheory. TheintegralsEquation6-Equation6representtheutilityfunctionsofa rstpriceallpayauctiongame [68].Duetothesensornetwork,thereiseitheranupper oralowerboundonPlayerA'sallocationvariable x A j dependingonwhether j 2 I 0 or j 2 I 1 .Iwillnowdescribeandanalyticallysolvethegeneralupperandlowerbounded auctiongamesthatcorrespondtheintegralsEquation6andEquation6.I canthenusetheseauctionsolutionstodescribetheequilibriumconditionstotheoverall sequentialColonelBlottogame. 6.2.3.1Auctiongamedescription Inarst-priceallpayauctiongame,PlayerAandPlayerBeachplaceabid, x A and x B respectively,inanattempttowintheauction.Aplayerwinstheauctionifthey placethehighestbid,andthewinnerisawardedaprize v i where i 2f A B g .Inthe caseofatie,PlayerBisawardedthewin.Eachplayerforfeitstheirbid x i regardlessof whetherornotitwins.ThemixedstrategiesofPlayerAandPlayerBarerepresented bytheunivariatecumulativeprobabilitydistributionfunctions F A x := P x A x and F B x := P x B x .Ialsodenetheprobabilitydensityfunctions f A x := dF A dx x and f B x := dF B dx 150

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Theremaybepointswherethesederivativesdonotexistwhichcorrespondtodiscrete jumpsin F A and F B .Toaccountforthese,Irepresentpointmasseswithintheplayer i 'sdensityfunctionas i x = y ,where x representsthelocationand y representsthe mass. TheutilityfunctionforPlayerA, U A ,andPlayerB, U B ,aredenedas U A F A F B = Z 1 0 v A F B x )]TJ/F39 11.9552 Tf 11.956 0 Td [(x dF A x dx and U B F A F B = Z 1 0 v B F A x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x dF B x dx IalsoallowforconstraintstobeimposedonPlayerA'sbiddingpolicy.Thesecould beupperboundaryconstraintssuchthat x A < c 2 ,lowerboundaryconstraintssuch that x A c 1 ,orboth c 1 x 1 < c 2 .Itwillbeshownthattheequilibriumsolutions areparticularlysensitivetotheorderof v A v B ,andtheimposedconstraints.Thisis because v A and v B sometimesactasinducedboundaryconstraintswithintheoptimal biddingpolicies.Also,theseboundariesareoftenthelocationofmasspointswithinthe equilibriumstrategies. 6.2.3.2RelationshiptotheColonelBlottogame InthecontextoftheColonelBlottogame,eachregion R i representsanauction gameinwhich v i = 1 n i representsPlayer i 'srewardforwinningthatregion.Thebidding variable x i representsPlayer i 'sresourceallocationtothatregion. RecallthattheinclusionofthedeterministicsensornetworkwithintheColonel BlottogameimposesaconstraintonPlayerA'sbiddingpolicyforeachregionalongthe battleeldforaparticularsensorvector s .Iftheauctiongamerepresentsthecontest ofregion R i where i 2 I 0 ,thenPlayerAhasanupperboundimposedonitsbidding policysuchthe x A < .Arelatedauctiongameisanalyzedin[69].Inmyresearch,Ido notrestrict tobestrictlylessthan v A and v B .Additionally,PlayerBdoesnothaveany restrictionsdirectlyimposedonitsbiddingpolicy.Forthecasewhen i 2 I 1 ,PlayerAhas 151

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alowerboundimposedonitsbiddingpolicysuchthat x A .Theinclusionofthese biddingconstraintsresultsinequilibriumsolutionsthatcanvarysignicantlyfromthe standardunconstrainedgame. 6.2.3.3Auctiongamesolutionnotation UsingtheutilityfunctionsEquation6-Equation6,Icannowdenea gameinwhicheachplayerstrivestomaximizeitsrespectiveutilityfunction: U A =max F A U A = Z 1 0 v A F B x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x dF A x dx and U B =max F B U B = Z 1 0 v B F A x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x dF B x dx Thesolutiontotheauctiongameconsistsofthepairofequilibriumbidding strategies, F A and F B aswellastheresultingequilibriumutilities U A and U B foreach oftheplayerssuchthat U A F A F B U A 8 F A and U B F A F B U B 8 F B Iwillnowpresentaparameterizedsolutiontothisgame.Inthisparameterized solution,theparameters i and i representthelowerandupperboundsofsupport forPlayer i 'smixedstrategy.Thefunctions H A x A and H B x B representthecdfof theplayerstrategieswithinthesebounds.Theparameters a 1 and b 2 representthe magnitudeofanymasspointsattheboundariesofthestrategies.InTheorem6.2and Theorem6.3,Iwilldenethevaluesoftheseparametersintermsof v A v B ,and X A foreachpossibleordering. 152

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Theequilibriumstrategiesandresultingequilibriumvaluesaredenedasfollows. F A x A = 8 > > > > < > > > > : 0 x < A H A x A A x < A 1 x A F B x B = 8 > > > > < > > > > : 0 x < B H B x B B x < B 1 x A andthemasspoints A y A = a 1 B y B = b 1 Theresultingequilibriumutilitiesaredenotedby U A = u A U B = u B Othercharacteristicsoftheequilibriumsolutionthatwillbeusedinlateranalysis includetheexpectedvaluesofPlayerAandPlayerB'sbids, E [ x A ] and E [ x B ] ,aswellas eachplayer'sprobabilityofwinningtheauction, W A and W B .Thesevaluesaredened as E [ x A ]= Z 1 0 x A dF A E [ x B ]= Z 1 0 x B dF B W A = Z 1 0 P x B < x A dF A = Z 1 0 F B x A dF A W B = Z 1 0 P x A x B dF A = Z 1 0 F A x B dF B 153

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6.2.3.4Auctiongamesolution Iwillnowstatetherstmainresultthatcharacterizesthesolutiontotherst-price allpayauction. Theorem6.2. ConsidertheFirst-PriceAllPayAuctiondenedinEquation6and Equation6.SupposethatPlayerAhasanupperboundconstraintplacedonits strategysuchthat x A < .Thenforaeachorderingof v A v B ,and speciedina particularcolumnofTable6-1,thesolutiontothegameisgivenbyequationsEquation 6-Equation6withparametersgiveninthecorrespondingcolumninTable6-2. Proof. Forthesakeofbrevity,Iwillonlyexplicitlyprovethecasewhere < v A < v B whichcorrespondstotherstcolumnofthetable.Theproofsfortheothercasesare similarinapproach,buthavesuitablemodicationsinconsideringthebiddingscenarios forboundarycasesat i and i Theproofofthetheoremiscomposedviaaseriesoflemmas.Irstprovidea generaloverviewoftheproofstrategy.Istartbyndingtheupperandlowerbounds ofsupportforbothplayersequilibriumstrategies.IshowbycontradictionthatPlayerA musthaveamasspointat 0 ,andPlayerBmusthaveamasspointat .Ialsoshowthat PlayerBdoesnotplaceamasspointat 0 .TheseresultsarecontainedinLemma12. SincePlayerAplays 0 withnonzeroprobability,Icanthenconcludethat u A = u A duetostandardpropertiesofequilibrium[70].Similarly,Icanconcludethat u B = u B ThesefactsarestatedinLemma13.Ithenprovethattherecannotbeanymass pointswiththerange inLemma14.Lemma15showsthat u A x A and u B x B are constantwithin .CombiningLemma13andLemma15,Icanthenconstructthe equilibriumstrategiesforeachplayer. Forthesakeofbrevity,Idene u A x A and u B x B astheexpectedpayoffwhen Player i implementsaction x i : u A x A := v A F B x A )]TJ/F39 11.9552 Tf 11.956 0 Td [(x A 154

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u B x B := v B F A x B )]TJ/F39 11.9552 Tf 11.955 0 Td [(x B Lemma11. Theboundariesofsupportare A = B =0 and A = B = Proof. Fromourassumptions,neitherplayercanbidlessthan 0 .Therefore, A 0 and B 0 .Supposeforthesakeofcontradiction A > 0 and B > 0 .If A < B ,PlayerA wouldbenetbymovinganyprobabilitymassbelow B tozerobecauseitwouldhave thesameprobabilityofwinningwithasmallerbiddingcost.Therefore, A > 0 wouldnot beinequilibrium.Similarlyif B < A ,PlayerBwouldbenetbymovingprobabilitymass tozero.Therefore, B > 0 wouldnotbeinequilibrium.If 0 < A = B ,PlayerAwould benetfrommovingasmallamountofdensityjustabove A tozero.Therefore, 0 < A and 0 < B cannotbeinequilibrium. WeknowthatPlayerAmustbidbelow .Therefore, A .Supposethat A < and B < .PlayerAwouldbenetbybiddingwithinthe B becausehewouldbe guaranteedtowin.Therefore,thissituationisnotinequilibrium.NowSupposethat A < and B = .PlayerBwouldbenetbymovingprobabilitymassabove A to A becauseitwouldstillhaveequalprobabilityofwinningbutwouldpaylessforthebid. Therefore,thissituationisnotinequilibrium.Therefore, A < isnotinequilibrium, whichimpliesthat A = .Asimilarargumentcanbemadethat B = Lemma12. Masspointswithintheplayers'equilibriumstrategiessatisfy A > 0 B =0 ,and B > 0 Proof. SupposethatbothPlayerAandPlayerBplaceamasspointat x i =0 .From Lemma,IknowthatPlayerAneverbidslessthan0.Therefore,abidof0byPlayerB willalwaysloseandresultinautilityof0.ButifPlayerBmovesitsmassto x B = > 0 itwillresultinautilityof u B = v B F A )]TJ/F25 11.9552 Tf 12.279 0 Td [( .FromtheassumptionthatPlayerAhasa massat x A =0 Iknowthat v B F A > 0 forall > 0 .Therefore,PlayerBcangeneratea utility u B > 0 for arbitrarilysmall.SincethereexistsabenecialdeviationforPlayer B,theassumptionthebothplayershavemasspointsat x i =0 inequilibriumisfalse. 155

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SupposethatPlayerBplacesamasspointat x B =0 .Iknowthatatmostone playercanplaceamasspointat x i =0 .SinceIhaveassumedthatPlayerBhasplaced amasspointat x B =0 ,PlayerAcannotplacemassat x A =0 .IfPlayerBplays x B =0 itcanexpectapayoffof u B = v B F A )]TJ/F22 11.9552 Tf 12.257 0 Td [(0=0 .Thisisstrictlydominatedbyplaying x B = ,whichwouldyieldapayoffof u B = v B F A )]TJ/F25 11.9552 Tf 12.193 0 Td [( = v B )]TJ/F25 11.9552 Tf 12.193 0 Td [(> 0 .Thus,playing x B =0 isadominatedstrategyandPlayerBwouldbenetbymovingthemasspoint from x B =0 to x B = .Therefore,PlayerBwouldnothaveamasspointat x B =0 in equilibrium. Lemma13. Theequilibriumutilitiessatisfy u A =0 and u B = V B )]TJ/F25 11.9552 Tf 11.955 0 Td [( Lemma14. Theequilibriumplayerstrategies, F A x and F B x ,containnomasspoints forall x 2 Lemma15. Foreach i 2f 1,2 g u i x i isconstantandequalto u i forall x i 2 Wecanseethat u A x = v A F B x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x =0 8 x 2 [0, ] and u B x = v B F A x )]TJ/F39 11.9552 Tf 11.955 0 Td [(x = v A )]TJ/F25 11.9552 Tf 11.955 0 Td [( 8 x 2 [0, ]. SolvingEquation6andEquation6for F A x and F B x yields F A x A = 8 > > > > < > > > > : 0 x < 0 v B )]TJ/F26 7.9701 Tf 6.587 0 Td [( + x A v B 0 x < 1 x F B x B = 8 > > > > < > > > > : 0 x < 0 x B v A 0 x < 1 x withmasspoints A = v B )]TJ/F25 11.9552 Tf 11.955 0 Td [( v B and B = v A )]TJ/F25 11.9552 Tf 11.955 0 Td [( v A 156

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IcanthenuseEquation6throughEquation6tosolvefor E [ x i ] and W i for i 2f A B g Theorem6.3. ConsidertheFirst-PriceAllPayAuctiondenedinEquation6and Equation6.SupposethatPlayerAhasalowerboundconstraintplacedonits strategysuchthat x A .Thenforagivenorderingof v A v B ,and speciedina particularcolumnofTable6-2,thesolutiontothegameisgivenbyEquation6Equation6withparametersgiveninthecorrespondingcolumninTable6-2. 6.2.4NumericalSolutionTechnique Inthissection,Iwilldiscussthemethodusedtosolvefortheadjointvariableswithin theplayerLagrangiansforaparticularsensorvector s .AlthoughInowhaveanalytic equilibriumsolutionsfortherst-priceallpayauctions,theoptimalstrategiesand equilibriumvaluesarecriticallydependentonthevaluesof v A v B ,and .Inparticular, thestructureoftheoptimalstrategiesareespeciallysensitivetotheorderofthesethree terms.Thismakesitdifculttocalculateaclosedformanalyticsolutionsincetheorder v A and v B isnotknownapriori.Inordertoeffectivelyobtainthesolutionforparticular valuesof X A X B n and k ,Icannumericallycomputethevaluesof v A and v B foreach ofthepossibleordersof v A v B ,and usingtheanalyticequilibriumsolutionsdescribed inSection6.2.3.Afterthevaluesfor v A and v B arecomputed,Icanthencheckthese valuesaswellastheresultingequilibriumstrategiesforconsistencywiththeparticular assumptionsthatwereused. RecallingtheplayerLagrangiansEquation6andEquation6,theadjoint variableswereincludedinordertoincorporatetheconstraintEquation6.IfI assumeaparticularorderfor v A v B ,and X A ,Icancalculatethevaluesof v A = 1 n A and v B = 1 n B bysolvingthefollowingsystemoftwoequations X A = N X j =1 E [ x A j ]= N )]TJ/F39 11.9552 Tf 11.956 0 Td [(k E U [ x A ]+ NE L [ x A ] 157

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X B = N X j =1 E [ x B j ]= N )]TJ/F39 11.9552 Tf 11.955 0 Td [(k E U [ x B ]+ NE L [ x B ]. Theterms E U [ x j ] and E L [ x j ] aretheexpecteddeploymentlevelsforupperandlower boundedregionsgivenequationsbyEquation6withparametervaluesstatedin Theorem6.2andTheorem6.3thatcorrespondtotheassumedorderof v A v B ,and Theresultingvaluesof v A and v B maynotnecessarilybeconsistentwiththeoriginal assumptionsonordering,andmustbeveriedaftercalculation. Onceaconsistentsolutionisfoundfor v A and v B ,Icanthencalculatetheexpected utilityforeachplayerwithintheColonelBlottogame: U A = N X j =1 W A j = 1 N N )]TJ/F39 11.9552 Tf 11.955 0 Td [(k W A U + kW A L U B = X j =1 N W B j = 1 N N )]TJ/F39 11.9552 Tf 11.955 0 Td [(k W B U + kW B L where W i U and W i L aretheprobabilitiesthatplayer i willwintheupperandlower boundedauctiongamerespectivelyandarecalculatedusingEquation6and Equation6. 6.3IllustrativeExample InthissectionInumericallysolvefortheparametersoftheequilibriumstrategies andutilityvaluesforseveralfourdifferentcasesagameparameters.Theresultsare showninTableIII.Inthesescenarios k representsthenumberofsensorsthatare triggeredandreturnavalueof1. 6.3.1Scenario1: n =5 X A =10 X B =15 =1 Thiscaserepresentsthescenariowherethethresholdsensorsareverysensitive andpickupevensmallattackingforces.Atrstthought,thiswouldappeartobethe mostdesirabletypeofsensor,butthehighsensitivityleadstoeasytriggeringbythe attackingforces.Infact,thisisexactlywhatthesolutionindicates.PlayerAselectsa mixedstrategythattriggerseverysensoreachtime.Thisnegatesthepotentialvalueof 158

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theinformationcontentpertainingtostrongandweakpointsinitsallocation.AllPlayerA nowknowsisthateachregionhasanallocationgreaterthanone,butitcouldstillbeat most10.Therefore,PlayerBstillneedstomixoverawiderangeofvalues. 6.3.2Scenario2: n =5 X A =10 X B =15 =9 Thisscenariorepresentstheoppositeextreme.Thesensorsinthisscenarioonly triggerwhenrelativelylargeattackingforcesarecoming.Withthisarrangement,Player Anowelectstodistributesmallforcesacrossallregionssothattheyallsneakthrough thesensorswithoutbeingdetected.Again,theinformationprovidedtoPlayerBdoesnot containmuchinformationregardingthethreatenedregions.Itnowonlyknowsthatthe attackingforceswillbewithinarangeof [0,7 ,whichisnotmuchbetterthantheoriginal rangeof [0,10] 6.3.3Scenario3: n =5 X A =10 X B =15 =3 Thisscenariorepresentsaselectivesensorwithasensitivitythatismoreappropriate fordistinguishingbetweenlargeandsmallattackingforces.NowPlayerBcaneffectively identifyregionswhichareunderrelativelyheavyattackandthenallocatetheappropriate levelofforce.Simultaneously,PlayerBalsoidentiesweakspotswithinPlayerB's forces.PlayerBnowhastootheroptionthantodivideitsforcesknowingthatthelarger groupswillbeidentied,whichthenlowerstheirchanceofwinningtheregiontheyare deployedto. 6.3.4Scenario4: n =20 X A =10 X B =20 =2 ThisscenariorepresentsthecasewherePlayerBmustattempttodefendalarge numberofregions.Althoughthesensorthresholdissetrelativelylow,PlayerAcanstill distributeisforcesthinenoughthattheycansneakbytheradar.TheresultisthatPlayer Bmuststillspreaditsforcesthinlyinordertoeffectivelycoverallregions. 159

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Table6-1.Solutionparameterstotheupperboundedrst-priceallpayauctions Parameters < v A < v B < v B < v A v A << v B v A < v B < v B << v A v B < v A < 0, 0, 0, v A 0, v A 0, v B 0, v B H A x A v B )]TJ/F26 7.9701 Tf 6.587 0 Td [( + x A v B v B )]TJ/F26 7.9701 Tf 6.587 0 Td [( + x A v B v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v A + x A v B v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v A + x A v B x A v B x A v B H B x B x B v A x B v A x B v A x B v A v A )]TJ/F40 7.9701 Tf 6.586 0 Td [(v B + x B v A v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v B + x B v A y A A y A 0 v B )]TJ/F26 7.9701 Tf 6.587 0 Td [( v B 0 v B )]TJ/F26 7.9701 Tf 6.587 0 Td [( v B 0 v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v A v B 0 v B )]TJ/F40 7.9701 Tf 6.586 0 Td [(v A v B 0 00 0 y B B y A v A )]TJ/F26 7.9701 Tf 6.587 0 Td [( v A v A )]TJ/F26 7.9701 Tf 6.587 0 Td [( v A 0 00 v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v B v A 0 v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v B v A u A 0000 v A )]TJ/F39 11.9552 Tf 11.955 0 Td [(v B v A )]TJ/F39 11.9552 Tf 11.955 0 Td [(v B u B v B )]TJ/F25 11.9552 Tf 11.956 0 Td [( v B )]TJ/F25 11.9552 Tf 11.955 0 Td [( v B )]TJ/F39 11.9552 Tf 11.955 0 Td [(v A v B )]TJ/F39 11.9552 Tf 11.955 0 Td [(v A 00 E [ x A ] 2 2 v B 2 2 v B v 2 A 2 v B v 2 A 2 v B v B 2 v B 2 E [ x B ] 2 v A )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 2 v A 2 v A )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 2 v A v A 2 v A 2 v 2 B 2 v A v 2 B 2 v A W A 2 2 v A v B 2 2 v A v B v A 2 v B v A 2 v B 2 v A v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v 2 B 2 v A v B 2 v A v B )]TJ/F40 7.9701 Tf 6.586 0 Td [(v 2 B 2 v A v B W B 2 v B v A )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 2 v B v A 2 v B v A )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 2 v B v A 2 v B v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v 2 A 2 v B v A 2 v B v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v 2 A 2 v B v A v 2 B 2 v B v A v 2 B 2 v B v A 160

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Table6-2.Solutionparameterstothelowerboundedrst-priceallpayauctions Parameters v A + v B v A + > v B < v B < v A v A + v B v A + > v B v B < A A v A + v B v B v A + v B n/a F A x A v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v A )]TJ/F40 7.9701 Tf 6.586 0 Td [(tau + x A v B x A v B x A v B v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(tau + x A v B x A v B n/a F B x B x B )]TJ/F26 7.9701 Tf 6.587 0 Td [( v A v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v B + x B v A v A )]TJ/F40 7.9701 Tf 6.586 0 Td [(v B + x B v A x B )]TJ/F26 7.9701 Tf 6.587 0 Td [( v A v A )]TJ/F40 7.9701 Tf 6.586 0 Td [(v B + x B v A n/a y A A y A v B )]TJ/F40 7.9701 Tf 6.586 0 Td [(v A v B v B v B v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v A v B v B ,1 y B B y B 0,0 v B v A )]TJ/F40 7.9701 Tf 6.586 0 Td [(v B + v A v B v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v B + v A 0,0 v B v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v B + v A 0,1 u A )]TJ/F25 11.9552 Tf 9.298 0 Td [( )]TJ/F25 11.9552 Tf 9.299 0 Td [( v A )]TJ/F39 11.9552 Tf 11.956 0 Td [(v B )]TJ/F25 11.9552 Tf 9.299 0 Td [( v A )]TJ/F39 11.9552 Tf 11.956 0 Td [(v B v A )]TJ/F25 11.9552 Tf 11.955 0 Td [( u B v B )]TJ/F39 11.9552 Tf 11.956 0 Td [(v A )]TJ/F25 11.9552 Tf 11.956 0 Td [( 00 v B )]TJ/F39 11.9552 Tf 11.955 0 Td [(v A )]TJ/F25 11.9552 Tf 11.955 0 Td [( 00 E [ x A ] v 2 A +2 v B 2 v B v 2 B + 2 2 v B v 2 B + 2 2 v B v 2 A +2 v B 2 v B v 2 B + 2 2 v B E [ x B ] v 2 A +2 v A 2 v A 2 v B v A )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 )]TJ/F40 7.9701 Tf 6.586 0 Td [(v 2 B +2 v B 2 v A v 2 B )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 2 v A v 2 A +2 v A 2 v A v 2 B )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 2 v A 0 W A v A 2 v B 2 v A v B )]TJ/F40 7.9701 Tf 6.586 0 Td [(v 2 B + 2 2 v A v B 2 v A v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v 2 B + 2 2 v A v B v 2 A 2 v A v B 2 v A v B )]TJ/F40 7.9701 Tf 6.587 0 Td [(v 2 B + 2 2 v A v B 1 W B 2 v B )]TJ/F40 7.9701 Tf 6.586 0 Td [(v A 2 v A v B v 2 B )]TJ/F26 7.9701 Tf 6.586 0 Td [( 2 2 v B v A v 2 B )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 2 v B v A 2 v B v A )]TJ/F40 7.9701 Tf 6.587 0 Td [(v 2 A 2 v A v B v 2 B )]TJ/F26 7.9701 Tf 6.587 0 Td [( 2 2 v B v A 0 161

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Table6-3.Scenarioresults ParametersScenario1Scenario2Scenario3Scenario4 n 5.005.005.0020.00 X A 10.0010.0010.0010.00 X B 15.0015.0015.0020.00 1.009.003.002.00 v A 4.006.800.700.30 v B 8.0020.002.700.70 A 0.050.030.290.17 B 0.030.010.070.07 U A 0.250.300.130.29 U B 0.750.700.870.71 k 5.000.002.002.00 162

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CHAPTER7 CONCLUSIONS Thepurposeofthisstudywastoapplygametheoreticmethodstowardthe developmentofdefensivestrategies.InmyresearchpresentedinChapter2and Chapter3,Iuseddifferentialgametheorytodevelopcoordinateddefensestrategies formulti-agentteamsofmobiledefendersinresponsetoasuperiorattacker.The solutiontothesegamesconsistedofcontrolstrategiesthatwereinNashEquilibrium. Thismeansthattheequilibriumdefensivestrategiesrepresenttheoptimalresponse whentheattackeremploystheoptimalstrategyfromitsownperspective.Theresulting valueofthegamethenrepresentsaworstcase,lowerboundfromthedefenders' perspective.Thecooperativebehaviorsexhibitedinthesolutiontothesegamesare qualitativelysimilartonumerousexamplesofpreystrategiesusedinresponseto attackingpredators. InChapter3,ahierarchicalattack-retreatgamewasconstructedthatincorporated theconceptofplayerintent.Inthistwo-playergame,oneplayerrepresentsanattacker, andtheotherplayerrepresentsadefensiveteamthatconsistsofmobile,high-value targetandseveralprotectiveagents.Theseprotectiveagentsinictacostonthe attackingagent.Forcertainconditions,itisshownthatitisoptimalforthedefensive teamtocooperatewiththeattackerinretreatsothatretreatbecomesamoreattractive optionthanengagementfromthatattacker'sperspective. ExpandingonthegamepresentedinChapter3,Idevelopedageneralizedattackor retreatgameinChapter4.Inthisgame,eachplayerisallowedtoswitchtheirrespective intentsatanypoint.However,itisshownthatisneveroptimalforeitherplayertodeviate fromtheirinitialintentselections.Additionally,theincorporationofavaluefunction constraintgeneratesconstrainedretreattrajectories.Alongthesetrajectories,the attackerisallowedtoretreatwithminimalcost,butthestatemusttakeapatharound 163

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regionswhereitisoptimaltoengage.Thesetrajectoriesarereferredtoasescort trajectories. MyresearchcontainedinChapter5focusedonthedevelopmentofageneral deceptivegameframework.Inthisgame,astochasticsensornetworkprovidesone playerwithaninformationadvantage.Simultaneously,theopposingplayeremploys deceptivetacticsinanattempttoneutralizetheaffectsofthesensornetwork.Usingthe strongdualitytheoremoflinearprogramming,Ishowedthatthisgamecanbeefciently solvedbysolvingapairofduallinearprograms.Thesolutiontothisgamesprovidesthe equilibriumstrategyforeachplayerwithinthegameaswellastheresultingequilibrium gamevalue.Aspecicexampleofthegeneraldeceptivegamewaspresentedand solvedforspecialcasesofparametervalues.Theresultingequilibriumplayerstrategies andutilityvaluescaptureawell-known,qualitativeprincipleinthedeceptioneldknown astheJones'Lemma. IdevelopedanewsequentialvariantoftheColonelBlottogameinmyresearch forChapter6.Thisgameconsistsoftwoplayers,PlayerAandPlayerB,whomust allocateniteresourcesamong N regionsofbattleeld.Akeyfeatureofmyproblem istheintroductionofasensornetworkemployedbyPlayerBtogainaninformational advantageoverPlayerA.Thesolutiontothisgameisrepresentedasasetofnecessary conditionsforNashequilibriumoptimalmixedstrategiesforthetwoplayersintermsof eachplayer'savailableresourcesandthesensornetworkcharacteristics. Insummary,Ihavedevelopedseveralnovelgame-theoreticmodelsthatcan representawiderangeofadversarialsituations.Thesemodelsweresolvedusingboth numericalandanalytictechniques,andtheresultingsolutionscaptureavarietyofreal worldbehaviors. 164

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REFERENCES [1]R.Isaacs, DifferentialGames .NewYork,NY:Wiley,1965. [2]A.W.Merz,TheHomicidalChauffeur-ADifferentialGame,GuidanceandControl Laboratory418,StanfordUniversity,Stanford,CA,Tech.Rep.,1971. [3]J.Lewin, DifferentialGames:TheoryandMethodsforSolvingGameProblemsWith SingularSurfaces .NewYork,NY:Springer-Verlag,1994. [4]J.Breakwell, DifferentialGamesandApplications ,ser.LectureNotesinControland InformationSciences.NewYork,NY:Springer-Verlag,1989,vol.119. [5]S.D.Bopardikar,F.Bullo,andJ.P.Hespanha,Acooperativehomicidalchauffeur game, Automatica ,vol.45,no.7,pp.1771,2009. [6]P.Cardaliaguet,M.Quincampoix,andP.Saint-Pierre,Somealgorithmsfor differentialgameswithtwoplayersandonetarget,CEREMADE,UniversityofParis -Dauphine,Paris,Tech.Rep.,1993. [7]M.PachterandY.Yavin,Astochastichomicidalchauffeurpursuit-evasion differentialgame, JournalofOptimizationTheoryandApplications ,vol.34, no.3,pp.405,1981. [8]K.HorieandB.A.Conway,Optimalghterpursuit-evasionmaneuversfoundvia two-sidedoptimization, JournalofGuidanceControlandDynamics ,vol.29,no.1, pp.105,2006. [9]M.PontaniandB.A.Conway,OptimalInterceptionofEvasiveMissileWarheads: NumericalSolutionoftheDifferentialGame, JournalofGuidance,Control,and Dynamics ,vol.31,no.4,pp.1111,2008. [10],Numericalsolutionofthethree-dimensionalorbitalpursuitevasiongame, JournalofGuidance,Control,andDynamics ,vol.32,no.2,pp.474,2009. [11]S.Bhattacharya,S.Hutchinson,andT.Basar,Game-theoreticanalysisofa visibilitybasedpursuit-evasiongameinthepresenceofobstacles,in Proceedings oftheAmericanControlConference ,St.Louis,MO,Jun.2009,pp.373. [12]S.Bhattacharya,T.Basar,andN.Hovakimyan,Singularsurfacesinmulti-agent connectivitymaintenancegames,in ProceedingsoftheIEEEConferenceon DecisionandControl ,Orlando,FL,Dec.2011,pp.261. [13]S.BhattacharyaandT.Basar,Spatialapproachestobroadbandjammingin heterogeneousmobilenetworks:Agame-theoreticapproach, SpecialIssuefor SearchandPursuit-Evasion,AutonomousRobots ,vol.31,pp.367,2011. 165

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BIOGRAPHICALSKETCH ZachariahElijiahFuchswasborninEvansville,Indianawherehegrewupand graduatedfromEvansvilleNorthHighSchoolin2003.AfterearningaB.S.inelectrical enginneringattheUniversityofEvansvillein2007,hemovedtoGainesville,Florida toattendgraduateschoolattheUniversityofFlorida.UponcompletionofhisPh.D. program,ZachwillcontinuehisresearchwiththeAirForceResearchLabAFRLat Wright-PattersonAirForceBaseinDayton,Ohio. 171