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