Working Paper LSEOR 09.116 ISSN 2041-4668 (Online)Optimizing Randomiz - PDF document

Working Paper LSEOR 09.116 ISSN 2041-4668 (Online)Optimizing Randomized Patrols Steve Alpern, Alec Morton, Katerina Papadakia. Department of Mathematics, London School of Economics and Political Science, London. b. Operational Research Group, Department of Management, London School of Economics and Political Science, London. OptimizingRandomizedPatrolsSteveAlperny,AlecMorton,KaterinaPapadakiNovember5,2009AbstractAkeyoperationalproblemforthosechargedwiththesecurityofvulnerablefacilities(suchasairportsorartgalleries)istheschedulinganddeploymentofpatrols.Motivatedbytheproblemofoptimizingrandomized,andthusunpredictable,patrols,wepresentaclassofpatrollinggamesongraphs.Thefacilitycanbethoughtofasagraphofinterconnectednodes(e.g.rooms,terminals)andtheAttackercanchoosetoattackanynodeofwithinagiventimeT:Herequiresconsecutiveperiodsthere,uninterruptedbythePatroller,tocommithisnefariousact(andwin).ThePatrollercanfollowanypathonthegraph.Thusthepatrollinggameisawin-losegame,wheretheValueistheprobabilitythatthePatrollersuccessfullyinterceptsanattack,givenbestplayonbothsides.Wedetermineanalyticallyoptimal(minimax)patrollingstrategiesforvariousclassesofgraphs,anddiscusshowourresultscouldsupportdecisionsabouthardeningfacilitiesorchangingthetopologyoftheterraintobepatrolled.Subjectclassi…cations:Games,noncooperative;Military,search/surveillance;Deci-sionAnalysis,riskAreaofreview:MilitaryandHomelandSecurity DepartmentofMathematics,LondonSchoolofEconomicsandPoliticalScience,London.OperationalResearchGroup,DepartmentofManagement,LondonSchoolofEconomicsandPoliticalScience,London. 1IntroductionAkeyoperationalproblemforthosechargedwiththesecurityofvulnerablefacilitiesistheschedulinganddeploymentofpatrols.Thisproblemisencounteredby,forexample:securityguardspatrollingamuseumorartgalleryantiterroristo¢cerspatrollinganairportorshoppingmallpoliceforcespatrollingacitycontaininganumberofpotentialtargetsfortheftsuchasjewelrystoressoldierspatrollinganoccupiedcityorterritoryairmarshalspatrollinganairlinenetworkinspectorspatrollingacontaineryardorcargowarehouseSuchproblemshavebeenstudiedindiverseliteratures.Forexample,awell-knownproblemincomputationalgeometrydealswiththepositionofsecurityguardsinartgalleries(Urrutia,2000)andaclassicalOperationsResearchliteratureexistsontheschedulingofpolicepatrols(seee.g.Larson(1972)andreferencestherein).Theimportanceofrandomizedpatrolshasbeenrecognizedinlawenforcementforsometime,butnotthenatureoftherandomization(e.g.ShermanandEck(2002,p.297)).Muchoftheoptimizationliteratureonthissubject(e.g.Chelst,1978)concentratesontheimportantproblemofhowtodeployrandomizedpatrolstomaximizetheprobabilityofinterceptingacrimeinprogress,whenthecrimefrequencyofdi¤erentlocationsistakenasgiven(oftenarealisticassumption,atleastintheshortterm).Suchmodelshoweverarenotgametheoreticanddonotcapturetheideaofapatrollingscheduleasastrategyselectedinthefaceofanintelligentandmalignadversary,forexampleanartthieforterrorist,whichisadistinctivefeatureoftheclassofmodelswestudyinthispaper.Althoughtheredoexistdi¤erentialgameformulationsoftherelationshipbetweenpoliceandcriminal (Isaacs,1999,Feichtinger,1983)thesetendtofocusonadynamic(andoftenstrategic)processofmutualadjustmentratherthanconfrontingtheproblemconfrontedbytheschedulerwhositsdowntodeterminethepathwhichthepatrolwilltake.GametheoreticanalyseshaverecentlyfeaturedprominentlyinORstudiesofhomelandsecurityandcounterterrorism(e.g.Brownetal,2006,BierandAzaiez,2009,Lindelaufetal.,2009).AnattractiveanduniquefeatureofgametheoreticformulationsinthecontextofpatrollingisthattheyprovideinsightintohowaPatrollershouldrandomizeherpatrols.Thereisaclearcommon-senserationaleforrandomization:apredictablePatrollerisanine¤ectiveone.Yetanaive"maximumentropy"heuristic(Foxetal,2005)maybenotfaremuchbetter:facedwithtargetsitmaynotmakesensetospend=noftheavailablepatrollingtimewitheachofthem.Thisdilemmahasattractedconsiderableattentionrecentlyamongstpractitionersandtheresearchcommunityhasrespondedtothischallenge:inparticular,theworkofParuchuriandcolleagues(2007)providesanumberofheuristicmodelswhichillustratehowequilibriumrandomizedstrategiescanbeapproximatedwhentheproblemisformulatedasaStackelberg(leader-follower)game,andsuchmodelshavefounduseinrealsecuritysituations(Gordon,2007;Newsweek,2007).Ourworkonthisproblemisinspiredbythetheoryofsearchgames,onwhichanextensivemathematicalliteraturedevelopedoverthelastfewdecades.ThistheorycapturessituationsinwhichaSearcheraimsminimizethetimetakento…ndastationaryormobileHiderwhodoesnotwanttobefound(AlpernandGal,2003).TherearealsorelatedliteraturesonInspectiongames(Avenhaus,vonStengelandZamir,2002),inwhichanInspectorwhoseekstocatchanInspecteered-handed,andIn…ltrationgames(Auger,1991;Garnaev,GarnaevaGoutal,1996;Garnaev,2000andAlpern,1992)inwhichaGuardseekstopreventanIn…ltratorfrompenetratingsomesensitivefacility.Similarattack/defencegameshavebeenstudiedinmilitaryoperationsresearch(Washburn,2003),datingasfarbackasMorseandKimball(1950).Manysuchgamesareofindependentmathematicalinterestandhavebeenstudiedin apurelymathematicalsettings(e.g.Baston,BostockandFerguson,1989).VariousresultsareavailableforhowtheSearcher/Inspector/Guard/Defendershouldproceed,dependingontheassumptionsaboutthestructureofthemathematicalspacewhichsheinhabits.Aparticularlyproductivelineofresearchinthesearchgameliteraturehasbeentoexplorethecasewherethesearchspacecanbethoughtofasagraph,aswedohere.InthispaperweformulateagamewhichwecallthePatrollingGame.UnliketheworkofParuchuriandcolleagues,ourproblemisazero-sumgame,andprovidesforadefenderwhoismobile,beingabletotravelbetweenlocationsinthecourseofhisshift(a"Patroller").Unlikesearchgames,our"Attacker"(theequivalentofthesearchgame"Hider")maycommencehisattackatanytimeandhastobedetectedwithinagiventime-windowinordertoforestalltheperformanceofsomemisdeed.Ourgameiswin-lose-agameoftyperatherthandegreeintheterminologyofIsaacs(1999).Ourproblemissu¢cientlyidealizedthatitispossibletoobtaininsightfulanalyticresults,butsu¢cientlyrealisticthatitisrecognizableasapracticalproblemfacedbypractitionersinvariousdomains.Inthispaperwepresentsomeanalyticresultsforthisgame,anddemonstratethatityieldspatrolling(andattacking)strategieswhicharenaturalandintuitive.Moreover,weshowhowthegamecanbeusedtoguidedecisionsaboutinvestmentinhardeningvulnerablesitesorinaddingadditionalpassagewaystoenablethePatrollertoshortenthetimerequiredtogobetweendi¤erentsitesthatmightbeattacked.Weareinthispaperunabletopresentgeneralanalyticresultsforallgamesofthistype,anditseemsunlikelythatsuchsolutionsexist.Indeed,evencomputingoptimalstrategiesmaybequitechallenging,becauseofthecombinatorialexplosioninthePatroller’sstrategyspace;inacompanionpaper(Alpern,MortonandPapadakiinprepa-rationa),wepresentsomealgorithmsfore¢cientlycomputingthevalueofthisgameformorecomplex(andrealistic)examples.Thispaperisorganizedasfollows.WepresentinSection2arigorousformulationofpatrollinggames,togetherwithsomeelementaryobservationsonpropertiesoftheValue.As thenumberofpurestrategiesfortheplayerscanbeverylarge,wegiveinSection3threemethodsforreducingthenumberthatwehavetoconsider:symmetrization,dominanceanddecomposition.Section4discussescertainclassesofstrategiesthattheplayerscanuseonanygraph,andwhichareoptimaloncertainclassesofgraphs.Section5solvespatrollinggamesoncertainclassesofgraphs:Hamiltonian,bipartiteandlinegraphs.Section6considershowthegamecanbeusedtoguidedecisionsininvestmentabouthardeningnodesoraddingedges.Section7presentsextensionsofthemodelandconcludes.2ThePatrollingGameInthissectionwegiveaformaldescriptionofthePatrollingGameQ;T;mwhereisthegraphwhosenodesareunderattack,isthetotalnumberoftimeunitsthegameisplayedover,and)isthenumberof(consecutive)periodsrequiredtosuccessfullycarryoutanattackonanode.Roughlyspeaking,theAttackerpicksanodetoattackandchoosessometimeinterval;+1;:::;oflengthinwhichtoattackit.ThePatrollerfollowsawalkonQ;thatis,hechoosesnodes(0);:::;w1)withconsecutivenodesthesameoradjacentinQ:ThePatrollerwinsifhiswalkisatthenodeinsomeperiodinwhichitisbeingattacked,thatis,if)=forsomeI:OtherwisetheAttackerwins.Wealsodemonstratesomesimplemonotonicityresultsandsomeboundsonthevalue,whichwillbeusefullateron.2.1FormulationMoreformally,thePatrollingGameQ;T;misawin-lose(andhencezero-sum)gamebetweenamaximizingPatroller(female)andaminimizingAttacker(male).Itcomesintwoforms,theone-o¤gameQ;T;mandtheperiodicgameQ;T;mTheone-o¤gameisplayedoutoveragiventimeinterval;:::;Toflengthonagraphwithnodesandedges.Wewilltendtoassumethatisconnectedunlessstated otherwise.ApurestrategyfortheAttackerisapairpairi;Iwhere2NiscalledtheattacknodeandTisan-intervalcalledtheattackinterval.ApurestrategyforthePatrollerisawalkT!calledapatrol.Ifwesaythatthepatrolinterceptstheattack,inwhichcasethePatrollerwinsandthepayo¤is=1;otherwisewehave=0Thusthepayo¤isgivenbyw;;i;I])=(Patrollerwins)if(Patrollerinterceptsattack),(Attackerwins)ifi=(attackissuccessful)TheValueofthisgameisthustheprobabilitythattheattackissuccessfullyintercepted.Exceptintrivialcases,optimalstrategiesmustbemixed.Theperiodicgameissimilarexceptthatthepatrols(Patrollerpurestrategies)arenowwalksofperiod(satisfying)=forallAttackintervalsarenow-intervalsinthetimecirclemod(soforexampleifis24andistheattackcouldbecarriedoutovernight,duringtheinterval2223(10o’clockto2inthemorning).Wecanalsoviewthepatrolsaswalks!N.Theperiodicgameissimplertoanalyzebecausetheattackcanbeassumedtotakeplaceequiprobablyinanytimeinterval,whichsimpli…estheanalysis(seeSubsection3.1).Whenthevaluesofthegamesdi¤er,wewillusethesuperscriptsandtodistinguishbetweentheValues,usingwhentheresultappliestobothcases.)canbeconsideredasparameterizedby,andjustasis,butmostofthetimewritingQ;T;misdistractingandconfusingandwewilltendtosuppresssomeorallofthesearguments.Wedenotebyi;ithedistancefunctiononthenodeset,theminimumnumberofedgesbetweenand2NThisformulationmakesanumberofassumptionswhicharenotinfactasrestrictiveastheymightappear.The…rstistheassumptionanattackwilltakeplace.Animmediateresponsetothisisthateventhoughattacksoccurveryrarely,oneshouldpatrolontheassumptionthatanattackwillhappen-otherwisewhatisthepointofpatrollingatall?Amoresophisticated Figure1.asaprojectionofLemma11.Q;T;misnondecreasinginm:2.Q;T;mcannotdecreaseifanadditionaledgeisaddedbetweentwononadjacentnodesofthegraph.Thatis,isnondecreasingin(withtheorderingonthelatterunderstoodinthesenseofsetinclusion).3.Q;T;mQ;T;m4.Ifisobtainedfrombynodeidenti…cation,Proof.The…rstpartfollowsfromtheobservationthatapatrolthatinterceptsanattackki;Ialsointerceptstsi;IifI:Thenexttwoarebasedonthefactthatinazerosumgameaplayercannotdoworseifhegetsadditionalstrategies.Thelastisbasedonthefollowingobservation:IfapatrolinterceptsanattackonanodeofthenthepatrolinterceptstheassociatedattackonthenodeofSothePatrollercanensurethattheexpectedpayo¤isatleastbychoosingpatrolsforaccordingtosomeoptimalmixedstrategy,andthenplayingtheprojectedpatrol ThenextresultgiveseasygeneralboundontheValue.Lemma2 nVm ,forequaltoorandanyparametersand.Moregenerally,!=n;whereisthemaximumnumberofnodesthatanypatrolcancover(dependsonwhethertheone-o¤orperiodicversionisbeingplayed).Proof.ThePatrollercanobtaintheleftinequalitybyrandomlypickinganodeandwaitingthere.TheAttackercanobtaintherightinequalitybyattackingarandomnodeduringsome…xedtimeinterval.Ofthesepurestrategies,thePatrollercaninterceptatmostjofthem,givingtheboundm=n;ormoregenerallythebound!=n;sincejbyde…nition. 8 therearenoattacksatpenultimatenodes(usingLemma3).Sincenodesandandnodesandareequivalentundersymmetry,wecanassumetheyareattackedwithequalprobability.SothemostgeneralAttackermixedstrategyisshown,where+6=1Thepatrol=(33)interceptsallattacksatnodeandtwoattacksatnode,sowinswithprobability+2similarly=(1anyinterceptsoneattackatnodeand…veattacksatnodesand,andwinswithprobability+5y:Thesetwopatrolstogetherdominateallothers.SotheAttackerminimaxeswhen+2+5y:Thisoccurswhen=110and=115withminimaxvalue=1330=43333:::.(AneasycalculationthenshowsthatthePatrollershouldadoptandwithprobabilitiesand i\t1 2 3 4 5 y x x x y y y y y 1/81/81/81/81/81/81/81/8 Figure3.Optimalattackingandpatrollingstrategiesforwith=5and=3Inthe(unrestricted)game3)itishardertoderivetheequilibriumstrategypair,butitisfairlyeasytodemonstratethattheValueis=38=0375whichshowsthatnotimeinvariantmixedAttackerstrategycanbeoptimal.Toseethat,considertheAttackstrategyshowninthemiddledrawing,andobservethatnopatrolcaninterceptmorethanthreeoftheeightequiprobableattacks.AnoptimalPatrollermixedstrategyistoadoptthefourstrategies(24)(26)andtheirre‡ections(53)(51)(drawninthinblueslantedlines)withprobabilityeach;andadoptthetwoequivalentstrategies(33)and(44)(drawninthickredlines)withprobabilityeach.Theassertionfollowsfromtheobservationthateveryoneoftheeighteenpossibleattacksisintercepted(hit,totherightof,totheleftof)byatleastthreeofthepatrols,countingthethickredonesas12 two.Itisalsointerestingtoobservethatalltenattackswhicharenotusedatallinthemiddledrawingareinterceptedbymorethanthreeofthesepatrols.Thus,theValueoftheone-o¤game3)isbutitrequirestheuseoftimedependentAttackerstrategies:themiddlenodeisonlytobeattackedinthemiddletimeintervalNextweanalyzetheperiodicversion,thegame3)Thisissimilartotherestrictedversionoftheone-o¤gamediscussedabove,exceptthatthemiddleoftheattackcanbeatanytime,socomparingwiththeleftdrawingofFigure3,the’sand’swouldextendthroughouttherows,andsowehave10+10=1.The(periodic)patrol=(33)interceptsalltenattacksatmiddlenodesandandwinswithprobability10y:The(periodic)patrol=(11)interceptsfouroftheattacksatnode(allexcepttheoneduringtimes2,3,4)andthreeoftheattacksatnodesowinswithprobability+3y:Thesetwo(togetherwiththeirsymmetrictranslations)dominateallotherpatrols.SotheAttackerminimaxeswhen10=4+3y:Thishassolution=7110and=4110withminimax=411ForthePatroller,(anditsequivalents)shouldbeusedwithprobability11;wwithprobability1011Tosummarize,forlinegraph,with=5and=3wehave36364V375V43333ThustheAttackerdoesbetterintheone-o¤game,andthustheboundstatedinLemma1Part3neednotbetight.Further,inthisinstance,theAttackerhastoadoptatimedependentstrategyinordertobene…tfully.3.4DecompositionSometimeswecanthinkofagraphasbeingmadeupofsimplergraphsandWecallthisadecompositionof.Thenodesoftheoriginalgrapharetheunionofthenodes13 3.5Example:thekitegraphTodemonstratetheuseofallofourstrategyreductiontechniques,weanalyzetheperiodicgameforthekitegraphillustratedinFigure5with=3.ThedominanceargumentofLemma3showedthattheAttackerwouldneverattacknode4,asitisalwaysbetterforhimtoattacktheadjacentleafnode5.Moreover,intheperiodiccasefor=3,thereisnofeasiblePatrollerstrategywhichvisitsbothnode5andanyoneof1,2,or3.Therefore,wecanremovenode4andbecon…dentthattheperiodicgameontheresultinggraphKThasthesameValueasthegameonKT: Q1Q23251 Figure5.DecompositionofKitegraphKTintoKTLemma4showsthatforandasinFigure5,wehaveKT =V)+1=V(1)Obviously)=1,anditcanbeeasilyshownthat)=1for=3.Henceby(1)wehaveKT)=KT 1+2 ThisisananotherexamplewherethePatrollerdoesstrictlybetterintheone-o¤game,inwhich=3Toseethis…rstnotethatbyLemma2theAttackercanensurethatm=n=3byattackingequiprobablyatthe…venodes.Thenobservethatbyusingthefourpatrols(23)(25)(35)(15)withrespectiveprobabilitiesthe15 leastoneofthepatrols.Thecoveringnumberistheminimumcardinalityofanycoveringset.De…nition9If,foranytwonodesand,anypatrolwhichinterceptsanattackatnodeinattackinterval,cannotalsointerceptanattackatinattackinterval,thenandwillbesaidtobeindependent.Intheone-o¤gamethisisequivalenttorequiringanytwonodestosatisfyi;iintheperiodicgame,theymustsatisfyi;iori;i(becausethePatrollerhastoreturntohisstartingpointbytheendoftheperiod).Theindependencenumberisthecardinalityofamaximalindependentset.ObviouslyIJObservethatbothanddependontheparametersQ;T;mandontheversionofthegamethatisplayed,orForexample,when=3and=3thenodesubsetofisindependentfortheperiodicgamebutnotfortheone-o¤game.FortheAttacker,theindependentstrategyisto…xanattackintervalandthenchoosetheattacknodeequiprobablyfromsomemaximalindependentset.ForthePatroller,thecoveringstrategyistochooseequiprobablyfromaminimalsetofcoveringpatrols.Notethatfor=2patrolscanbeidenti…edwithedgesof,sothesede…nitionsreducetotheusualnotionofanindependentsetnothavingadjacentnodesandacoveringsetconsistingofedges.Lemma10 JV1 (with=1when).Proof.TheAttacker’sindependentstrategygivestheupperboundandthePatroller’scoveringstrategygivesthelowerbound. Thecaseswheredealwithmanypatrollinggames.Forexample,wecanusethistechniquetogiveanothersolutiontothekitegraphKTofFigure2fortheperiodicgamewith=3Herethenodes2,3,and5formanindependentset(becausei;i)=43=andinterceptingpatrolsonthetopleft,toprightandbottomedges(periodpatrolsofthe18 Proof.Firstobservethatineithercasewehavem=nbyLemma2.Intheone-o¤game,supposethePatrolleradoptsarandomHamiltonianpatrol.ThenforanyattackintervalI;isarandom-arcoftheHamiltoniancycle,andassuchcontainstheattacknodewithprobabilitym=n;asclaimed.Ifisamultipleofn;thisstrategyisalsofeasibleintheperiodicgame.Toobtainthelimitingresult,notethatifmod=0theperiodicPatrollercanmodifytherandomHamiltonianpatrolbywaitingatarandomnodeduringarandominterval.Thiswillnothurthimunlesstheattackintervaloverlapsthewaitinginterval,whichhasprobability1)=T;so Tm nVpVo=m andso (2) Sincetheaboveresultappliestothecyclegraph,wecanuseittosolvethegameonsomegraphswhichcanbeobtainedfromthecyclegraphbyidenti…cationofnodes.WenowsolvetheperiodicPatrollingGamefortheeightnodegraphshownbelowontheleftofFigure6inthecase=10and=4Firstnotethatsincethediameteris=5wehavefromTheorem7thatthediametricalAttackstrategyensuresthatm==410Byviewingthegraphasaprojectionof10(withValuem=10=410fromTheorem11)weconcludefromLemma1Part4that10so=410 Figure6.Agraphshownasprojectionof1020 ingeneralandwithasimilarlimitingresultfortheperiodiccase Inallthesecases,informallyspeaking,anoptimalstrategyfortheAttackeristo…xanattackintervalandchoosetheattacknodeequiprobablyfromthelargerhalfset;anoptimalstrategyforthePatrolleristorandomizeoveracollectionofstrategieswhichvisitthelargerhalfseteverysecondtimeperiod.Inthecaseof=2,Patrollerchoosesanedgejoiningthehalfsets;theAttacker’sandPatroller’sstrategiescanbeseenasarandomchoicefromanindependenceandcoveringsetrespectively;inthiscasetheTheoremcanbeunderstoodasaversionofKönig’sTheorem(Harary,1971,Theorem10.2)inourcontext,sinceKönig’sTheoremstatesthattheindependenceandcoveringnumbersofabipartitegraphareidentical.Toillustratetheproof,considerthespecialcaseofthestargraph;nconsistingofacentralnodeconnectedtoextremenodes.ThismodelsthesituationwherethePatrollerhasresponsibilityforthesafekeepingofabuildingwhichhasmultiplewings,accessiblethroughacommonlobbyarea.Wecanviewasobtainedfromtheevencyclegraph2(1)byidentifying(say)allevennumberednodes,asinFigure7. Figure7.obtainedfrombynodeidenti…cation.ThismodeofreasoningleadsustodiscoveradditionalequilibriumpairsfortheHamiltoniancase.Considerthecyclegraphforevenandamultipleofn:WesawearlierthattheuniformstrategywasoptimalfortheAttacker.Butsincen=;n=isbipartiteTheorem12nowgivestheadditionaloptimalstrategyofattackingequiprobablyontheodd(oreven)nodes.InfactthereisonemoreoptimalAttackerstrategy:sincethediameterisn=thediametricalstrategyalsogivesm=m=nbyTheorem7.22 probability foranoverallprobabilityofinterceptionof 43 7+1 7
=3 .Allotherpatrolsyieldalowerexpectedpayo¤andso3) Nextconsider2)Weclaimthatthevalueofthisgameis .Wehave=4andsofromTheorem10wehaveToensurewinningwiththisprobability,thePatrollermustuse"biasedoscillations"onedgesi;ioftheformi;i;i;i;i,whichwedenoteas ,witharandomtimerotation.Clearly interceptsanyattackonandinterceptsanyattackonwithprobability(thatisunlesstheattackcoincideswitharepeatedTheoptimalprobabilitiesofthebiasedoscillationsonconsecutivenodesareshownbelow.161616161616Attacksonanynodeareinterceptedwithprobabilityatleastwithequalityexceptforthecentralnode(whichshouldneverbeattacked).Forexampleanattackonnodeisinterceptedwithprobabilityifeither or isadoptedbythePatroller,thatiswithprobability(416+516)=516Soitisinterceptedwithprobability(45)(516)=16HardeningNodesorAddingEdgesUptonowwehavetakenthenetworkasgiven.ButtheagencythatcontrolsthePatroller(the"Defender")maybeabletopaytoeither‘harden’asitesothatitisimmunetoattackortobuildapassagewaytohelpthePatrollermovemorequicklybetweensites.InourformulationthiscorrespondseithertoreducingtheAttacker’sstrategyspacebyremovinganattacknode,oraddinganedgetothegraph.Inthissectionwegivesomeverysimpleexamplesintheeasycasewhereallsites(nodes)areequallyexpensivetohardenandnewpassageways(edges)areequallyexpensivetobuild(morecomplexexampleswillbedealtwithinourforthcomingpaperonthecomputationalaspectsofthisproblem).6.1Hardeningnodes24 6.2AddingedgesWeobservedearlier(Lemma1)thataddingedgescannothurtthePatroller,andusedthistoanalyzecertainnetworks.Herewetakeadi¤erentpointofviewandaskwhetheritpaysforthePlayerthatcontrolsthePatrollertospendmoneytoaddanedgebetweentwonodesinordertoincreasethee¢cacyofthePatroller.WeconsideragainthenetworkandA;3)andaskhowtheinterceptionprobabilityincreases,ifatall,byaddingedgesbetweennonadjacentnodesofA:TheindependentsetsareindicatedinallcasesbydisksatnodesinFigure9.Thecoveringsetsareeitheridenti…edwithedgesorcycles,whicharethickened.Weobservethataddingedgesasinorachievesnoincreaseatallintheinterceptionprobability(theValue,butasinorincreasesittofromTheintuitionisthattheseedgescreatecycles,whicharemoree¢cientforpatrolsthanoscillationsonedgeswitharepeatednode. A: I=J=3, V=1/3 B: I=J=3, V=1/3C: I=J=2, V=1/2 D: I=J=2, V=1/2 E: I=J=3, V=1/3 Figure9.Thegraphandfourwaysofaddinganedge7ConclusionInthispaperwehavedescribedasimple,intuitivemodelwhichcanserveasthebasisforobtainingoptimalrandomizedpatrols.Theassumptionsofthepresentpapercanberelaxedinthefollowingextendedmodels:26 Thenodescanhavedi¤erentvalues(e.g.paintingsofvaryingartisticmeritinanartgallery).ThesimplestwaytomodelthisistokeepthesamestrategiesbutviewthePatrolleraswantingtominimizethevalueofasuccessfullyattackednode.Sothisversioncomeswithacostvectorc;wheredenotesthecostofasuccessfulattackonnodei:Somenodesmaybeunequallyhard,orevenimpossible,toattack.WethusreplacetheparameterbyavectorM;wheredenotesthenumberofperiodsrequiredtosuccessfullyattacknodei:Inarelatedway,edgesmayhavelengthsattachedtothem.(Twonodeswithaninterveningnodewhichcannotbeattackedareinsomesensetwotimeunitsapart.)Theremaybemultiplepatrollersand/orattackers.Someoftheresultsofthecurrentpapertransferovereasilytothissituation-forexampleonecansimply"multiplyup"thenumeratorofLemma10tohandlethesituationwherethereareseveralpatrollers-butingeneralthisseemstoberathermorecomplex.PerhapsthePatrollermuststartataknownnode=0intheone-o¤game.Ofcourse,inthiscasethediameterofcannotbelargewithrespectto,otherwisetheAttackerwillalwayswin.AnaturalconjectureisthattheAttackerwouldattackearlierinordertotakeadvantageofhisgreaterknowledgeofwherethePatrollerwillbe,butisthisalwaysso?Itmaybenaturaltoconsideracontinuoustimeformulationofthisproblem.Anattacktakesplaceatanypointofthenetwork(notnecessarilyanode)onacontinuoustimeintervalof…xedlength.ThePatrollerusesaunitspeedpathandwinsifheisattheattackedpointatsometimeduringtheattackinterval.Thiswouldmodel,forexample,thedefenseofapipelinesystem.ThePatrollermaybealerted(perhapsnoisilyandwithsomeerror)tothepresenceofanAttacker;andtheAttackermaybealertedbyaconfederatewhocanidentifywhena27 Patrollerleavesaparticularnode(forexample,ifthePatrollerisinamarkedpolicecar).Manyoftheseproblemsarenotanalyticallytractable,andsomeofthemwillbediscussedinourforthcomingpaperoncomputationalaspectsofthesegames(Alpern,MortonandPapadakiinpreparationa).Acknowledgement14WewouldliketothankDelofvonWinterfeldtforsuggestingthisprob-lemtous,andtoMilindTambeforinterestingdiscussions.AMandKParealsogratefultotheCentreforRiskandEconomicAnalysisofTerroristEvents(CREATE)attheUniversityofSouthernCaliforniafortheirsupportandhospitality.SAwassupportedbyNATOCollaborativeLinkageGrant983583onDefenseAgainstTerrorism.8ReferencesAlpern,S.(1992).In…ltrationgamesonarbitrarygraphs.J.Math.Anal.Appl.163,no.1,286-288.Alpern,S.andM.Asic(1985)."TheSearchValueofaNetwork."Networks15(2):229-238.Alpern,S.andS.Gal(2003).TheTheoryofSearchGamesandRendezvous.Dordrecht,Kluwer.Alpern,Morton,andPapadaki(inpreparationa).Computingoptimizedpatrols.Alpern,Morton,andPapadaki(inpreparationb).Patrollingachannelforaslowin…ltrator.Auger,J.(1991)Anin…ltrationgameonkarcs.Nav.Res.Logist.38,511-530.Avenhaus,R.,B.VonStengel,etal.(2002).Inspectiongames.HandbookofGameTheorywithEconomicApplications.R.J.AumannandS.Hart.AmsterdamElsevier.Baston,V.,F.Bostock,etal.(1989)."TheNumberHidesGame."ProceedingsoftheAmericanMathematicalSociety107(2):437-447.Bier,V.M.andM.N.Azaiez(eds)(2009).GameTheoreticRiskAnalysisofSecurityThreats.NewYork,Springer.28 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