Towards an Iterated Game Model with Multiple Adversaries in Smart-World Systems

28 Pages·2017·1.11 MB·English

sensors
Article
Towards an Iterated Game Model with Multiple
†
Adversaries in Smart-World Systems
XiaofeiHe1,‡ ID,XinyuYang1,*,WeiYu2,*,‡,JieLin1,‡ andQingyuYang3,‡
1 DepartmentofComputerScienceandTechnology,Xi’anJiaotongUniversity,Xi’an710049,China;
[email protected](X.H.);[email protected](J.L.)
2 DepartmentofComputerandInformationSciences,TowsonUniversity,Towson,MD21252,USA
3 SKLMSELab,SchoolofElectronicandInformationEngineering,Xi’anJiaotongUniversity,
Xi’an710049,China;[email protected]
* Correspondence:[email protected](X.Y.);[email protected](W.Y.);
Tel.:[phone](X.Y.);[phone](W.Y.)
† ThispaperisanextendedversionofourpaperpublishedinAGame-TheoreticModelonCoalitional
AttacksinSmartGrid.InProceedingsofthe2016IEEETrustcom/BigDataSE/ISPA,Tianjin,China,
23–26August2016.
‡ Theseauthorscontributedequallytothiswork.
Received:29December2017;Accepted:15February2018;Published:24February2018
Abstract: Diverseandvariedcyber-attackschallengetheoperationofthesmart-worldsystemthatis
supportedbyInternet-of-Things(IoT)(smartcities,smartgrid,smarttransportation,etc.) andmust
becarefullyandthoughtfullyaddressedbeforewidespreadadoptionofthesmart-worldsystemcan
befullyrealized. Althoughanumberofresearcheffortshavebeendevotedtodefendingagainst
thesethreats,amajorityofexistingschemesfocusonthedevelopmentofaspecificdefensivestrategy
todealwithspecific,oftensingularthreats. Inthispaper,weaddresstheissueofcoalitionalattacks,
whichcanbelaunchedbymultipleadversariescooperativelyagainstthesmart-worldsystemsuchas
smartcities. Particularly,weproposeagame-theorybasedmodeltocapturetheinteractionamong
multiple adversaries, and quantify the capacity of the defender based on the extended Iterated
Public Goods Game (IPGG) model. In the formalized game model, in each round of the attack,
aparticipantcaneithercooperatebyparticipatinginthecoalitionalattack, ordefectbystanding
aside. Inourwork,weconsiderthegenericdefensivestrategythathasaprobabilitytodetectthe
coalitionalattack. Whenthecoalitionalattackisdetected,allparticipatingadversariesarepenalized.
The expected payoff of each participant is derived through the equalizer strategy that provides
participantswithcompetitivebenefits. Themultipleadversarieswiththecollusivestrategyarealso
considered. Viaacombinationoftheoreticalanalysisandexperimentation,ourresultsshowthatno
matterwhichstrategiestheadversarieschoose(randomstrategy,win-stay-lose-shiftstrategy,oreven
theadaptiveequalizerstrategy),ourformalizedgamemodeliscapableofenablingthedefenderto
greatlyreducethemaximumvalueoftheexpectedaveragepayofftotheadversariesviaprovisioning
sufficient defensive resources, which is reflected by setting a proper penalty factor against the
adversaries. Inaddition,weextendourgamemodelandanalyzetheextortionstrategy,whichcan
enableoneparticipanttoobtainmorepayoffbyextortinghis/heropponents. Theevaluationresults
showthatthedefendercancombatthisstrategybyencouragingcompetitionamongtheadversaries,
andsignificantlysuppressthetotalpayoffoftheadversariesviasettingtheproperpenaltyfactor.
Keywords: InternetofThings(IoT);security;gametheory;zero-determinantstrategy;iteratedpublic
goodsgame(IPGG)
Sensors2018,18,674;doi:10.3390/s18020674 www.mdpi.com/journal/sensors
Sensors2018,18,674 2of28
1. Introduction
Therapiddevelopmentofthesmart-worldsystemssupportedbyInternet-of-Things(IoT)suchas
smartcities,smartgrid,smarttransportation,etc. hasgivenrisetovarioussecurityissues,whichhave
becomeoneofthemajorbarrierstowidespreadadoption[1–8]. Smart-worldsystemscovernumerous
smart-worldresearchareasthatourdailylifedependson,includingsmartcities,smartgridsystems,
smart transportation systems, smart medical systems, smart manufacturing systems, etc. In these
smart-worldsystems,thegeographicallydistributedsensors,actuators,andcontrollersareclosely
incorporatedthroughcommunicationnetworksandcomputationalinfrastructures,enablingsecured,
efficient,andremoteoperationsofphysicalsystems.
Withtherapiddevelopmentofsmart-worldsystems,massivenumbersofmonitoringsensors
andactuators(alsocalledIoTdevices)aredeployedtoenablemonitoringandcontrollingacrossa
varietyofdomains. ThenumberofIoTdeviceshasgrownto8.4billionintheyearof2017,andwill
continuetogrowmassivelyinthenearfuture[9].Nonetheless,cyber-threatsposeseriousthreatstoIoT
devicesandthesmart-worldsystemsthattheyoperateupon. Smartdeviceshavebeendemonstrated
to be vulnerable, as evidenced by a recent attack on 21 October 2016, which led to many popular
sitesbecomingunreachable[10]. Behindthisattackwasanetworkofunknowinglycompromised,
mass-produced smart devices (webcams and other similar products). In addition, an extended
functionality attack was investigated [11], which can compromise the smart lights and exfiltrate
datafromahighlysecureofficebuildingbyacovertcommunicationsystemoreventriggerepileptic
seizureswithstrobedlight.
As a typical smart-world system, the smart cities that integrate energy, transportation and
othersmart-worldcomponents,potentialadversariesmaylaunchmaliciousattacksviacontrolling
smart meter and sensor devices, and may manipulate critical information, including energy
consumption/supply, the state of power transmission and distribution links, electricity prices,
transportationroutes,andsoon[3,12–16]. Assmartmetersinthepowergridsubsystem,whichis
anessentialcomponentinthesmartcities,areoftendeployedintheopenenvironment,thepowergrid
maysuffergreaterrisksthanthetraditionalpowergrid. Unlikethecyber-attacksoncommunication
networksalone, thepotentialattacksinthepowergridcanleadtoseriouseconomicandphysical
damages[7,13–15,17].
Inaddition,forasmarthealthcaresystem,whichisalsoanessentialcomponentinsmartcities,
thedataintegrityinvolvesauthentication,accesscontrolandsecurecommunication[18]. Threatsto
thehealthcaresystemcandamagethetrackingofpatients’identificationandauthenticationofpeople,
patientmobility,andautomaticsensingandcollectionofdata,whichconstitutesreal-timeinformation
onpatients’healthindicatorsasabasisformedicaldiagnosis. Forthesmarthome,theappliances
integratedwithIoTarevulnerabletocyberattacksandtheadversarycaninstallmaliciousfirmware
onthecompromisedIoTdevices. Forexample,Hernandezetal. [19]showedthatacompromised
thermostatcouldactasabeachheadtoattackothernodeswithinalocalnetworkandanyinformation
storedwithinthenodeisavailabletotheadversaryaftermalicioussoftwareisinstalledintothenode.
Therehavebeenanumberofresearcheffortsdevotedtostudyingtheimpactsofcyber-attacksin
smart-worldsystems[4–7,11–15,17–25]. Nonetheless,mostoftheexistingeffortsfocusonstrategiesof
attackordefenseinasingularorspecificallyuniquesecurityissue,ofteninwhichonlyoneadversary
launchesanattackatatime. Inaddition,multipleadversariescouldexistinthesmart-worldsystem,
cooperativelylaunchingcoalitionalattackstodisrupttheoperationofthesmart-worldsystemmore
effectively. For each participant in a coalitional attack, he/she can choose either cooperation or
defectionineveryround. Thus,aniteratedgamemodelcanbeusedtoinvestigatetheinteractions
amongadversaries. Noticethat,inthegamemodelthatweinvestigateinthispaper,alladversaries
arereferredtoasactiveparticipants,whilethedefenderenforcesapenalty(determinedbypenalty
factor)toaffectthepayoffsofadversaries.
Because the strategies of one participant can affect the others, different strategies adopted by
the participants result in different outcomes. Thus, the interaction between the outcomes and the
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strategiesusedbyadversariesiscriticalinthegamemodel. Furthermore,mostexistingresearchefforts
onthedefensivestrategiesagainstthreatsalsofocusheavilyonthespecificsecurityissuesratherthan
evaluatingthecostfordeployingthedefensivemechanism. Toachievebetterdetection,thedefender
oftenneedstodeployexpensivecountermeasurestodealwiththethreatslaunchedbyadversaries.
Thus,howtoquantifytheinteractionbetweenthecostandeffectivenessofdefensivemechanismsis
acriticalproblemthatneedstoberesolved.
Toaddresstheseissues,ourpapermakesseveralcontributionsasfollows.
• Game Theory-Based Model. We propose a game theory-based model to investigate the
interaction among multiple adversaries who launch coalitional attacks against the system.
WeestablishanextendedIteratedPublicGoodsGame(IPGG)modeltoanalyzetheinteractions
amongadversariesandeachadversaryissubjectedbyapenaltyfactorenforcedbythedefender
viathedefensivecapability. Ineachround,eachadversarymustchooseeithertocooperateby
participatinginthecoalitionalattack,ortodefectbystandingaside. Theparticipatingadversaries
contributetheirownendowmentandthegainobtainedthroughtheattackisdistributedtoall
adversaries. Onlyparticipatingadversarieswillsufferthepenaltyfromthedefenderwhenthe
coalitional attack is detected. Our proposed game model reveals the expected payoff of the
participants through the equalizer strategy. The equalizer strategy can help a participant to
choose cooperation or defection according to the last round outcomes, in order to control the
payoff of his/her opponents to be a fixed value. In this paper, we present two typical cases:
Foranaltruisticparticipant,he/shewillsetthepayoffofhis/heropponentstothemaximum
value. For an adaptive participant, he/she will set the payoff of his/her opponents to be the
sameashis/herowndynamically,meaningallparticipantsobtainthesamepayoff. Inaddition,
wefurtherstudythegamemodelwithmultipleparticipantsandacollusivestrategy,whichhas
thesameobjectiveastheequalizerstrategy,butthestrategyadoptedbyparticipantsistotally
different. Thecollusivestrategyrequiresmorethanoneparticipanttocolludewitheachotherto
controlthepayoffoftheiropponentstobeafixedvalue,makingitmoredifficulttobedetected.
With our proposed game model, we can quantify the capacity of the defender to reduce the
expectedpayoffofadversaries.
• Theoretical Analysis and Evaluation. Via a combination of comprehensive analysis and
performance evaluation on our developed game model, we show the maximum payoff of
adversariesindifferentcases.Forexample,withtheincreaseoftherateofattackgain,theexpected
averagepayoffcanreachthemaximumvalue. Withtheaidofthepenaltyfactorintroducedby
defensivemechanisms,themaximumvalueoftheexpectedaveragepayoffcanbereducedto
theminimumvalue. Thismeansthattheparticipatingadversariescanobtainlittlegainfromthe
coalitionalattack,whichreducesincentivetoparticipateintheattack. Meanwhile,ourproposed
game model can help the defender set a proper defense level based on the affordable cost to
reducetheattackconsequenceraisedbytheattack,improvingtheeffectivenessofthedefense.
• ExtortionStrategy. Weextendourdevelopedgamemodeltoconsidertheextortionstrategyas
well. Inthisstrategy,aselfishparticipantcanextorthis/heropponents,seekingtoalwaysobtain
agreaterpayoffthanhis/heropponents,evenifthetotalpayoffdecreases. Viathecombined
theoretical analysis and evaluation results, we find that the penalty of the defender can lead
to more severe competition among the participants in the game. Therefore, it is difficult for
adversariestoachieveglobaloptimaloutcomes,limitingtheimpactscausedbyadversaries.
Notice that this paper is an extension of our prior work [26]. Based on the much shorter
conferenceversion,thissubmittedjournalversionconsistsofaboutsubstantialnewlyaddedmaterials
in comparison with the shorter conference version. The important new materials include a new
gamemodelthatconsiderscollusiveadversaries,anewgamemodelconsideringanadversarywith
anextortionstrategy,theproofforNashequilibrium,asetofnewperformanceevaluationresultswith
adaptiveequalizerstrategy,additionaldiscussion,newliteraturereview,andothers.
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The remainder of this paper is organized as follows: in Section 2, we give a literature review
aboutthesmart-worldsystemsandgametheory;inSection3,weintroducetheiteratedgamemodel
andthreatmodel;inSection4,wepresentourproposedgameformalizationindetail;inSection5,
weconductthetheoreticalanalysisoftheformalizedgamewithrespecttotheinteractionbetweenthe
expectedpayoffofadversaries,andthepenaltyfactorenforcedbythedefender;inSection6,weshow
theexperimentalresultstovalidatetheeffectivenessofourproposedscheme;weenhancetheproposed
game model to include adversaries with the extortion strategy in Section 7; we discuss possible
extensionsofourdevelopedgamemodelinSection8;finally,weconcludethepaperinSection9.
2. RelatedWork
Wenowreviewtheexistingresearcheffortsrelevanttoourstudy. Inthesmart-worldsystems
(e.g.,smartcities,smartgrid,smarttransportation),anumberofeffortshavebeendevotedtostudying
theimpactsofcyberattacksaswellasthedevelopmentofdefensiveschemes[5–7,11,13,17–25,27–32].
For example, Ericsson et al. [33] presented some important issues on the cyber security and
informationsecurityintheenergy-basedcyber-physicalsystems. Moetal. [34]establishedascienceof
cyber-physicalsystemsecuritybyintegratingsystemtheoryandcybersecurity.Particularly,therehave
beenanumberofresearcheffortsdevotedtodataintegrityattacksagainstkeyfunctionalmodulesin
theenergy-basedcyber-physicalsystems,aswellasdefensethereof[7,14,15,32,35,36]. Forexample,
Yangetal. [13]developedanoptimalattackstrategyagainstthestateestimationprocessthatenables
a minimum set of compromised sensors to launch a successful attack. Yang et al. [35] developed
mechanismsforoptimalPMU(PhasorMeasurementUnit)placementtodefendagainstdataintegrity
attacks. Li[37]proposedalightweightkeyestablishmentprotocolforsmarthomeenergymanagement
systemsandpresentedtheimplementationdetailsofthedesignedprotocol.
Game theory has been widely studied in a broad range of areas as well. For example,
someresearcheffortsfocusonapplyinggametheorytonetworksecurityandsecurityinavarietyof
systems [38–50]. For example, Xiao et al. [41] investigated an indirect reciprocity security game
for mobile wireless networks. Zhang et al. [44] applied the game theory to carry out a path
selectionalgorithmtoprotecttheanonymityofprivacy-preservingcommunicationnetworkssuchas
Tor. Yuetal. [43]appliedthegametheorymodeltoinvestigatetheinteractionsbetweentheintelligent
adversariesthatinstigatewormpropagationovertheInternetanddefenderswithasetofstrategies.
Hilbe et al. [51] showed the evolution of direct reciprocity in a group of multiple players and the
instructiveness of the zero-determinant strategies. Zhang et al. [52] presented an iterated game
model for resource sharing among a variety of participants. In this model, an administrator of
cooperation(AoC)isresponsibleformaintainingthesocialwelfare,whiletheregularparticipants
ofcooperation(PoCs)areselfishparticipants. Guo[53]investigatedzero-determinantstrategiesfor
multi-strategygames.
For cyber-physical and smart-world systems such as energy-based cyber-physical systems,
game theory has strong potential to provide solutions for pertinent problems [18,48,54–56].
For example, Saad et al. [55] presented an overview of applying game theory in three emerging
areas, including microgrid systems, demand-side management, and smart grid communications.
Furthermore, a growing number of research efforts have adopted game theory-based models to
addresssecurityissues. Forexample,Zhuetal. [54]proposedaniteratedzero-sumgametomodel
security policies at the cyber-level with corresponding optimal control response at the physical
layer. Maetal. [57] developed a zero-sum game with a mixed strategies model to formulate the
survivabilityforcyber-physicalsystems,inwhichtheadversaryanddefenderplayoverresources
beingdisruptedandmaintained/restored,respectively. Sieveletal. [58]formulatedtheplacementand
utilizationofunifiedpowerflowcontrollers(UPFCs)inapowertransmissionsystemasaniterative
game. Inresponsetotrippingtransmissionlinesfromtheadversary,thedefendercouldoptimizethe
installationlocationsoftheUPFCstomaximizetheamountofpowerdeliveredwhenthesystemis
underattack. Lawetal. [58]proposedagame-theoryformulationoftheriskdynamicsoffalsedata
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injectionattackstargetingautomaticgenerationcontrol,whichadoptsazero-sumMarkovsecurity
gamemodel. Inthismodel,riskstatesaredefinedasfunctionsoftheprobabilityofattackandthe
potential impact corresponding to the attack. Esmalifalak et al. [48] presented a zero-sum game
betweentheadversaryandthedefendertomodelthescenarioinwhichthepriceofelectricitycanbe
manipulatedbytheadversaryintheelectricitymarket. Abieetal. [18]describedarisk-basedadaptive
securityframeworkforIoTsineHealththatcouldbeusedtoestimateandpredictriskdamageand
futurebenefitsusinggametheoryandcontext-awarenesstechnology.
Distinct from existing research efforts, which have not taken into account cooperation and
competitionamongthemultipleadversaries,inthiswork,wefocusonthepayoffsthattheadversaries
canobtainintheircoalitionalattacksandpresenttheroleofthedefender. Viatheoreticalanalysis,
ourproposedgametheorymodelcanquantifythepayoffsofadversarieswithdifferentstrategiesunder
differentpenaltyfactors,whichcanbeimposedbythedefender.Thus,ourpaperestablishesaniterated
gametheory-basedgamethatdemonstratesthecooperationandcompetitionrelationshipsamong
adversaries,andprovidesaguideforselectingtheappropriatedefensivestrengthofthedefender.
3. Model
Inthissection,wefirstintroducetheiteratedgamemodel,andthenpresentthethreatmodel.
3.1. IteratedGameModel
Theiteratedgamemodelhasbeenwidelyusedinthegame-theorystudyandhasbeenapplied
indifferentfields[59]. Particularly,inaniteratedgame,theselfishbehaviorofparticipantscanlead
toalossforboththeiropponentsandthemselves. Thereareanumberofresearcheffortsfocusedon
theiteratedgame[25,38,60–65]. Theiteratedgameproblemhasbeenconsideredtohavenounilateral
ultimatesolutionastheresultsofthegamearejointlydeterminedbyallparticipants. Forinstance,
Pressetal.[60]proposedthezero-determinantstrategy,showingthat,inaniteratedgame,aparticipant
canunilaterallydeterminetheexpectedpayoffofhis/heropponentsbythepinningstrategy,orobtain
a higher payoff than his/her opponents by the extortion strategy. Furthermore, Pan et al. [63]
investigatedamulti-playeriteratedgamestrategy,whichextendsthezero-determinantstrategyto
solvetheIPGGproblem[66].
InaconventionalIPGGmodel,allparticipantshavetheirownendowmentatthebeginningof
eachroundofthegameplayed. Then,eachparticipantmustchooseeithertocooperatebycontributing
his endowment or to defect by standing aside. At the end of each round, the endowment will be
multipliedbyarateofgaintoobtaintherewardorpayoff,whichwillbeequallydistributedtoall
participants. Generally speaking, the strategies of participants often depend on the last move of
his/her opponents, which can be represented as the condition probability. The main issue is how
participantscooperatewitheachother,andavoidtheobviousNashequilibriumatzero[67].
3.2. ThreatModel
Inthesmart-worldsystemsuchassmartcitiesthatintegrateenergy,transportationandother
critical infrastructures in cities, the adversaries can obtain the economic benefits or achieve their
maliciousobjectivebylaunchingvariouscyber-attacks. Forexample,dataintegrityattacks[7,13,20]
could be used to disrupt the key functional modules in the power grid operation, including the
integrationofdistributedenergyresources,stateestimation,energypricing,andothers. Dataintegrity
attacks [4] can be launched to disrupt the efficiency of the smart transportation. Furthermore,
data integrity attacks can also be launched in the smart home automation system, so that the
adversarycanmakeunauthorizedaccesstosystemorevenperformsystemmanipulationanddata
leakage[68]. Generallyspeaking,adversariesneedtousetheirresourcestolaunchattackstoinfluence
theeffectivenessofthesmartIoTsystemandobtainsomegainfromtheattackslaunched. Forexample,
Farrajetal. [25]presentedananalysisofacyber-attackinwhichanadversarycanusethestorage
resourcestoaffecttherotationspeedofsynchronousgeneratorsinthepowergrid. Ronenetal. [11]
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alsopresentedfourtypesofattackingbehaviorandsomeofthemcanbringbenefitstoadversaries
fromtheattacks,suchasforminganumberofcompromisedIoTdevicesintoabotnetinordertosend
spamortominebitcoins.
Thegame-theorymodelthatconsidersmultipleadversariesandadefenderinthesmart-world
systemcanbeformalizedasanextendedIPGGmodel. Whentheadversariesattempttolaunchattacks,
participantswhochoosetocooperatewillcontributetheirownresources,andtakeariskbeingdetected
bythedefender, inordertoobtainthegainfromincreasingtheattackdamagetothesmart-world
system. Furthermore, when the launched attacks are detected by the defender, the participating
adversarieswillsufferapenalty. Theparticipantswhochoosetodefectcansharetheattackgainfrom
theimpairedsmart-worldsystem,butwillnottakeanycostorsufferanyrisk.
Theobjectiveoftheadversariesistomaximizetheirpayoffintheiteratedgame,whichissimilar
totheIPGGmodel. Thepenaltyfactorreflectstheintensityofthedefender,meaningthatalarger
penaltyfactorcorrespondstoastrongdefensivemechanismanditsdeployment,whichcommonly
incurs a higher cost. Notice that, in this paper, we consider a generic defensive strategy that can
captureasetofdefensiveschemestodetectthecoalitionattacktosomeextent,determinedbythe
probabilityofdetectionintroducedinthegamemodel. Inaddition,anotherobjectiveofthisstudyisto
investigatetherelationshipbetweentheeffectivenesstomitigateattacksandthecostassociatedwith
thedefense.
4. OurApproach
In this section, we introduce our proposed game-theory model to investigate the interaction
among multiple adversaries, and quantify the capacity of the defender. In the following, we first
introducethebasicidea,thenshowthetwokeycomponentsindetail,andfinallydiscussthescenario
withmultiplecollusiveparticipants.
4.1. BasicIdea
In the paper, we propose a game-theory model to deal with the coalitional attack that can be
launchedbymultipleadversariescooperativelyinthesmart-worldsystem. Basedontheextended
IPGGmodel,wedesignagame-theorymodelthatconsistsofmultipleadversariesandonedefender.
Inourmodel,weintroduceapenaltyfactorthatreferstothepenaltytoadversarieswhenthelaunched
attacksaredetectedbythedefender. Again,itisworthnotingthatweconsideragenericdefensive
strategy, which captures a set of defensive schemes to detect the coalition attack to some extent,
determinedbyaprobabilityofdetectionintroducedinthegamemodel. Thus,thepenaltyfactorcan
generallyreflectthecapacityofthedefender.
At the beginning of each round in the game played, some adversaries will contribute their
endowmenttolaunchacoalitionalattackwhiletheothersdonotjointhecoalitionalattack. Ifthe
coalitional attack is successful, the obtained attack gain will be distributed to all adversaries who
participate in the coalitional attack. In a similar way, only the involved adversaries will suffer
the penalty when the coalitional attack is detected. We assume that the probability of the attack
being detected will increase when the number of participating adversaries increases, which is
areasonableassumption.
Inaddition,weadoptthezero-determinantstrategytoderivetheexpectedpayoffofparticipants
andunderstandtherelationshipbetweentheexpectedpayoffandthepenaltyfactorenforcedbythe
defender. Bydoingthis,thedefendercanreducethemaximumexpectedpayoffofadversariessothat
thecoalitionalattackcanbedefeatedwhenanadequatepenaltyfactorisselected.
Our proposed game-theory model consists of the following two key components. First,
theextendedIPGGmodelisestablishedtomodelthepayoffoftheadversariesintheiteratedgame,
in which the defender can affect their payoff via the penalty factor. When the coalitional attack
is detected, the participating adversaries will pay for the penalty. Second, the expected payoff of
participantsisderivedbytheequalizerstrategy. Withtheequalizerstrategythatbelongstoonekind
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ofthezero-determinantstrategy,aparticipantcancontroltheexpectedpayoffofhis/heropponents.
Finally,wepresentthecasewherethemultiplecolludingparticipantsareinvolvedinthegame.Thekey
notationsusedinthispaperareshowninTable1.
Table1.Notation.
X ParticipantX
pX StrategyofparticipantX
pX ProbabilityforparticipantXtocooperateundertheithoutcomeinthelastround
i
N Numberofalltheparticipantsintheiteratedgame
uX PayoffvectorobtainedbyparticipantX
r Rateofgainfromthecoalitionalattack
Probabilityforparticipant1tocooperateinthecurrentroundifhe/shechoosescooperation
p1
C,n (C)andhis/hernopponentschoosecooperationinthelastround
Probabilityforparticipant1tocooperateinthecurrentroundifhe/shechoosesdefection(D)
p1
D,n andhis/hernopponentschoosecooperationinthelastround
ps Probabilitythatasingleadversaryattemptstolaunchanattackwithoutbeingdetected
α ,α Coefficientsforlinearcombinationinzero-determinantstrategy
0 X
β Penaltyfactorwhentheattackisdetected
γ Parameterthatcontrolsthetotalpayofffortheopponents
µ,ξ Coefficientssatisfyingthelinearrelationshipintheequalizerstrategy
EX ExpectedpayoffobtainedbytheopponentsofparticipantX
L Numberofthecolludingparticipantsinthecollusivestrategy
χ Extortionatefactorintheextortionstrategy
Φ Freeparameterintheextortionstrategy
4.2. AnExtendedIPGGModel
Inthispaper,weconsideranextendedIPGGmodelforNparticipants,inwhicheachparticipant
obtains an initial endowment c = 1 at the beginning of each round [69,70]. Each participant has
twochoices: (i)Cooperation(C);or(ii)Defection(D).Here,Cooperationreferstothechoicethatthe
participantchoosestocooperateandcontributehis/herownendowmentintothecoalitionalattack,
whileDefectionreferstothechoicethattheparticipantwillkeephis/herownendowment,anddoes
notparticipateinthecoalitionalattack. Attheendofeachroundinthegameplayed,ifthecoalitional
attackissuccessful,theendowmentwillbemultipliedbyarateofattackgainrandtheobtainedgain
willbedistributedtoallNparticipants.Ifthecoalitionalattackisdetected,theparticipatingadversaries
willsufferapenalty,whichisrepresentedasthepenaltyfactorβenforcedbythedefender. Wedenote
thesuccessfulprobabilitythatasingleadversarylaunchesanattackas p ,andtheprobabilitythatthe
s
coalitionalattackisdetectedas1−pn,wherenisthenumberofparticipatingadversarieswhochoose
s
tocooperateintheattack.
For an arbitrary participant, he/she will first obtain the positive gain, which represents the
gain from launching the attack, similar to the conventional IPGG model. This positive gain is the
benefitfromtheattackbehavior, includingtheillegallygainedfinancialincome, physicaldamage
ofthetargeteddevices,etc. Nonetheless,iftheattackisdetectedbythedefender,theparticipating
adversarieswillbepenalized,withthedetectedprobabilitybeing1−pn.Thus,inthispaper,weextend
s
theabovegamemodelbyaddingthenegativepayoff,whichrepresentsthepotentialpenaltyincurred
whentheattackisdetectedbythedefender. Thisnegativepayoffcanincludeafine,thelimitationof
furtherparticipationorotherbehavior,etc.
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Then,wehaveuX = r(n+hX) +(1−hX),uX = −β(1−pn+1)·hX,whereuX isthepositive
pos N neg s pos
payoff,uX isthenegativepayoff,nisthenumberofcooperatorsamongthetotalN−1opponentsof
neg
participantXinthecurrentroundofthegameplayed. IfaparticipantXchoosestocooperate,wehave
hX = 1. Otherwise,wehavehX = 0. Inaddition,ristherateofattackgain, βisthepenaltyfactor
whenthecoalitionalattackisdetected, and p ∈ [0,1] referstotheprobabilitythataparticipating
s
adversarylaunchesthesuccessfulattackwithoutbeingdetected.
Then,thetotalpayoffofparticipantXcanberepresentedas
uX = uX +uX
pos neg
r(n+hX) (1)
= +(1−hX)−β(1−pn+1)×hX.
N s
Thus, theconventionalIPGGmodelisaspecialcaseinEquation(1)when β = 0. Noticethat
theobjectiveoftheadversaryXistomaximizehis/herpayoffuX viausingvariousavailableattack
strategies. Next,wepresentthepayoffoftheequalizerstrategiesindetail.
4.3. ExpectedPayoffofEqualizerStrategy
Thezero-determinantstrategywasproposedbyPressandDyson[60]. Inthisstrategy,wecan
makeaparticipantunilaterallysetthepayoffofhis/heropponenttoafixedvalueintheprisoner’s
dilemma.Toachievethecompetitivebenefit,theparticipantsmayintendtoadoptthezero-determinant
strategies. Pan et al. [63] extended it to the multi-player IPGG problem and demonstrated that,
inaninfiniterepeatedgame,thelong-memoryplayerhasnoadvantagesovershort-memoryplayers
(i.e., the length of memory does not affect the results). Thus, we can assume that the choices of
participantsinthecurrentroundonlydependontheoutcomesinthelastround. Becausethereare2N
possibleoutcomesineachround,thestrategyofparticipantXcanbedenotedbya2N-dimensionvector,
(cid:104) (cid:105)
pX = pX,··· ,pX,··· ,pX , (2)
1 i 2N
whereiisthesequencenumberofallpossibleoutcomes,and pX istheconditionalprobabilitythat
i
participantXchoosestocooperateundertheithoutcomeinthelastround.
In the multi-player repeated game process, a participant does not need to know the accurate
choicesofhis/heropponentsineachround. Thismeansthatitissufficientforaparticipanttoknow
howmanyofhis/heropponentschoosetocooperate,whichisdenotedasn. Iftheparticipant’slast
moveisC(cooperation)orD(defection),theprobabilitythattheparticipantchoosestocooperatein
thecurrentroundis p or p ,respectively. AsshowninFigure1,theprobability p and p are
C,n D,n C,n D,n
thekeyparametersintheiteratedgame. Tosimplifytheproblem,weignorethespecificchoicesof
theopponentsineachroundandjustfocusonthenumberofcooperatorsamongtheopponentsof
participant X. Bydoingso,weonlyneedtoanalyze2N outcomes,insteadof2N outcomes. Inthe
iteratedgameprocess,theprobabilitiesreflectthelikelihoodthatparticipantXandhis/heropponents
willchooseCooperationbasedonthelastmoveoutcome. Obviously,theprobabilitythatparticipant
X and his/her opponents will choose Defection are 1− p or 1− p , depending on their last
C,n D,n
moveoutcome.
Asdescribedin[63],along-memoryplayercanbeconsideredasamemory-oneplayer. Then,
the game can be characterized by a Markov Chain with a state transition matrix M. Denoting the
stationary vector of M as vT, we have vT ·M = vT. For this Markov model, Pan et al. [63] have
demonstratedthat,forparticipant1,thereexistsaspecialcolumninthedeterminantvT·u1,which
canbedeterminedbyonlytheparticipant’sstrategyp1(Noticethat,astheparticipantsaresymmetric,
Sensors2018,18,674 9of28
we use participant 1 as an example for the analysis). We denote this special column as p˜1. If the
participantscanproperlysetp1,wehave
N
p˜1 = ∑ α uX+α 1, (3)
X 0
X=1
whereuX = [uX,··· ,uX,··· ,uX ]isthepayoffvector,anduX isthepayoffofparticipant X inthe
1 i 2N i
ithoutcome.
Number of cooperators
among the opponents !)#&'(
!)#%
!)#$
!"#&'(
N-1 n 0 !"#% N-1 n 0
Choice of C C C !"#$ C C C
Participant X
N-1 n 0 N-1 n 0
D D D D D D
Last move outcome This move
Figure1.Theiteratedprocessinginthegamemodel.
Then,theexpectedpayoffofallparticipantssatisfiesthelinearrelationship,andwehave
N
∑ α EX+α =0. (4)
X 0
X=1
Here,EX denotestheexpectedpayoffforparticipantXandα ,α ,··· ,α arethecoefficientsfor
0 1 X
linearcombination.Then,participant1’sstrategyp1,whichleadstothelinearrelationshipEquation(4),
isdenotedastheequalizerstrategyofmultipleparticipants.
Tosimplifytheproblem,weassumethataparticipantwiththeequalizerstrategywillattemptto
controltheaveragepayoffofhis/heropponents,whichreferstotheequalizerstrategy.Forparticipant1,
he/shecanchoosetheproperstrategyp1suchthat
N
p˜1 = µ ∑ uX+ξ1. (5)
X=2
Here,theparticipantonlyneedstosetα1 =0andαX(cid:54)=1 = µ. Noticethatp˜1isthespecialcolumn,
whichcanonlybedeterminedbyparticipant1’sstrategyp1.
Withtheabovestrategyp˜1andEquation(4),thelinearrelationshipbetweentheexpectedpayoff
ofalltheopponentscanbeestablishedbytheparticipant1asfollows:
N
µ ∑ EX+ξ =0. (6)
X=2
Without loss of generality, we omit the sequence number of the participant 1 to simplify the
expression. Equation(5)isequivalenttoaseriesof2N linearequations. Then,wehave
µ (cid:104) (cid:105) µ
p =1+ rN−r−N−β(1−pn+1) n+ [(r+N)(N−1)]+ξ, (7)
C,n N s N
µ (cid:104) (cid:105) µ
p = rN−r−N−β(1−pn+1) n+ [N(N−1)]+ξ, (8)
D,n N s N
wheren =0,1,...,N−1.
Sensors2018,18,674 10of28
The above 2N probabilities pC,n and pD,n can be represented by pC,N−1 and pD,0, which are
reflectedtobetheprobabilitiesformutualcooperationandmutualdefection,respectively.Accordingto
Equations(7)and(8),wehave
µ (cid:104) (cid:105)
pC,N−1 =1+ N rN−β(1−psN) (N−1)+ξ, (9)
p = µ(N−1)+ξ. (10)
D,0
Theparametersµandξ mustsatisfytheprobabilityconstraints0≤ pC,N−1 ≤1and0≤ pD,0 ≤1.
Thus,therangeofµandξ canbeobtained. Denoteµandξ asfollows:
µ = − (1−pC,N−1+pD,0)N , (11)
[(r−1)N−β(1−pN)](N−1)
s
ξ = (1−pC,N−1+rpD,0)N−β(1−psN)pD,0. (12)
(r−1)N−β(1−pN)
s
Wecanseethatthesignofµdependson(r−1)N−β(1−pN). Withrespecttoµ,wewillconduct
s
furtheranalysisinSection5.
Finally,substitutingEquations(11)and(12)intoEquation(6),theparticipantcansettheexpected
payoffofhis/heropponentstoafixedvalue. Then,wehave
∑N EX = −ξ = (N−1)+ (N−1)[(r−1)N−β(1−psN)], (13)
µ N(1+γ)
X=2
whereγ = 1−ppDC,,N0−1 denotesthelinearrelationship pC,N−1+γpD,0−1=0between pC,N−1and pD,0.
Tosummarize,wecanseethatthetotalexpectedpayoffoftheopponentsdependsonthenumber
ofplayersN,therateofattackgainr,andtheparameterγ. Then,participant1cansettheexpected
payoffofhis/heropponentsbysettingtheparameterγtovariousvalues.
4.4. CollusiveStrategy
As mentioned above, we have already studied the game scenario with multiple adversaries,
inwhichonlyoneparticipantattemptstocontrolthepayoffofhis/heropponents. Nonetheless,itis
possiblethatmorethanoneadversarycooperatescollusivelytocontrolthepayoffoftheiropponents,
whichisdenotedascollusivestrategy. Thiscollusivestrategyisdifferentfromtheequalizerstrategy
mentionedinSection4.3. Nonetheless,itwillachieveasimilarperformancebecausebothstrategies
havethesameobjectives(i.e.,controllingthepayoffoftheiropponents).
InSection4.3,itisshownthat,inthedeterminantvT·u1,therealsoexistsomecolumns,whichcan
bedeterminedbymultipleparticipants’strategies. Thus,someparticipantscancollusivelychoose
theproperstrategies,andenforcealinearrelationshipbetweentheirownexpectedpayoffandtheir
opponents’,whichissimilartoEquation(3),asfollows:
N
p˜(cid:48) = ∑ α uX+α 1, (14)
X 0
X=1
wherep˜(cid:48) isthespecialcolumninthedeterminantvT·u1,uX = [uX,··· ,uX,··· ,uX ]isthepayoff
1 i 2N
vector,uX isthepayoffofparticipantXintheithoutcome,andα ,α ,··· ,α arethecoefficientsfor
i 0 1 X
linearcombination.
Toextendourmodeltoageneralcase,weassumethatLadversariescolludetogetherandattempt
tosetthepayoffoftheirN−Lopponentstoafixedvalue. Wecanseetheobjectiveofthiscollusive
strategyissimilartotheequalizerstrategyinSection4.3. Noticethatthisstrategyonlyexistswhenthe
collusivegroupsizeL = N−1[63].