Power Allocation in Team Jamming Games in Wireless Ad Hoc Networks

2.1 MB·English

by  Sourabh Bhattacharya

Power Allocation in Team Jamming Games in Wireless Ad Hoc
Networks
Sourabh Bhattacharya Ali Khanafer Tamer Bas¸ar
[email protected] [email protected] [email protected] ∗
1
1 ABSTRACT ation breaks down, leading to more complex songs. Specifically,
0 ithasbeenreportedthatfemalesrespondtounpairedsexualrivals
2 Inthiswork, westudytheproblem ofpower allocationinteams. byjammingthesignalsoftheirownmates,whointurnadjusttheir
Each team consists of two agents who try to split their available signalstoavoidtheinterference[27].
n powerbetweenthetasksofcommunicationandjammingthenodes
a oftheotherteam.Theagentshaveconstraintsontheirtotalenergy Manydefencestrategieshavebeenproposedbyresearchersagainst
J
andinstantaneouspowerusage. Thecostfunctionisthedifference jamminginwirelessnetworks.Abriefsurveyofvarioustechniques
1 betweentheratesoferroneouslytransmittedbitsofeachteam.We injammingrelevanttoourresearchisprovidedin[5]. Inthepast,
3 model the problem as a zero-sum differential game between the networkswithmultipleattackershavealsobeenconsideredinthe
twoteamsanduseIsaacs’approachtoobtainthenecessarycondi- literature. In[15,16],theauthorsconsidertheinteractionbetween
T] tionsfortheoptimaltrajectories.Thisleadstoacontinuous-kernel asource-destination pair, an eavesdropper, and friendly jammers.
G power allocationgameamongtheplayers. Basedonthecommu- Thesource can buy "jamming power" fromthe friendlyjammers
nications model, we present sufficient conditions on the physical whichdisguisetheeavesdropper.Thisallowsthesourcetoachieve
s. parametersoftheagentsfortheexistenceofapurestrategyNash increasedsecrecyrate. Theauthorsstudytheprobleminthecon-
c equilibrium(PSNE).Finally,wepresentsimulationresultsforthe textofaStackelberggameandshowthatatrade-offexistsbetween
[ casewhentheagentsareholonomic. thepriceannouncedbythejammersandtheresultingperformance.

Asimilarproblemwastackledin[9]whererelaynodes canhelp
1
v 1. INTRODUCTION thesourceinthepresenceofmultipleeavesdroppers. Theauthors
0 proposedifferentrelayingschemesandstudytwodesignproblems:
3 minimizingthetransmissionpowersubjecttoaminimumsecrecy
Thedecentralizednatureofwirelessadhocnetworksmakesthem
0 rateandmaximizingthesecrecyratesubjecttoatotalpowercon-
vulnerabletosecuritythreats.Aprominentexampleofsuchthreats
6 straint.Analyticalanalysesshowthatrelayingyieldsimprovedper-
is jamming: a malicious attack whose objective is to disrupt the
. formancewhencomparedtodirecttransmissioninmaliciousenvi-
1 communicationofthevictimnetworkintentionally,causinginter-
ronments. Differentfromtheaforementionedreferences,ourwork
0 ferenceorcollisionatthereceiverside. Jammingattackisawell-
hereconsidersnon-friendlyjammingteams,i.e.,thesecuritybot-
1 studiedandactiveareaofresearchinwirelessnetworks. Unautho-
tleneckconsideredhereisjammingandnoteavesdropping. More-
1 rizedintrusionofsuchkindhasstartedaracebetweentheengineers
over, to the best of our knowledge, pursuit-evasion strategies for
v: andthehackers;therefore,wehavebeenwitnessingasurgeofnew jammingteamswerenotstudiedbefore.
i smartsystemsaimingtosecuremoderninstrumentationandsoft-
X warefromunwantedexogenousattacks.
Adhocnetworksconsistofmobileenergy-constrainednodes.Mo-
r bility affects all layers in a network protocol stack including the
a Theproblemunderconsiderationinthispaperisinspiredbyrecent
physical layer as channels become time-varying [13]. Moreover,
discoveries of jamming instances in biological species. In a se-
nodessuchassensorsdeployed inafieldormilitaryvehiclespa-
riesofplaybackexperiments,researchershavefoundthatresident
trolling in remote sitesare often equipped withnon-rechargeable
pairs of Peruvian warbling antbirds sing coordinated duets when
batteries. Powercontrolplays,therefore, acrucialroleindesign-
responding torivalpairs. Butunder othercircumstances, cooper-
ingrobustcommunicationssystems. Atthephysicallayer,power
∗AllthreeauthorsarewiththeCoordinatedScienceLab,Univer- control can be used to maximize rate or minimize the transmis-
sityofIllinoisatUrbana-Champaign,Urbana,IL61801.Thiswork sion error probability, see [18, 3] and the references therein. In
wassupportedinpartbygrantsfromAROMURIandAFOSR. addition,inmulti-usernetworks,powercontrolcanbeusedtoreg-
ulatetheinterferencelevelattheterminalsofotherusers[24,28,
10]. Duetothelackofacentralizedinfrastructureinadhocnet-
works, distributedsolutionsareessential. Inthiswork, similarto
[3,24,28],wemodelthepowerallocationproblemasanoncoop-
erativegame,thusallowingustodeviseanon-centralizedsolution.
Asadeparturefrompreviousresearch,howeverthepowercontrol
mechanismweproposesplitsthepowerbudgetofeachplayerinto
twoportions: aportionusedtocommunicatewithteam-matesand
aportionusedtojamtheplayersoftheotherteam. Moreimpor-
tantly,theobjectivefunctionischosentobethedifferencebetween 1a,2a andTeamBiscomprisedofthetwoplayers 1b,2b .The
{ } { }
the cumulative bit error rate (BER) of each team; this allows for agentsmoveonaplaneandtherefore,havetwodegreesoffreedom
increasedfreedominchoosingphysicaldesignparameters,besides (x,y). The dynamics of the players are given by the following
thepowerlevel,suchasthesizeofmodulationschemes. equations:
Inthepastyear, therehavebeenreportsofpredator dronesbeing
hacked[14,20],resultinginintrudersgainingaccesstoclassified TeamA:
•
data being transmittedfrom an aircraft. Motivated fromsuch in-
actitdaecnktss,awmeonhgavaeutaonnaolmyzoeudsvaagreinotuss. sIcnetnhaericoasseofofevaasdiinngglejajmammminegr xy˙˙iaia==ffxyaaii((xxaiai,,uuaiai,,tt)) (cid:27)i∈{1,2} (1)
tryingtointrudethecommunicationlinkbetweenatransmitterand
areceiver, theproblemcanbeformulatedasmultiplayer(specifi- TeamB:
cally,threeplayer)pursuit-evasiongame[17,2]. In[5],weinves- •
taiugtaotneodmthoeupsrovbehleimcleosf(fiUnAdVinsg)mtooteivoandsetrajatemgmieisnfgorintwtoheunpmreasnennecde xy˙˙ibbi ==ffyxbbii((xxbibi,,uubibi,,tt)) (cid:27)i∈{1,2} (2)
of anaerialintruder. Weconsidered adifferential gametheoretic
approach to compute optimal strategies by a team of UAVs. We
formulatedtheproblemasazero-sumpursuit-evasiongame. The Intheaboveequations, xi andui denotevectorsrepresentingthe
costfunctionwaspickedastheterminationtimeofthegame. We state and control input of agent i, with the superscript (a or b)
usedIsaacs’approachtoderivethenecessaryconditionstoarriveat identifying thecorresponding team. Thestatespaceof theentire
theequationsgoverningthesaddle-point strategiesoftheplayers. system is represented by X R2 R2 R2 R2. Moreover,
In [7], we extended the previous analysis to a team of heteroge- ui i φ : [0,t] ≃ i ×φ()×isme×asurable , where
neousvehiclescontainingUAVsandautonomousgroundvehicles i ∈RUpi.≃f :{R2 i R→ ARis|unifo·rmlycontinuous,}bounded
(AGVs). In[4],weanalyzedtheproblemofmultiplejammersin- Aand⊂Lipschitzcont×inAuo×us in→xi forfixedui. Consequently, given
truding the communication network in a formation of UAVs. In a fixed ui() and an initial point, there exists a unique trajectory
·
[6],weanalyzedtheproblemofconnectivitymaintenanceinmulti- solving(1)and(2)[1].
agent systems inthepresence of ajammer. Inthiscurrent work,
we study a scenario where a team of malicious nodes launch a Now,wedescribethephysicallayercommunicationsmodelinthe
jammingattackonanother team,whichiscapableofjammingas presenceofajammerwhichismotivatedby[26]. Foreachtrans-
well. Weagainusedifferential gametheory tostudy thepursuit- mitterandreceiverpair,weassumethefollowingcommunications
evasionstrategiesoftheteams,whichconstituteofunmannedde- model. Giventhatthetransmitterandthereceiverareseparatedby
cisionmakers(UDMs). adistanced,andthetransmittertransmitswithconstantpowerPT,
thereceivedsignalpowerPRisgivenby
Themaincontributionsofthispaperareasfollows. Tothebestof
ourknowledge,thisisthefirstworkthatconsiderstheproblemof PR =ρPTd−α, (3)
twoteamsofmobileautonomousagentsjammingeachother. Our
whereρdependsontheantennas’gainsand,accordingtothefree
analysis takes intoconsideration constraints inenergy and power
spacepathlossmodel,isgivenby:
among the agents. Moreover, we relate the problem of optimal
pnoicwateironalmloocadteilobneftowreceonmthmeuangiecnattsio.nFiannadllyj,amwempinrogvtiodethaescuoffimcmieun-t ρ= GTGRλ2,
(4π)2
conditionforexistenceofanoptimaldecisionstrategyamongthe
agentsbasedonthephysicalparametersoftheproblems. whereλisthesignal’swavelengthandGT,GRarethetransmitand
receiveantennas’gains,respectively,inthelineofsightdirection.
The rest of the paper is organized as follows. We formulate the Inrealscenarios,ρisverysmallinmagnitude. Forexample,using
probleminSection2andexplaintheunderlyingnotation. InSec- nondirectionalantennasandtransmittingat900MHz,wehaveρ=
tion3,weintroduceandsolveanassociatedoptimalcontrolprob- 1·1·0.33 =6.896 10−4.
lem. The Nash equilibrium properties of the team power con- (4π)2 ×
trolproblemarestudiedinSection4,andthespecificexampleof
Thesignal-to-interferenceratio(SINR)sisgivenby
systemsemployinguncodedM-quadratureamplitudemodulations
(QAM) follows in Section5. Simulationresults are presented in s= PR , (4)
Section6. Weconcludethepaperandprovidefuturedirectionsin I+σ
Section7.AnAppendixattheendincludesexplicitexpressionsfor
someofthevariablesintroducedinthepaper. where σ istheambient noiselevel. TheBitError Rate(BER)is
givenbythefollowingexpression:
2. PROBLEMFORMULATION p(t)=g(s), (5)
whereg()isadecreasingfunctionofs. Explicitexpressions for
·
Considertwoteamsofmobileagents. Eachagentiscommunicat- g()areprovidedinSectionVwhereweconsidertheexampleof
·
ingwithmembersoftheteamitbelongstoandatthesametime, M-QAM. Each player uses its power for the following purposes:
jammingthecommunicationbetweenmembersoftheotherteam. (1)Communicatingwiththeteam-mate,and(2)Jammingthecom-
Weconsiderascenariowhereeachteamhastwomembers,though munication of the other team. Weassume that fa and fb arethe
at a conceptual level our development applies to higher number frequencies at which Team A and Team B communicate, respec-
ofteammembersaswell. TeamAiscomprisedofthetwoplayers tively,andfa =fb.
6
Foraninitialpositionx X,theoutcomeofthegameπ,isgiven
0
∈ Table1:Decisionvariablesanddistancesamongagents.
bythefollowingexpression:
T 1b 2b 1a 2a
π(x ,ua,ua,ub,ub)= [pa(t)+pa(t) pb(t) pb(t)]dt,
0 1 2 1 2 1 2 − 1 − 2
Z0
L
1a γ1,d1 γ1,d1 γ12,d12
wherepa(t)andpb(t)arethe|BERsforagen{tziinteamAand}team 1 1 2 2
i i
B,respectively, andT isthetimeof terminationof thegame. pi
depends in si, i.e, the SINR perceived by agent i. From (3), si 2a γ2,d2 γ2,d2 γ21,d21
depends onthemutual distancesbetween theplayers. Therefore, 1 1 2 2
wecanseethattheoutcomefunctional,π,dependsonthestateof
theplayersandhence,theircontrolinputs.Theoutcomefunctional
1b δ ,d δ1,d1 δ2,d2
modelsthedifferenceintheerroneouscommunicationpacketsex- 12 12 1 1 1 1
changedbetweenthemembersofthesameteamduringtheentire
courseofthegame. TheobjectiveofteamAistominimizeπand
2b δ ,d δ1,d1 δ2,d2
theobjectiveofteamBistomaximizeit. 21 21 2 2 2 2
Let Pa(t) and Pb(t) denote the instantaneous power levels for
i i
communication usedbyplayer iinTeamAand TeamB,respec-
A
tively. Since the agents are mobile, there are limitations on the
amountofenergyavailabletoeachagentthatisdictatedbytheca- 1 δ11Pb B
pacityof thepower sourcecarriedbyeach agent. Wemodel this 1
restrictionasthefollowingintegralconstraintforeachagent γ1 a 1
1P
T 1
The game is said toZte0rmPinia(tte)dwth≤enEa.ny one agent runs out(6o)f γ21P2a γ12P1a γ21Paδ21P2b δ21P1γb12Pa
power,thatis(6)isviolated. 1 2
mInaaxdimdiutimonptoowtehreleevneelrgoyfcthoensdtreaviinctess, tthhearteaarereulsiemditoantiboonasrdoneatchhe δ21P2b δ12P1b
astgreaninttfoisrmthoedpeulerdpobsyetohfecfoomllomwuinnigcasteiotno.fFinoerqeuaaclhitipelsa:yer,thiscon- 2 γ22P2a
0≤Pia(t),Pib(t)≤Pmax. (7) δ22P2b
At every instant, each agent has to decide on the fraction of the
2
powerthatneedstobeallocatedforcommunicationandjamming.
Table 1 provides a list of decision variables for the players that
Figure1: Powerallocationamongtheagentsforcommunica-
models this allocation. Each decision variable is a non-negative
tionaswellasjamming.
real number and liesintheinterval [0,1]. Thedecisionvariables
belonging to each row add up to one. The fraction of the total
power allocated by the player in row i to theplayer incolumn j
Inthenextsection,weanalyzetheproblemofcomputingtheopti-
is given by thefirst entry in the cell (i,j). This allocated power
malcontrolsforeachagent.
isusedforjammingiftheplayerincolumnj belongstotheother
team; otherwise, it is used to communicate with the agent in the
3. OPTIMALCONTROLPROBLEM
sameteam. Similarly,thedistancebetweentheagentinrowiand
the agent in column j is given by the second entry in cell (i,j).
Sincedistanceisasymmetricquantity,dij =djiandd =d .
ij ji From the problem formulation presented in the previous section,
wecanconcludethattheobjectivefunctionsofthetwoteamsare
Figure1summarizesthepowerallocationbetweenthemembersof inconflict. Thetuple(ua∗,ua∗,ub∗,ub∗)issaidtobeoptimal(or,
1 2 1 2
thesameteamaswellasbetweenthemembersofdifferentteams.
inpair-wisesaddle-pointequilibrium)fortheplayersifitsatisfies
thefollowingconditions:
Intheabovegame,eachagenthastocomputethefollowingvari-
ablesateveryinstant: π[x0,ua1∗,ua2∗,ub1,ub2]≤π[x0,ua1∗,ua2∗,ub1∗,ub2∗] (8)
π[x ,ua∗,ua∗,ub∗,ub∗] π[x ,ua,ua,ub∗,ub∗] (9)
0 1 2 1 2 ≤ 0 1 2 1 2
1. Theinstantaneouscontrol,ui(t).
In simple terms, the above equations imply that agents in Team
2. Theinstantaneouspowerlevel,Pi(t). Aaresolvingajointoptimizationproblemofminimizingtheout-
come. Similarly,agentsinTeamBaresolvingajointoptimization
3. Allthedecisionvariablespresent intherowcorresponding problemofmaximizingtheoutcome. Moreover,thetwoteamsare
totheagentinTable1. playingazero-sumgameagainstoneanother.Inthiscase,thevalue
ofthegame,denotedbythefunctionJ : X R,canbedefined Additionally, thegradientofthevaluefunctionsatisfiestheretro-
→
asfollows: gressive path equations (RPE)given bythefollowing partial dif-
J(x)=π[x ,ua∗,ua∗,ub∗,ub∗] (10) ferentialequation:
0 1 2 1 2
∂ J ∂H
The value of the game is unique at a point X in the state-space. ∇ = ,
∂τ ∂x
An important property satisfied by the value of the game is the
Nash equilibrium property. The tuple(ua∗,ua∗,ub∗,ub∗) issaid whereτ istheretrogradetimeortimeleftfortermination.
1 2 1 2
tobeinNashequilibriumifnounilateraldeviationinstrategyby
aplayercanleadtoabetteroutcomeforthatplayer. Hence,there TheRPEleadstothefollowingequationsfortheplayers.
isno motivation for the players todeviate fromtheir equilibrium
strategies. In terms of the outcome of the game, the strategies Team A
(ua1∗,ua2∗,ub1∗,ub2∗)areinNashequilibrium(forthe4-playergame) ∂fa(x) sag′(sa)(xa xa)
iftheysatisfythefollowingproperty: J˚xai =∇J· ∂xxia +α k (kd12)j2− i +
πππ[x[[0xx,00u,,a1uu∗a1a1,∗∗u,,a2uu∗a2a2,∗∗u,,b1uu∗b1b1,∗,u,ub2ub2∗∗b2]]]≤(cid:27)(cid:26)≤πππ[[[xxx000,,,uuua1a1a1∗,∗,u,ua2ua2∗a2,,∗uu,b1b1u∗∗b1,,∗uu,b2b2u∗∗b2]]∗] (11) J˚yia =∇J· ∂f∂yiyi(axs)bk−g′(αsbkkX=)(Pd1,ikm2)a(cid:2)−xs(γαakki+g(′2x()sai(akd−)1(2yx)jabk2)−(cid:3)yia) +
i kX=1,2(cid:2)
Ithnagteanreerainl,Nthaesrheemqauyilibberimumul.tipAlesssuemtsinogfsthtreateexgiisetsenfcoertohfeaplvaaylueres, sbkg′(sbk()dPim)a−x(γαk+i(2y)ia−ykb)
asdefinedin(10),andtheexistenceofauniqueNashequilibrium, k
(cid:3)
wecanconcludethattheNashequilibriumconcept ofperson-by-
personoptimalitygivenin(11)isanecessaryconditiontobesatis- Team B
fiedforthevalueofthegame. Further,obtainingthesetofstrate-
giesthatareinNashequilibriumyieldstheoptimalstrategiesfor J˚ = J ∂fxbi(x) α sbkg′(sbk)(xbj−xbi) +
tthioenepdlacyoenrsd.itiIonnsthienfoorldloewrtiongcoamnapluytseist,hewoepatismsuamlsetrtahteegaiefos.remen- xbi ∇ · ∂xbi − kX=1,2(cid:2) (d12)2
sakg′(sak)Pmaxδik(xbi −xak)
The Hamiltonian of the system isgiven by the following expres- (dk)−(α+2)
i
sion: ∂fi(x) sbg′(sb)(yb (cid:3)yb)
J˚ = J y +α k k j − i +
H == pLa1+(t)∇+Jp·a2f((tx))−pb1(t)−pb2(t)+∇J·f(x) (12) yib ∇ · ∂yibsakg′(sakkX=)P1,m2a(cid:2)xδik(yib(d−12y)ka2)
(dk)−(α+2)
i
Inordertocomputetheoptimalcontroloftheplayers,wewilluse (cid:3)
Here,(˚.)denotesderivativewithrespecttoretrogradetime. Since
theIsaacs’conditions[17]whicharethefollowing:
terminationisonlyafunctionofthepowerofeachplayer,J isin-
dependentofthepositionoftheplayersontheterminalmanifold.
1. Therefore, J = 0attermination. Thisformstheboundarycon-
HH[[xx00,,uua1a1∗∗,,uua2a2∗∗,,uub1b1∗,,uub2∗b2]] (cid:27)≤H[x0,ua1∗,ua2∗,ub1∗,ub2∗] dInititohnenfoerxtth∇seecRtiPoEn.,weaddresstheproblemofpowerallocation.
H[x0,ua1∗,ua2∗,ub1∗,ub2∗]≤(cid:26) HH[[xx00,,uua1a1∗,,uua2a2∗,,uub1b1∗∗,,uub2b2∗∗]] 4. POWER ALLOCATION
2. H[x ,ua∗,ua∗,ub∗,ub∗]=0
0 1 2 1 2 From(4),theSINRreceivedbyeachagentintermsofthepower
levelsoftheotheragentsaswellastheirmutualdistancesisgiven
TheagentsinTeamAwanttominimizetheHamiltonianatevery
bythefollowingexpressions
instant, and the agents in Team B want to maximize it. The dy-
namicsoftheagentsaredecoupled. Therefore,theHamiltonianis sa= P2a(t)γ21(d12)−α
separableinitscontrolsand,hence,theorderoftakingtheextrema 1 σ +Pb(t)δ1(d1)−α+Pb(t)δ1(d1)−α
ρ 1 1 1 2 2 2
becomesinconsequential. Asaconsequence, theoptimalcontrols Pa(t)γ12(d12)−α
of theplayers aregivenbythefollowingexpression fromIsaacs’ sa= 1
2 σ +Pb(t)δ2(d2)−α+Pb(t)δ2(d2)−α
firstcondition. ρ 1 1 1 2 2 2
(ua1∗,ua2∗,ub1∗,ub2∗)=argmub1,auxb2uma1,iuna2H sb1 = σρ +P1a(t)γP112b((dt11))δ−21α(d+12P)2−a(αt)γ12(d21)−α
Pb(t)δ (d )−α
Theoptimalcontroluiisobtainedbythefollowingexpression: sb2 = σ +Pa(t)γ11(d1)−12α+12Pa(t)γ2(d2)−α (13)
ρ 1 2 2 2 2 2
uuaibi∗∗==mmianxuuaibi∂∂∂∂xJxJaibi ··ffiiab((xxaibi,,uubiai,,tt)) ) i=1,2 Fcartoimontahmeoenxgprtehsesiaognenints(o1n2l)y,awfefeccatsnthcoentecrlumdeLtihnatthteheHpamowiletornailalon-.
Therefore,Isaacs’firstconditionleadstothefollowingpowerallo- Thenextcorollarythenfollowsregardingthetimehorizonofthe
cationproblemamongtheagents. differentialgame.
4.1 Team A
COROLLARY 1. TheentiregameterminatesinafixedtimeT =
TheobjectiveofeachagentistominimizeL. E irrespectiveoftheinitialpositionoftheagents.
Pmax
1. Player1: PROOF. From Theorem 1 and (6), we conclude that the T =
P1a,γm11,iγn21,γ12L⇒P1a,γm11,iγn21,γ12(pa2−pb1−pb2) (14) tPhmeEasxamfoertailmlpe,laiy.ee.r,sT..ATllhtehreefpolraey,etrhsecgoanmsuemeendthseaitrteinmtiereTe.nergyat
La1
subjectto: | {z } Now weconsider theproblemof computing theoptimal valueof
Pa(t) P the decision variables for the players. In order to do so, we use
1 ≤ max preexistingresultsfromcontinuouskernelgamesthatarepresented
γ1+γ1+γ12 =1, γ1,γ1,γ12 0 γ ∆3 hereinTheorem2andTheorem3andarestatedwithoutproof.
1 2 1 2 ≥ ⇒ ∈
2. Player2:
P2a,γm12,iγn22,γ21L⇒P2a,γm12,iγn22,γ21(pa1−Lpa2b1−pb2) (15) fifunniTtcetH-iodEniOmaRlesEnJMsiion(2ia.l∈[a2Nc]t)iAoannrNesp-cpaoecnretssionUnuoinuo(sinoz∈enrNUo-)1sua×mre·g·ca·om×mepUainNctwaahdnimcdhicttoshsaet
mixedstrategyNashequilibrium(MSNE).
subjectto: | {z }
Pa(t) P
2 ≤ max Fromtheabovetheorem, wecanconclude that thepower alloca-
γ2+γ2+γ21 =1, γ1,γ1,γ21 0 γ ∆3 tiongamehasaNashequilibriuminmixedstrategiessincethede-
1 2 1 2 ≥ ⇒ ∈
cisionvariablesofeachplayerlieonasimplexwhichiscompact.
4.2 Team B Moreover, Lisacontinuous functionofthedecisionvariablesof
alltheplayers.Therefore,thegameadmitsaMSNE.Although,the
TheobjectiveofeachagentistomaximizeL.
MSNEhasbeencomputedforsomegamesbyexploitingsomespe-
cialcharacteristicsinthecostfunctions,therearenostandardtech-
1. Player1: niquestocomputeMSNEforgeneralcontinuous-kernelgames[22,
max L max (pa+pa pb) (16) 2]. Therefore,wesearchfortheconditionsunderwhichthepower
P1b,δ11,δ21,δ12 ⇒P1b,δ11,δ21,δ12 1 2− 2 allocationgameadmitsapurestrategyNashEquilibrium(PSNE).
Lb1
subjectto: | {z } THEOREM 3. [2]LetU beaclosed,boundedandconvexsub-
P1(t) P setofRm,andforeachi NthecostfunctionalJi : U Rbe
b ≤ max continuousonUandconve∈xinuiforeveryuj / Uj,j N→,j / i.
δ11+δ12+δ12 =1, δ11,δ12,δ12 ≥0 Then,theassociatedN-personnonzero-sumga∈meadmit∈saPSN∈E.
2. Player2:
The above theorem provides the conditions under which we can
max L max (pa+pa pb) (17)
P2b,δ21,δ22,δ21 ⇒P2b,δ21,δ22,δ21 1 L2b2− 1 1guaa.rTanhteeeexepxriesstesinocnesofforaSPISNNREp.rLovetiduesdcionn(s1id3e)rrethleevacnastetootfhaegoepn-t
timizationproblembeingsolvedby1a canbewritteninaconcise
subjectto: | {z }
formasshownbelow:
P2(t) P
b ≤ max
δ1+δ2+δ =1, δ1,δ2,δ 0
2 2 21 2 2 21 ≥ sa =a γ12, sb = b1 , sb = d1 ,
2 1 1 c +γ1 2 e +γ1
1 1 1 2
Since the players do not communicate, they possess information
where
only about their own decision variables. This makes the power
allocationproblemacontinuouskernelnon-zerosumgameamong 1
a = ,
theplayers. 1 σ +δ2 d21 −α+δ2 d22 −α
Pmaxρ(d12)−α 1 d12 2 d12
d −α (cid:16) (cid:17) (cid:16) (cid:17)
THEOREM 1. Theoptimalvalueofthepowerconsumptionfor b1 = δ21 d112 ,
eachplayerisPmax. (cid:18) 1 (cid:19)
σ d2 −α
c = +γ2 1 ,
PROOF. ConsiderPlayer1a. Theratep2a isadecreasingfunc- 1 Pmaxρ(d11)−α 1(cid:18)d11(cid:19)
tionofPa1(t). Also,pb1 andpb2 areincreasingfunctionsofPa1(t). d −α
ttThhheeeooretphftoeimrrepa,ll(apvyaa2el−rusepleob1af−dPspa1tb2o()tt)ihse=acdoPencmcraelxua.ssiUionsngintfhguanttchttehioesnaomopfetiPmaar1ug(mut)m.leeHvneetlnfcooefr d1 = δ12(cid:18)dσ1122(cid:19) , d2 −α
e = +γ2 2 .
powerconsumptionofeveryplayerisPmax. 1 Pmaxρ(d12)−α 2(cid:18)d12(cid:19)
Notethat a ,b ,c ,d ande areindependent of thedecision of Tothisend,weobtain:
1 1 1 1 1
1a.
a g′(sa)
THEOREM 4. The power allocation team game has a unique ∇La1 = (bbc1111gg+′′((γss11b2b12)))2 ,∇h(γ¯)= 111 
Nashequilibriuminpurestrategiesifthefollowingconditionshold  (c1+γ21)2   
forTeamA:  
g′′(sa)>0, (18) 1 0 0
i h (γ¯)= 0 , h (γ¯)= 1 , h (γ¯)= 0
1 2 3
g′′(sb1)+ b2i(ci+γ1i)g′(sb1)<0, (19) ∇  0  ∇  0  ∇  1 
g′′(sb2)+ d2i(ei+γ2i)g′(sb2)<0, (20) gSiivnecneγpo∈int∆. 3H,eantcme,otshtethgrreaedoiefntthoefctohnestcroainnsttsracianntsbaetaacntyivfeeaatsiabnlye
pointarealwayslinearlyindependent.
andequivalentconditionsholdforTeamB:
Iftwoofthethreeconstraintsamong h ,h ,h areactive,thenγ¯
g′′(sb)>0, (21) { 1 2 3}
i hasauniquesolutionthatisgivenbythevertexofthesimplexthat
g′′(sa1)+ l2i(mi+δi1)g′(sa1)<0, (22) saht1is,fihe2s,thh3etwisoacctoivnes,trtahienntsw. eIfhoanvleythoenefoolflotwheincgocnasstreasindtespeanmdoinngg
{ }
g′′(sa2)+ n2i(oi+δi2)g′(sa2)<0, (23) ontheactiveconstraint
wherei 1,2 . 1. h (γ¯1) = 0: γ¯1 = (0,γ1∗,1 γ1∗)satisfiesthefollowing
∈{ } 1 1 − 1
equations
Theconstantsb2,c2,d2,ei,li,mi,ni,andoiareobtainedbyre- g′(sb) d1 =g′(sb) b1 (25)
writingtheSINRexpressionsasdoneabove;theirexpressionscan 2 [e1+(1−γ11∗)]2 1 [c1+γ11∗]2
befoundintheAppendix.
2. h (γ¯2) = 0: γ¯2 = (1 γ1∗,0,γ1∗)satisfiesthefollowing
2 − 2 2
equations
PROOF. Letus consider thecaseof 1a. FromTheorem2, we
canconcludethatapurestrategyNashequilibriumexistsifLa1 is a g′(sa)= d1g′(sb2) (26)
convex in its arguments when the decision variables of the other 1 2 (e +γ1∗)2
playersarefixed. From[8],Laisconvexifandonlyif 2La >0 1 2
(HFeosrsiTaenam2BL,aLbiisisgicvoenncainve(2if14)a.ndTohnislyciofn∇sti2tLutbies<th0a)t,∇gw′′h(es1rae)th>e 3. h3(γ¯3) = 0: γ¯3 = (1−γ11∗,γ11∗,0)satisfiesthefollowing
0,g′′(sb)∇+21(c +γ1)g′(sb)<0,andg′′(sb)+ 2 (e +γ1)g2′(sb)< equations
∇0.2LTha2e>1th0e,ob∇r1em2L1tb1he<n10fo,lalnodw1∇s b2yLb2fo<llo0w.ing2similda1r st1eps2to veri2fy a1g′(sa2)= (cb11+g′(γs11b1∗))2 (27)
Ifnoneoftheinequalityconstraintsareactive,then
ApplyingtheKKTconditions[19],inadditiontotheassumptions
upsrotvhiedefodllionwthinegtheeqouraetmionthsatthgautanreaendteteosbtreicstactiosnfiveedxibtyyothfeLga1l,ogbiavlelys γ¯4 =(1−γ11∗−γ21∗,γ11∗,γ21∗),
uniqueoptimalsolution(γ¯): γ12∗
isthesolutiontothefoll|owing{ezquatio}ns:
3
∇La1(γ¯)+ λi∇hi(γ¯)+η∇h(γ¯)=0 a g′(sa) b1 g′(sb)=0
Xi=1 1 2 − [c1+γ11∗]2 1
λiλhii,(ηγ¯)≥=00 (cid:27) i∈{1,2,3} a1g′(sa2)− [e1+d1γ21∗]2g′(sb2)=0 (28)
where
Here,γ¯liesintheset (1,0,0),(0,1,0),(0,0,1),γ¯1,γ¯2,γ¯3,γ¯4 .
{ }
h (γ¯) = γ12 0
1
− ≤ Animportantpointtonoteisthata ,b ,c ,d ande dependon
h (γ¯) = γ1 0 1 1 1 1 1
2 − 1 ≤ the decision of the other players. Therefore, the computation of
h3(γ¯) = −γ21 ≤0 thedecisionvariablesdependonthevalueofthedecisionvariables
h(γ¯) = γ12+γ1+γ1 1=0 of therestof theplayers. Apossible waytodeal withthisprob-
1 2 − lemistouse iterativeschemes for computation of strategies. [2]
Now,wepresentthenecessaryandsufficientconditionsfortheso- providessomeinsightsintotheefficacyofsuchschemesfromthe
lutiontotheoptimizationproblemfortheagents. Letusconsider pointofviewofconvergenceandstability.Inthiswork,weassume
thecaseof1a.TheassumptionsinTheorem4regardingstrictcon- thateachagenthasenoughcomputationalpowersoastocomplete
vexityofLarendertheKKTconditionstobenecessaryaswellas theseiterationsinanegligibleamountoftimecomparedtothetotal
1
sufficientconditionsfortheuniqueglobalminimum. horizonofthegame.
a2g′′(sa) 0 0
1 2
∇2La1 = 00 −(c1+b21γ11)4[g′′(sb1)+0b21(c1+γ11)g′(sb1)] d21 [g′′(sb)+0 2 (e +γ1)g′(sb)]  (24)
 −(e1+γ21)4 2 d1 1 2 2 
 
Inthenextsection,weexpresstheconditionsfortheexistenceof or:
PSNEintermsoflimitationsimposedbythephysicalcommunica-
tionslayer. Pmσaxρ > β3(d12)−α. (33)
5. EXISTENCE OF PSNE UNDER M-QAM Byfollowingsimilarsteps,wecanshowthat(33)issufficientfor
(20)tohold. Infact, condition(33)isalsosufficientforthecon-
MODULATIONSCHEMES
vexityofLa. ForTeamB,asufficientconditionfortheconcavity
2
ofLb andLb is
1 2
Tschheembiet,earnrdorthreaeterro(BrcEoRn)trodlepceonddinsgosnchtehmeeSuItNilRiz,edth.eCmomodmuulantiicoan- σ > β(d12)−α, (34)
Pmaxρ 3
tionsliteraturecontainsclosed-formexpressionsandtightbounds
thatcanbeusedtocalculateg(s)whenthenoiseandinterference which can be derived following similar steps to the above. The
areassumedtobeGaussian[12]. Forexample,usinguncodedM- theoremfollowsfrom(33)and(34).
QAM,whereGrayencodingisusedtomapthebitsintothesym-
bolsoftheconstellation,theBERcanbeapproximatedby[23]
Note that the left hand side of inequality (30) depends entirely
ζ
g(s) βs , (29) onphysicaldesignparameters;thisisofparticularimportancefor
≈ log(M)Q design purposes. Moreover, sufficient conditions for Theorem 5
(cid:16)p (cid:17)
where ζ = 4(1 1/√M), β = 3/(M 1), and (.) is the can be expressed in terms of the received SNRs for all players,
− − Q which could be more insightful from a communication systems
tailprobabilityofthestandardGaussiandistributionwhichcanbe
prospective. Consider, for example, Player 1a, and let SNRx =
expressedintermsoftheerrorfunction: y
1 1 x Pmaxγyxσρ(dxy)−α and SNRxy = Pmaxδxyσρ(dyx)−α. Expression
(x)= erf . (32)canthenbewrittenas
Q 2 − 2 √2
(cid:18) (cid:19)
3
The conditions of Theorem 4 depend primarily on the employed SNR < (SNR2+1).
21 β 1
modulationandcodingschemes.
Similarly,condition(20)holdsif
THEOREM 5. WhenallplayersemployuncodedM-QAMmod- SNR < 3(SNR2+1).
ulation schemes, the power allocation team game has a unique 12 β 2
PSNEsolutionifthefollowingconditionissatisfied:
Yetanotherusefulwaytointerpretcondition(30)isregardingitas
βρPmax min d12,d12 −α<3σ. (30) aminimumratecondition:
{ }
(cid:0) (cid:1) ρPmax min d12,d12 −α
R>log 1+ { } ,
(cid:0) σ (cid:1) !
PROOF. TheconditionsofTheorem4needtobesatisfiedfora
uniquepurestrategiessolutiontoexist.Wefirstverifythosecondi- whereR=log(M).
tionsforPlayer1a whenuncodedM-QAMmodulationsareused.
Tothisend,wedifferentiate(29)intoobtain: The specific conditions for Player 1a corresponding to (25)-(27)
whenM-QAMmodulationsareutilizedare:
′ ζ√β β
g (s)= exp s ,
−2log(M)√2πs (cid:18)−2 (cid:19) sb1 23 exp β(sb sb) b1 = 0,
g′′(s)= ζ√β(1+β√s2)exp βs . (31) (cid:18)sb2(cid:19) (cid:18)−2 1− 2 (cid:19)− d1
4log(M)√2πs3 (cid:18)−2 (cid:19) sb2 12 exp β(sa sb) a1d1 = 0,
aF1roamnd(3γ11)2,.wCeocnodnitciolund(e1t9h)ahtoclodnsdgitiivoenn(t1h8a)tholdsforanyvalueof (cid:18)ssa2b(cid:19)12 (cid:18)−β2 2− 2 (cid:19)− (e1a+bγ21)2
c1 > β3b1−γ11, (cid:18)sa21(cid:19) exp(cid:18)−2(sa2−sb1)(cid:19)− (c1+1γ111)2 = 0.
Also,(28)inthiscasebecomes
orequivalently
c1 > β3b1, (cid:18)ssa2b2(cid:19)12 exp(cid:18)−β2(sa2−sb2)(cid:19)− (e1a+1dγ!21)2 = 0,
whichwecanre-writeas 1
sb 2 β a b
Pmσaxρ +γ12(d21)−α > β3δ21(d12)−α, (32) (cid:18)sa21(cid:19) exp(cid:18)−2(sa2−sb1)(cid:19)− (c1+1γ111)2 = 0.
Player 1 (Team A) Player 2 (Team A)
80 1 1
40 d11 0.8 γ1 0.8 γ2
2300 2 6700 ddd121212 000...246 γγ12112 000...246 γγ11221
d2
10 1 ance50 d212 00 50 100 150 200 00 50 100 150 200
0 1 Dist40 1 Player 1 (Team B) 1 Player 2 (Team B)
−10 2 30 0.8 δ11 0.8 δ21
−20 0.6 δ12 0.6 δ22
−30 20 00..24 δ12 00..24 δ21
−4−040 −30 −20 −10 0 10 20 30 40 100 50 100 150 200 00 50 100 150 200 00 50 100 150 200
Time
(a)Trajectoriesoftheagents (b)Distancesbetweentheagents (c)Valuesofthedecisionvariables
Figure2:
100 40 40 40
80 30 30 30
60
40 20 2 20 2 1 20 2
200 1 2 1 100 1 1 100 1 100 1 1
−20 2 −10 −10 2 −10
−40 2 2
−20 −20 −20
−60
−80 −30 −30 −30
−1−00100 −50 0 50 100 −4−040 −30 −20 −10 0 10 20 30 40 −4−040 −30 −20 −10 0 10 20 30 40 −4−040 −30 −20 −10 0 10 20 30 40
(1) (2) (3) (4)
Figure3:Trajectoriesoftheagentsontheplane
6. SIMULATIONRESULTS 1. DifferentSpeeds:ua =5,ua =1,ub =3,andub =4
1 2 1 2
2. DifferentModulation:M =16
Inthissection,wepresentsomesimulationresults.First,wepresent
simulationsforarbitrarilychosenvaluesofmaximumpowerPmax, 3. Differentterminalconditions.
frequenciesf ,f ,modulationschemesizeM,andspeedsofthe
1 2 4. Frequencyexchange: f =100MHzandf =300MHz
players usatisfying theconvexity conditions intheprevious sec- 1 2
tion. Wethenchangethevalueofoneparameter,whilefixingthe
rest,andpresentsimulationresultsforthevariantsoftheoriginal Ineachcase,wepresentthesimulationresultswhenalltheparam-
problem. etersarefixedexceptfortheonelistedabove.
Figure 2 shows the trajectories of the agents, distances between 7. CONCLUSION ANDFUTURE WORK
agents,andthevaluesofthedecisionvariablesover200timesteps.
Thekinematicsofalltheagentsaregivenbythefollowingequa-
tions Thispaperhasstudiedthepowerallocationproblemforjamming
teams.Themotionoftheteamswasmodelledusingtheframework
x˙i=uicosθi, y˙i =uisinθi ofpursuit-evasion gamesandtheoptimalstrategieswerederived.
Weusedthefollowingvaluesfortheparametersinthesimulations: An underlying static game was used toobtain the optimal power
allocation, where the power budget of each user is split between
communication and jamming powers. This work focused on the
P =100
• max analysisofteamsconsistingoftwoplayersonly. Potentialfuture
f =300MHzandf =100MHz directionsinclude:
1 2
•
TypicalvaluesofMare2,4,16,64,and256.Forsimulation
•
purposes,wefixthevalueofM at2. Computation of Singular Surfaces: In this work, we have
•
computedthetrajectoriesbasedonthenecessaryconditions
ua =ua =ub =ub =1
• 1 2 1 2 of optimality imposed by the Isaacs’ conditions. In order
to complete the construction of the optimal trajectories of
Figures3, 4, and5contain four subfigures eachrepeating one of the agents, we have to identify the singular surfaces in the
theabovesimulationsforavariationoftheoriginalparametersas statespace[2]. Thisisaninterestingfutureresearchdirec-
follows: tionsincetheconstructionandnatureofthesingularsurfaces
Player 1 (Team A) Player 2 (Team A) Player 1 (Team A) Player 2 (Team A)
1 1 1 1
0.8 0.8 0.8 0.8
γ1 γ2 γ1 γ2
0.6 1 0.6 1 0.6 1 0.6 1
γ1 γ2 γ1 γ2
0.4 2 0.4 1 0.4 2 0.4 1
γ12 γ21 γ12 γ21
0.2 0.2 0.2 0.2
0 0 0 0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
Player 1 (Team B) Player 2 (Team B) Player 1 (Team B) Player 2 (Team B)
1 1 1 1
0.8 0.8 0.8 0.8
δ1 δ1 δ1 δ1
0.6 1 0.6 2 0.6 1 0.6 2
δ2 δ2 δ2 δ2
0.4 1 0.4 2 0.4 1 0.4 2
δ δ δ δ
0.2 12 0.2 21 0.2 12 0.2 21
0 0 0 0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
(1) (2)
Player 1 (Team A) Player 2 (Team A) Player 1 (Team A) Player 2 (Team A)
1 1 1 1
0.8 0.8 0.8 0.8
γ1 γ2 γ1 γ2
0.6 γ11 0.6 γ12 0.6 γ11 0.6 γ12
0.4 γ212 0.4 γ121 0.4 γ212 0.4 γ121
0.2 0.2 0.2 0.2
0 0 0 0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
Player 1 (Team B) Player 2 (Team B) Player 1 (Team B) Player 2 (Team B)
1 1 1 1
0.8 0.8 0.8 0.8
00..46 δδ1112 00..46 δδ2122 00..46 δδδ1112 00..46 δδδ2122
δ δ 12 21
0.2 12 0.2 21 0.2 0.2
0 0 0 0
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
(3) (4)
Figure4:Variationsofthedecisionvariablesasafunctionoftime.
200 80 70 80
Distance11111024688000000 dddddd111212122122 Distance45670000 dddddd111212122122 Distance456000 dddddd111212122122 Distance45670000 dddddd111212122122
30
60 30 30
40
20 20 20
20
0 10 10 10
0 50 100 150 200 0 50 100 150 200 0 50 100 150 200 0 50 100 150 200
Time Time Time Time
(1) (2) (3) (4)
Figure5:Distancesamongtheagents
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