Adaptive Resource Allocation in Jamming Teams Using Game Theory

Adapti v e Resource Allocation in Jamming T eams Using Game Theory* Ali Khanafer , Sourabh Bhattac harya and T amer Bas ¸ar Coordinated Science Laborato ry , University of Illinois at Urbana-Cham paign { khan afe2, sbhattac, basar1 } @illinois.edu Abstract —In this work, we study the problem of power allocation and adaptive modulation in teams of decision makers. W e consider the special case of two teams with each team consisting of two mobile agents. Agents belonging to the same team communicate ov er w ireless ad h oc networks, and they try to split their av ailable power between the tasks of communication and jamming the nodes of th e other team. The agents h av e constraints on th eir total en ergy an d instantaneous p ower usage. The cost function adopted is the difference b etween the rates of erroneously transmitted b its of each team. W e model the adaptive modulation problem as a zero-sum matrix game which in turn gives rise to a a contin uous ker nel game to hand le power control. Based on the communications model, we present sufficient cond itions on the physical parameters of the agents fo r the existence of a pure strategy saddle-point equilibrium (PSSPE). I . I N T RO D U C T I O N The decen tralized nature of wireless ad h oc networks makes them vulne rable to security threats. A prominent example of such threats is jamming: a m alicious attack whose objectiv e is to disrupt the c ommun ication of the victim network inten - tionally , causing interf erence or collision at the receiver side . Jamming attack is a well-studied and active area of research in wireless n etworks. Un author ized intru sion of such kin d has in itiated a rac e between the eng ineers and th e hackers; therefor e, we h av e been witnessing a surge of n ew smart systems aiming to secure modern instrumentation and software from un wanted exogeno us attack s. The prob lem un der co nsideration in this paper is inspired by recent d iscoveries of jamming instances in b iological species. I n a ser ies of p layback experiments, research ers have found that resident pair s of Per uvian warbling an tbirds sing coordin ated duets when respondin g to riv al pair s. But un der other circumstances, cooper ation breaks down, leading to more complex son gs. Sp ecifically , it has been reported that females respond to u npaired sexual riv als by jamming the signals of their own mates, who in tur n ad just their sign als to a void the interferen ce [ 14]. Ad ho c networks co nsist of mobile energy-co nstrained nodes. Mobility affects all layers in a network proto col stack including the physical layer as channels become time-varying [7]. More over , n odes such as sensor s deployed in a field o r military vehicles patr olling in remo te sites a re o ften equipp ed *Researc h supported in part by grants from AFSOR and AFSOR MURI. with non -rechargeab le batteries. Power contr ol a nd adaptive resource allocation (RA) play , ther efore, a crucia l ro le in designing robust commun ications systems. At th e ph ysical layer, power control can be u sed to m aximize rate o r min i- mize the transm ission erro r pro bability , see [2], [ 8] an d the referenc es therein. In addition, in multi-user networks, p ower control can be used to regulate the interfer ence level a t the terminals of oth er users [ 4], [1 2], [ 15]. Due to th e lack o f a cen tralized infrastru cture in ad h oc network s, distributed solutions are essential. In this work , similar to [2 ], [12], [15], we mod el the power allocation problem as a non cooper ativ e game, which allows us to devise a non-centralize d solution. As a departure from previous research, howe ver , the power control mechanism we propose splits th e power budget of each p layer into tw o portions: a portion u sed to comm unicate with team- mates and a portion used to jam the players of the other team. More im portantly , the ob jectiv e function is cho sen to be the difference between the cumulative bit err or rate (BER) of each team; this allo ws for in creased freedom in choosing physical design parameters, besides the p ower level, such as the size of m odulation schem es. Adaptive RA mechanisms involv e varying ph ysical layer parameters accord ing to chann el, interfer ence, an d noise con - ditions in or der to optimize a specific metric, such as spec- tral efficiency . Adapting the modulation schem e, choosing coding schem es, a nd co ntrolling the tran smitter power level are examples of ad aptive RA schem es. Gold smith and Chua demonstra ted in [6] that adaptive RA p rovides five times m ore the spectral efficiency of nonadaptive schemes. In th is work, we propose an adaptive modulatio n scheme based on a zero- sum ma trix g ame p layed by both teams. The conflicting objectiv es of the tw o teams entails the use of game-theo retic framework to study this problem. W e identify three m ain tasks that ea ch team needs to perform: 1) Optimal trajectory: computing the optimal motion path for the ag ents 2) P ower Allocation: di viding power between internal com- munication s and jam ming 3) Adaptive Mo dulation : choosin g an appr opriate modu la- tions schem e W e addr essed T ask 1 in [3] by posing the problem as a pursuit-evasion game . The optimal strategies of the player s are obtained b y using tech niques fro m differ en tial game theory . W e also addressed T ask 2 in [3] using con tinuous kernel static games ; we will gen eralize the formulation in this work to include a minimum ra te constra int. This will lead to a m ore practical scheme as it gua rantees a n on-zer o co mmun ications rate. T ask 3 will be addr essed in this work using static ma trix games . T he saddle-po int eq uilibrium o f the power allocation problem is parametr ized by the modulation schemes of the two teams. W e theref ore introduc e a thir d ga me in o rder to ar rive at the equilib rium modu lation schemes. I n fact, this giv es rise to a ga mes-within-ga mes structure: the op timal tr ajectory is first found, power allocation is then performed , and finally the optimal modulatio n is com puted at each time instant. The main contributions of th is paper are a s follows. W e introdu ce a third layer of gam es to o ur re cent work [ 3] in order to per form ad aptive mo dulation . W e also generalize the power allocatio n problem in troduc ed in [3] to en sure non-ze ro co mmun ications rate. W e introduce an optim ization framework ta king into consid eration constraints in energy and power a mong the agen ts. Moreover, we relate the prob lem of optimal power allocatio n for commu nication and jam ming to the commu nication m odel between th e agen ts. Finally , we provide a sufficient condition for existence of an o ptimal decision strategy amon g the ag ents based o n the physical parameters o f the p roblems. The rest of the pa per is organized as follows. W e formu - late the problem in Section II and explain the under lying notation. The saddle-p oint e quilibrium prope rties of th e te am power control pro blem are stud ied in Section I II with the specific examp le of systems employing uncod ed M-quadr ature amplitude modu lations (QAM) . W e intro duce our adaptive modulatio n scheme in Section IV. Simu lation results are presented in Section V. W e conclu de the paper and provide future d irections in Section V I. I I . P RO B L E M F O R M U L AT I O N Consider two teams of mo bile agents. Each agen t is com- municating with m embers o f the team it belon gs to, and , at th e same time, jamming the commun ication between mem bers of the oth er team . In particu lar , e ach team attempts to minimiz e its own BER while maxim izing th e BER of the other team. W e con sider a scenario where each team has two memb ers, though at a concep tual le vel our development applies to h igher number of team mem bers as well. T eam A is c omprised of the two players { 1 a , 2 a } an d T eam B is comp rised of the two players { 1 b , 2 b } . W e a ssume th at f a and f b are th e fr equencie s at which T eam A a nd T eam B comm unicate, respectiv ely , and f a 6 = f b . Naturally , for an initial position x 0 ∈ X , the outcom e of the game π , is given by the difference in the BERs of both teams during the en tire cou rse of the ga me. Formally: π ( x 0 , u a i , u b j ) = N · Z T 0 [ p a 1 ( t ) + p a 2 ( t ) − p b 1 ( t ) − p b 2 ( t )] | {z } L dt, where p a i ( t ) and p b j ( t ) are the BERs o f agent i in T eam A and and ag ent j in T eam B, r espectively; u a i and u b j are like wise the control inpu ts of agents i and j in te ams A and B, respectively; N is the total nu mber of transmitted bits which remains co nstant throu ghou t the game; and T is th e time of termination of the g ame. W e co nclude that the objec ti ve of T e am A is to minimiz e π , wher eas that of T eam B is to maximize it. Since th e agents are mob ile, th ere a re lim itations o n the amount o f energy available to each agent that is dictated by the cap acity of the power sour ce carried by each ag ent. The gam e is said to termin ate when any a gent ru ns o ut of power . Let P a i ( t ) and P b j ( t ) denote the instantaneo us power for commu nication used b y player i in T eam A and play er j in T eam B, respectively . W e mo del this restriction as the following integral constraint for ea ch agent: Z T 0 P i ( t ) dt ≤ E (1) For each tr ansmitter and receiver pair, we assume the following com municatio ns mo del in the presence of a jammer which is motiv a ted b y [13]. Given th at the tran smitter and the receiver are separated by a distan ce d , and the tran smitter transmits with co nstant power P T , th e received sig nal power P R is given by P R = ρP T d − α , (2) where α is the p ath-loss exponen t and ρ d epends on th e antennas’ gain s. T ypical values o f α a re in the range of 2 to 4 . Accord ing to the fr ee space path loss mo del, ρ is given by: ρ = G T G R λ 2 (4 π ) 2 , where λ is th e signal’ s wavelength and G T , G R are the transmit and receiv e antennas’ gains, respectively , in the line of sight d irection. I n real scenario s, ρ is very small in magn itude. For example , using no ndirectio nal antenn as and tran smitting at 9 00 MHz, we h av e ρ = (1)(0 . 33) (4 π ) 2 = 6 . 896 × 10 − 4 . The re ceiv ed sig nal-to-inte rference r atio (SI NR) s is given by s = P R I + σ 2 , (3) where σ 2 is the power o f the no ise added at the r eceiver , and I is the total r eceived interference power due to jam ming an d is de fined a s in ( 2). Let P a i ( t ) and P b j ( t ) deno te the instantaneou s power levels for comm unication used by player i in T eam A and p layer j in T eam B, respecti vely . Since the agents a re mobile, th ere are limitations on th e amount of en ergy a vailable to each agen t that is d ictated by the capacity o f the power source carr ied by each agent. W e mod el this restriction as the following in tegral constraint fo r each agen t Z T 0 P i ( t ) dt ≤ E . (4) In addition to th e energy constraints, there are limitations on the m aximum power lev el of the devices that are used onbo ard each agen t fo r the purpose of co mmun ication. F or ea ch player, this c onstraint is m odelled b y 0 ≤ P a i ( t ) , P b i ( t ) ≤ P max . W e also assume that playe rs of each team hav e access to different M-QAM m odulation schemes. W e den ote th e set of av a ilable modu lation sizes to the players in T eam A by M a and that available to players of T eam B b y M b . The sizes of the employed QAM modu lation by the teams are M a ∈ M a and M b ∈ M b . W e assume that T eam A can choose among n different mo dulation scheme s, and T eam B ch ooses from a set o f m different schem es, i.e., |M a | = n, |M b | = m . The instantaneo us BER depend s on the SINR, the m odu- lation scheme, and the erro r con trol co ding scheme utilized. Communica tions liter ature con tains closed- form exp ressions and tight b ound s that can be u sed to calculate the BER when the n oise a nd inter ference are assumed to be Gaussian [5]. For uncod ed M-QAM, where Gray encoding is used to map the b its into the symbols o f the co nstellation, the BER can b e approx imated by [11] p ( t ) = g ( s ) ≈ ζ r Q  p β s  , (5) where r = log ( M ) , ζ = 4(1 − 1 / √ M ) , β = 3 / ( M − 1) , and Q (.) is th e tail probability of the standar d Gaussian distribution. T o en sure a non-zer o communication rate between the agents of eac h tea m, we im pose a minim um rate co nstraint for each agent: R a i ( t ) , R b i ( t ) ≥ ˜ R , where ˜ R > 0 is a thresh old design rate, wh ich we assume is the sam e fo r all agents. The results can b e readily extend ed to networks of player s having a different value of th e min imum design rate. At e very instant, each agent has to decide o n the fr action of the power th at n eeds to be allocated for comm unication and jammin g. T able I p rovides a list of decision variables f or the players, wh ich models th e po wer a llocation. Each decisio n variable is a n on-negative real n umber and lies in th e interval [0 , 1] . Th e decision variables belon ging to each row add up to one. The fra ction of the total power allocated b y the pla yer in row i to the p layer in column j is given by th e first entry in the cell ( i, j ) . This alloc ated power is u sed for jamming if the play er in colum n j belon gs to the o ther team; other wise, it is used to commun icate with the agen t in th e same team. Similarly , the distance between the agent in row i and the agent in column j is given by the secon d en try in cell ( i, j ) . Since distance is a sy mmetric quan tity , d ij = d j i and d ij = d j i . Fig. 1 depicts the power allocatio n between the mem bers of the same team as well as between members o f d ifferent teams. In addition to power allocation , each team has to decide on the size of the QAM modu lation to b e used. In sum mary , each age nt has to compu te th e following decision variables at each in stance in accorda nce with the above tasks: (i) the instantaneou s contro l (T ask 1); (ii) the in stantaneou s power lev el, P i ( t ) (T ask 2); (iii) all the decision variables pr esent in the row co rrespon ding to the agent in T able I (T ask 2); an d (iv) the size of th e QAM schem es, M a or M b (T ask 3). T ABLE I D E C I S I O N V A R I A B L E S A N D D I S TA N C E S A M O N G AG E N T S . 1 b 2 b 1 a 2 a 1 a γ 1 1 , d 1 1 γ 1 2 , d 1 2 — γ 12 , d 12 2 a γ 2 1 , d 2 1 γ 2 2 , d 2 2 γ 21 , d 21 — 1 b — δ 12 , d 12 δ 1 1 , d 1 1 δ 2 1 , d 2 1 2 b δ 21 , d 21 — δ 1 2 , d 1 2 δ 2 2 , d 2 2 P P P P P A B a a a P P a P b b b b 1 1 P a 1 1 2 2 1 1 2 2 2 2 2 δ δ δ γ γ γ γ 1 a P 1 2 γ γ 12 δ P 1 b δ 21 12 1 2 δ 2 1 1 1 1 1 P 2 b 2 1 2 2 2 2 2 1 Fig. 1. Po wer allocat ion among the agents for communicatio n as well as jamming. I I I . P O W E R A L L O C A T I O N From (3), th e received SINR an d th e r ate achieved by eac h agent are given by th e following expressions s a i = ρ a P a j ( t ) γ j i ( d ij ) − α σ 2 + ρ b P b 1 ( t ) δ i 1 ( d i 1 ) − α + ρ b P b 2 ( t ) δ i 2 ( d i 2 ) − α s b i = ρ b P b j ( t ) δ j i ( d ij ) − α σ 2 + ρ a P a 1 ( t ) γ 1 i ( d 1 i ) − α + ρ a P a 2 ( t ) γ 2 i ( d 2 i ) − α R a i = log(1 + s a j ) , R b i = log(1 + s b j ) (6) Agents of the same team embar k in a team p roblem which eliminates their need to exchange inf ormation abo ut their decision variables. Furthe r , since agents in different teams do n ot commun icate, they possess info rmation only ab out their own de cision variables. This m akes the power allocation problem a con tinuous kernel zero-sum gam e between the teams: T eam A: Th e ob jectiv e of each agen t is to minimize L . min P a i ,γ i 1 ,γ i 2 ,γ ij L ( M a , M b ) ⇒ min P a i ,γ i 1 ,γ i 2 ,γ 12 ( p a j − p b 1 − p b 2 | {z } L a i ( M a ,M b ) ) (7) subject to : 0 ≤ P a i ( t ) ≤ P max , R a i ≥ ˜ R γ i 1 + γ i 2 + γ ij = 1 , γ i 1 , γ i 2 , γ ij ≥ 0 T eam B: Th e ob jectiv e of each agen t is to maximize L . max P b i ,δ 1 i ,δ 2 i ,δ ij L ( M a , M b ) ⇒ max P b i ,δ 1 i ,δ 2 i ,δ ij ( p a 1 + p a 2 − p b j | {z } L b i ( M a ,M b ) ) (8) subject to : 0 ≤ P i b ( t ) ≤ P max , R b i ≥ ˜ R δ 1 i + δ 2 i + δ ij = 1 , δ 1 i , δ 2 i , δ ij ≥ 0 Note that th e p ower allocation vector fo r 1 a denoted by γ = ( γ 12 , γ 1 1 , γ 1 2 ) belongs to th e inter section be tween the three-dim ensional simplex ∆ 3 and the plan e r 0 , w here r 0 = { γ | γ 12 = 1 a 1 (2 ˜ R − 1) } . Th e p ower allocation vectors of other players belong to similar sets. In [3], we showed that the op timal value of the power consump tion for each player is P max . W e a lso showed th at the entire game termina tes in a fixed time T = E P max irrespective of the initial p osition of the ag ents. More over we provide d a sufficient con dition for the existence o f a pure-strategy saddle-po int eq uilibrium (PSSPE) f or the p ower allo cation game when uncod ed M -QAM schemes are used by all agents. Here, we modif y the cond ition to allow teams to use different modulatio n sch emes as made formal b y the n ext theorem. Theor em 1: When all pla yers emp loy uncod ed M -QAM modulatio n schemes, the power allocatio n team g ame f ormu- lated above has a unique PSSPE solution if the following condition is satisfied: P max · max  ρ a ( d 12 ) − α M a − 1 , ρ b ( d 12 ) − α M b − 1  < σ 2 . (9) For the special case of M a = M b = M a nd ρ b ≈ ρ a = ρ , the condition becomes β ρP max  min { d 12 , d 12 }  − α < 3 σ 2 . (10) The pr oof is similar to that presented in [ 3] and is omitted here. Note that the left hand side of ineq uality ( 10) dep ends entirely on phy sical design para meters; this is of p articular importan ce for design pu rposes. Moreover , we sh owed in [3] that this cond ition can be expr essed in terms of th e received signal-to-n oise-ratios (SNRs) fo r all players, which could b e more in sightful f rom a commu nication sy stems perspective. Consider, for example, 1 a , and let SNR x y = P max γ x y ρ ( d x y ) − α σ 2 and SNR xy = P max δ xy ρ ( d yx ) − α σ 2 . W e then have: SNR ij < 3 β ( SNR 2 j + 1) i, j ∈ { 1 , 2 } ; i 6 = j Y et another useful way to interpre t con dition (10) is regarding it as a min imum rate co ndition: r > log 1 + ρP max  min { d 12 , d 12 }  − α σ 2 ! . Assuming (10) holds, the o bjective f unction is strictly conv ex in the decision variables o f 1 a , 2 a and strictly concave in the d ecision variables o f 1 b , 2 b . A un ique globally optim al solution ( ¯ γ ) ther efore exists, which we char acterize using the KKT co ndition s [9]. Conside r , for example, the case of 1 a . The expressions for SINR provided in (6) relev ant to the optimization prob lem bein g solved b y 1 a can be written in a concise f orm as shown below: s a 2 = a 1 γ 12 , s b 1 = b 1 c 1 + γ 1 1 , s b 2 = d 1 e 1 + γ 1 2 , where a 1 = 1 σ P max ρ a ( d 12 ) − α + ˜ ρδ 2 1  d 2 1 d 12  − α + ˜ ρδ 2 2  d 2 2 d 12  − α , b 1 = ˜ ρδ 21  d 12 d 1 1  − α , d 1 = ˜ ρδ 12  d 12 d 1 2  − α , c 1 = σ P max ρ a ( d 1 1 ) − α + γ 2 1  d 2 1 d 1 1  − α , e 1 = σ P max ρ a ( d 1 2 ) − α + γ 2 2  d 2 2 d 1 2  − α , ˜ ρ =  ρ b ρ a  . The KKT c ondition s can then be written as: ∇ L a 1 ( ¯ γ ) + 4 X i =1 λ i ∇ h i ( ¯ γ ) + η ∇ h ( ¯ γ ) = 0 , (11) λ i h i ( ¯ γ ) = 0 , λ i , η ≥ 0 , i ∈ { 1 , 2 , 3 } where h 1 ( ¯ γ ) = − γ 12 + min ( 2 ˜ R − 1 a 1 , 1 ) ≤ 0 h 2 ( ¯ γ ) = − γ 1 1 ≤ 0 , h 3 ( ¯ γ ) = − γ 1 2 ≤ 0 h ( ¯ γ ) = γ 12 + γ 1 1 + γ 1 2 − 1 = 0 Now , we present the ne cessary and sufficient con ditions for the solution to the o ptimization prob lem for the agen ts. Let us consider th e case of 1 a . The a ssumptions in Th eorem 4 regarding strict co n vexity o f L a 1 render the KKT condition s to b e n ecessary as we ll as su fficient conditions fo r the u nique global min imum. T o this end, we obtain: ∇ L a 1 =     a 1 g ′ ( s a 2 ) b 1 g ′ ( s b 1 ) ( c 1 + γ 1 1 ) 2 b 1 g ′ ( s b 2 ) ( c 1 + γ 1 2 ) 2     , ∇ h ( ¯ γ ) =   1 1 1   ∇ h 1 ( ¯ γ ) =   − 1 0 0   , ∇ h 2 ( ¯ γ ) =   0 − 1 0   , ∇ h 3 ( ¯ γ ) =   0 0 − 1   . Since γ ∈ ∆ 3 ∩ r o , at most three of the constraints can be acti ve at any given p oint. Hence, the gradient of the constraints at any feasible point ar e always linearly indep endent. If two of the three constraints among { h 1 , h 2 , h 3 } are activ e, then ¯ γ has a uniqu e solution tha t is giv en by the vertex of the simplex th at satisfies the two co nstraints. If only one of th e constraints among { h 1 , h 2 , h 3 } is ac ti ve, the n we have the following cases d ependin g on the active constraint A = M b (1) M b (2) ... M b ( m − 1) M b ( m ) M a (1) L ( M a (1) , M b (1)) L ( M a (1) , M b (2)) ... L ( M a (1) , M b ( m − 1)) L ( M a (1) , M b ( m )) M a (2) L ( M a (2) , M b (1)) L ( M a (2) , M b (2)) ... L ( M a (2) , M b ( m − 1)) L ( M a (2) , M b ( m )) . . . . . . . . . . . . . . . . . . T e am A M a ( n ) L ( M a ( n ) , M b (1)) L ( M a ( n ) , M b (2)) ... L ( M a ( n ) , M b ( m − 1)) L ( M a ( n ) , M b ( m )) T e am B (12) 1) h 1 ( ¯ γ 1 ) = 0 : ¯ γ 1 = ( ˜ γ 12 , γ 1 ∗ 1 , 1 − γ 1 ∗ 1 − ˜ γ 12 ) , where ˜ γ 12 = min n 2 ˜ R − 1 a 1 , 1 o , satisfies th e equ ation g ′ ( s b 2 ) d 1 [ e 1 + (1 − γ 1 ∗ 1 − ˜ γ 12 )] 2 = g ′ ( s b 1 ) b 1 [ c 1 + γ 1 ∗ 1 ] 2 (13) 2) h 2 ( ¯ γ 2 ) = 0 : ¯ γ 2 = (1 − γ 1 ∗ 2 , 0 , γ 1 ∗ 2 ) satisfies the following equation a 1 g ′ ( s a 2 ) = d 1 g ′ ( s b 2 ) ( e 1 + γ 1 ∗ 2 ) 2 (14) 3) h 3 ( ¯ γ 3 ) = 0 : ¯ γ 3 = (1 − γ 1 ∗ 1 , γ 1 ∗ 1 , 0) satisfies the following equation a 1 g ′ ( s a 2 ) = b 1 g ′ ( s b 1 ) ( c 1 + γ 1 ∗ 1 ) 2 (15) If non e of th e inequ ality constrain ts are active, th en ¯ γ 4 = (1 − γ 1 ∗ 1 − γ 1 ∗ 2 − ˜ γ 12 | {z } γ 12 ∗ , γ 1 ∗ 1 , γ 1 ∗ 2 ) is the solution to : a 1 g ′ ( s a 2 ) − b 1 [ c 1 + γ 1 ∗ 1 ] 2 g ′ ( s b 1 ) = 0 a 1 g ′ ( s a 2 ) − d 1 [ e 1 + γ 1 ∗ 2 ] 2 g ′ ( s b 2 ) = 0 (16) Here, ¯ γ lies in the set { (1 , 0 , 0) , (0 , 1 , 0) , (0 , 0 , 1) , ¯ γ 1 , ¯ γ 2 , ¯ γ 3 , ¯ γ 4 } . An importan t point to note is that a 1 , b 1 , c 1 , d 1 and e 1 depend on the decisions o f the other player s. Ther efore, the co mputatio n of the d ecision variables dep end o n the value of the dec ision variables of the rest o f the playe rs. A possible way to d eal with this problem is to use iterative sche mes for compu tation of strategies. [1] provides some insights into th e efficac y of such schemes fr om the poin t of view of conver gence a nd stability . In this work , we assume that each agent has en ough computatio nal power so as to co mplete these iteratio ns in a negligible amo unt o f time co mpared to the total ho rizon of the game . The specific condition s for 1 a correspo nding to (13)-(15) wh en M- QAM mo dulations ar e utilized ar e:  s b 1 s b 2  3 2 exp  − β 2 ( s b 1 − s b 2 )  − b 1 d 1 = 0 (17)  s b 2 s a 2  1 2 exp  − β 2 ( s a 2 − s b 2 )  − a 1 d 1 ( e 1 + γ 1 2 ) 2 = 0 (18)  s b 1 s a 2  1 2 exp  − β 2 ( s a 2 − s b 1 )  − a 1 b 1 ( c 1 + γ 1 1 ) 2 = 0 (19) Also, (16) in this ca se correspond s to solving (18) and (19) jointly . I V . A DA P T I V E M O D U L AT I O N The time-varying n ature of the chann els du e to mo bility emphasizes th e n eed f or ro bust communic ations. Adaptive modulatio n is a widely used technique as it allo ws for choosing the design p arameters of a commu nications system to better match the physical character istics of th e chann els in order to optimize a gi ven m etric such as: minimizin g BER o r maximizing spectral efficiency . In this work , we model the adaptive modu lation as a matrix zero-su m game betwee n the two teams. W e ther efore look for a n equ ilibrium solution which would d ictate wh at modulatio ns should be adopted by the teams at each time instant. T he com petitiv e n ature of the jamming teams makes our appr oach to the problem most p ractical as any oth er non-e quilibrium solution canno t produ ce an impr oved ou tcome, relative to th at yield ed b y the equilibriu m, f or a ny of the teams. The matrix game is giv en in (12). The rows are all the possible actio ns for p layers of T eam A, an d the column s are the different optio ns available for T eam B. T he ( i, j ) - th element of the matr ix is the value of the objective function L when T eam A employs M a = M a ( i ) , a nd T eam B employs M b = M b ( j ) . A PSSPE do es no t alw ays exist for the power alloc a- tion game. The condition for the existence of a PSSPE is min i max j A ij = max j min i A ij [1]. In case a PSSPE d oes not exist, we n eed to look f or a solution in the larger c lass of mixed-strategies. A p air of strategies { M a ∗ , M b ∗ } is said to be a a m ixed-strategy saddle point equilibriu m (MSSPE) fo r the matr ix ga me if [1] ( M a ∗ ) T A M b ≤ ( M a ∗ ) T A M b ∗ ≤ ( M a ) T A M b ∗ For an n × m m atrix g ame, the following theor em fr om [1] , which we state without proo f, estab lishes th e existence of an MSSPE f or th e ada ptiv e modu lation gam e. Theor em 2: Th e adaptive modulation game admits an MSSPE. In case multiple MSSPEs exist, th e following corollary becomes essential [1]. Cor ollary 1 : If {M a ( i 1 ) , M b ( j 1 ) } and {M a ( i 2 ) , M b ( j 2 ) } are two MSSPEs of the adaptive modulatio n game, then {M a ( i 1 ) , M b ( j 2 ) } a nd {M a ( i 2 ) , M b ( j 1 ) } are also MSSPEs. This is termed the or der ed interc hangea bility pr op erty and its importan ce lies in th at it removes any ambig uity associated with the existence of mu ltiple eq uilibrium solutions as the teams do not need to commu nicate to each other which equilibriu m solu tion they will b e adopting. Literature contains different efficient low-complexity algo rithms th at co mputes MSSPEs for matr ix games, such as Gambit [10]. W e refer the interested read er to [1] fo r a discussion of some of these app roaches. Section V illustrates these co ncepts and shows how the ch oice of mo dulation size ch anges with the characteristics o f the environment. Finally , it is assume d that players of each team comm unicate th eir modu lation ch oices among th emselves thro ugh a reliable side channe l. V . S I M U L AT I O N S R E S U LT S T o better understand the adaptive mod ulation scheme, we presen t the following example. Let M a = M b = { 16 , 64 , 265 } , d 1 1 = 17 . 786 4 , d 1 2 = 15 . 337 6 , d 2 1 = 19 . 8951 , d 2 2 = 14 . 11 28 , d 12 = 20 . 63 09 , d 12 = 26 . 32 24 , ρ a = 0 . 057 0 , ρ b = 0 . 0 517 , P max = 10 0 , σ 2 = 10 − 3 , ˜ R = 1 , and α = 2 . The matrix A cor respond ing to these values was found to b e: A = 16 64 256 16 0 . 0158 0 . 0533 0 . 1229 64 − 0 . 03 56 0 . 0091 0 . 0728 T eam A 256 − 0 . 115 5 − 0 . 067 7 0 . 0040 T e am B Note that the thir d row d ominates the other rows strictly , and there is a un ique PSSPE g iv en by { 256 , 256 } in this case. Fig. 2 depicts ho w the teams adap t their modulation schem e relativ e to the SNR, wh ich we define as P max /σ 2 . The set of modulatio ns av ailable to each team is M = { 16 , 20 , 24 , 28 } . Players 1 a , 2 a , and 1 b were p laced close to each oth er , while 2 b is far f rom all of them; in particular: d 1 1 = 2 . 203 6 , d 1 2 = 33 . 683 0 , d 2 1 = 2 . 421 1 , d 2 2 = 33 . 63 9 3 , d 12 = 4 . 560 7 , d 12 = 33 . 202 2 . W e also let ρ a = 0 . 0 570 , ρ b = 0 . 0 517 , P max = 1 , ˜ R = 1 , α = 3 , an d v aried the noise variance at all receivers to simulate the p resented SNR range. W e observe that both teams switch to a con stellation of a smaller size at SNR = 50 dB. This is due to both teams switching from pure commun ications to perfor m b oth co mmun ication and jammin g. I n order to do so, th ey both switch to a smaller co nstellation size wh ich will guaran tee robust commun ications for them as they will allocate some p ower to jam. V I . C O N C L U S I O N A N D F U T U R E W O R K This paper has studied th e power alloc ation pro blem f or jamming teams. 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