Game theory and the frequency selective interference channel - A tutorial

1 Game theory and the f requenc y selecti v e interfer ence channel - A tutorial Amir Leshem 1 , 2 and Ephraim Zehavi 1 Abstract This paper provides a tutorial o verview of game theoretic techniques used for communication over frequen cy s electiv e interference channe ls. W e discuss both com petitiv e and co operative techniques. Ke ywords: Game th eory , comp etiti ve g ames, co operative games, Nash Equ ilibrium, Nash bargaining solution, Generalized Nash games, Spe ctrum optimization, distributed coor dination, interf erence ch annel, multiple access chan nel, iterative water-filling. I . C O M M U N I C A T I O N OV E R I N T E R F E R E N C E L I M I T E D C H A N N E L S The su ccess o f unlicens ed b roadband c ommunication has led to very rapid d eployment of communica- tion networks that work inde penden tly of e ach other using a relati vely n arro w spectrum. For example the 802.11g standard is using the ISM band which ha s a total bandwidth of 60 MHz. This ba nd is divided into 12 partially overlapping bands of 2 0 MHz. The su ccess of these tech nologies might beco me the ir own limiting factor . Th e relatively sma ll number of channe ls a nd the massive use of the techno logy in densely populate d urban metropolitan areas will c ause significant mutual interference. Th is is espec ially important for high quality real ti me distrib ution of multi-media con tent that is intolerant to e rrors as we ll as latency . Existing 802.11 (W iFi) networks h av e very limited means to coordina te spectrum with other interfering sys tems. It would be highly d esirable to improve the interference envir onment by d istrib u ted spectral co ordination b etween the different a ccess points. An other s cenario is that of c entralized acce ss points such as 802.16 (W iMax) whe re the resources are allocated centrally by a single base station. Similar situation is facing the ad vanced DSL sy stems su ch as ADSL2 + and VDSL. The se sys tems are currently limited by crosstalk between the lines. As suc h the DSL envir onment is another example of highly frequency selective interference ch annel. While the need to operate over interferenc e limited frequency selecti ve chann els is clear in many of the current and future communica tion tec hnologies, the theoretical situation is much less satisfying. Th e capacity region of the interference channel is still open (see [1] for short overvie w) even for the fixed c hannel two user c ase. In rece nt yea rs great ad vances in understand ing the situation for fla t c hannels und er weak interference have be en ach iev e d. It ca n be 1 School of Engineering, Bar -Ilan Uni versity , Ramat-Gan, 52900, Israel. 2 Faculty of EEMCS, Delft Univ ersity of T echnology . This work was supported by Intel Corporation and by the Netherlands Foundation of Science and T echnology . e-mail: leshema@eng.biu.ac.il . 2 shown tha t in this case treating the interference as no ise is almost optimal. On the o ther han d for the medium strength interferenc e as is typ ical in the wireless en v ironment, the simplest strategy is b y using orthogonal sign aling, e. g. TDMA/FDMA, for high spe ctral ef ficiency networks, or CDMA for very strong interference with low sp ectral efficiency per user . Moreover , seque ntial c ancelation techniqu es that are required for the be st known capac ity region in the medium interferenc e case [2] are only p ractical for small numb er of interferers. The interference c hannel is a conflict situa tion, a nd not ev ery achievable rate pair (from informational point of view) is actua lly reason able o perating po int for the users. This conflict is the reason to an alyze the interference ch annel us ing g ame the oretical tools. Much work has been done on c ompetiti ve game theory ap plied to frequency s electi ve interferenc e channel, with the early works of Y u et a l. [3] an d su bseque nt works of Scutari et al. (see [4] a nd the references therein). A particularly interes ting topic is the u se of g eneralized Nash g ames to the weak interference chan nel [5], and the algorithm in [6] whic h extends the FM-IWF to iterativ e p ricing under fi xed rate co nstraint. The fact tha t competiti ve strategies can result in significant degradation due to the p risoner’ s dilemma has be en called the price of ana rchy [7]. In the interference channel c ase, a simple c ase ha s be en analyz ed by La ufer and Les hem [8], wh o cha racterized the case s of prisoner’ s dilemma in interference limited channe ls. T o overcome the sub-optimality of the comp etiti ve ap proach we have two alternatives: Us ing repeated games or u sing cooperativ e game theory . Since mo st works on repeate d ga mes c oncentrated on flat fading chan nels, we will ma inly con centrate on coope rati ve game theoretic ap proaches . One of the earliest solutions for cooperative ga mes is the Nas h ba r gaining solution [9]. Many papers in recent years were d ev oted to analyzing the Nash ba r gaining solution for the frequency flat interferenc e chan nel in the SISO [10], [11], MISO [12], [13] an d MIMO c ases [14]. Interes ting extensions for log-con vex utility functions ap peared in [15]. Anothe r interesting application of the bar gaining technique s discusse d here is for multimedia distribution n etworks. Park and V an der Sch aar [16] used both Nash bargaining and the generalized Kalai-Smorodinski s olutions [17] for multimedia resource mana gement. An other a lternati ve cooperative mod el was explored in [18] wh ere, the coo peration b etween rational wireless play ers was studied u sing coa litional game theo ry by allowing the receivers to cooperate us ing joint d ecoding. In the co ntext of frequency se lecti ve interference chan nels much less res earch ha s be en d one. Han et al. [19 ] in a pioneering work, stud ied the Nas h B ar gaining und er FD M/TDM strategies and total power constraint. Unfortunately , the algorithms propos ed were only sub-optimal. Iterati ve su b-optimal algorithms to achieve Nash bargaining solution for spe ctrum allocation under average power c onstraint have been applied in [20]. Only rece ntly , we have man aged to overcome the difficulties by imposing a total PSD mask constraint [21], in order to obtain compu tationally efficient so lutions to the b ar gaining problem in the freque ncy selectiv e SISO and MIMO cas es under TDM/FDM strategies. Furthermore, it can be shown [22] that the PSD limited cas e ca n be used to deri ve a computationally efficient con verging algorithm also in the total power constraint case. A very interes ting problem is allowing the u sers to treat the interference as noise in s ome bands, while using orthogonal F DM/TDM strategies in others. This is a very c hallenging tas k, since the Na sh b ar gaining solution in volv e s a h ighly non-con vex power allocation problem, instea d of the simpler orthogona l signaling. 3 As disc ussed be fore, the frequency selec ti ve interference channel is very interesting both from practical point of view a nd from information theo retic p oint of view . W e ca n show that it h as many interesting aspec ts from game theoretic point of vie w , and that various le vels o f interference admit dif ferent types o f game theoretic technique. The pu rpose of this paper is to provide an overview of the v arious g ame theoretic techniques in volv e d in the ana lysis and algorithms u sed for frequ ency s electiv e interference chann els. W e de monstrate h ow game theory ca n be applied in spec ific scena rios and discuss signal processing aspec ts of the req uired ga me theoretic solutions . Our main g oal is to d iscuss in a tutorial manner s ome new applications of game theory to frequen cy selective interference ch annels. First, we outline various interference cha nnel sce narios with emph asis on frequ ency selectiv e channels . T hen we d iscuss the basic concep ts of game the ory required for analyzing thes e channe ls: Nas h equilibrium, strategies, Generalized Nash games a nd Nash b argaining theory . Using the se conce pts we discus s the s pecific translations into working strategies for the communica tion models a nd discu ss important signal proces sing aspec ts su ch as sp ectrum sens ing and centralized and distributed strategies. T o wrap up the discus sion with real life applications, end up with two ca se studies : DSL and W iMax where we de monstrate the gains o f the techniques o n real chann els. W e end up with c onclusions and future res earch directions. I I . I N T RO D U C T I O N T O I N T E R F E R E N C E C H A N N E L S Computing the ca pacity region of the interference ch annel is an ope n problem in information theory [23]. A go od overview of the results until 198 5 is given by van d er Meulen [1] a nd the references therein. The capacity region o f ge neral interference chan nel is not known ye t. However , in the last forty five yea rs of research some progres s has been made. T he b est known achievable region for the g eneral interference channe l is due to Han and K o bayash i [2]. The computation o f the Han and K ob ayashi formula for a general disc rete memoryles s channel is in general too c omplex. Recently large ad vance in obtaining upp er bounds o n the rate region hav e been o btained espe cially for the c ase of weak interference. A 2x 2 Gaus sian interference c hannel in standard form (after suitable normalization) is given by: x = H s + n , H = " 1 α 1 α 2 1 # (1) where, s = [ s 1 , s 2 ] T , and x = [ x 1 , x 2 ] T are samp led v a lues of the input a nd output signals, respectively . The noise vector n represents the ad diti ve Gaussian noises with zero mean and unit v ariance. The p owers of the inpu t sign als are con strained to b e less than P 1 , P 2 , respectively . Th e off-diagonal elements of H , α 1 , α 2 represent the d egree of interference prese nt. The major difference be tween the interference channe l a nd the multiple acces s channel is that both encoding and decoding of each chan nel are performed separately an d indepe ndently , with n o information sharing b etween receivers. The capacity region of the Gaussian i nterference channel with very strong interference (i.e., α 1 ≥ 1+ P 1 , α 2 ≥ 1 + P 2 ) is given by [24] R i ≤ log 2 (1 + P i ) , i = 1 , 2 . (2) 4 This surprising res ult shows that very strong interference dos e n ot reduc e the capa city of the u sers. A Gaussian interference ch annel is sa id to h av e strong interference if min { α 1 , α 2 } > 1 . Sato [25 ] deriv ed an achiev able c apacity region (inner bo und) of Gaussian interference chan nel as intersec tion o f two multiple acces s Gau ssian capac ity regions e mbedded in the interference channe l. The achiev able region is the intersection of the rate pa ir of the rectangular region of the very strong interference (2) and the following region: R 1 + R 2 ≤ log 2 (min { 1 + P 1 + αP 2 , 1 + P 2 + β P 1 } ) . (3) While the two u ser flat interferenc e ch annel is a well stud ied (although no t solved) p roblem, much les s is known in the frequen cy selective c ase. An N × N frequency s electiv e Gaus sian interference chan nel is given by: x k = H k s k + n k k = 1 , ..., K H k =     h 11 ( k ) . . . h 1 N ( k ) . . . . . . . . . h N 1 ( k ) . . . h N N ( k )     . (4) where, s k , and x k are sampled values of the input and output signal vectors at frequency k , respecti vely . The noise vector n k represents an additive wh ite Gaussian noise with z ero mean and un it v ariance. The power spectral den sity (PSD) of the input s ignals a re con strained to be less than p 1 ( k ) , p 2 ( k ) res pectiv ely . Alternati vely , only a total p ower co nstraint is given. The off-diagonal elements of H k , repres ent the degree of interference present at frequency k . The main difference between interference channel and a multiple acces s chan nel (MA C) is that in the interference c hannel, each component of s k is coded independen tly , and eac h receiv er has acces s to a single element of x k . Therefore, iterativ e dec oding schemes are muc h more limited, and typically impractical for large numb er of u sers. T o overcome this problem there are two simple strategies. When the interference is sufficiently weak, the common wisdom is to treat the interference as noise , and cod e a t a rate c orresponding to the total noise. When the interference is stronger , i.e, Signal to Interference Ratio (SIR) is s ignificantly lo wer than Signal to ad diti ve Noise Ratio (SNR), treating the interference as noise can b e highly ine f fi cient. On e of the s implest ways to d eal w ith me dium to strong interference chan nels is through orthogon al signaling. T wo extremely simple orthogon al scheme s are us ing FDM or TDM strategies. These tech niques a llo w a s ingle use r d etection (which will be as sumed througho ut this pa per) without the ne ed to complicated multi-user detection. Th e loss of thes e techniqu es compared to tec hniques requiring joint deco ding h as been thorough ly studied, e. g., [24 ] in the co nstant channe l cas e, showing degradation compared to the techniques requiring joint or seque ntial decod ing. Howev er , the wide spread us e of FDMA/TDMA as well as collision avoidance medium acc ess control (CSMA) tec hniques, make the an alysis of thes e techniques very important from prac tical point of v ie w as well. For frequency s electiv e channels (also kno wn a s ISI channe ls) we can combine both strategies by a llo w ing time v a rying allocation of the fr equency bands to the d if feren t use rs as shown in fig ure 1(b). 5 1 s 2 s 2 1 1 n 2 n 1 x 2 x (a) Interference channel (b) TDMA and j oint TDMA/OF DMA Fig. 1. (a) Standard form i nterference channel. (b) TDMA and joint TDMA/OFDMA In this paper we limit ourselves to joint FDM and T DM scheme wh ere an assign ment of disjoint portions of the freque ncy band to the several transmitters is ma de at eac h time instanc e. This tech nique is widely used in practice becau se s imple filtering can be used at the receivers to eliminate interference. All of these schemes as to op erate un der physical and regulation constraint like A verage power constraint or/and PSD mas k constraint. I I I . B A S I C C O N C E P T S O F C O O P E R A T I V E A N D C O M P E T I T I V E G A M E T H E O RY In this section we review the bas ic co ncepts of game theory in an abs tract s etting. Our focu s is o n concep ts that have been found to be relev a nt to the frequency selec ti ve interference channe l. W e begin with competiti ve game theory and then continue to describe the cooperative solutions. The reader is referred to the excellent books of [26], [27] and [28] for more de tails and for proofs of the main results mentioned h ere. A. Static co mpetitive games and the Nas h equilibrium An sta tic N player g ame in s trategic form is a three tuple ( N , A, u ) compose d of a set of p layers { 1 , ..., N } , a set of possible combinations of actions of each player deno ted by A = Q N n =1 A n , where A n is the s et of ac tions for the n ’ th player and a vector uti lity function u = [ u 1 , ..., u N ] , where u n ( a 1 , ..., a N ) : Q N n =1 A n →R is the utility of the n ’th play er w hen strategy vector a = ( a 1 , ..., a N ) has been played. The interpretation of u n is that player n rece i ves a pay of f of u n ( a 1 , ..., a N ) when the players have c hosen actions a 1 , ..., a N . Th e game is finite when for a ll n , A n is a finite set. A sp ecial type of c ompetiti ve game s a re the co nstant su m games. A g ame is cons tant s um, if for all action vectors a , P N n =1 u n ( a ) = c for some constant c . When the game is constant sum we ca n subtract c/ N from e ach utility and obtain a ze ro sum game that has the same p roperties as the original game . A 6 two players zero s um game is strictly compe titi ve s ince anything gaine d by one player leads to a loss to the o ther player . An impo rtant notion of s olution relev ant to ga mes is that o f a Na sh eq uilibrium. Definition 3.1: A vector of a ctions a = ( a 1 , ..., a N ) ∈ A is a Nas h equilibrium in pure s trategies if and on ly if for eac h pla yer 1 ≤ n ≤ N and for every a ′ = ( a ′ 1 , ..., a ′ N ) suc h tha t a ′ i = a i for all i 6 = n and a ′ n 6 = a n we h av e u n ( a ′ ) < u n ( a ) , i.e., ea ch playe r can only loose b y d eviati ng by itself from the equilibrium. The Na sh equ ilibrium in pu re strategies doe s not a lw ays exist as the follo wing examp le shows: Example I - A g ame with no pure strate gy Na sh equilibrium : C onsider the two p layers game d efined by the follo wing: A i = { 0 , 1 } . u i ( a 1 , a 2 ) = a 1 ⊕ a 2 ⊕ ( i − 1) , i.e., the first play er payo f f is 1 when actions are iden tical and 0 o therwise, while the s econd player’ s pay of f is 1 when the actions a re diff erent and 0 otherwise . Clearly , this game a lso kn own a s ma tching penn ies h as no Nash equilibrium in pu re strategies, since a lw a ys on e of the playe rs can improve his situation by c hanging his choice. Even wh en it exists, the Nash equ ilibrium in pu re strategies is not neces sarily unique, as the following example shows: Example II - A c ommunication game with infinitely many pu r e s trate gy NE . As sume that two u sers are sharing an A WGN multiple acce ss chann el (i.e., the accs s point can perform joint de coding of the us ers) y = x 1 + x 2 + z , (5) where z ∼ N (0 , σ 2 ) is a Ga ussian n oise random variable. Each u ser has power P . It is well known [23] that the rate region of this multiple ac cess chan nel is given by a pentago n defined by: R 1 ≤ 1 2 log  1 + P σ 2  = C max R 2 ≤ 1 2 log  1 + P σ 2  = C max R 1 + R 2 ≤ 1 2 log  1 + 2 P σ 2  = C 1 , 2 . (6) The corne rs A, B (see fig ure 2(a)) of the pentagon are A =  C max , C min  and B =  C min , C max  , whe re C min = 1 2 log  1 + P P + σ 2  , is the rate achiev a ble by as suming that the other user’ s signal is interference. Note tha t any point on the line co nnecting the points A, B is a chieved by time sharing be tween these two points. Each playe r n = 1 , 2 ca n choose a strategy 0 ≤ α n ≤ 1 that is the time s haring ratio b etween coding a t its rate at point A o r B . T he pay of f in this ga me is given by u n ( α 1 , α 2 ) = ( α n C max + (1 − α n ) C min if α 1 + α 2 ≤ 1 0 otherwise. (7) The reas on that the utilit y is 0 wh en α 1 + α 2 > 1 is that no reli able communication is pos sible, since the rate pair ac hiev ed is o utside the rate region. In this game a ny valid strategy point such that α 1 + α 2 = 1 is a Nash e quilibrium. If us er n reduc es its α n obviously its rate is lo wer since he transmits larger fraction of the time at the lower rate. If on the other hand he increa ses α n then α 1 + α 2 > 1 and bo th players achieve 0 . Hence the A WGN MA C ga me has infinitely many Nash equilibrium points. Similar game has 7 been use d in [29] where the fact that infi nitely many NE points exist is sh own. It is interesting to note that a similar MA C game for the fading ch annel has a un ique Nas h equilibrium p oint [30]. T o better understan d this g ame, we might loo k at the bes t respo nse dynamics . Best respons e a ction is the attempt of a play er to ma ximize its utility ag ainst a g i ven strategy vector . It is a well establishe d mean of distrib utively achieving the Nash Equ ilibrium. In the context of information theory , this strategy has been termed Iterati ve W ater-Filli ng (IWF) [3]. If in the multiple a ccess game the players use the b est response simultaneo usly , the first step would b e to transmit at C max . Ea ch player then receives 0 utility and in the next ste p re duces its ra te to C min , an d vice versa. The iteration never c on verges a nd the utility of each player is gi ven by 1 2 C min , worse than transmitting constantly at C min . Interestingly in this case, the sequential b est response lea ds to one of the po ints A, B , which are the (non axis) corners of the rate region. The moral of this example is that using the be st response strate gy should be done carefully even in multiple acce ss scen ario’ s s uch as in [31]. B. Pure a nd mixed strate gies T o overcome the first problem o f n o equilibrium in p ure strategies, the notion of mixed strategy has been propo sed. Definition 3.2: A mixed strategy π n for player n is a p robability distributi on over A n . The interpretation of the mixed strategies is that play er n choose s his a ction rand omly from A n according to the distrib ution π n . The p ayoff o f player n in a g ame where the mixed strategies π 1 , ..., π N are p layed by the playe rs is the expec ted v a lue of the utility u n ( π 1 , ..., π N ) = E π 1 × ... × π N [ u n ( x 1 , ..., x N )] . (8) Example III: Mixed strate gies in random acc ess ga me over multiple acc ess chann el T o d emonstrate the notion of mixed strate gy , we no w extend the multiple acces s game, into a random multi ple access game, where the players can choose with probability p n of working at rate C min and 1 − p n working at C max . This replaces the synch ronized TDMA strate gy in the previous game, with slotted random acc ess protocol. This formulation, a llo ws for two pure s trategies correspond ing to the corne r points A, B an d the mixed strategies amount to randomly choos ing between thes e points. T his game is a sp ecial cas e of the chicken dilemma (a termed propose d by B. Russe l, [32]), sinc e for e ach u ser it is better to chic ken out, then to obtain z ero rate, when both players cho ose the tough s trategies. Obviously from the pre viou s disc ussion, the points  C max , C min  and  C min , C max  are Nash equilibria. Simple computation sho ws tha t there is a u nique Na sh equilibrium in mixed strategy co rresponding to p 1 = p 2 = C min /C max . Interes tingly the rates ach iev e d by this random a ccess (mixed strategy) approac h is exactly  C min , C min  , i.e., the price paid for rand om ac cess is that bo th p layers a chieve their minimal rate, so s imple p -persistent rando m acces s p rovides no gain for the multiple acc ess chan nel. Follo wing Papa dimitriou [7 ] we ca n call this the p rice of random a ccess. 8 T ABLE I P A Y O FF S I N T H E M U LT I C H A N N E L R A N D O M A C C E S S G A M E I \ II 0 1 0 ` C min , C min ´ ` C min , C max ´ 1 ` C max , C min ´ (0 , 0) m ax C ]\ 2 R 1 R mi n m in ( , ) C C m ax C m in C max max ( , ) C C ]\ m in C A B (a) Multiple Access game −30 −20 −10 0 10 20 30 40 50 60 −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 h lim1 , h lim2 Vs. SNR SNR [dB] h [dB] T>R>P>N ; PD region T>P>R>N ; Deadlock region T>R>N>P ; Chicken region h lim1 h lim2 (b) Prisoner’ s dilemma r egion s Fig. 2. (a) Multiple Access Game with infinitely many pure strategy equilibria. Note that the parallel best response dynamics leads to unstable dynamics while the serial best respon se leads t o one of two ex tremal NE points. (b) Graph of h lim 1 , h lim 2 Vs. SNR, The solid line corresponds to h lim 1 and the dashed line corresponds to h lim 2 [8]. C. con v ex g ames An important sp ecial type of games which is important for the spe ctrum management problem is that of co n vex competiti ve ga mes. Definition 3.3: An N player game ( N , A, U ) is conv ex if each A n is compac t and co n vex and eac h u n ( x 1 , ..., x n ) is a conv ex function of x n for every choice of { x j : j 6 = n } . Con vex game s always ha ve Nash equilibrium [33]. A simple proo f c an be found in [27]. Con vex competitiv e games are espec ially important in the context o f s pectrum man agement, since the ba sic Gaussian interferenc e game forms a conv ex g ame. D. The pr isoner’ s d ilemma The prisoner’ s dilemma game is the major p roblem of the competiti ve app roach. It was first described by Flood a nd Dreshe r in 1950 a lmost immediately after the co ncept of Nas h equilibrium was publishe d [9]. For an overview of the prisoner’ s dilemma and its h istory s ee the excellent book of Poundstone [32] or in this jou rnal [34]. It turns ou t that this game has a uniqu e Nash equilibrium which is the stable po int of the game. Moreover this ou tcome is su boptimal to all player . T he e mer gence o f the prison er’ s dilemma 9 in simple symmetric interference channe ls ha s been discuss ed in [8]. In [10], [21] c haracterization of cases where c ooperativ e solutions are better for ge neral interferenc e chann els have been demon strated. W e briefly describe a simple case where the prisoner’ s dilemma oc curs [8]. W e assume a simplified two players game. Th e game is played over two frequency bands ea ch with symmetric interference cha nnel. The ch annel matrices o f this chan nel are H (1) = H (2) = H where | h 12 (1) | 2 = | h 21 (1) | 2 = | h 12 (2) | 2 = | h 21 (2) | 2 = h and h ii ( k ) = 1 . W e limit the d iscussion to 0 ≤ h < 1 . In our symmetric game both users have the s ame power constraint P a nd the power is a llocated by p 1 (1) = (1 − α ) P , p 1 (2) = αP, p 2 (1) = β P , p 2 (2) = (1 − β ) P . W e a ssume that the decod er treats the interference as noise a nd ca nnot dec ode it. The u tility for use r I giv en power allocation parame ters α, β is given by its achiev able rate C 1 = 1 2 log 2  1 + (1 − α ) S N R − 1 + β · h  + 1 2 log 2  1 + α S N R − 1 + (1 − β ) · h  (9) and simil arly for user II , we replace α, β . The set of strategies in thi s simplified ga me is { α, β : 0 ≤ α, β ≤ 1 } . There a re four alternativ es outcomes : • Both u sers se lect FDM resu lting in α = β = 0 . This is the reward (R) for c ooperation. • Player I s elects FDM while user II selects c ompetiti ve approach (IWF) res ulting in α = 0 , β = (1 − h ) / 2 is the resu lt of waterfilling by use r II when α = 0 ). This is wh en p layer I is naive (N) and the temptation for player II (T) • Player I s elects IWF wh ile us er II selec ts FDM resulting in α = (1 − h ) / 2 , β = 0 . This is the temptation for playe r I (T) a nd player II is the naive (N). • Both p layers se lect IWF res ulting in α = β = 1 2 . Th is is the pen alty for not co operating (P). T able II d escribes the payoffs of u sers I at fou r different levels of mutua l co operation (The pa yoffs of us er II a re the same with the in version of the coo perati ve/competetiv e roles). A prisoner’ s dilemma situation T ABLE II U S E R I PAYO FF S AT D I FF E R E N T L E V E L S O F M U T UA L C O O P E R A T I O N user II is fully cooperati ve user II is fully competing ( β = 0) “ β = (2 α − 1) h +1 2 ” user I is fully cooperati ve ( α = 0) 1 2 log 2 ` 1 + 1 S N R − 1 ´ 1 2 log 2 „ 1 + 1 S N R − 1 + (1 − h ) 2 h « user I is fully competing “ α = (2 β − 1) h +1 2 ” 1 2 log 2 „ 1 + 1+ h 2 S N R − 1 « + 1 2 log 2 „ 1 + 1 − h 2 S N R − 1 + h « log 2 “ 1 + 1 2 S N R − 1 + 1 2 h ” is de fined by the follo wing p ayoff relati ons for b oth pla yers - T > R > P > N . It is easy to sh ow that the Nash equilibrium point in this case is that both players will defec t ( P ). This is caus ed b y the fact that giv e n the other user act the be st res ponse will be to defect (sinc e T > R and P > N ). Obviously a better strategy (which makes this game a dilemma) is mutual cooperation (since R > P ). W e can no w analyze 10 this simple ga me. It turns out that there are two func tions, h lim 1 ( S N R ) , h lim 2 ( S N R ) as described in Figure 2(b) an d only three pos sible situations [8]: • (A) T > P > R > N , for h < h lim 1 . • (B) T > R > P > N , for h lim 1 < h < h lim 2 . • (C) T > R > N > P , for h lim 2 < h . The payoff relations in (A) c orresponds to a game called ”Deadlock” . In this game there is no d ilemma, since no matter wh at the othe r player does, it is better to defect ( T > R and P > N ), s o the Nash equilibrium point is P . Since P > R thu s there is no reason to co operate. The ma ximum sum rate is also P because 2 · R > T + N and P > R . The payoff relations in (B) correspo nd to the prisoner’ s dilemma s ituation. While the Nash equilibrium point is P , eac h user’ s maximum payo f f is ac hiev ed by R . In this region the FDM strategy will achieve the individual maximum rate. The last p ayoff relations (C) correspo nds to a game c alled ”Chicken”. This game ha s two Na sh equilibrium points, T an d N . This is ca used by the fact tha t for e ach of the o ther playe r’ s strategies the op posite respons e is preferred (if the other cooperates it is be tter to defect since T > R , while if the o ther defects it is better to cooperate since N > P ). The maximum rate sum point is s till at R (since R > P an d 2 · R > T + N ) thus, a gain FDM will ach iev e the maximum rate s um while IWF will not. E. Generalized Nash games Games in s trategic forms are very important part o f game theory , and have many a pplications. Howe ver , in some case s the notion of a g ame does no t capture all the c omplications in volved in the interaction between the players. Arrow and Deb reu [35] defi ned the co ncept of a g eneralized Na sh game and gen eral- ized Nash equilibrium. In strategic form games, each player has a set of s trategies that is independent from the actions of the other players. However , in reality sometimes the ac tions of the players are c onstrained by the actions of the other players . The gene ralized Nash game or abs tract econo my conc ept, captures exactly this de penden ce. Definition 3.4: A ge neralized Nas h game with N playe rs, is de fined as follo ws: For each player n we h av e a set of poss ible ac tions X n and a set function K n : Q m 6 = n X m → P ( X n ) where P ( X ) is the power se t of X , i.e, K n ( h x m : m 6 = n i ) ⊆ X n defines a sub set o f poss ible actions for player n whe n other p layers play h x m : m 6 = n i . u n ( x ) is a utility func tion de fined o n all tuples ( x 1 , ..., x N ) such that x n ∈ K n ( x m : m 6 = n ) . Similarly to the defin ition of a Nash equilibrium, we can defin e a g eneralized Nash e quilibrium: Definition 3.5: A gene ralized Nash eq uilibrium is a point x = ( x 1 , ..., x N ) suc h that for a ll n x n ∈ K n ( h x m : m 6 = n i ) , and for all y = ( y 1 , ..., y n ) s uch that y n ∈ K n ( h x m : m 6 = n i ) a nd y m = x m for m 6 = n u n ( x ) ≥ u n ( y ) . Arro w and Debreu [35] proved the existence of a generalized N ash equ ilibrium under very limiting conditions. Their result was ge neralized, and currently the best result is by Ichiishi [36]. Th is result can be use d to a nalyze certain fixed rate a nd mar gin versions of the iterati ve water-filli ng a lgorithm [5] as will be shown in the next s ection. 11 F . Nas h bargaining theory The ”Prisoner Dilema” h ighlights the drawbacks of competition, du e to mutual mistrust of play ers. W e therefore ask ourselves whe n w ould a coope rati ve s trategy be p referable to a competiti ve strategy . It is ap parent that the e ssential condition for coope ration is that it should gen erate a s urplus, i.e. an extra gain which can be divided betwee n the pa rties. Bargaining is es sentially the proc ess of distrib uting the surplus. Th us, bargaining is foremost, a process in which pa rties see k to rec oncile contradictory interests and values. The main qu estion is if a ll players commit to following the rules , wha t reasona ble outcome is acce ptable by all parties. Therefore, the players have to agree on a ba r gaining mechanism which they will no t aba ndon during the negotiations. The bargaining results ma y be aff ected b y several f a ctors like the power of eac h party , the amount of information av ailable to each of the players, the de lay response of the players, etc. Nash [9 ], [37] introduced an axiomatic a pproach that is based o n properties that a v a lid outcome of the bargaining should satisfy . This a pproach proved to b e very u seful since it succe eded in choosing a unique s olution through a sma ll number of simple cond itions (axioms), thus saving the need to pe rform the complicated ba r gaining proces s, once all pa rties accep t these cond itions. W e now define the bargaining problem. An n-playe r bargaining problem is described a s a pair h S, d i , where S is a c on vex s et in n -dimensiona l Euclidea n spac e, con sisting of a ll feas ible sets of n -players utilities that the players c an receive whe n coope rating. d is an element of S , called the disa greement point, represen ting the outco me if the p layers fail to reach an a greement. d can also can be viewed a s the u tiliti es resulting from non-co operativ e strategy (compe tition) betwe en the p layers, whic h is a Nash Equilibrium o f a compe titi ve game. The as sumption that S is a conv ex se t, is a reas onable ass umption if b oth players select c ooperative strategies, since, the p layers can u se alternating or mixed strategies to achieve co n vex combinations of pu re bargaining outcomes . Gi ven a bargaining problem we say that the vector u ∈ S is Pareto optimal if for all w ∈ S if w ≥ u (coordina te wise) then w = u . A s olution to the bargaining problem is a fun ction F defin ed on all ba r gaining problems s uch tha t F ( h S, d i ) ∈ S . Nash’ s axiomatic approac h is b ased o n the following fou r a xioms that the so lution function F s hould satisfy: Linearity (LIN). As sume that we consider the bargaining problem h S ′ , d ′ i o btained from the problem h S, d i by transformations: s ′ n = α n s n + β n , n = 1 , ..., N . d ′ n = α n d n + β n . The n the solution satisfies F i ( h S ′ , d ′ i ) = α n F n ( h S, d i ) + β n , for all n = 1 , ..., N . Symmetry (SYM) . If two players m < n a re iden tical in the s ense that S is symmetric with res pect to cha nging the m ’th and the n ’ th coordinates, then F m ( h S, d i ) = F n ( h S, d i ) . Equ i valently , play ers which h av e identical ba r gaining preferenc es, get the same outcome at the e nd of the bar gaining process . P areto optimality (P AR ) . If s is the outcome of the bargaining then no other state t exists such that s < t (coo rdinate wise). Independ ence of irrele vant a lternatives (IIA) . If S ⊆ T and if F ( h T , d i ) = ( u ∗ 1 , u ∗ 2 ) ∈ S , then F ( h S, d i ) = ( u ∗ 1 , u ∗ 2 ) . 12 Surprisingly these simply axioms fully characterize a bargaining solution titled Nash’ s Bargaining solution. Nas h’ s theorem ma y be stated a s follows [9]. Based on this axioms a nd defi nitions we n ow can state Na sh’ s theorem. Theorem 3.1: Assume that for all S is compact and co n vex, then there is a unique bargaining solution F ( h S, d i ) , sa tisfying the axioms INV , SYM, IIA, and P AR, which is given by F ( h S, d i ) = ( s 1 , .., s N ) = arg max ( d 1 ,..,d N ) ≤ ( s 1 ,..,s N ) ∈ S N Y n =1 ( s n − d n ) . (10) Before c ontinuing the study o f the ba r g aining solution, we a dd a ca utionary word. While, Nash ’ s axioms are mathematically appealing, they may not be accep table in some scena rios. Indeed vari ous alternati ves to these axioms have bee n propose d that lea d to o ther solution concep ts. Extens i ve survey of these solutions ca n b e found in [17]. In the communica tion context, the axioms propose d by Boche at. el. [11] lead to a generalized NBS s olution. More resu lts of ge neralized ba r g aining for frequen cy selective channels will be discus sed in [22]. T o demon strate the use of the Nas h ba r gaining solution to interference ch annels, we begin with a simple example for flat chann els. Example III : Co nsider tw o players c ommunicating over a 2x2 memoryless Ga ussian interference channe l with band width W = 1 , as desc ribed in (1. As sume for simplicity that P 1 = P 2 = P . W e assume that no rece i ver ca n perform joint deco ding, a nd the utility of player n , U n , is giv en by the achiev able rate R n . Similarly to the prisoner’ s dilemma examp le, if the playe rs choo se to c ompete then the compe titi ve strategies in the interference game are g i ven by flat power allocation and the resu lting rates a re given by R nC = 1 / 2 log 2  1 + P 1+ α 2 n P  . Since the rates R nC are a chieved b y competitiv e strategy , player n would coo perate only if he will obtain a rate higher than R nC . The ga me theoretic rate region is define d by pair rates high er than the compe titi ve rates R nC (see figure 3(b)). Ass ume that the playe rs agree on us ing only F DM coope rati ve strategies. i.e. p layer n uses a fraction of 0 ≤ ρ n ≤ 1 of the ba nd. If we consider only Pareto optimal s trategy vectors, then obviously ρ 2 = 1 − ρ 1 . Th e rates obtained by the two use rs are given by R n ( ρ n ) = ρ n 2 log 2  1 + P ρ n  . The two us ers will b enefit from FDM type of coope ration as long as R nC ≤ R n ( ρ 1 ) , n = 1 , 2 . (11) The FDM Nas h bargaining solution is giv en by solving the problem max ρ F ( ρ ) = max ρ ( R 1 ( ρ ) − R 1 C ) ( R 2 ( ρ ) − R 2 C ) . (12) The co operativ e solution for this fl at ch annel model was deriv ed in [10], [21]. I V . A P P L I C A T I O N O F G A M E T H E O RY T O F R E Q U E N C Y S E L E C T I V E I N T E R F E R E N C E C H A N N E L S In this section we ap ply the ideas pres ented in previous sec tions to analyz e the frequency sele cti ve interference ga me. W e provide examples for bo th competitiv e and coope rati ve game theoretic con cepts. 13 A. W ate rfilling base d solutions an d the Nash eq uilibrium T o analyze the competitiv e approa ch to frequency selec ti ve interference cha nnels, we first defin e the discrete-frequency Ga ussian interferenc e ga me whic h is a discrete version of the game de fined in [3]. Let f 0 < · · · < f K be a n increa sing seq uence o f freque ncies. Let I k be the clos ed interval giv e n by I k = [ f k − 1 , f k ] . W e now define the ap proximate Gaus sian interference game deno ted by GI { I 1 ,...,I K } . Let the players 1 , . . . , N operate o ver K pa rallel chan nels. As sume that the K cha nnels hav e coupling functions h ij ( k ) . As sume that user i is a llo wed to transmit a total power of P i . Eac h play er can transmit a power vector p n = ( p n (1) , . . . , p n ( K )) ∈ [0 , P n ] K such that p n ( k ) is the power transmitted in the interval I k . Therefore we have P K k =1 p n ( k ) = P n . Th e equa lity follows from the fact that in a non -cooperativ e scena rio all use rs w ill use all the a vailable power . T his impli es that the se t of power distributions for all users is a closed c on vex subset of the hy percube Q N n =1 [0 , P n ] K giv e n by: B = N Y n =1 B n (13) where B n is the set of admissible power distributi ons for player n given by: B n = [0 , P n ] K ∩ ( ( p (1) , . . . , p ( K )) : K X k =1 p ( k ) = P n ) (14) Each p layer cho oses a PSD p n = h p n ( k ) : 1 ≤ k ≤ N i ∈ B n . Le t the payoff for us er i b e given by: C n ( p 1 , . . . , p N ) = K X k =1 log 2  1 + | h n ( k ) | 2 p n ( k ) P | h nm ( k ) | 2 p m ( k ) + σ 2 n ( k )  (15) where C n is the capa city av ailable to play er n given power distrib utions p 1 , . . . , p N , chan nel response s h n ( f ) , crosstalk coup ling functions h mn ( k ) and σ 2 n ( k ) > 0 is external no ise p resent at the i ’t h c hannel receiv er at freque ncy k . In cases whe re σ 2 n ( k ) = 0 capacities might become infin ite us ing FDM strategies, howe ver this is non-ph ysical s ituation d ue to the rec ei ver no ise that is always pres ent, ev en if sma ll. Eac h C n is c ontinuous in all variables. Definition 4.1: The Gaussian Interference game GI { I 1 ,...,I k } = { N , B , C } is the N players n on-cooperative game with payoff vector C = ( C 1 , . . . , C N ) where C n are de fined in (15) and B is the strategy set defi ned by (13). The interference game is a conv ex non-co operativ e N-pe rsons game , since e ach B i is compac t an d con vex and each C n ( p 1 , ..., p N ) is co ntinuous and con vex in p n for any value of { p m , m 6 = n } . The refore it always ha s a Na sh equilibrium in p ure strategies. A pres entation of the proof in this case using waterfilling interpretation is given in [38]. A much harde r problem is the un iqueness o f N ash equilibrium points in the water -filling game. This is very important to the stability of the waterfilling strategies. A first result in this direction h as bee n giv e n in [39]. A more g eneral an alysis o f the con vergence has been giv e n in [3], [40],[41],[42] a nd [43]. Even the uniqu eness of the Nash equilibrium, does not imply a stable dyn amics. Scutari et al. [4] provided co nditions for co n vergence. Bas ically , they us e the Ban ach fixed p oint the orem, and require that 14 the waterfilling res ponse will be a co ntracti ve mapping . The waterfilling proc ess has s ev eral versions: The sequential IWF is performed by a sing le player a t each step. Th e parallel IWF is performed simultaneously by all p layers a t each step, a nd the a synchron ous IWF is performed in a n arbitrary order . For good disc ussion of the con ver g ence of these tech niques see [4]. It s hould be emph asized that some cond itions on the interference chann el matrices a re indeed required. A simple condition is s trong diagonal d ominance [3], a nd other papers relaxed the se assump tions. In all typ ical DSL sce narios, the IWF algorithms c on verge. Howe ver , the c on vergence conditions are no t a lw ays met, even in very simple cases , as the follo wing examp les shows: Example IV - Divergence of the parallel IWF . W e consider a Gauss ian interference game with 2 tones and 3 players. Each player ha s total p ower P . The signa l rece i ved by each rec eiv e r is just y n ( k ) = P 3 m =1 x m ( k ) + z n ( k ) where z n ( k ) N  0 , σ 2 ( k )  , where the noise in the s econd ba nd is stronger satisfying σ 2 (2) = P + σ 2 (1) . W e assume simultaneous waterfilli ng is performed (Similar examples ca n be co nstructed for the sequen tial IWF a lgorithm, but they are omitted due to spa ce limitations). At the first stage, all us ers put all the ir p ower into frequen cy 1 , by the first ine quality . At the sec ond s tage all users see noise an d interference power o f 2 P + σ 2 (1) at the fi rst frequency , wh ile the interferenc e at the second freque ncy is σ 2 (2) = P + σ 2 (1) . Since even when all power is pu t into freque ncy 2 the total power + n oise is below the noise level at fr equency 1 all users will mov e their total power to frequency 2 . This will con tinue, with all use rs alternating betwee n frequenies simultaneously . The a verage rate o btained by the s imultaneous iterativ e waterfilling is 1 4 log  1 + P 2 P + σ 2 (1)  + 1 4 log  1 + P 3 P + σ 2 (1)  Nash equilibrium exists in this c ase, for example, two users us e frequency 1 and o ne use r us es frequency 2 resulting in a rate 1 2 log  1 + P P + σ 2 (1)  for eac h use r . The previous examp le demonstrated a simple condition where one of the w ater- filling sche mes di verges. Howe ver , the re a re multiple NE points. The situation ca n be e ven worse. Th e following ch annel is a frequency selective chann el, with a single NE in the Gaussian interferenc e game. Howe ver non of the water filling scheme s con ver ges. Example V - Divergence of all waterfilling app r oaches . W e provide now a se cond example, where both the s equential and the parallel IWF div er ge, even thou gh there is a uniqu e NE point. As sume that we have two chann els where the chann el matrices H ( k ) , k = 0 , 1 a re equa l and given by: H ( k ) =    1 0 2 2 1 0 0 2 1    (16) and the noise at the first tone is σ 2 and a t the se cond tone σ 2 + P . Each us er has total power P . This situation might occur when the re is a strong interferer at tone 2 while the receivers a nd transmitters are 15 T3 R2 T2 T1 R1 R3 Omni Jammer Tone 2 Skyscraper (a) Channel where the IWF fails to con ver ge 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 FDM rate region, NE and NBS Rate 1 Rate 2 FDM boundary Nash Equilibrium Nash bargaining solution (b) Game theoretic rate region Fig. 3. (a) A frequenc y selective channel with unique NE, where both sequential and parallel IWF di ver ge. (b)The Game Theoretic Rate region is defined by the set of rates which are better t han the competitive equilibrium. The boundary of the region is exactly the set of Pareto-optimal points. The Hyperbola tangent to the rate region defines the Nash bargaining solution. located on the sides o f a triangle, e ach use r transmitting to a receiver n ear the next trans mitter a s in figure 3(a). Wh en the first user allocates its power it puts all it s power at the first frequency . The second user chooses tone 2 . The third p layer puts a ll its p ower at frequency 1 , but this g enerates interference to user 1 which migrates to frequency 2 and we obtain that the us ers chan ge their transmission tone at eac h step. In the simultane ous IWF all u sers will choos e ch annel 1 and then migrate together to cha nnel 2 and back. It is interesting that this ga me has a u nique eq uilibrium, whe re ea ch use r allocate s two thirds of the p ower to frequen cy 1 and one third of the power to frequency 2. Still all iterative scheme s di verge. B. Pricing me chanisms for re gulating distributed solutions T o o vercome some of the problems of the competiti ve be havior , re gulation can play an important role. A generalization of the RA-IWF a lgorithm is the band preference (BPSM) algo rithm [44] in which each user so lves the following problem in pa rallel (or s equentially) to the o ther use rs: max p n (1) ,...,p n ( K ) P K k =1 c n ( k ) log 2  1 + | h nn ( k ) | 2 p n ( k ) P m 6 = n | h nm ( k ) | 2 p m ( k )+ σ 2 n ( k )  Subject to P n = P K k =1 p n ( k ) . (17) In the BPSM algo rithm the total rate is rep laced by a weighted s um of the rates at each freque ncy . The weights ca n be provided by the regulator to limit the use of ce rtain bands by s trong users, s o that us ers that suf fers sev ere interference b ut d o no t a f fec t o ther use rs will be p rotected. T his resu lts in waterfilling against a compe nsated noise level. Alternati ve a pproach to the BPSM is u sing gene ralized Nas h ga mes. T he basic approa ch h as been proposed in [3] an d termed Fixed Margin IWF . Each user has both power constraint, a target noise margin and a de sired rate. The user minimizes its power as long a s it a chiev es its target rate. This is a 16 generalized Nash g ame, firs t analyzed by Pang et al. [5], who p rovided the first conditions f or con vergence. Noam an d Leshe m [6] propose d a gene ralization of the FM-IWF termed Iterati ve Power Pricing (IPP). In their solution, a weighted power is minimized, whe re frequency bands wh ich have higher capac ity are ”reserved” to players with longer lines, through a line d epende nt pricing mec hanism. The users iteratively optimize their power allocation so as their r ate and total power constraint are maintained while minimi zing the total weighte d power . It ca n be shown that the cond itions of Pang et al. ca n also be use d to analyze the IPP algorithm. For bo th the BPSM and the IPP approa ches, s imple pricing schemes that a re adapted to the DSL sc enario h av e b een propose d. The ge neral que stion of finding go od pricing sch emes is still open, but would require to combine physical modeling o f the cha nnels as well a s insight into the utiliti es as func tion of the des ired rate. Even the a utonomous spe ctrum balanc ing algorithm (ASB) [45] c an be viewed as a gen eralized Nas h game, where the utility is giv en by the rate of a fi ctitious reference line, and the s trategy sets sh ould satisfy b oth rate an d power constraint. Th e drawback of the ASB a pproach here, is find ing a referen ce line which serves as a good utility function. C. Bargaining o ver frequency selective channels un der mask c onstraint In this sec tion we de fine the co operativ e game correspo nding to the joint FDM/TDM ach iev able rate region for the frequency s electi ve N − user interference cha nnel. For simplicity of pres entation we limit the deriv a tion to the two playe r ca se und er PSD mask constraint. In [19] Nash b argaining solution was used for resou rce alloca tion in OFDMA systems. The goal was to maximize the overall sy stem rate, under cons traints on ea ch use rs minimal rate requiremen t and total trans mitted p ower . Howe ver , in that paper the solution was used only a s a measure of f a irness. Therefore, the point of disagreemen t was not taken as the Nash e quilibrium for the competiti ve game, but an arbitrary R min was us ed. T his ca n resu lt in a non-feas ible problem, and the propos ed a lgorithm might beco me uns table. An alternative approac h is based on PSD ma sk c onstraint [21] in co njunction with gene ral bargaining theory originally developed by Nash ([9],[37]). Ba sed on the solution for the PSD limited case, computing the Nash B ar gaining solution under total power constraint can then be s olved efficiently as well [22]. In orde r to keep the discus sion simple w e conc entrate the discu ssion on the two user PS D mask limited c ase. In real ap plications, the regulator limits the PS D mask and not only the total power constraint. Let the K chan nel matrices at frequencies k = 1 , ..., K be g i ven by h H k : k = 1 , ..., K i . Playe r is allowed to transmit at ma ximum power p n ( k ) in the k ’t h frequency bin. In competitiv e scen ario, und er mask constraint, all players transmit at the maximal power they c an use. Thus, player n choos e the PSD, p n = h p n ( k ) : 1 ≤ k ≤ K i . The p ayoff for user n in the non-co operativ e game is therefore giv en by: R nC ( p ) = K X k =1 log 2 1 + | h nn ( k ) | 2 p n ( k ) P m 6 = n | h nm ( k ) | 2 p m ( k ) + σ 2 n ( k ) ! . (18) Here, R nC is the capacity av ailable to playe r n gi ven a PSD mask co nstraint distri buti ons p . σ 2 n ( k ) > 0 is the no ise pres ents a t the n ’th receiver at frequency k . Note that without loss o f generality , and in order to simplify notation, we ass ume that the width of e ach bin is normalized to 1. W e n ow define the cooperative game G T F (2 , K, p ) . 17 Definition 4.2: The FDM/TDM game G T F (2 , K, p ) is a game betwee n 2 players transmitting over K frequency bins un der c ommon PSD mas k co nstraint. Ea ch player ha s full knowledge of the chann el matrices H k and the followi ng conditions hold: 1) Player n transmits using a PS D limited by h p n ( k ) : k = 1 , ..., K i . 2) The play ers are us ing coordinated FDM/TDM strategies. The strategy for player 1 is a vector α 1 = [ α 1 (1) , ..., α 1 ( K )] T where α 1 ( k ) is the proportion of time playe r 1 us es the k ’th freque ncy channe l. Similarly , the strategy for p layer 2 is a vector α 2 = [ α 2 (1) , ..., α 2 ( K )] T . 3) The u tilit y of the players is given by R n ( α n ) = K X k =1 α n ( k ) R n ( k ) = K X k =1 α n ( k ) log 2  1 + | h 11 ( k ) | 2 p n ( k ) σ 2 n ( k )  . (19) By Pareto optimality of the Na sh Bargaining s olution for ea ch k , α 2 ( k ) = 1 − α 1 ( k ) , so we will only refer to α = α 1 as the strategy for b oth players. Note, that interference is av oided by time sharing at each frequen cy b and, i.e only o ne player transmits with maximal power at a giv en frequency bin at any time. The a llocation o f the spec trum using the vector α induces a simple con vex op timization p roblem that ca n be posed a s follows max ( R 1 ( α ) − R 1 C ) ( R 2 ( α ) − R 2 C ) subject to: 0 ≤ α ( k ) ≤ 1 ∀ k , R nC ≤ R n ( α ) ∀ n . (20) since the log of the Nash function (20) is a con vex function the overall problem is con vex. Hen ce, it can be solved e f ficiently u sing KKT cond itions [21]. Assuming that a fea sible s olution exists it follows from the KK T conditions that the allocation is d one a ccording to the following rules: 1) The two players are sha ring the frequency bin k , ( 0 < α ( k ) < 1 ) if R 1 ( k ) R 1 ( α ) − R 1 C = R 2 ( k ) R 2 ( α ) − R 2 C . (21) 2) Only player n is using the freque ncy bin k , ( α n ( k ) = 1 ), if R n ( k ) R n ( α ) − R nC > R 3 − n ( k ) R 3 − n ( α ) − R 3 − n,C . (22) These rules can b e further simplified. Let L k = R 1 ( k ) /R 2 ( k ) be the ra tio be tween the rates at each frequency bin. W e can sort the freque ncy bins in de creasing o rder a ccording to L k . F r om now o n we assume that when k 1 < k 2 then L k 1 > L k 2 . If all the values o f L k are dis tinct then there is at most a s ingle frequency bin tha t ha s to be shared between the two play ers. Since o nly one bin c an satisfy equation (21), let us denote this frequency bin as k s , the n all the frequency bins 1 ≤ k < k s will only be used by pla yer 1 , wh ile all the frequen cy bins k s < k ≤ K will be use d by p layer 2 . The freque ncy bin k s has to be shared a ccording to the rules. W e now have to find the frequency bin that has to be shared betwe en the playe rs if there is a solution. Le t us defin e the surplus of players 1 an d 2 when using Na sh ba r g aining solution as A = P K m =1 α ( m ) R 1 ( m ) − R 1 C , and B = P K m =1 (1 − α ( m )) R 2 ( m ) − R 2 C , respectiv ely . No te that the 18 ratio, Γ = A/B is independen t of frequ ency an d is set by the optimal assign ment. A-priori Γ is un known and ma y not exists. W e are now ready to d efine the optimal assignme nt of the α ’ s. Let Γ k be a moving thres hold defi ned by Γ k = A k /B k . where A k = k X m =1 R 1 ( m ) − R 1 C , B k = K X m = k + 1 R 2 ( m ) − R 2 C . (23) A k is a monotonica lly inc reasing seque nce, while B k is monotonically decrea sing. He nce, Γ k is also monotonically increas ing. A k is the surplus of p layer 1 when frequ encies 1 , ..., k are alloca ted to player 1 . Similarly B k is the surplus of p layer 2 when freque ncies k + 1 , ..., K are a llocated to p layer 2 . Let k min = min k { k : A k ≥ 0 } , and k max = min k { k : B k < 0 } . Since we a re interested in feasible NBS , we must have positiv e su rplus for both users. Th erefore, us ing the allocation rules, we obtain k min ≤ k max and L k min ≤ Γ ≤ L k max . The sequen ce { Γ m : k min ≤ m ≤ k max − 1 } is s trictly inc reasing, and always p ositi ve. While the thresho ld Γ is unknown, o ne c an u se the sequen ces Γ k and L ( k ) to find the c orrect Γ . If there is a Nash ba r gaining solution, let k s be the frequency bin that is shared by the players. Then, k min ≤ k s ≤ k max . Sinc e, both players must h av e a positiv e gain in the game ( A > A k min − 1 , B > B k max ). Let k s be the sma llest integer such that L ( k s ) < Γ k s , if such k s exists. Otherwise let k s = k max . Lemma 4 .1: The following two stateme nts provide the solution 1 If a Nash b ar gaining solution exists for k min ≤ k s < k max , then α ( k s ) is giv en by α ( k s ) = max { 0 , g } , where g = 1 + B k s 2 R 2 ( k s )  1 − Γ k s L ( k s )  . (24) 2 If a Nash bargaining solution exists an d there is no s uch k s , the n k s = k max and α ( k s ) = g . Based on the pervious lemmas the algorithm is des cribed in table III. In the first s tage, the algo rithm computes L ( k ) and sorts them in a non increasing o rder . Then k min , k max , A k , and B k are computed. In the s econd stage the algo rithm computes k s and α . T o de monstrate the algorithm we c ompute the Nash bargaining for the follo wing examp le: Example V : Cons ider two play ers c ommunicating over 2x2 memoryless Gaus sian interference channel with 6 freque ncy bins . The players’ rates if they are not c ooperate is R 1 C = 15 , an d R 2 C = 10 . The feasible rates R 1 ( k ) and R 2 ( k ) in each freque ncy bin with no interference are given in T a ble IV after sorting the frequency bins with res pect to L k . W e n ow have to compute the surplus A k and B k for e ach players. If NBS exists then the playe rs mu st have postiv e s urplus, thus, k min = 2 , and k max = 4 . S ince, k = 4 is the first bin such that Γ k > L k , we can conclude that k s = 4 , and α = 0 . 33 (using lemma 4.1) . Thus, players 1 is using frequency bins 1 , 2 , and 3 , and using 1 / 3 of the time frequency bin 4 . Th e total rate of players 1 and 2 are 40 1 3 and 48 respectively . W e can also gi ve a geo metrical interpretation to the solution. In Figure 4(a) we draw the feasible total rate that play er 1 can obtain as a function of the total rate o f player 2 . The en closed a rea in blue, is the a chiev a ble utilities set. Since, the freque ncy bins a re so rted acc ording to L k the s et is conv ex. The 19 T ABLE III A L G O R I T H M F O R C O M P U T I N G T H E 2 X 2 F R E Q U E N C Y S E L E C T I V E N B S : Initialization: Sort the ratios L ( k ) in decreasing order . Calculate t he values of A k , B k and Γ k , k min , k max , If k min > k max no NBS exists. Use competitive solution. Else For k = k min to k max − 1 if L ( k ) ≤ Γ k . Set k s = k and α ′ s according to the lemmas-This is NBS. St op End End If no such k exists, set k s = k max and calculate g . If g ≥ 0 set α k s = g , α ( k ) = 1 , for k < k max . Stop. Else ( g < 0 ) There is no NBS. Use competitiv e solution. End. End k 1 2 3 4 5 6 R 1 14 18 5 10 9 3 R 2 6 10 5 15 19 19 L ( k ) 2.33 1.80 1.00 0.67 0.47 0.16 A k -1 17 22 32 41 44 B k 58 48 43 28 9 -10 Γ k -0.02 0 .35 0.51 1.14 4.56 -4.40 T ABLE IV T H E R AT E S O F T H E P L AY E R S I N E AC H F R E Q U E N C Y B I N A F T E R S O RT I N G . point R C = ( R 2 C , R 1 C ) = (10 , 15) is the point of disagreement. If the point R C is inside the ac hiev ab le utility se t there is a solution. The slope of the bou ndaries of the ac hiev ab le utilities s et with respec t to the − x ax is is L k . Th e vector R C − B conn ects the point R C and the point B is with the sa me slope with respe ct the x a xis, this is the g eometrical interpretation of (21). The area o f the wh ite rec tangular is the value of Na sh’ s produc t function. The res ults ca n be generalized in several d irections: First, if the values { L ( k ) : k = 1 , ..., K } are n ot all distinct then if there is a solution one ca n al ways find alloca tion such that at most a single frequency has to be shared. Second , in the general cas e of N players the op timization problem h as similar KKT con ditions a nd can be solved using con vex optimization algorithm. Moreover , the optimal solution has at most  K 2  frequencies that are shared be tween diff erent players . This sugg ests, that the optimal FDM NBS is very 20 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 80 R 2 R 1 R C B (a) The FDM/TDM rate region (b) Price of Anarchy for 32 parallel Rayleigh fading channels Fig. 4. (a) Feasible FDM rate region (blue area), Nash Barg aining Solution (the area of the white rectangular) (b) P er user price of anarchy for frequenc y selecti ve Rayleigh fading channel. SNR=30 dB. close to the joint F DM/TDM s olution. It is obtained by allocating the commo n frequencies to one of the users. Th ird, while the method de scribed above fits well to stationary chann els, the method is also us eful when only fading statistics is k nown. In this cas e the cod ing strategy will chang e, and the a chiev ab le rate in the competitiv e cas e and the coop erati ve ca se are given by ˜ R nC ( p i ) = P K k =1 E h log 2  1 + | h nn ( k ) | 2 p n ( k ) P m 6 = n | h nm ( k ) | 2 p m ( k )+ σ 2 n ( k ) i ˜ R n ( α n ) = P K k =1 α n ( k ) E h log 2  1 + | h nn ( k ) | 2 p n ( k ) σ 2 n ( k ) i , (25) respectively . All the re st of the d iscussion is unch anged, replac ing R nC and R n ( α n ) by ˜ R nC , ˜ R i ( α n ) respectively . This is particularly attractiv e , when the comp utations are done in distributed way . In this ca se only channe l state distributions ar e se nt between the un its . Hence the time scale for this data exch ange are much longer . This implies that method can be use d without a c entral con trol, by exchange of parameters between the units at a very low rate. Fourth, computing the NBS u nder total power co nstraint is more difficult to solve. Several ad-hoc techniques h av e bee n propos ed in the literature. Rec ently , it was shown that for this cas e there exist a n algorithm which can find the o ptimal solution [22]. V . S I G N A L P R O C E S S I N G A N D C O M P U TA T I O N A L I S S U E S The bas ic requireme nt behind any ada pti ve a pproach for s pectrum manag ement and co-existenc e o f communication networks, is the capab ility to ada pti vely s hape the sp ectrum. While ten yea rs ago this was beyond the capabilities of c ommercial commun ication sys tems, this is n o long er the situation. The signal proce ssing hardware av ailable at mos t of the modern chips is sufficient for this purpose, and the mar gina l c ost of this spe ctrum sen sing and shap ing capab ility is rapidly diminishing. Th e main ingredients requ ired to implement g ame theoretic a pproaches , are v arying . T ypically we distinguish three 21 lev e ls of managemen t: Autono mous, whe re e ach unit ope rated with no exc hange of information with other un its or management centers, d istrib uted , whe re a ma nagemen t ce nter provides some parameters to the units, which then d esign the spe ctrum a nd transmit by the mselves, c entralized where all the s pectrum design is performed by a spe ctrum ma nagement center that collects information from units, an d pe rforms centralized proces sing. The competiti ve solutions correspo nd to the autonomo us operation. For these, each communication u nit (playe r) is required to have a s pectral an alysis mo dule, c apable of estimating channel transfer functions an d no ise s pectrum. Th e regulated versions of the co mpetiti ve games such as ASB, IPP and FM-IWF belon g to this sec ond f a mily of distributed solutions where the s pectrum management center provides s ome a-priori network information or pricing functions as well as a list o f the desired rates. For the distributed solutions, a lo w rate commu nication with a cen ter o r with adjac ent units is required like in the Na sh b argaining solution of the p re vious se ction whe re only cha nnel distributi ons are excha nged between use rs. Finally , an important system and signal proce ssing issu e is sync hronization o f the various play ers. Th is is espec ially important for multi-carrier sys tems, where inter chan nel interference ca n be severe wh en the various units are n ot locked to a common time a nd frequen cy referen ce. V I . A P P L I C A T I O N S A. W e ak interference: The DSL c ase The DSL cha nnel is an interesting example for tes ting algorithms emerging from g ame the oretic considerations . The iterative waterfilling a lgorithm [3] has been succ essfully implemented for distributed spectrum c oordination of DSL lines. Howe ver the d rawbacks caused by the prisoner’ s dilemma sugg est that the strictly comp etiti ve app roach (RA-IWF) is inapp ropriate for real life applications. Several amend- ments have been propos ed. Th e first is the fixed margin iterativ e waterfilling [3]. In this algorithm the players are provided with a fixed target rate and each us er , ind epende ntly minimizes its total transmit power . As sh own by Pang et al. [5] this is a gene ralized Nash ga me that conv e r ges if the interference is sufficiently we ak. In [6] a gen eralization o f the FM-IWF is prop osed, tha t fa vors weak us ers us ing a pricing mecha nism termed iterativ e power pricing. This pricing mechan ism impro ves the pe rformance of the FM-I WF . The game theoretic app roaches have very good performance whe n co mpared to o ptimal spectrum ma nagemen t tech niques, as shown in figu re 5. B. Medium and str ong interference - W ir eless technologies The rapid ado ption of wireless services by the public, have ca used a remarkab le increase in dema nd for reliable h igh data rate Internet acces s. T his proces s moti vated the dev elopment of new technologies. New gene ration of c ellular systems li ke L TE and W iMax ope rating in the license d band will b e lunched in the nea r future. In the unlicens ed band, 802.1 1N with MIMO techn ology are g oing to beco me p art of our da ily life. The ca pacity o f future wireless da ta ne tworks will inevitably be interference limited due, the the limited radio spe ctrum. It is clear tha t a ny coop eration between the dif ferent networks or bas e s tations sharing the same sp ectral resource ca n offer sign ificant improvement in the utilization 22 (a) Simulation setup 0 10 20 30 40 50 60 70 10 12 14 16 18 20 22 24 26 28 RT users averaged rate [Mbps] CO users averaged rate [Mbps] IPP IWF OSB (b) Rate region Fig. 5. ( a) Simulation setup. 3 users at each location. (b) Rate regions of FM I WF , IPP and OS B. The optimal OSB i s centralized and computationally very expe nsi ve. It is giv en as a reference to the optimal performance. of the radio resources. E ven in the same ce ll, coo peration be tween sec tors c an improve the o ver all spectral efficiency (bit/Hz/sec./sector). OFDMA tec hnology is cap able to allocate e f ficiently freque ncy bins base d on the channel response of the use r . In [46], a n oncoope rati ve g ame approach was employed for distributed sub-c hannel as signment, adaptive mo dulation, a nd power co ntrol for mu lti-cell OFDM networks. T he goal was to minimize the overall transmitted power un der maximal power an d per use r minimal rate c onstraints. Ba sed on simulation res ults, the p roposed d istrib uted algo rithm red uces the overall transmitted power in compa rison with pure water-fil ling scheme for a se ven-cell cas e. Kwon and Lee [47] presented a dis trib uted resource allocation algorithm for multi-cell OFDMA s ystems relying on a nonc ooperativ e game in wh ich e ach base station tries to max imize the s ystem performance while minimizing the c ochanne l interference. They p roved that there exists a Nas h eq uilibrium point for the noncoo perati ve game an d the e quilibrium is un ique in s ome c onstrained en v ironment. Howev er , Na sh equilibrium a chieved by the distri buted algorithm may n ot b e a s efficient as the resource allocation obtained throug h centralized optimization. T o de monstrate the a dvantage of the Nash b argaining s olution over c ompetiti ve a pproaches for a frequency selec ti ve interference ch annel we as sume that two u sers are sharing a frequency s electiv e Rayleigh fading channe l. The direct cha nnels have unit fading variance and SNR of 3 0 dB. The us ers su f fer from cross interferenc e. Th e cros s chan nels fading variance was varied from -10 d B to 0 dB ( σ 2 h ij = 0 . 1 , ... 1 ). The spectrum cons isted of 32 parallel frequency b ins with indepe ndent fading matrices. At e ach interference lev el of interference σ 2 1 = σ 2 h 21 , σ 2 2 = σ 2 h 12 we randomly picked 25 c hannels (each co mprising of 32 2x 2 random matrices). The results of the minimal relati ve improvement (26) are depicted in figure 4(b). ∆ min = min  R N B S 1 /R C 1 , R N B S 2 /R C 2  . (26) The relative ga in of the Nash bargaining solution over the competitiv e solution is 1.5 to 3 .5 times, w hich clearly demo nstrates the merits of the method. 23 R E F E R E N C E S [1] E. van der Meulen, “Some reflections on t he interference channel, ” in Communication s and Cryptogr aphy: T wo Sides of One T apestry (R. Blahut, D. J. Costell, and T . Mittelholzer, eds.), pp. 409–421 , Kluwer , 1994. [2] T . Han and K. Kob ayashi, “ A new achie vable rate region for the interference channel, ” IEEE T rans. on Information Theory , vol. 27, pp. 49–60, Jan. 1981. [3] W . Y u, G. Ginis, and J. Cioffi, “Distributed multi user powe r control f or digital subscriber lines, ” IEE E J ournal on Selected ar eas i n Communications , vol. 20, pp. 1105–11 15, June 2002. [4] G. Scutari, D. Palomar , and S. Barbarossa, “ Asynchronous iterativ e wa ter-filling for Gaussian frequency -selectiv e interference channels, ” IEEE T ransactions on Information T heory , vol. 54, pp. 2868–287 8, July 2008. [5] J.-S. Pang, G. Scutari, F . Facchinei, and C. W ang, “Distributed po wer allocation with rate constraints in Gaussian parallel interference channels, ” IEEE T ransactions on Information T heory , vol. 54, pp. 3471–348 9, Aug. 2008. [6] Y . Noam and A. Leshem, “Iterati ve po wer pricing for distri buted spectral coordin ation in DSL networks, ” T o ap pear in IEEE T rans. on Communications , 2009. [7] C. Papadimitriou, “ Algorithms, games and the i nternet, ” in P r oc. of 34’ th ACM symposium on theory of computing , pp. 749–75 3, 2001. [8] A. Laufer and A. Leshem, “Distributed coordination of spectrum and the prisoner’ s dilemma, ” in Proc. of the First I EEE International Symposium on New F r ontiers in Dynamic Spectrum A ccess Networks - DySP AN 2005 , pp. 94 – 100, 2005. [9] J. Nash, “T he barg aining problem, ” Econometrica , vol. 18, pp. 155–162 , Apr . 1950 . [10] A. Leshem and E . Zehavi, “Bargaining ov er the interference channel, ” in Proc. IEEE ISIT , pp. 2225–2229, July 2006. [11] H. Boche, M. Schubert, N. V ucic, and S . Naik, “Non-Symmetric Nash Bargaining Solution for Resource Allocation in Wireless Networks and Connection to I nterference Calculus, ” in Pr oc. 15th Euro pean Signal P r ocessing Confere nce (EUSIPCO-2007) , ( Poznan, Poland), Sept. 2007. [12] E. Jorswieck and E . Larsson, “The MISO interference channel from a game-theoretic perspecti ve: A combination of selfishness and alt ruism achiev es pareto optimality , ” Acoustics, Speech and Signal Pr ocessing , 2008 . ICASSP 2008. IEEE International Confer ence on , pp. 5364–53 67, 31 2008-April 4 2008. [13] J. Gao, S. A. V orobyov , and H. Jiang, “Game t heoretic solutions for precoding strategies over the interference chann el, ” Global T elecommunications Confer ence, 2008. IEEE GLOBECOM 2008. IEEE , pp. 1–5, 30 2008-Dec. 4 2008. [14] M. Nokleby , A. Swindlehurst, Y . Rong, and Y . Hua, “Cooperati ve power scheduling for wireless MIMO networks, ” Global T elecommunications Confer ence, 2007. GLOBECOM ’07. IEE E , pp. 2982–2986, Nov . 2007. [15] M. Schubert and H. Boche, “Nash bargaining and proportional fairness for log-con vex utility sets, ” A coustics, Speech and Signal Pr ocessing, 2008. ICA SSP 2008. IEEE International Confer ence on , pp. 3157–31 60, 31 2008-April 4 2008. [16] H. Park and M. V . D. S chaar , “Bargaining strategies for networked multimedia resource management, ” IEEE T ransactions on Signal Processing , vol. 55, pp. 3496–35 11, July 2007. [17] E. Kalai, “Solutions to the bargaining problem, ” Di scussion Papers 556, Northwestern Univ ersity , Center for Mathematical Studies i n Economics and Management Science, Mar . 1983. [18] S. Mathur , L. Sankaranarayanan, and N. Mandayam, “Coalitional games in gaussian interference channels, ” i n Pro c. IEE E ISIT , pp. 2210 – 2214, July 2006. [19] Z. Han, Z. Ji, and K. Liu, “Fair multiuser channel allocation for OFDMA networks using the Nash bargaining solutions and coalitions, ” IE EE T rans. on C ommunications , vol. 53, pp. 1366– 1376, Aug. 2005. [20] Y . Z hao, L. kang Zeng, G. Xie, Y . an L iu, and F . Xiong, “Fairness based resource all ocation for uplink OF DMA systems, ” The Journal of China Universities of P osts and T elecommunications , vol. 15, no. 2, pp. 50 – 55, 2008. [21] A. L eshem and E. Zehavi, “Cooperati ve game theory and the Gaussian interference channel, ” IEEE Journal on Selected Ar eas in Communications , vol. 26, pp. 1078– 1088, Sept. 2008. [22] E. Z ehav i and A. Leshem, “Bargaining ov er the i nterference channel with t otal power constraints, ” in Pr oc. of GameNets 2009 , ( Istanbu l), May 2009. [23] T . Cover and J. Thomas, Elements of informa tion theory . Wiley series in telecommunications, W i ley , 1991. [24] A. B. Carleial, “Interference channels, ” IEEE T ransactions on Information T heory , vol. 24, pp. 60–70, Jan. 1978. 24 [25] H. S ato, “The capacity of the Gaussian interference channel under st rong interference, ” IEE E T rans. on Information Theory , vol. 27, pp. 786–78 8, Nov . 198 1. [26] G. Owen, Game theory . Academic Press, third ed., 1995 . [27] T . Basar and G. Olsder , Dynamic non-cooper ative game theory . Academic Press, 1982. [28] M. Osborn, Introdu ction to game theory . Oxford Univ ersi ty P ress, 1995. [29] V . Gajic and B. Rimoldi, “Game theoretic considerations for the Gaussian multiple access channel, ” i n ISIT’08 - IEEE International Symposium on Information Theory . [30] L. Lai and H. El-Gamal, “The water-filling game in fading multiple-access channels, ” IEEE Tr ansactions on Information Theory , vol. 54, no. 5, pp. 2110–212 2, 20 08. [31] W . Y u, W . Rhee, S. Boyd, and J. Cioffi, “Iterative waterfilling for Gaussian vector multi ple-access channels, ” IEEE T ransactions on Information Theory , vol. 50, no. 1, pp. 145–152, 2004. [32] W . P oundstone , Prisoner’ s Dilemma: Jo hn V on Neumann, Game Theory and the Puzzle of the Bomb . New Y ork, NY , USA: Doubleday , 1992. [33] H. Nikaido and K. Isoida, “Note on non-cooperati ve con vex games, ” P acific Jo urnal of Matematics , vol. 5, pp. 807–815, 1955. [34] F . Meshkati, H. Poor , and S. Schwartz, “E nergy -ef ficient resource al location in wireless networks, ” Signal Pr ocessing Maga zine, IEEE , vol. 24, pp. 58–68 , May 2007. [35] K. Arrow and G. Debreu, “The existenc e of an equilibrium f or a competitive economy , ” Econometrica , vol. XXII, pp. 265– 290, 1954. [36] T . Ichishi, Game theory for economic analysis . Academic Press,, 1983. [37] J. Nash, “T wo-person coope rativ e games, ” Econometrica , vo l. 21, pp. 128–140, Jan. 1953. [38] A. Laufer, A. Leshem, and H . Messer , “Game theoretic aspects of distributed spectral coordination with application to DSL networks. ” arXiv:cs/0 602014, 2006. [39] S. Chung, J. L ee, S . Kim, and J. Cioffi, “On the con verge nce of iterative waterfilling in the frequency selective Gaussian interference channel, ” Prep rint , 2002. [40] S. Chung and J. Cioffi, “Rate and powe r control in a two-user multicarrier channel with no coordination: the optimal scheme versus a suboptimal method, ” IEE E T ransa ctions on C ommunications , vol. 51,, pp. 1768 – 1772, Nov . 2003. [41] Z. Luo and J. Pang, “ Analysis of iterative waterfilling algorithm for multiuser po w er control in digital sibcriber lines, ” EURASIP J ournal A pplied Signal Processin g , vol. 2006, pp. 1–10 , 2006. [42] G. Scutari, D. Palomar , and S. Barbarossa, “ Asynchronou s iterative water -fi lling for Gaussain frequency -selectiv e interference channles: A unified framew ork, ” in IEE E SP A WC-2006 , July 2006. [43] K. W . S hum, K.-K. Leung, and C. W . Sung, “Con verg ence of i terativ e waterfilli ng algorithm for Gaussian interference channels, ” IEE E Journ al on Selected Ar eas in Communications , vol. 25, pp. 1091–1 099, August 2007. [44] W . Lee, Y . Kim, M. H. Brady , and J. M. Cioffi, “Band-preference dynamic spectrum management in a DSL en vironment, ” in P r oc. IEEE Globecom 2006 , (San Francisco, CA), IEEE, Nov . 200 6. [45] R. Cendrillon, J. Huang, M. Chiang, and M. Moonen, “ Autonomous spectrum balancing for digital subscriber lines, ” IEE E T rans. on Signal Pr ocessing , vo l. 55, pp. 4241 – 4257, August 2007. [46] Z. Han, Z. Ji, and K. Liu, “Po wer minimization for multi-cell OFDM networks using distributed non-cooperati ve game approach, ” Global T elecommunications Confer ence, 200 4. GLOBECOM ’04 . IEEE , vol. 6, pp. 3742– 3747 V ol.6, No v .-3 Dec. 2004 . [47] H. Kwon and B. G. Lee, “Distributed resource allocation through nonco operati ve game approach in multi-cell OFDMA systems, ” Commun ications, 2006. IC C ’06. IEEE International Confer ence on , vol. 9, pp. 4345–4350 , June 2006.