Closed form solutions for symmetric water filling games

apport   de recherche ISSN 0249-6399 ISRN INRIA/RR--6254--FR+ENG Thème COM INSTITUT N A TION AL DE RECHERCHE EN INFORMA TIQUE ET EN A UTOMA TIQUE Closed form solutions f or symmetric wate r filling games Eitan Altman — K onstantin A vrachenkov — Andrey Garnae v N° 6254 July 2007 Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France) Téléphone : +33 4 92 38 77 77 — Téléco pie : +33 4 92 38 77 65 Closed form solutions for symmetri w ater lling games Eitan Altman ∗ , K onstan tin A vra henk o v † , Andrey Garnaev ‡ Thème COM  Systèmes omm unian ts Pro jets MAESTR O Rapp ort de re her he n ° 6254  July 2007  23 pages Abstrat: W e study p o w er on trol in optimization and game framew orks. In the opti- mization framew ork there is a single deision mak er who assigns net w ork resoures and in the game framew ork users share the net w ork resoures aording to Nash equilibrium. The solution of these problems is based on so-alled w ater-lling te hnique, whi h in turn uses bisetion metho d for solution of non-linear equations for Lagrange m ultiplies. Here w e pro- vide a losed form solution to the w ater-lling problem, whi h allo ws us to solv e it in a nite n um b er of op erations. Also, w e pro due a losed form solution for the Nash equilibrium in symmetri Gaussian in terferene game with an arbitrary n um b er of users. Ev en though the game is symmetri, there is an in trinsi hierar hial struture indued b y the quan tit y of the resoures a v ailable to the users. W e use this hierar hial struture to p erform a suessiv e redution of the game. In addition, to its mathematial b eaut y , the expliit solution allo ws one to study limiting ases when the rosstalk o eien t is either small or large. W e pro vide an alternativ e simple pro of of the on v ergene of the Iterativ e W ater Filling Algorithm. F ur- thermore, it turns out that the on v ergene of Iterativ e W ater Filling Algorithm slo ws do wn when the rosstalk o eien t is large. Using the losed form solution, w e an a v oid this problem. Finally , w e ompare the non-o op erativ e approa h with the o op erativ e approa h and sho w that the non-o op erativ e approa h results in a more fair resoure distribution. Key-w ords: wireless net w orks, p o w er on trol, symmetri w ater-lling game, Nash equi- librium, prie of anar h y This w ork w as supp orted b y BioNets Europ ean pro jet and b y join t RFBR and NNSF Gran t no.06-01- 39005. ∗ INRIA Sophia An tip olis, Altmansophia.inria.fr † INRIA Sophia An tip olis, K.A vra henk o vsophia.inria.fr ‡ St.P etersburg State Univ ersit y , agarnaevram bler.ru Solution analytique des jeux de w ater-lling symétriques Résumé : Nous étudions le on trle de puissane dans le adre de l'optimisation et dans elui de la théorie des jeux. Dans le premier, il y a un seul agen t qui assigne les ressoures du réseau tandis que dans le deuxième, les utilisateurs se partagen t les ressoures du réseau selon l'équilibre de Nash. La solution de es problèmes est basée sur la métho de du w ater- lling. On alule des m ultipliateurs de Lagrange en utilisan t une métho de de bisetion p our resoudre des équations non linéaires. Nous fournissons ii une solution analytique au problème du w ater-lling, qui nous p ermet de le résoudre en un nom bre ni d'op érations. En outre, nous pro duisons une solution analytique de l'équilibre de Nash dans le adre de la théorie des jeux. Nous étudions un jeu symétrique en terme d'in terférene a v e un nom bre arbitraire d'utilisateurs. Quoique le jeu soit symétrique, il y a une struture hiérar-  hique induite par la quan tité des ressoures disp onibles p our les utilisateurs. Nous utilisons ette struture p our eetuer une rédution suessiv e du jeu. En plus de son éléguene mathématique, la solution analytique p ermet d'étudier des as limites quand le o eien t d'in terférene est p etit ou grand. Nous fournissons une preuv e simple de la on v ergene de l'algorithme itératif de w ater-lling (l'algorithme de meilleur rép onse). Il s'a v ère que la on v ergene de l'agorithme est ralen tie quand le o eien t d'in terférene est pro  he de l'unité. En utilisan t la solution analytique, nous p ouv ons éviter e problème. Aussi, nous omparons l'appro  he non o op érativ e à l'appro  he o op érativ e et mon trons que l'appro  he non o op érativ e fournit une distribution des ressoures plus équitable. Mots-lés : réseaux sans ls, on trle de puissane, jeu w ater-lling symétrique, équilibre de Nash, oût de l'anar hie Close d form solutions for symmetri water l ling games 3 1 In tro dution In wireless net w orks and DSL aess net w orks the total a v ailable p o w er for signal transmis- sion has to b e distributed among sev eral resoures. In the on text of wireless net w orks, the resoures ma y orresp ond to frequeny bands (e.g. as in OFDM), or they ma y orresp ond to apait y a v ailable at dieren t time slots. In the on text of DSL aess net w orks, the resoures orresp ond to a v ailable frequeny tones. This sp etrum of problems an b e on- sidered in either optimization senario or game senario. The optimization senario leads to W ater Filling Optimization Problem [ 3 , 6 , 14 ℄ and the game senario leads to W ater Filling Game or Gaussian In terferene Game [ 8, 11 , 12 , 15 ℄. In the optimization senario, one needs to maximize a ona v e funtion (Shannon apait y) sub jet to p o w er onstrain ts. The Lagrange m ultiplier orresp onding to the p o w er onstrain t is determined b y a non-linear equation. In the previous w orks [3 , 6, 14 ℄, it w as suggested to nd the Lagrange m ultiplier b y means of a bisetion algorithm, where omes the name W ater Filling Problem. Here w e sho w that the Lagrange m ultiplier and hene the optimal solution of the w ater lling problem an b e found in expliit form with a nite n um b er of op erations. In the m ultiuser on text, one an view the problem in either o op erativ e or non-o op erativ e setting. If a en tralized on troller w an ts to maximize the sum of all users' rates, the on troller will fae a non-on v ex optimization problem with man y lo al maxima [ 13 ℄. On the other hand, in the non-o op erativ e setting, the p o w er allo ation problem b eomes a game problem where ea h user p ereiv es the signals of the other users as in terferene and maximizes a ona v e funtion of the noise to in terferene ratio. A natural approa h in the non-o op erativ e set- ting is the appliation of the Iterativ e W ater Filling Algorithm (IWF A) [ 16 ℄. Reen tly , the authors of [10 ℄ pro v ed the on v ergene of IWF A under fairly general onditions. In the presen t w ork w e study the ase of symmetri w ater lling game. There is an in trinsi hier- ar hial struture indued b y the quan tit y of the resoures a v ailable to the users. W e use this hierar hial struture to p erform a suessiv e redution of the game, whi h allo ws us to nd Nash equilibrium in expliit form. In addition, to its mathematial b eaut y , the expliit solution allo ws one to nd the Nash equilibrium in w ater lling game in a nite n um b er of op erations and to study limiting ases when the rosstalk o eien t is either small or large. As a b y-pro dut, w e obtain an alternativ e simple pro of of the on v ergene of the Iterativ e W ater Filling Algorithm. F urthermore, it turns out that the on v ergene of IWF A slo ws do wn when the rosstalk o eien t is large. Using the losed form solution, w e an a v oid this problem. Finally , w e ompare the non-o op erativ e approa h with the o op era- tiv e approa h and onlude that the ost of anar h y is small in the ase of small rosstalk o eien ts and that the the deen tralized solution is b etter than the en tralized one with resp et to fairness. Appliations that an mostly b enet from deen tralized non-o op erativ e p o w er on trol are ad-ho  and sensor net w orks with no predened base stations [ 4, 9, 7 ℄. An in terested reader an nd more referenes on non-o op erativ e p o w er on trol in [2 , 8 ℄. W e w ould lik e to men tion that the w ater lling problem and jamming games with transmission osts ha v e b een analyzed in [1℄. The pap er is organized as follo ws: In Setion 2 w e reall the single deision mak er setup of the w ater lling optimization problem and pro vide its expliit solution. Then in Setions 3-7 RR n ° 6254 4 A ltman, A vr ahenkov & Garnaev w e form ulate m ultiuser symmetri w ater lling game and  haraterize its Nash equilibrium, also w e giv e an alternativ e simple pro of of the on v ergene of the iterativ e w ater lling algorithm and suggest the expliit form of the users' strategy in the Nash equilibrium. In Setion 8 w e onrm our nding with the help of n umerial examples and ompare the deen tralized approa h with the en tralized one. 2 Single deision mak er First let us onsider the p o w er allo ation problem in the ase of a single deision mak er. The single deision mak er (also alled user or transmitter) w an ts to send information using n indep enden t resoures so that to maximize the Shannon apait y . W e further assume that resoure i has a w eigh t of π i . P ossible in terpretations: (i) The resoures ma y orresp ond to apait y a v ailable at dieren t time slots; w e assume that there is a v arying en vironmen t whose state  hanges among a nite set of states i ∈ [1 , n ] , aording to some ergo di sto  hasti pro ess with stationary distribution { π i } n i =1 . W e assume that the user has p erfet kno wledge of the en vironmen t state at the b eginning of ea h time slot. (ii) The resoures ma y orresp ond to frequeny bands (e.g. as in OFDM) where one should assign dieren t p o w er lev els for dieren t sub-arriers [ 14 ℄. In that ase w e ma y tak e π i = 1 /n for all i . The strategy of user is T = ( T 1 , . . . , T n ) with P n i =1 π i T i = ¯ T , T i ≥ 0 , π i > 0 for i ∈ [1 , n ] and ¯ T > 0 . As the pa y o to user w e tak e the Shannon apait y v ( T ) = n X i =1 π i ln  1 + T i / N 0 i  , where N 0 i > 0 is the noise lev el in the sub-arrier i . W e w ould lik e to emphasize that this generalized desription of the w ater-lling problem an b e used for p o w er allo ation in time as w ell as p o w er allo ation in spae-frequeny . F ollo wing the standard w ater-lling approa h [ 3, 6 , 14 ℄ w e ha v e the follo wing result. Theorem 1 L et T i ( ω ) =  1 /ω − N 0 i  + for i ∈ [1 , n ] and H T ( ω ) = P n i =1 π i T i ( ω ) . Then T ( ω ∗ ) = ( T 1 ( ω ∗ ) , . . . , T n ( ω ∗ )) is the unique optimal str ate gy and its p ayo is v ( T ( ω ∗ )) wher e ω ∗ is the unique r o ot of the e quation H ( ω ) = ¯ T . (1) In the previous studies of the w ater-lling problems it w as suggested to use n umerial (e.g., bisetion) metho d to solv e the equation ( 1). Here w e prop ose an expliit form approa h for its solution. INRIA Close d form solutions for symmetri water l ling games 5 Without loss of generalit y w e an assume that 1 / N 0 1 ≥ 1 / N 0 2 ≥ . . . ≥ 1 / N 0 n . (2) Then, sine H ( · ) is dereasing, w e ha v e the follo wing result: Theorem 2 The solution of the water-l ling optimization pr oblem is given by T ∗ i =       ¯ T + k X t =1 π t ( N 0 t − N 0 i ) . k X t =1 π t  , i ≤ k , 0 , i > k , wher e k  an b e found fr om the fol lowing  ondition: ϕ k < ¯ T ≤ ϕ k +1 , wher e ϕ t = t X i =1 π i ( N 0 t − N 0 i ) for t ∈ [1 , n ] . Th us, on trary to the n umerial (bisetion) approa h, in order to nd an optimal resoure allo ation w e need to exeute only a nite n um b er of op erations. 3 Symmetri w ater lling game Let us no w onsider a m ulti-user senario. Sp eially , w e onsider L users who try to send information through n resoures so that to maximize their transmission rates. The strategy of user j is T j = ( T j 1 , . . . , T j n ) sub jet to n X i =1 π i T j i = ¯ T j , (3) where ¯ T j > 0 for j ∈ [1 , L ] . The elemen t T j i is the p o w er lev el used b y transmitter j when the en vironmen t is in state i . The pa y o to user j is giv en as follo ws: v j ( T 1 , . . . , T L ) = n X i =1 π i ln 1 + α j i T j i N 0 i + g i P k 6 = j α k i T k i ! , where N 0 i is the noise lev el and g i ∈ (0 , 1) and α j i are fading  hannel gains of user j when the en vironmen t is in state i . These pa y os orresp ond to Shannon apaities. The onstrain t (3) orresp onds to the a v erage p o w er onsumption onstrain t. This is an instane of the W ater Filling or Gaussian In terferene Game [ 8 , 11 , 12 , 15 , 16 ℄. In the imp ortan t partiular ases of OFDM wireless net w ork and DSL aess net w ork, π i = 1 /n, i = 1 , ..., n . RR n ° 6254 6 A ltman, A vr ahenkov & Garnaev W e will lo ok for a Nash Equilibrium (NE) of this problem. The strategies T 1 ∗ ,. . . , T L ∗ onstitute a NE, if for an y strategies T 1 ,. . . , T L the follo wing inequalities hold: v 1 ( T 1 , T 2 ∗ , . . . , T L ∗ ) ≤ v 1 ( T 1 ∗ , T 2 ∗ . . . , T L ∗ ) , · · · v L ( T 1 ∗ , . . . , T ( L − 1) ∗ , T L ) ≤ v L ( T 1 ∗ , . . . , T ( L − 1) ∗ , T L ∗ ) . F or nding NE of su h game usually the follo wing n umerial algorithm is applied. First, a strategy of L − 1 users (sa y , user 2,. . . , L ) are xed. Then, the b est reply of user 1 is found solving the W ater Filling optimization problem. Then, the b est reply of user 2 on these strategies of the users is found solving the optimization problem and so on. It is p ossible to pro v e that under some assumption on fading  hannel gains this sequene of the strategies on v erge to a NE [10 ℄. In this w ork w e restrit ourselv es to the ase of symmetri game with equal rosstalk o eien ts. This situation an for example orresp ond to the senario when the users are situated at ab out the same distane from the base station. Namely , w e assume that α 1 i = . . . = α L i and g i = g for i ∈ (0 , 1) . So, in our ase the pa y os to users are giv en as follo ws v j ( T 1 , . . . , T L ) = n X i =1 π i ln 1 + T j i N 0 i + g P k 6 = j T k i ! , where N 0 i = N 0 /α i , i ∈ [1 , n ] and without loss of generalit y w e an assume that the  hannels are arranged in su h a w a y that the inequalities (2) hold. W e w ould lik e to emphasize that the dep endane of N 0 i on i allo ws us to mo del an en vironmen t with v arying transmission onditions. F or this problem w e prop ose a new algorithm of nding the NE. The algorithm is based on losed form expressions and hene it requires only a nite n um b er of op erations. Also, explaining this algorithm w e will pro v e that the game has the unique NE under assumption that g ∈ (0 , 1) . Sine v j is ona v e on T j , the Kuhn-T u k er Theorem implies the follo wing theorem. Theorem 3 ( T 1 ∗ , . . . , T L ∗ ) is a Nash e quilibrium if and only if ther e ar e non-ne gative ω j , j ∈ [1 , L ] (L agr ange multipliers) suh that ∂ ∂ T j i v j ( T 1 ∗ , . . . , T L ∗ ) = 1 T j ∗ i + N 0 i + g X k 6 = j T k ∗ i ( = ω j for T j ∗ i > 0 , ≤ ω j for T j ∗ i = 0 . (4) It is lear that all ω j are p ositiv e. The assumption that g < 1 is ruial for uniqueness of equilibrium as it is sho wn in the follo wing prop osition. INRIA Close d form solutions for symmetri water l ling games 7 Prop osition 1 F or g = 1 the symmetri water l ling game has innite numb er ( ontinuum) of Nash e quilibria. Pr o of . Supp ose that ( T 1 ∗ , . . . , T L ∗ ) is a Nash equilibrium. Then, b y Theorem 3 , there are non-negativ e ω j , j ∈ [1 , L ] su h that 1  N 0 i + L X k =1 T k ∗ i  ( = ω j for T j ∗ i > 0 , ≤ ω j for T j ∗ i = 0 . Th us, ω 1 = . . . = ω L = ω . So, T 1 ∗ i , . . . , T L ∗ i , i ∈ [1 , n ] ha v e to b e an y non-negativ e su h that L X k =1 T k ∗ i = π i [1 /ω − N 0 i ] + , and n X i =1 π i T k ∗ i = ¯ T k for k ∈ [1 , L ] , where ω is the unique p ositiv e ro ot of the equation n X i =1 [1 /ω − N 0 i ] + = L X k =1 ¯ T k . It is lear that there are innite n um b er of su h strategies. F or example, if T a ∗ i and T b ∗ i , i ∈ [1 , n ] ( a 6 = b ) is the one of them and T a ∗ k , T b ∗ k > 0 and T a ∗ k , T b ∗ m > for some k and m . Then, it is lear that the follo wing strategies for an y small enough p ositiv e ǫ are also optimal: ˜ T a ∗ i =      T a ∗ i for i 6 = k , m, T a ∗ i + ǫ for i = k , T a ∗ i − ǫπ k /π m for i = m, ˜ T b ∗ i =      T b ∗ i for i 6 = k , m, T b ∗ i − ǫ for i = k , T b ∗ i + ǫπ k /π m for i = m. This ompletes the pro of of Prop osition 1 . 4 A reursiv e approa h to the symmetri w ater lling game Let ω 1 ,. . . , ω L b e some parameters whi h in the future will at as Lagrangian m ultiplies. Us- ing these parameters w e in tro due some auxiliary notations. Assume that these parameters RR n ° 6254 8 A ltman, A vr ahenkov & Garnaev are arranged as follo ws (this assumption do es not redue the generalit y of our forthoming onlusions): ω 1 ≤ . . . ≤ ω L . (5) Also denote ¯ ω = ( ω 1 , . . . , ω L ) . In tro due the follo wing auxiliary sequene: t r = 1 1 − g   1 + ( r − 1) g ω r − g r X j =1 1 ω j   for r ∈ [1 , L ] . It is lear that b y (5) t r +1 = 1 + ( r − 1) g 1 − g  1 ω r +1 − 1 ω r  + t r ≤ t r . Th us, t L ≤ t L − 1 ≤ . . . ≤ t 1 , and 1 ω r +1 − 1 ω r = 1 − g 1 + ( r − 1) g ( t r +1 − t r ) . (6) Hene, for j ∈ [ k + 1 , L ] w e ha v e: 1 ω k − 1 ω j = j − 1 X r = k 1 − g 1 + ( r − 1) g ( t r − t r +1 ) . (7) Then, sequenes { ω r } and { t r } has the follo wing reurren t relations: 1 ω 1 = t 1 , 1 ω 2 = (1 − g ) t 2 + g t 1 , 1 ω r +1 = 1 − g 1 + ( r − 1) g t r +1 + r X j =2 (1 − g ) g (1 + ( j − 1) g )(1 + ( j − 2) g ) t j + t 1 , (8) where r ≥ 1 . If w e kno w sequene { t r } w e an restore sequene { ω r } . Th us, these t w o sequenes are equiv alen t. In tro due one more auxiliary sequene as follo ws: τ k r = 1 1 − g   1 + ( L − 1 − r + k ) g ω k − g L − r + k X j =1 1 ω j   , INRIA Close d form solutions for symmetri water l ling games 9 where r ∈ [ k , L ] , k ∈ [1 , L ] . There is a simple relation b et w een sequenes { ω k } and { t k } and { τ k r } : τ k L = t k , (9) and τ k r = 1 + ( L − 1 − r + k ) g 1 − g  1 ω k − 1 ω L − r + k  + t L − r + k . (10) So, b y (7), olleting terms whi h dep ends on t k w e obtain τ k r = b k,r t k + A k,r , (11) where b k,r = 1 + ( L − 1 − r + k ) g 1 + ( k − 1) g , and A k,r = g L − r + k − 1 X j = k +1 1 + ( L − 1 − r + k ) g (1 + ( j − 1) g )(1 + j g ) t j − g (1 + ( L − 2 − r + k ) g t L − r + k . Th us, A k,r dep ends only on { t j } with j > k . Finally in tro due the follo wing notation: (a) for N 0 i < t L T k i ( ¯ ω ) = 1 1 + ( L − 1) g ( τ k k − N 0 i ) , (b) t L + k +1 − r ≤ N 0 i < t L + k − r where r ∈ [ k + 1 , L ] T k i ( ¯ ω ) = 1 1 + ( L − 1 − r + k ) g ( τ k r − N 0 i ) , () for t k ≤ N 0 i T k i ( ¯ ω ) = 0 . F or others om binations of relations b et w een ω j , j ∈ [1 , L ] , T k i are dened b y symmetry . By Theorem 3 w e ha v e the follo wing result. Theorem 4 Eah Nash e quilibrium is of the form ( T 1 ( ¯ ω ) , . . . , T L ( ¯ ω )) . The next lemma pro vides a nie relation b et w een L and L − 1 p erson games whi h sho ws that the in tro dution of a new user in to the game leads to a bigger omp etition for the b etter qualit y  hannels mean while users prefer to k eep the old struture of their strategies for w orse qualit y  hannels. RR n ° 6254 10 A ltman, A vr ahenkov & Garnaev Lemma 1 L et ( T 1 ,L ( ω 1 , . . . , ω L ) , . . . , T L,L ( ω 1 , . . . , ω L )) given by The or em 4 (her e we adde d the se  ond sup er-sript index in the notation of the str ate gies in or der to emphasize that the str ate gies dep end on the numb er of users). Then, we have T k,L i ( ω 1 , . . . , ω L ) =    τ k k − N 0 i 1 + ( L − 1) g for N 0 i < t L , T k,L − 1 i ( ω 1 , . . . , ω L − 1 ) for t L ≤ N 0 i , wher e k ∈ [1 , L − 1] and T L,L i ( ω 1 , . . . , ω L ) =    t L − N 0 i 1 + ( L − 1) g for N 0 i < t L , 0 for t L ≤ N 0 i . 5 A w ater-lling algorithm In this setion w e desrib e a v ersion of the w ater-lling algorithm for nding the NE and supply a simple pro of of its on v ergene based on some monotoniit y prop erties. Let H k ( ¯ ω ) = n X i =1 π i T k i ( ¯ ω ) for k ∈ [1 , L ] . T o nd a NE w e ha v e to nd ¯ ω su h that H k ( ¯ ω ) = ¯ T k for k ∈ [1 , L ] . (12) It is lear that H k ( ¯ ω ) has the follo wing prop erties, olleted in the next Lemma, whi h follo w diretly from the expliit form ulas of the NE. Lemma 2 (i) H k ( ¯ ω ) is nonne gative and  ontinuous, (ii) H k ( ¯ ω ) is de r e asing on ω k , (iii) H k ( ¯ ω ) → ∞ for ω k → 0 , (iv) H k ( ¯ ω ) = 0 for enough big ω k , say for ω k ≥ 1 / N 0 1 , (v) H k ( ¯ ω ) is non-inr e asing by ω j wher e j 6 = k . This prop erties giv e a simple pro of of the on v ergene of the follo wing iterativ e w ater lling algorithm for nding the NE. Let ω k 0 for all k ∈ [1 , L ] b e su h that H k ( ¯ ω 0 ) = 0 , for example ω k 0 = 1 / N 0 1 . Let ω k 1 = ω k 0 for all k ∈ [2 , L ] and dene ω 1 1 su h that H 1 ( ¯ ω 1 ) = ¯ T 1 . Su h ω 1 1 exists b y Lemma 2(i)-(iii). Then, b y Lemma 2(i),(v) H k ( ¯ ω 0 ) = 0 for k ∈ [2 , L ] . Let ω k 2 = ω k 1 for all k 6 = 2 and dene ω 2 2 su h that H 2 ( ¯ ω 2 ) = ¯ T 2 . Then, b y Lemma 2(i),(v) H k ( ¯ ω 0 ) = 0 for k > 2 and H k ( ¯ ω 0 ) ≤ ¯ T k for k = 1 and so on. Let ω k L = ω k L − 1 for all k 6 = L and dene ω L L su h that H L ( ¯ ω L ) = ¯ T L . Then, b y Lemma 2(i),(v) H k ( ¯ ω L ) ≤ ¯ T k for k 6 = L and so on. So w e ha v e non-inreasing p ositiv e sequene ω k . Th us, it on v erges to an ¯ ω ∗ whi h pro dues a NE. INRIA Close d form solutions for symmetri water l ling games 11 6 Existene and uniqueness of the Nash equilibrium In this setion w e will pro v e existene and uniqueness of the Nash equilibrium for L p erson symmetri w ater-lling game. Our pro of will ha v e onstrutiv e  harater whi h allo ws us to pro due an eetiv e algorithm for nding the equilibrium strategies. First note that there is a monotonous dep endene b et w een the resoures the users an apply and Lagrange m ultipliers. Lemma 3 L et ( T 1 ( ¯ ω ) , . . . , T L ( ¯ ω )) b e a Nash e quilibrium. If ¯ T 1 ≥ . . . ≥ ¯ T L (13) then (5 ) holds. Pr o of. The result immediately follo ws from the follo wing monotoniit y prop ert y implied b y expliit form ulas of the Nash equilibrium, namely , if ω i < ω j then H i ( ¯ ω ) > H j ( ¯ ω ) . Without loss of generalit y w e an assume that (13 ) holds. Th us, b y Lemma 3, (5) also holds. Let ¯ ω b e the p ositiv e solution of ( 12). Then, b y Lemma 3 , the relation (5) holds. T o nd ¯ ω w e ha v e to solv e the system of non-linear equations ( 12). It is quite bulky system and it lo oks hard to solv e. W e will not solv e it diretly . What w e will do w e express ω 1 ,. . . ω L b y t 1 , . . . , t L , substitute these expression in to (12 ). The transformed system will ha v e a triangular form, namely ˜ H L ( t L ) = ¯ T L , ˜ H L − 1 ( t L − 1 , t L ) = ¯ T L − 1 , · · · ˜ H 1 ( t 1 , . . . , t L − 1 , t L ) = ¯ T 1 . (14) The last system, b eause of monotoniit y prop erties of ˜ H k on t k , an b e easily solv ed. No w w e an mo v e on to onstrution of ˜ H L ( t L ) , . . . , ˜ H 1 ( t 1 , . . . , t L − 1 , t L ) . First w e will onstrut ˜ H L ( t L ) and nd the optimal t L . Note that, H L ( ¯ ω ) = X N 0 i . . . > ¯ T L . W e will onstrut the optimal strategies T L ∗ , . . . , T 1 ∗ sequen tially . Step for  onstrution of T L ∗ . Sine ˜ H L ( · ) is stritly inreasing w e an nd an in teger k L su h that ˜ H L ( N 0 k L ) < ¯ T L ≤ ˜ H L ( N 0 k L +1 ) . or from the follo wing equiv alen t onditions: ϕ L k L < ¯ T L ≤ ϕ L k L +1 , where ϕ L k = 1 1 + ( L − 1) g k X i =1 π i ( N 0 k − N 0 i ) , RR n ° 6254 14 A ltman, A vr ahenkov & Garnaev for k ≤ n , and ϕ L n +1 = ∞ . Then, sine ˜ H L ( t L ∗ ) = ¯ T L , w e ha v e that t L ∗ = (1 + ( L − 1) g ) ¯ T L + P k L i =1 π i N 0 i P k L i =1 π i . Th us, the optimal strategy of user L is giv en as follo ws T L ∗ i = ( 1 1 + ( L − 1) g ( t L ∗ − N 0 i ) if i ∈ [1 , k L ] , 0 if i ∈ [ k L + 1 , n ] . Step for  onstrution of T ( L − 1) ∗ . Sine t L − 1 ∗ is the ro ot of the equation ˜ H L − 1 ( · , t L ∗ ) = ¯ T L − 1 there is k L − 1 su h that k L − 1 ≥ k L and N 0 k L − 1 +1 ≥ t L − 1 ∗ > N 0 k L − 1 . Th us, t L − 1 ∗ =  ¯ T L − 1 + 1 1 + ( L − 2) g k L − 1 X i = k L +1 π i N 0 i + 1 1 + ( L − 1) g k L X i =1 π i ( g t ∗ L 1 + ( L − 2) g + N 0 i )  . 1 1 + ( L − 2) g k L − 1 X i =1 π i  . Here and b ello w w e assume that P y x 1 = 0 for y < x . So, k L − 1 ≥ k L an b e found as follo ws: (i) k L − 1 = k L if ¯ T L − 1 ≤ ϕ L − 1 k L − 1 +1 , (ii) otherwise k L − 1 is giv en b y the ondition: ϕ L − 1 k L − 1 < ¯ T L − 1 ≤ ϕ L − 1 k L − 1 +1 , where ϕ L − 1 k = k X i = k L +1 π i 1 + ( L − 2) g ( N 0 k − N 0 i ) + k L X i =1 π i 1 + ( L − 1) g ×  1 + ( L − 1) g 1 + ( L − 2) g N 0 k − N 0 i − g 1 + g t L − 1 ∗  , for k ∈ [ k L − 1 + 1 , n ] and ϕ L − 1 n +1 = ∞ . INRIA Close d form solutions for symmetri water l ling games 15 Th us, the optimal strategy T ( L − 1) ∗ of user L − 1 is giv en b y T ( L − 1) ∗ i =                    t L − 1 ∗ 1 + ( L − 2) g − g 1 + g t L ∗ + N 0 i 1 + ( L − 1) g , i ∈ [1 , k L ] , 1 1 + ( L − 2) g ( t L − 1 ∗ − N 0 i ) , i ∈ [ k L + 1 , k L − 1 ] , 0 , i ∈ [ k L − 1 + 1 , n ] . Step for  onstrution of T M ∗ wher e M < L . W e ha v e already onstruted T L ∗ , . . . , T ( M +1) ∗ and no w w e are going to onstrut T M ∗ . Sine t M ∗ is the ro ot of the equation ˜ H M ( · , t M +1 ∗ , . . . , t L ∗ ) = ¯ T M there is k M su h that k M ≥ k M +1 and N 0 k M +1 ≥ t M ∗ > N 0 k M . Th us, t M ∗ =  ¯ T M + 1 1 + ( L − 1) g k M X i =1 π i ( A k,k − N 0 i ) + L X r = M +1 k p − 1 X i = k p +1 π i ( A p,r − N 0 i ) 1 + ( L − 1 − r + p ) g ) . 1 1 + ( M − 1) g k M X i =1 π i  . So, k M ≥ k M +1 an b e found as follo ws: (i) k M = k M +1 if ¯ T M ≤ ϕ M k M +1 , (ii) otherwise k M is giv en b y the ondition: ϕ M k M < ¯ T M ≤ ϕ M k M +1 where ϕ M k = 1 1 + ( L − 1) g k X i =1 π i ( b k,k N 0 k + A k,k − N 0 i ) + L X r = M +1 k p − 1 X i = k p +1 π i ( b p,r N 0 k + A p,r − N 0 i ) 1 + ( L − 1 − r + p ) g . RR n ° 6254 16 A ltman, A vr ahenkov & Garnaev Th us, the optimal strategy of user L is giv en as follo ws T M ∗ i =                τ M M − N 0 i 1 + ( L − 1) g , i ∈ [1 , k L ] , τ M r − N 0 i 1 + ( L − 1 − r + M ) g , i ∈ [ k r + 1 , k r − 1 ] , r ∈ [ M + 1 , L ] 0 , i ∈ [ k M + 1 , n ] . In partiular for t w o and three p erson games ( L = 2 and L = 3 ) w e ha v e the follo wing results. Theorem 6 L et ¯ T 1 > ¯ T 2 . Then, the Nash e quilibrium str ate gies ar e given by T 1 ∗ i =        t 1 ∗ − g t 2 ∗ + N 0 i 1 + g if i ∈ [1 , k 2 ] , t 1 ∗ − N 0 i if i ∈ [ k 2 + 1 , k 1 ] , 0 if i ∈ [ k 1 + 1 , n ] , T 2 ∗ i = ( 1 1 + g ( t 2 ∗ − N 0 i ) if i ∈ [1 , k 2 ] , 0 if i ∈ [ k 2 + 1 , n ] , wher e (a) k 2 , t 2 ∗ ar e given by t 2 ∗ = (1 + g ) ¯ T 2 + P k 2 i =1 π i N 0 i P k 2 i =1 π i , k 2  an b e found fr om the  ondition ϕ 2 k 2 < ¯ T 2 ≤ ϕ 2 k 2 +1 , wher e ϕ 2 k = 1 1 + g k X i =1 π i ( N 0 k − N 0 i ) , for k ≤ n , and ϕ 2 n +1 = ∞ , (b) k 1 and t 1 ∗ ar e given by t 1 ∗ = ¯ T 1 + k 1 X i = k 2 +1 π i N 0 i + 1 1 + g k 2 X i =1 π i ( g t ∗ 2 + N 0 i ) k 1 X i =1 π i , k 1 ≥ k 2  an b e found as fol lows: INRIA Close d form solutions for symmetri water l ling games 17 (i) k 1 = k 2 if ¯ T 1 ≤ ϕ 1 k 2 +1 (ii) otherwise k 1 is given by the  ondition: ϕ 1 k 1 < ¯ T 1 ≤ ϕ 1 k 1 +1 , wher e ϕ 1 k = k X i = k 2 +1 π i ( N 0 k − N 0 i ) + 1 1 + g k 2 X i =1 π i  (1 + g ) N 0 k − N 0 i − g t 2 ∗  for k ∈ [ k 2 + 1 , n ] , and ϕ 1 n +1 = ∞ . Theorem 7 L et ¯ T 1 > ¯ T 2 > ¯ T 3 . Then, the Nash e quilibrium str ate gies ar e given by T 1 ∗ i =                  t 1 ∗ − g t 2 ∗ 1 + g − g t 3 ∗ 1 + g + N 0 i 1 + 2 g if i ∈ [1 , k 3 ] , t 1 ∗ − g t 2 ∗ + N 0 i 1 + g if i ∈ [ k 3 + 1 , k 2 ] , t 1 ∗ − N 0 i if i ∈ [ k 2 + 1 , k 1 ] , 0 if i ∈ [ k 1 + 1 , n ] , T 2 ∗ i =            t 2 ∗ 1 + g − g 1 + g t 3 ∗ + N 0 i 1 + 2 g if i ∈ [1 , k 3 ] , 1 1 + g ( t 2 ∗ − N 0 i ) if i ∈ [ k 3 + 1 , k 2 ] , 0 if i ∈ [ k 2 + 1 , n ] , T 3 ∗ i = ( 1 1 + 2 g ( t 3 ∗ − N 0 i ) if i ∈ [1 , k 3 ] , 0 if i ∈ [ k 3 + 1 , n ] , wher e (a) k 3 , t 3 ∗ ar e given by t 3 ∗ = ((1 + 2 g ) ¯ T 3 + k 3 X i =1 π i N 0 i ) / ( k 3 X i =1 π i ) , ϕ 3 k 3 < ¯ T 3 ≤ ϕ 3 k 3 +1 , and ϕ 3 k = 1 1 + 2 g k X i =1 π i ( N 0 k − N 0 i ) , RR n ° 6254 18 A ltman, A vr ahenkov & Garnaev for k ≤ n , and ϕ 3 n +1 = ∞ , (b) k 2 , t 2 ∗ ar e given by t 2 ∗ =  ¯ T 2 + 1 1 + g k 2 X i = k 3 +1 π i N 0 i + 1 1 + 2 g k 3 X i =1 π i ( g t 3 ∗ 1 + g + N 0 i ) . 1 1 + g k 2 X i =1 π i  , (i) k 2 = k 3 if ¯ T 2 ≤ ϕ 2 k 3 +1 , (ii) otherwise k 2 is given by the  ondition: ϕ 2 k 2 < ¯ T 2 ≤ ϕ 2 k 2 +1 and ϕ 2 k = k X i = k 3 +1 π i 1 + g ( N 0 k − N 0 i ) + k 3 X i =1 π i  1 1 + g N 0 k − N 0 i + g t 3 ∗ / (1 + g ) 1 + 2 g  . for k ∈ [ k 3 + 1 , n ] and ϕ 2 n +1 = ∞ () k 1 , t 1 ∗ ar e given by t 1 ∗ =  ¯ T 1 + k 1 X i = k 2 +1 π i N 0 i + k 2 X i = k 3 +1 π i g t 2 ∗ + N 0 i 1 + g + k 3 X i =1 π i  g t 2 ∗ 1 + g + g t 3 ∗ 1 + g + N 0 i 1 + 2 g . k 1 X i =1 π i . So, k 1 ≥ k 2  an b e found as fol lows: (i) k 1 = k 2 if ¯ T 1 ≤ ϕ 1 k 2 +1 , (ii) otherwise k 1 is given by the  ondition: ϕ 1 k 1 < ¯ T 1 ≤ ϕ 1 k 1 +1 wher e ϕ 1 k = k X i = k 2 +1 π i ( N 0 k − N 0 i ) + k 2 X i = k 3 +1 π i  N 0 k − g t 2 ∗ + N 0 i 1 + g  + k 3 X i =1 π i  N 0 k − g t 2 ∗ 1 + g − g t 3 ∗ 1 + g + N 0 i 1 + 2 g  . 8 Numerial examples Let us demonstrate the losed form approa h b y n umerial examples. T ak e n = 5 , N 0 i = κ i − 1 , κ = 1 . 7 , π i = 1 / 5 for i ∈ [1 , 5] . W e onsider the ases 1, 2 and 3 users senari. Single user s enario . Let ¯ T = 5 . Then, b y Theorem 2 as the rst step w e alulate ϕ t for t ∈ [1 , 5] . In our ase w e get (0, 0.14, 0.616, 1.8298, 4.58108). Th us, w e ha v e k = 5 and the optimal w ater-lling strategy is T ∗ = (7 . 771 , 7 . 071 , 5 . 881 , 3 . 858 , 0 . 419) with pa y o 1.11. INRIA Close d form solutions for symmetri water l ling games 19 Two users s enario . Let also g = 0 . 9 , ¯ T 1 = 5 , ¯ T 2 = 1 . Then, b y Theorem 6 as the rst step w e alulate ϕ 2 t for t ∈ [1 , 5] . In our ase w e get (0, 0.074, 0.324, 0.963, 2.411). Th us, k 2 = 4 and t 2 ∗ = 5 . 0 01 . Then w e alulate ϕ 1 t for t = 5 . In our ase w e get 6.994052. Th us, k 1 = 4 and t 1 ∗ = 0 . 0 10 . Therefore, w e ha v e the follo wing equilibrium strategies T 1 ∗ = (7 . 1 06 , 6 . 737 , 6 . 111 , 5 . 046 , 0) and T 2 ∗ = (2 . 1 06 , 1 . 737 , 1 , 11 1 , 0 . 0462 , 0) with pa y os 0.801 and 0.116, resp etiv ely . Thr e e users s enario . Let us in tro due the third pla y er with the a v erage p o w er onstrain t ¯ T 3 = 0 . 5 . Then, b y Theorem 7 w e an nd that T 1 ∗ = (6 . 4 19 , 6 . 169 , 5 . 744 , 4 . 900 , 1 . 769 ) , T 2 ∗ = (1 . 8 61 , 1 . 611 , 1 . 186 , 0 . 342 , 0) and T 3 ∗ = (1 . 1 42 , 0 . 892 , 0 . 467 , 0 , 0) are equilibrium strategies with pa y os 0.728, 0.113 and 0.055, resp etiv ely . The equilibrium strategies of all three ases are sho wn in Figure 1 . When a new user omes in to omp etition, it leads to a bigger riv alry for using go o d qualit y  hannels and it results in the situation when bad qualit y  hannels turn out to b eome more attrativ e for users than they w ere when there w ere smaller n um b er of users. Figure 1: Optimal strategies for 1, 2 and 3 user games W e ha v e run IWF A, whi h pro dued the same v alues for the optimal strategies and pa y os. Ho w ev er, w e ha v e observ ed that the on v ergene of IWF A is slo w when g ≈ 1 . In Figure 2, for the t w o users senario, w e ha v e plotted the total error in strategies || T 1 k − T 1 ∗ || 2 + || T 2 k − T 2 ∗ || 2 , where T i k are the strategies pro dued b y IWF A on the k -th iteration and T i ∗ are the Nash equilibrium strategies. Our approa h instan taneously nds the Nash RR n ° 6254 20 A ltman, A vr ahenkov & Garnaev equilibrium for all v alues of g . Also, it is in teresting to note that b y Theorems 6 and 7 the quan tit y of  hannels as w ell as the  hannels themselv es used b y w eak er user (with smaller resoures) is indep enden t from the b eha vior of the stronger user (with larger resoures). Of ourse, ea h user allo ates his/her resoures among the  hannels taking in to aoun t the opp onen t b eha vior. In Figures 3 and 4 , w e ompare the non-o op erativ e approa h with the o op erativ e approa h. Sp eially , w e ompare the transmission rates and their sum under Nash equi- librium strategies and under strategies obtained from the en tralized optimization of the sum of users' rates. The main onlusions are: the ost of anar h y is nearly zero for g ∈ [0 , 1 / 4] and then it gro ws up to 22% when g gro ws from 1 / 4 to 1 ; the user with more resoures gains signian tly more from the en tralized optimization. Hene, the non-o op erativ e ap- proa h results in a more fair resoure distribution. In Figure 4 w e plot the total transmission rate under Nash equilibrium strategies and under strategies obtained from the en tralized optimization for the ases of 2 and 3 users. As exp eted the in tro dution of a new user inreases the ost of anar h y . F urthermore, in the ase of the en tralized optimization with the in tro dution of a new user the total rate inreases, and on on trary in the game setting the total rate dereases. 0 20 40 60 80 100 120 140 160 180 200 0 0.5 1 1.5 2 2.5 number of iterations error g=0.9 g=0.99 g=0.999 Figure 2: Con v ergene of IWF A INRIA Close d form solutions for symmetri water l ling games 21 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 g Rate of Transmitter 1 (Game) Rate of Transmitter 2 (Game) Sum of Rates (Game) Sum of Rates (Optim.) Rate of Transmitter 1 (Optim.) Rate of Transmitter 2 (Optim.) Figure 3: Cen tralized Optimization vs. Game 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Optim. 2 Users Game 2 Users Optim. 3 Users Game 3 Users Figure 4: The eet of a new user 9 Conlusion W e ha v e onsidered p o w er on trol for wireless net w orks in optimization and game frame- w orks. Closed form solutions for the w ater lling optimization problem and L users symmet- RR n ° 6254 22 A ltman, A vr ahenkov & Garnaev ri w ater lling games ha v e b een pro vided. Namely , no w one an alulate optimal/equilibrium strategies with a nite n um b er of arithmeti op erations. This w as p ossible due to the in trin- si hierar hial struture indued b y the quan tit y of the resoures a v ailable to the users. W e ha v e also pro vided a simple alternativ e pro of of on v ergene for a v ersion of iterativ e w ater lling algorithm. It had b een kno wn b efore that the iterativ e w ater lling algorithm on- v erges v ery slo w when the rosstalk o eien t is lose to one. F or our losed form approa h p ossible pro ximit y of the rosstalk o eien t to one is not a problem. W e ha v e sho wn that when the rosstalk o eien t is equal to one, there is a on tin uum of Nash equilibria. Fi- nally , w e ha v e demonstrated that the prie of anar h y is small when the rosstalk o eien t is small and that the deen tralized solution is b etter than the en tralized one with resp et to fairness. Referenes [1℄ E. Altman, K. A vra henk o v, A. Garnaev, A jamming game in wireless net w orks with transmission ost. in Pr o . of NET-COOP 2007. L e tur e Notes in Computer Sien e , v.4465, pp.1-12, 2007. [2℄ E. Altman, K. A vra henk o v, G. Miller and B. Prabh u, Disrete p o w er on trol: o op er- ativ e and non-o op erativ e optimization, in Pro eedings of IEEE INF OCOM 2007 . An extended v ersion is a v ailable as INRIA Resear h Rep ort no.5818. [3℄ T. Co v er and J. Thomas, Elements of Information The ory , Wiley , 1991. [4℄ W. R. Heinzelman, A. Chandrak asan, and H. Balakrishnan, Energy-eien t omm uni- ation proto ol for wireless mirosensor net w orks, in Pr o . of the 33r d A nnual Hawaii International Confer en e on System Sien es , v.2, Jan. 2000. [5℄ A. Garnaev, Se ar h Games and Other Appli ations of Game The ory , Springer, 2000. [6℄ A.J. Goldsmith and P .P . V araiy a, Capait y of fading  hannels with  hannel side infor- mation, IEEE T r ans. Information The ory , v.43(6), pp.1986-1992, 1997. [7℄ T. J. K w on and M. Gerla, Clustering with p o w er on trol, in Pr o . IEEE Military Com- muni ations Confer en e (MILCOM'99) , v.2, A tlan ti Cit y , NJ, USA, 1999, pp.1424 1428. [8℄ L. Lai and H. El Gamal, The w ater-lling game in fading m ultiple aess  hannels, submitted to IEEE T r ans. Information The ory , 2005. [9℄ C. R. Lin and M. Gerla, A daptiv e lustering for mobile wireless net w orks, IEEE JSA C , v.15, no.7, pp.12651275, 1997. [10℄ Z.-Q. Luo and J.-S. P ang, Analysis of iterativ e w aterlling algorithm for m ultiuser p o w er on trol in digital subsrib er lines, EURASIP Journal on Applie d Signal Pr o-  essing , 2006. INRIA Close d form solutions for symmetri water l ling games 23 [11℄ O. P op esu and C. Rose, W ater lling ma y not go o d neigh b ors mak e, in Pr o  e e dings of GLOBECOM 2003 , v.3, pp.17661770, 2003. [12℄ D.C. P op esu, O. P op esu and C. Rose, In terferene a v oidane v ersus iterativ e w ater lling in m ultiaess v etor  hannels, in Pr o  e e dings of IEEE VTC 2004 F al l , v.3, pp.20582062, 2004. [13℄ K.B. Song, S.T. Ch ung, G. Ginis and J.M. Cio, Dynami sp etrum managemen t for next-generation DSL systems, IEEE Communi ations Magazine , v.40, pp.101109, 2002. [14℄ D. T se and P . Visw anath, F undamentals of Wir eless Communi ation , Cam bridge Uni- v ersit y Press, 2005. [15℄ W. Y u, Comp etition and  o op er ation in multi-user  ommuni ation envir onements , PhD Thesis, Stanford Univ ersit y , June 2002. [16℄ W. Y u, G. Ginis and J.M. Cio, Distributed m ultiuser p o w er on trol for digital sub- srib er lines, IEEE JSA C , v.20, pp.11051115, 2002. Con ten ts 1 In tro dution 3 2 Single deision mak er 4 3 Symmetri w ater lling game 5 4 A reursiv e approa h to the symmetri w ater lling game 7 5 A w ater-lling algorithm 10 6 Existene and uniqueness of the Nash equilibrium 11 7 Closed form solution for L p erson game 13 8 Numerial examples 18 9 Conlusion 21 RR n ° 6254 Unité de recherche INRIA Sophia Antipolis 2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France) Unité de reche rche INRIA Futurs : Parc Club Orsay Uni versité - ZAC des V ignes 4, rue Jacques Monod - 91893 ORSA Y Cedex (France ) Unité de reche rche INRIA Lorraine : LORIA, T echnopôle de Nancy- Brabois - Campus scientifique 615, rue du Jardin Botani que - BP 101 - 54602 V illers-lè s-Nancy Cedex (France) Unité de reche rche INRIA Rennes : IRISA, Campus univ ersitai re de Beaulie u - 35042 Rennes Cedex (Franc e) Unité de reche rche INRIA Rhône-Alpes : 655, aven ue de l’Europe - 38334 Montbonnot Saint-Ismier (France) Unité de recherch e INRIA Rocquen court : Domaine de V oluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France) Éditeur INRIA - Domaine de V olucea u - Rocquenco urt, BP 105 - 78153 Le Chesnay Cede x (France) http://www.inria.fr ISSN 0249 -6399