Closed form solutions for water-filling problems in optimization and game frameworks

Transcription

1 Closed form solutons for water-fllng problems n optmzaton and game frameworks Etan Altman INRIA BP93 24 route des Lucoles 692 Sopha Antpols FRANCE altman@sopha.nra.fr Konstantn Avrachenkov INRIA BP93 24 route des Lucoles 692 Sopha Antpols FRANCE k.avrachenkov@sopha.nra.fr Andrey Garnaev St. Petersburg State Unversty Unverstetsk pr 35 Peterhof St Petersburg 954 RUSSIA agarnaev@rambler.ru ABSTRACT We study power control n optmzaton and game frameworks. In the optmzaton framework there s a sngle decson maker who assgns network resources and n the game framework players share the network resources accordng to Nash equlbrum. The soluton of these problems s based on so-called water-fllng technque, whch n turn uses bsecton method for soluton of non-lnear equatons for Lagrange multples. Here we provde a closed form soluton to the water-fllng problem, whch allows us to solve t n a fnte number of operatons. Also, we produce a closed form soluton for the Nash equlbrum n symmetrc Gaussan nterference game. In addton, to ts mathematcal beauty, the explct soluton allows one to study lmtng cases when the crosstalk coeffcent s ether small or large. We provde an alternatve smple proof of the convergence of the Iteratve Water Fllng Algorthm. Furthermore, t turns out that the convergence of Iteratve Water Fllng Algorthm slows down when the crosstalk coeffcent s large. Usng the closed form soluton, we can avod ths problem. Fnally, we compare the non-cooperatve approach wth the cooperatve approach and show that the non-cooperatve approach results n a more far resource dstrbuton.. INTRODUCTION In wreless networks and DSL access networks the total avalable power for sgnal transmsson has to be dstrbuted among several resources. In the context of wreless networks, the resources may correspond to frequency bands (e.g. as n OFDM), or they may correspond to capacty avalable at dfferent tme slots. In the context of DSL access networks, the resources correspond to avalable frequency tones. Ths spectrum of problems can be consdered n ether optmza- Ths work was partly supported by BoNets European project and by the jont RFBR and NNSF Grant no Permssontomakedgtalorhardcopesofallorpartofthsworkfor personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bearthsnotceandthefullctatononthefrstpage.tocopyotherwse,to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. GameComm 7, October 22, 27, Nantes, France Copyrght 27 ICST ton scenaro or as a result of a non-cooperatve game scenaro. The optmzaton scenaro leads to Water Fllng Optmzaton Problem [4, 6, 3] and the game scenaro leads to Water Fllng Game or Gaussan Interference Game [,, 2, 4]. In the optmzaton scenaro, one needs to maxmze a concave functon (Shannon capacty) subject to power constrants. The Lagrange multpler correspondng to the power constrant s determned by a non-lnear equaton. In the prevous works [4, 6, 3], t was suggested to fnd the Lagrange multpler by means of a bsecton algorthm, comes the name Water Fllng Problem. Here we show that the Lagrange multpler and hence the optmal soluton of the water fllng problem can be found n explct form wth a fnte number of operatons. In the multuser context, one can vew the problem n ether cooperatve or non-cooperatve settng. If a centralzed controller wants to maxmze the sum of all users rates, the controller wll face a non-convex optmzaton problem [3]. On the other hand, n the non-cooperatve settng, the power allocaton problem becomes a game problem each user perceves the sgnals of the other users as nterference and maxmzes a concave functon of the nose to nterference rato. In [, 4] the spectrum of avalable resources was contnuous, here as n [,, 2] we consder the dscrete spectrum of avalable resources. A natural approach n the non-cooperatve settng s the applcaton of the Iteratve Water Fllng Algorthm (IWFA) [5]. Recently, the authors of [] proved the convergence of IWFA under farly general condtons. In the present work we consder the case of symmetrc water fllng game wth two users. The restrcton to the symmetrc scenaro allows us to fnd Nash equlbrum n explct form. In addton, to ts mathematcal beauty, the explct soluton allows one to fnd the Nash equlbrum n water fllng game n a fnte number of operatons and to study lmtng cases when the crosstalk coeffcent s ether small or large. As a by-product, we obtan an alternatve smple proof of the convergence of the Iteratve Water Fllng Algorthm. Furthermore, t turns out that the convergence of IWFA slows down when the crosstalk coeffcent s large. Usng the closed form soluton, we can avod ths problem. Fnally, we compare the non-cooperatve approach wth the cooperatve approach and conclude that the cost of anarchy s small n the case of small crosstalk coeffcents. We also show that the non-cooperatve approach results n a more far resource dstrbuton. Applcatons that can mostly beneft from decentralzed non-cooperatve power control are ad-hoc and sensor

2 networks wth no predefned base statons [5, 9, 7]. An nterested reader can fnd more references on non-cooperatve power control n [2, ]. We would lke to menton that the water fllng problem and jammng games wth transmsson costs have been analyzed n []. The structure of the paper s as follows: In the next Secton 2, we recall the sngle decson maker setup of the water fllng optmzaton problem. Then, n Secton 3 we provde ts explct soluton. In Secton 4 we formulate two user symmetrc water fllng game and characterze ts Nash equlbrum. In Secton 5 we gve an alternatve smple proof of the convergence of the teratve water fllng algorthm. In Secton 6 we gve the explct form of the players strategy n the Nash equlbrum. In Secton 7 we confrm our fndng wth the help of numercal example. In that secton we also show that the cost of anarchy s small when the crosstalk coeffcent s small. We make conclusons n Secton. 2. SINGLE DECISION MAKER We consder the followng power allocaton problem n the case of a sngle decson maker. There s a sngle decson maker (also called Transmtter ) who wants to send nformaton usng n ndependent resources so as to maxmze the Shannon capacty. We further assume that resource has a weght of π. Possble nterpretatons: () The resources may correspond to capacty avalable at dfferent tme slots; we assume that there s a varyng envronment whose state changes among a fnte set of states [, n], accordng to some ergodc stochastc process wth statonary dstrbuton {π } n =. We assume that both players have perfect knowledge of the envronment state at the begnnng of each tme slot. () The resources may correspond to frequency bands (e.g. as n OFDM) one should assgn dfferent power levels for dfferent sub-carrers [3]. In that case we may take π = /n for all. P The strategy of Transmtter s T = (T,..., T n) wth n = πt = T, T, π > for [, n] and T >. As the payoff to Transmtter we take the Shannon capacty v(t) = π ln = + T N, N > s the nose level n the sub-carrer. Defne also the functon H T(ω) = π T (ω). = We would lke to emphasze that the above generalzed descrpton of the water-fllng problem can be used for power allocaton n tme as well as power allocaton n space-frequency. Followng the standard water-fllng approach [4, 6, 3], whch assumes applcaton Kuhn-Tacker Theorem, we have the followng result. Theorem. Let T (ω) = ˆ/ω N for [, n]. + Then T(ω ) = (T (ω ),..., T n(ω )) s the unque optmal strategy and ts payoff s v(t(ω )) ω s the unque root of the equaton H T(ω) = T. () We observe that the above result can be easly proved wthout applcaton of the Kuhn-Tacker condtons. In the appendx we supply ths proof. 3. CLOSED FORM SOLUTION FOR WATER FILLING PROBLEM In the prevous studes of the water-fllng problems t was suggested to use numercal (e.g., bsecton) method to solve equaton (). Here we propose an explct form approach for the soluton of equaton (). Wthout loss of generalty we can assume that the subcarrers are arranged by the nose level as follows: N N Nn Theorem 2. The soluton of the water-fllng optmzaton problem s gven by k T + π t(nt N ) >< T t=, f k, = k (2) π t t=, f > k, k can be found from the followng condtons: ϕ t = ϕ k < T ϕ k+, (3) t π (Nt N ) for t [, n]. = Let us demonstrate the closed form approach by a numercal example. Take n = 5, T =, N = κ, κ =.7, π = /5 for [, 5]. Then, as the frst step we calculate ϕ t for t [, 5]. In our case we get (..4,.66,.29, 4.5). Then, by (3), k = 3. Thus, by (2), the optmal water-fllng strategy s T = (2.53,.3,.64,, ) wth payoff SYMMETRIC WATER FILLING GAME Two Transmtters try to send nformaton through n resources so as to maxmze the qualty of the transmtted nformaton. The strategy of Transmtter j s T j = (T j,..., T j n) wth T j and wth n = π T j = T j, (4) T j > for j =, 2. The payoffs to Transmtters are gven as follows v (T, T 2 T ) = π ln + gt 2 = +, N v 2 (T, T 2 T 2 ) = π ln + gt +, N = g (, ). These payoffs correspond to Shannon capactes. Ths s an nstance of the Water-Fllng or Gaussan Interference Game [, 4, 5]. In the mportant partcular cases of OFDM wreless network and DSL access network,

3 π = /n, =,..., n. In ths work we restrct ourselves to the case of symmetrc game wth equal crosstalk coeffcents. Ths stuaton can correspond for example to the scenaro when the transmtters are stuated at about the same dstance from the base staton. We shall characterze a Nash Equlbrum of ths problem. The strateges (T, T 2 ) consttute a Nash Equlbrum, f for any strateges (T, T 2 ) the followng nequaltes hold: v (T, T 2 ) v (T, T 2 ), v 2 (T, T 2 ) v 2 (T, T 2 ). Snce v and v 2 are concave n T and T 2 respectvely, the Kuhn-Tucker condtons mply the followng theorem. Theorem 3. (T, T 2 ) s a Nash equlbrum f and only f there are non-negatve ω and ω 2 (Lagrange multplers) such that ( = ω j for T j >, gt m + T j + N j, m {, 2} and m j. ω j for T j =, We would lke to emphasze that the postvty of the Lagrange multplers ω and ω 2 s specfc for our problem. In general, the Lagrange multplers correspondng to equalty constrants do not have any restrctons on the sgn. Let us ntroduce the followng sets: I (ω, ω 2 ) = { [, n] : ω N, ω 2 N }, I (ω, ω 2 ) = { [, n] : N < ω and ether N ω 2 or N < ω 2 and ω 2 N g( ω N )}, I (ω, ω 2 ) = { [, n] : N < ω 2 and ether N ω or N < ω and ω N g( ω 2 N )}, I (ω, ω 2 ) = { [, n] : < g( ω 2 N ) < ω N < g ( ω 2 N )}. The next result characterzes the forms that the Nash equlbrum can take. Lemma. Let (T, T 2 ) be a Nash equlbrum, then () f T () f T () f T (v) f T = and T 2 = then I (ω, ω 2 ), > and T 2 = and T 2 > and T 2 = then I (ω, ω 2 ) and T = /ω N, > then I (ω, ω 2 ) and T 2 = /ω 2 N, > then I (ω, ω 2 ) and T = (/ω N ) g(/ω 2 N ) g 2, T 2 = (/ω2 N ) g(/ω N ) g 2. (5) (6) Although the game has symmetrc nature there are some non-symmetrc features mpacted by the fact that the Lagrange multples are dfferent as a rule. Ths dfference wll allow us to smplfy the structure of the sets I and the strateges. For ths purpose frst ntroduce the followng auxlary notatons for postve ω and ω 2 : () f ω < ω 2, so /ω 2 < /ω then let I (ω, ω 2 ) = { [, n] : ω N }, I (ω, ω 2 /ω 2 g/ω ) = { [, n] : N < g ω }, I (ω, ω 2 ) =, I (ω, ω 2 ) = { [, n] : N < /ω2 g/ω }, g () f ω 2 < ω, so /ω < /ω 2 then let I (ω, ω 2 ) = { [, n] : I (ω, ω 2 ) =, I (ω, ω 2 ) = { [, n] : ω 2 N }, /ω g/ω 2 g N < ω 2 }, I (ω, ω 2 ) = { [, n] : N < /ω g/ω 2 }, g () f ω 2 = ω then let I (ω, ω 2 ) = { [, n] : ω 2 N }, I (ω, ω 2 ) =, I (ω, ω 2 ) =, I (ω, ω 2 ) = { [, n] : N < ω 2 }. The next lemma asserts that the sets I are concdes wth the sets I. Lemma 2. There are the followng relatons between sets I and I: I (ω, ω 2 ) = I (ω, ω 2 ), I (ω, ω 2 ) = I (ω, ω 2 ), I (ω, ω 2 ) = I (ω, ω 2 ) and I (ω, ω 2 ) = I (ω, ω 2 ) for postve ω and ω 2. Now we ntroduce some strateges, whch the Nash Equlbrum wll have the form of. Namely, for postve ω and ω 2 such that ω ω 2 and for [, n] we ntroduce the followng notatons: T (ω, ω 2 ) = B + ω g ω 2 g N C A f N < >< ω N f f f ω 2 g ω g ω N, ω 2 g ω g, N < ω, T 2 (ω, ω 2 ) = B >< + ω 2 g ω g N C A f N < /ω2 g/ω g, ω 2 g ω g N,

4 ether n the followng equvalent form as follows T 2 (ω, ω 2 ) ( `t2 = + g N f N < t 2, f t 2 N, T (ω, ω 2 ) >< + g `( + g)t gt 2 N f N < t 2, = t N f t 2 N < t, f t N, It s clear that t 2 = /ω2 g/ω, t = g ω. (7) /ω = t 2, /ω 2 = gt + ( g)t 2. () For the case ω > ω 2, T (ω, ω 2 ) and T 2 (ω, ω 2 ) can be defned by symmetry. The next result smplfes the form of the Nash equlbrum, gven by Lemma and t shows that the strateges are not so symmetrc as t could be expected and ther nonsymmetrc structure motvated by dfference n Lagrange multplers and so n the power of the sgnals the players have to transfer. Theorem 4. Each Nash equlbrum s of the form (T (ω, ω 2 ), T 2 (ω, ω 2 )) for some postve ω and ω 2. The next result shows that there s a monotonous dependence between the power of the sgnals the players have to transfer and Lagrange multplers. Corollary. Let (T (ω, ω 2 ), T 2 (ω, ω 2 )) be a Nash equlbrum. If T > T 2, then ω < ω 2. To fnd the equlbrum strateges we have to fnd ω and ω 2 such that the followng condtons hold H j (ω, ω 2 ) = H j (ω, ω 2 ) = T j for j =, 2 (9) π T j (ω, ω 2 ) for j =, 2. = The next Lemma shows that ths system has the unque soluton and moreover ts proof supples a smple method of ts determnaton. Lemma 3. The system of non-lnear equatons (9) has unque postve soluton (ω, ω 2 ). Proof. Wthout lost of generalty we can assume that T T 2. Let (ω, ω 2 ) be the postve soluton of (9). Then, by Corollary, ω < ω 2. Thus, nstead of the system of equaton (9) wth varables ω and ω 2 we can consder the followng equvalent system of equaton () wth varables t and t 2 < t 2 t : H 2 (t 2 ) = T 2, H (t, t 2 ) = T, () H (t, t 2 ) = H 2 (t 2 ) = + g {:t 2 N <t } + {:N <t2 } {:t 2 >N } π (t 2 N ), π (t N ) π + g `( + g)t gt 2 N. It s clear that H 2 ( ) s contnuous n (, ), H2 (τ) = for τ N, H2 (+ ) = + and H 2 ( ) s strctly ncreasng n (N, ). Then, there s the unque postve t 2 such that H 2 (t 2 ) = T 2. () It s clear that H (, t 2 ) s contnuous and ncreasng n (t 2, ), H (, t 2 ) = + and H (t 2, t 2 ) = H 2 (t 2 ) = T 2 T. So, there s the unque postve t such that H (t, t 2 ) = T. (2) So, the system () has the unque soluton (t, t 2 ). Thus, (9) also has the unque soluton and t can be found by (). Ths completes the proof of Lemma 3. Lemmas 3 mply the followng man result. Theorem 5. The symmetrc water fllng game has the unque Nash equlbrum (T (ω, ω 2 ), T 2 (ω, ω 2 )) for g (, ), ω, ω 2 can be fount through t and t 2 from (7) whch are the unque soluton of the trangular system of equatons (). The assumpton that g < s essental for the unqueness of Nash Equlbrum as t s shown n the followng Proposton. Proposton. For g = the symmetrc water fllng game has a contnuum of Nash equlbra. 5. CONVERGENCE OF AN IWFA In ths secton we descrbe a verson of the water-fllng algorthm for fndng the Nash Equlbrum and supply a smple proof of ts convergence based on some monotonsty propertes. It s clear that H (ω, ω 2 ) =, 2 have the followng propertes, collected n the next Lemma, whch follow drectly from the explct formulas of the Nash Equlbrum. Lemma 4. () H (ω, ω 2 ), =, 2 are nonnegatve and contnuous, () H (ω, ω 2 ) s non-ncreasng on ω, () H (ω, ω 2 ) for ω, (v) H (ω, ω 2 ) = for enough bg ω, say for ω /N, (v) H (ω, ω 2 ) s non-decreasng by ω j (j ). These propertes gve a smple proof of the convergence of the followng water-fllng algorthm for fndng the Nash Equlbrum. Let ω and ω 2 be such that H (ω, ω 2 ) = H 2 (ω, ω 2 ) =, for example ω = ω 2 = /N. Let ω 2 = ω 2 and defne ω such that H (ω, ω 2 ) = T. Such ω exsts by Lemma 4()- (). Then, by Lemma 4(),(v) H 2 (ω, ω 2 ) =. Let ω 2 = ω and defne ω 2 2 such that H 2 (ω 2, ω 2 2) = T 2. Then, by Lemma 4(v) H (ω 2, ω 2 2) T and so on. So we have nonncreasng postve sequence (ω k, ω 2 k). Thus, t converges to an (ω, ω 2 ) whch produces a Nash Equlbrum.

5 6. CLOSED FORM SOLUTION FOR SYM- METRIC WATER FILLING GAME In ths secton, based on the proof of Lemma 3, we propose the soluton of the two players symmetrc water fllng game n the closed form. Wthout lost of generalty we can assume that T > T 2. Let k 2 be such that N k 2 + t 2 > N k 2. Then, snce H 2 (t 2 ) = T 2, we have that t 2 = ( + g) T 2 + P k 2 = πn P k2 = π. (3) Snce H 2 ( ) s strctly ncreasng, k 2 can be found from the condton H 2 (N k 2) < T 2 H 2 (N k 2 +). Hence, k 2 can be found from the followng equvalent condtons: ϕ 2 k 2 < T 2 ϕ 2 k 2 +, (4) ϕ 2 k = + g k π (Nk N ), = for k n, and ϕ 2 n+ =. Snce t s the root of the equaton H (, t 2 ) = T there s k k 2 such that N k + t > N k. So, () f k > k 2 then t = T + k =k 2 + () f k = k 2 then t = π N + + g π (gt 2 + N ) k 2 =, (5) k π = T + + g π (gt 2 + N ) k 2 =. (6) k π = Thus, k k 2 can be found as follows: () k = k 2 f T ϕ k 2 +, () otherwse k s gven by the condton: ϕ k = k =k g ϕ k < T ϕ k +, (7) π (N k N ) k 2 = π `( + g)n k N gt 2 for k [k 2 +, n], and ϕ n+ =. We can summarze the obtaned results n the followng theorem. Theorem 6. Let T > T 2. Then, the Nash equlbrum strateges are gven by >< t T gt2 + N + g f [, k 2 ], = t N f [k 2 +, k ], f [k +, n], () ( T 2 = + g (t2 N ) f [, k 2 ], f [k 2 +, n], k 2, t 2, k and t are gven by (4), (3), (7) and (5). 7. NUMERICALEAMPLE Let us demonstrate the closed form approach by a numercal example. Take n = 5, g =.9, T = 5, T 2 =.5, N = κ, κ =.7, π = /5 for [, 5]. Then, as the frst step we calculate ϕ 2 t for t [, 5]. In our case we get (,.74,.324,.963, 2.4). Then, by (4), k 2 = 3. Thus, by (3) t 2 = Then we calculate ϕ t for t [4, 5]. In our case we get (.3,4.3). So, by (7), k = 5. Usng (5), we fnd t = Thus, by () we have the followng equlbrum strateges T = (7.62, 6.694, 6.67, 4.3,.69) and T 2 = (.2,.99,.293,, ) wth payoffs.99 and.62. We have run IWFA, whch produced the same values for the optmal strateges and payoffs. However, we have observed that the convergence of IWFA s slow when g. In Fgure we have plotted the total error n strateges T k T 2 + T 2 k T 2 2, T k are the strateges produced by IWFA on the k-th teraton and T are the Nash equlbrum strateges. Our approach nstantaneously fnds the Nash equlbrum for all values of g. Also, t s nterestng to note that by () the quantty of channels as well as the channels themselves used by weaker player (wth smaller resources) s ndependent on behavor of the stronger player (wth bgger resources) but of course each player allocatng hs/her resources among these channels take nto account the opponent behavour. In Fgure 2, we compare the non-cooperatve approach wth the cooperatve approach. Specfcally, we compare the transmsson rates and ther sum under Nash equlbrum strateges and under strateges obtaned from the centralzed optmzaton of the sum of transmtters rates. The man conclusons are: the cost of anarchy s nearly zero for g [, /4] and then t grows up to 22% when g grows from /4 to ; the transmtter wth more resources gans sgnfcantly more from the centralzed optmzaton. Hence, the non-cooperatve approach results n a more far resource dstrbuton. In Table we gve strateges of both users obtaned n the case of the centralzed optmzaton for dfferent values of the crosstalk coeffcent g. Frst, we observe that when the crosstalk coeffcent s large, the users occupy dfferent resources. The user wth the larger average power takes better resources. When the crosstalk coeffcent s below.7, the users start to share the resources. As the value of the crosstalk coeffcent decreases, the 2nd user wth the smaller average power begns to occupy better resources. As expected, when the crosstalk coeffcent s very small, the optmal strateges start to look lke strateges whch are optmal n the case of no nterference.

6 error g=.9 g=.99 g= number of teratons g Table : Centralzed optmzaton Users strateges st user nd user st user nd user st user nd user st user nd user st user nd user st user nd user st user nd user Fgure : Convergence of IWFA centralzed wth respect to farness. We expect that the present approach can be generalzed to the case of more than two users. A detal analytcal study of the centralzed optmzaton s another future research topc Rate of Transmtter (Game) Rate of Transmtter 2 (Game) Sum of Rates (Game) Sum of Rates (Optm.) Rate of Transmtter (Optm.) Rate of Transmtter 2 (Optm.) g Fgure 2: Centralzed Optmzaton vs. Game. CONCLUSION We have consdered power control for wreless networks n optmzaton and game frameworks. Closed form solutons for the water fllng optmzaton problem and two players symmetrc water fllng games have been provded. Namely, now one can calculate optmal/equlbrum strateges wth a fnte number of arthmetc operatons. We have also provded a smple alternatve proof of convergence for a verson of teratve water fllng algorthm. It had been known before that the teratve water fllng algorthm converges very slow when the crosstalk coeffcent s close to one. For our closed form approach possble proxmty of the crosstalk coeffcent to one s not a problem. We have shown that when the crosstalk coeffcent s equal to one, there s a contnuum of Nash equlbra. Fnally, we have demonstarted that the prce of anarchy s small when the crosstalk coeffcent s small and that the decentralzed soluton s better than 9. APPENDI Proof of Theorem. Let T = (T,..., T n) be the optmal strategy. Snce P n = πt = T and T for [, n] there s a m [, n] such that Tm >. Let ǫ be any small enough postve number and k m. Let T ǫ,k = (T ǫ,k,..., Tn ǫ,k ) be such that >< Tm ǫ/π m for = m, T ǫ,k = Tk + ǫ/π k for = k, for {m, k}. T It s cleat that T ǫ,k also s a strategy for any enough small postve ǫ. Then, snce T s the optmal strategy we have that v(t ) v(t ǫ,k ). Thus, π k ln + T k N k + π m ln + T m π k ln + T k ǫ/π k Nk So, puttng ǫ we have that N m + π m ln + T m + ǫ/π m. Nm Tm + Nm Tk + for any m k. N k Thus, there s a postve ω such that ( = ω, for T >, T + N ω, for T =. So, the optmal strategy T s of the form T(ω) = (T (ω),..., T n(ω)) T (ω) = [/ω N ] + for [, n].

7 Remndng that the optmal strategy s non-negatve vector satsfyng the condton P n = πt = T we obtan that ω has to be found as a soluton of the equaton H T(ω) = T. It s clear that H T(+) = +, H T( ) s contnuous n (, ), H T(ω) = for ω [max (/N ), ) and H T( ) s strctly decreasng n (, max (/N )). Thus, there s unque postve ω such that H T(ω) = T. Ths completes the proof of Theorem. Proof of Theorem 2. Frst note H(ω) = for ω /N, H(ω) s strctly postve and decreasng n (, /N ). Let k [, n] be such that N k > ω, Nk+ Nn+ =. Then, ˆ/ω N = + /ω N for [, k] and ˆ/ω N = [k +, n]. So, + H(ω ) = k π (/ω N ). = Snce H(ω ) = T we have that ω = T + k = π. (9) k π N Because H s strctly decreasng on (, /N ) we can fnd k from the followng condton Snce = H(/N k) < T H(/N k+). k k+ π (Nk+ N ) = π (Nk+ N ), = the nteger k can be found from the followng equvalent condton ϕ t = = ϕ k < T ϕ k+, (2) t π (Nt N ) for t [, n]. = Therefore, Theorem, (9) and (2) mply Theorem 2. Proof of Theorem 3. The Lagrangan correspondng to mnmzaton of v j subject to the constrant (4) and nonnegatvty constrants on T j s gven by! L j T j n! k = π k ln + + ω j π k T j k T j k= gt m k + N k +ν j ( T j ), k= wth m j. Dfferentatng the Lagrangan wth respect to T j and equatng the dervatve to zero, we obtan gt m + T j + N + νj π = ω j. (2) Now, usng the complmentary slackness condton ν j T j =, we obtan condton (5). Snce the left hand sde of equaton (2) s postve, the Lagrange multpler ω j s postve as well. Proof of Lemma. () follows drectly from (5) = T 2 =. T () Let T Thus, that Thus, > and T 2 T =. Then by (5) we have that + N = ω. ω > N and T = ω N. Then, by (5) we have ω 2 gt and the result follows. + N = g(. ω N ) + N g( ω N ) ω 2 N () can be proved smlarly to (). (v) Let T > and T 2 >. Then, by (5)we have that ω > N and ω 2 > N. Also, by (5) we have that T and T 2 are gven by (6). Then, snce T > and T 2 > we have that I (ω, ω 2 ). Ths completes the proof of the lemma. Proof of Lemma 2. Let, for example, ω < ω 2. It s clear that I (ω, ω 2 ) = I (ω, ω 2 ). Let I (ω, ω 2 ). Then ether /ω 2 N < /ω or /ω2 g/µ g N < mn{/ω, /ω 2 }. Snce ω < ω 2, mn{/ω, /ω 2 } = /ω 2 and /ω2 g/ω g < /ω 2. Thus, I (ω, ω 2 ) = I (ω, ω 2 ). Let I (ω, ω 2 ). Then ether /ω N < /ω 2 or /ω g/ω 2 g N < mn{/ω, /ω 2 }. Snce ω < ω 2, then /ω g/ω 2 g > /ω 2. So, I (ω, ω 2 ) = = I (ω, ω 2 ). Let I (ω, ω 2 ). Then N < mn{ /ω g/ω 2 g, /ω2 g/ω g } = /ω2 g/ω g. Thus, I (ω, ω 2 ) = I (ω, ω 2 ). Ths completes the proof of Lemma 2. Proof of Corollary. Assume that ω ω 2. Then /ω g/ω 2 /ω 2 g/ω. Thus, T (ω, ω 2 ) T 2 (ω, ω 2 ) for [, n]. So, T = π T (ω, ω 2 ) π T 2 (ω, ω 2 ) = T 2. = = Ths contradcton completes the proof of Corollary. Proof of Proposton. Suppose that (T, T 2 ) be a Nash equlbrum. Then, smlarly to Lemma, we have to consder three cases ()-() at least one of components of the vector (T, T 2 ) s postve.

8 () Let T > and T 2 =. Then, by (5), we have that Thus, Then, by (5) ω 2 () Let T 2 have that and () Let T T + N = ω. ω > N and T = ω N. T + N = > and T T 2 ω N + N = ω. =. Then, smlarly to (), we = ω 2 N ω 2 > N, ω ω 2. > and T 2 T >. Then, by (5), have that + T 2 + N = ω = ω 2. Assume that ω > ω 2 then () does not hold, so T = for each whch contradcts to (4). Smlarly, the case ω < ω 2 cannot hold. Thus, ω = ω 2 = ω. So, T be any non-negatve such that and = T + T 2 = [/ω N ] + π T = T, = and T 2, [, n] have to π T 2 = T 2, ω s the unque postve root of the equaton π [/ω N ] + = T + T 2. = It s clear that there s a contnuum of such strateges. For example f (T, T 2 ) s the one of them, and let Tk, Tk 2 > and Tm, Tm 2 > for some k and m. Then, t s clear that the followng strateges for any enough small postve ǫ are also optmal: >< T for k, m, T = T + ǫ for = k, T ǫπ k /π m for = m, >< T 2 for k, m, T 2 = T 2 ǫ for = k, T 2 + ǫπ k /π m for = m. Ths completes the proof of Proposton. [2] E. Altman, K. Avrachenkov, G. Mller and B. Prabhu, Dscrete power control: cooperatve and non-cooperatve optmzaton, n Proceedngs of IEEE INFOCOM 27. An extended verson s avalable as INRIA Research Report no.5. [3] S.T. Chung and J.M. Coff, Rate and power control n a two-user multcarrer channel wth no coordnaton: the optmal scheme vs. suboptmal methods, n Proceedngs of IEEE VTC 2-Fall, v.3, pp , 22. [4] T. Cover and J. Thomas, Elements of Informaton Theory, Wley, 99. [5] W. R. Henzelman, A. Chandrakasan, and H. Balakrshnan, Energy-effcent communcaton protocol for wreless mcrosensor networks, n Proc. of the 33rd Annual Hawa Internatonal Conference on System Scences, v.2, Jan. 2. [6] A.J. Goldsmth and P.P. Varaya, Capacty of fadng channels wth channel sde nformaton, IEEE Trans. Informaton Theory, v.43(6), pp , 997. [7] T. J. Kwon and M. Gerla, Clusterng wth power control, n Proc. IEEE Mltary Communcatons Conference (MILCOM 99), v.2, Atlantc Cty, NJ, USA, 999, pp [] L. La and H. El Gamal, The water-fllng game n fadng multple access channels, submtted to IEEE Trans. Informaton Theory, November 25, avalable at helgamal/. [9] C. R. Ln and M. Gerla, Adaptve clusterng for moble wreless networks, IEEE JSAC, v.5, no.7, pp , 997. [] Z.-Q. Luo and J.-S. Pang, Analyss of teratve waterfllng algorthm for multuser power control n dgtal subscrber lnes, EURASIP Journal on Appled Sgnal Processng, 26. [] O. Popescu and C. Rose, Water fllng may not good neghbors make, n Proceedngs of GLOBECOM 23, v.3, pp , 23. [2] D.C. Popescu, O. Popescu and C. Rose, Interference avodance versus teratve water fllng n multaccess vector channels, n Proceedngs of IEEE VTC 24 Fall, v.3, pp , 24. [3] D. Tse and P. Vswanath, Fundamentals of Wreless Communcaton, Cambrdge Unversty Press, 25. [4] W. Yu, Competton and cooperaton n mult-user communcaton envronements, PhD Thess, Stanford Unversty, June 22. [5] W. Yu, G. Gns and J.M. Coff, Dstrbuted multuser power control for dgtal subscrber lnes, IEEE JSAC, v.2, pp.5 5, 22.. REFERENCES [] E. Altman, K. Avrachenkov, and A. Garnaev, Jammng game n wreless networks wth transmsson cost. Lecture Notes n Computer Scence, v.4465, pp.-2, 27.