Auction-Based Spectrum Sharing

Transcription

1 Aucton-Based Spectrum Sharng Janwe Huang, Randall Berry and Mchael L. Hong Department of Electrcal and Computer Engneerng Northwestern Unversty, Evanston IL 60208, USA Abstract. We study aucton-based mechansms for sharng spectrum among a group of users, subject to a constrant on the nterference temperature at collocated recevers. The users access the channel usng spread spectrum sgnalng and thus generate nterference wth each other. Each user receves a utlty that s a functon of the receved sgnal-tonterference plus nose rato. We propose two aucton mechansms for allocatng the receved power. The frst s an SINR-based aucton, whch, when combned wth logarthmc utltes, leads to a weghted max-mn far SINR allocaton. The second s a power-based aucton that maxmzes the total utlty when the bandwdth s large enough. Both aucton mechansms acheve socal optmalty n a large system lmt where bandwdth, power and the number of users are ncreased n a fxed proporton. We also gve suffcent condtons for global convergence of a dstrbuted updatng algorthm and dscuss the convergence speed. 1 Introducton There has been growng nterest n makng more effcent use of spectrum by shftng from the conventonal command-and-control spectrum usage models to more flexble Exclusve Use and Commons models (e.g., see [1]). In the Exclusve Use model, the lcensee has exclusve rghts to the spectrum, but can allow other users access as long as they keep the nterference temperature under some threshold. Here the nterference temperature s defned as the RF power avalable at the recevng antenna per unt bandwdth. In the Commons model, an unlmted number of users can share spectrum wth usage rghts governed by techncal standards, but wth no explct protecton from nterference. In ether model, a basc queston s how to share the avalable spectrum effcently and farly. In ths paper we focus on a spectrum allocaton problem for the Exclusve Use model. A group of spread spectrum users transmttng to collocated recevers want to share a fxed bandwdth. A manager (owner or regulator) must allocate the spectrum subject to a constrant on the nterference temperature at the recevers. We model ths as a constrant on the total receved power. The manager can then be vewed as allocatng the receved power to the users. Each user has a utlty, whch s a functon of the receved Sgnal-to-Interference plus Nose Rato (SINR), reflectng hs desred Qualty of Servce (QoS). The nterference a user receves s the total receved power of all other users scaled by the bandwdth. We consder aucton mechansms to allocate the receved power as a functon of bds the users submt. We model the resultng problem as a noncooperatve game [2], and characterze the Nash equlbra and related propertes for two dfferent aucton mechansms. Our approach s smlar to a share aucton (see [3 7] and the references theren), or dvsble aucton, where a perfectly dvsble good s splt among bdders whose payments depend solely on the bds. A common form of bds n a share aucton s for each user to submt hs demand curve (e.g., [3 5]),.e., the amount of goods a user desres as a functon of the prce. The auctoneer can then compute a market clearng prce based on the set of demand curves. However, n our problem, a user s utlty depends on hs SINR level, whch n turn depends on the power assgned to other users, makng the users demand curves dependent on each other. Instead, we adopt a sgnalng system smlar to [6, 7], where users submt one dmensonal bds for the resource. We assume a weghted proportonal allocaton rule n whch a user s power allocaton s proportonal to hs bd. Ths type of allocaton rule has been studed n a wde range of applcatons (e.g., see [8, 9]), ncludng network resource allocaton (e.g., [6, 7]). Gven ths allocaton, the users partcpate n a game wth the objectve of maxmzng ther own beneft. It s well known that the Nash Equlbrum (NE) of Ths work was supported by the Northwestern-Motorola Center for Communcatons, and by NSF CAREER award CCR

2 a game typcally does not maxmze the total system utlty [10]. Ths has been referred to as the prce of anarchy (e.g., [6]). In order to acheve a more desrable system operatng pont, we allow the manager to announce a unt prce (e.g., [11, 12]) ether for receved SINR or receved power. SINR prcng wth logarthmc utltes leads to a weghted max-mn far SINR allocaton. Power prcng maxmzes the total utlty for a large enough bandwdth. Both prcng schemes maxmze the total utlty n a large enough system f the total power and bandwdth are ncreased n fxed proporton to the number of users. Related work on uplnk power control for CDMA has appeared n [12 15]. A key dfference here s that there s a constrant on the total receved power at all tmes 1. Because of ths, a user s nterference depends on hs own power allocaton, whch can make the problem non-convex. Ths also allows us to vew the receved power as a dvsble good, whch leads naturally to the precedng share aucton mechansms. We assume the user populaton s statonary,.e., the users and ther correspondng utltes stay unchanged durng the tme perod of nterest. On a larger tme-scale one can vew tme dvded nto perods, durng whch the number of users and each user s utlty are fxed and the proposed aucton algorthm s used. When a new perod begns, users may jon or leave the system. Also remanng users may update ther utlty functons to reflect changes n ther QoS requrements. For example, a user wth data that must be delvered by a deadlne mght ncrease hs utlty (as a functon of SINR) as the deadlne approaches. Here we do not consder the assocated mechansms and the dynamcs of the auctons over multple perods. The remander of the paper s organzed as follows. After ntroducng the aucton mechansms n Sect. 2, we analyze the performance for a fnte system and for a lmtng large system n Sect. 3 and 4, respectvely. In Sect. 5 we gve suffcent condtons for global convergence of a myopc bd updatng algorthm and analyze the convergence speed. Numercal results are gven n Sect. 6 and conclusons n Sect. 7. Several of the man proofs are gven n the Appendx. 2 Aucton Mechansms A spectrum wth bandwdth B s to be shared among M spread spectrum users. User s valuaton of the spectrum s characterzed by a utlty functon U (θ, γ ), where γ s the user s receved SINR and θ s a user-dependent parameter. As a partcular example, we consder the logarthmc utlty U (θ, γ ) = θ ln (γ ). To smplfy the notaton, we let U (θ, γ ) = U (θ, γ ) / γ and U (θ, γ ) = 2 U (θ, γ ) / γ 2, and at tmes we omt the user ndex. Each user s utlty satsfes the followng assumpton: Assumpton 1. U (θ, γ ) s ncreasng, strctly concave and twce contnuously dfferentable n γ. For each, the SINR s gven by γ = γ (p ) = p n 0 + p /B, where n 0 s the background nose power densty, p s user s receved power allocaton, and p = M j p j. The total power allocaton must satsfy: p = p + p P. (1) Notce that the precedng constrant s on the total receved power at the collocated recevers, thus the lnk attenuatons need not be consdered n the model. However, once allocated the receved power, the users must adjust ther transmsson power to compensate for the lnk attenuaton between the transmtter and the recever. A power allocaton s Pareto optmal f no user s utlty can be ncreased wthout decreasng another user s utlty. Proposton 1. Assume for each, U (θ, γ ) s strctly ncreasng n γ. A power allocaton scheme s Pareto optmal f and only f the total power constrant s tght,.e., M p = P. Ths follows because f the power constrant s not tght, then each user can ncrease ther power by P/ M p, whch ncreases the SINR for every user. 1 We assume that any transmsson power constrant for each user s large enough so that t can be gnored.

3 Assume the manager s nterested n a socally optmal allocaton, where the total utlty over all users s maxmzed. Note that socal optmalty mples Pareto optmalty, but the reverse s not true. Thus, the manager should always allocate the receved power up to the constrant. A socally optmal soluton can be vewed as one type of far allocaton. The manager may also consder other farness objectves such as a max-mn far allocaton. We consder the followng aucton mechansms, whch operates n dscrete tme-slots n = 1, 2,... Aucton Algorthm. 1. Intalzaton (durng tme slot 0): (a) The manager announces a constant reserve bd β 0 and a postve unt prce π s (n a SINR-based aucton) or π p (n a power-based aucton). (b) User {1,..., M} submts an ntal bd b (0) Iteraton (durng tme slot n 1): (a) After observng bds b (n 1) n a SINR γ (n) wth where b (n) = β + M j b(n) j, for = 1,..., M, the manager allocates each user power p (n), resultng p (n) =. If b (n) b (n) b (n) + b (n) + b (n) P, and γ (n) = = 0, then p(n) = 0. p (n) n 0 + P p(n) B. (2) (b) In an SINR-based (power-based) aucton, user pays C = π s γ (n) (C ( = π p p (n) ). ) (c) Each user submts a bd b (n) 0 to maxmze hs surplus functon S b (n) ; b (n 1) = U (θ, γ ) C. The aucton process stops when the bds do not change n two consecutve tme slots. Ths means the system reaches a Nash Equlbrum (NE), whch s defned as a set of bds {b } 1 M such that S (b ; b ) S (ˆb ; b ) for any ˆb and any. Defne user s best response bd as the b that maxmzes S (b ; b ), assumng b s fxed. At the NE every user submts hs best response bd,.e., no one has the ncentve to devate unlaterally. The exstence and unqueness of a NE depends on the choce of β and π. If the system does not converge for gven values of β and π, the manager should change these values and restart the aucton process. We show later that the only requrement on β s that t s postve, whle the condton on π s more complex. Our aucton mechansms dffer from prevous aucton-based network resource allocaton schemes (e.g., [6, 7]) n that the bds here are not the same as the payments. Instead, the bds are sgnals of wllngness to pay, and the manager can reach the desred NE by settng the rght β and π. Ths allevates the typcal neffcency of the NE, and allows us to reach Pareto optmal or even socally optmal solutons. 3 Fnte System Analyss 3.1 SINR-based Aucton ( ) In ths case C (γ ) = π s γ = π s p / n 0 + P p B, so that each user s payment depends on the receved power, as well as the nterference he receves. Defne k = θ (P + Bn 0 ) B (π s P θ n 0 ). (3) Theorem 1. In an SINR-based aucton wth logarthmc utlty, 1. For β > 0, a unque Nash Equlbrum exsts f k > 0 for all and M Equlbrum exsts. 2. For β = 0, an nfnte number of Nash Equlbra exst f k > 0 for all and M no Nash Equlbrum exsts. k 1+k < 1, otherwse no Nash k 1+k = 1, otherwse The proof s gven n the appendx; as shown there, user s best response bd s b = k b. The bddng and power profles at the NE are: b = k 1+k 1 β and p M k l = k P for 1 M. (4) 1 + k l=1 1+k l

4 In order to have a unque Nash Equlbrum, the manager has to announce a postve reserve bd (β > 0). Otherwse, there ether exsts no NE or an nfnte number of NEs. However, snce the users bds at the NE are proportonal to β, the power allocaton p s ndependent of β. Thus, the manager only needs to announce an arbtrary postve constant at the ntal stage of the aucton. To have k > 0 requres π s > θ n 0 /P. Also, π s should be set hgh enough so that M k 1+k < 1. Note that such π s can always be found, because k 0 as π s. It may seem from Theorem 1 that the manager needs to know the utlty parameters {θ } 1 M n order to set the rght prce. However, ths requrement may be nether practcal nor necessary. It s not practcal when the utlty functons are prvate nformaton of the users. It s not necessary because the manager can adaptvely fnd the rght prce by observng the users bddng behavors: f π s n θ 0 P for any or M k 1+k 1, at least one user s bd wll quckly ncrease towards nfnty, ndcatng that the prce needs to be ncreased. An allocaton {x } 1 M s weghted max-mn far wth weghts {w } 1 M f no x can be ncreased wthout decreasng some x j such that x j /w j x /w. The SINR allocaton at the NE s γ = p n 0 + (P p )/B = θ π s, (5) and user pays C (γ ) = πs γ = θ. It follows that both the SINR allocaton and payments are weghted max-mn far wth the weghts {θ } 1 M. In [16], Kelly et al. showed that logarthmc utlty functons lead to a weghted proportonal far rate allocaton n a network rate control problem. Ther problem s convex and uncoupled across users snce there s no externalty effect (.e. nterference) among dfferent users. Here, due to the nterference among users, the problem may not be convex and the relaton between the utlty and the constraned resource (receved power) s qute dfferent from [16]. Indeed, n ths case, a socally optmal soluton s typcally not proportonal far. Nevertheless, we acheve a weghted max-mn far allocaton. The nformaton exchange durng the aucton s mnmal,.e., each user submts a bd to the manager and observes only hs own power allocaton. There s no need for the user to know the bds or power allocatons of any other users; the only nformaton a user needs to update hs bd s the summaton of all the other bds, whch can be easly calculated from hs own power allocaton and bd. Also, snce the power/sinr allocaton at the NE only depends on a user s local varable and global system varables, t s easy for the user to check that he receves the correct allocaton, whch may prevent the manager from cheatng. We call a system stable f there exsts a unque NE. In a stable system, defne the system usage effcency as η = M p P = k 1 + k. (6) For Pareto optmalty, η = 1, but the necessary condton for stablty s η < 1. Thus Pareto optmalty and stablty are conflctng objectves. We defne an ε-system as one wth parameters (P ε, B ε, M ε, n ε 0) = (P (1 ε), B, M, n 0 + εp/b), where ε (0, 1). An ε-pareto optmal allocaton s defned as a Pareto optmal soluton for the ε-system. Proposton 2. In an SINR-based aucton wth logarthmc utlty, for any ε (0, 1), there exsts a unque prce π sε, such that the system s stable and acheves an ε-pareto optmal soluton (.e., η = 1 ε n the orgnal system). Proof. From (3), t can be seen that as π s ncreases from max {θ n 0 /P } to, η = M k 1+k contnuously and monotoncally decreases from υ > 1 to 0. Thus, there must exsts a unque prce π sε (max {θ n 0 /P }, ) that acheves any η = 1 ε (0, 1). In practce, the manager can acheve a target η by adjustng π s after observng the usage effcency at the current NE: f t s too low, the prce should be decreased. Note f the prce s decreased too much, the stablty condtons n Theorem 1 may be volated. Next we consder the revenue collected by the manager. Compared wth lnear power-based prcng, where the payment for user s α p, and α s a user-dependent constant, we have the followng result: Proposton 3. As M, the revenue collected n the SINR-based aucton wth logarthmc utlty, M θ, s the maxmum revenue acheved by any power-based, user-dependent prcng scheme.

5 The proof s gven n the appendx. When there are enough users n the system, each user does not expect to affect the receved nterference by changng hs own power. Thus each user maxmzes hs surplus functon assumng that the receved nterference s fxed. In ths large populaton scenaro, the SINR-based aucton collects as much revenue as any other lnear power-based prcng scheme. 3.2 Power-based Aucton In ths case C (p ) = π p p. For users wth logarthmc utlty functons, Theorem 1 stll holds wth a more complcated expresson for k. The bddng and power profles at the NE are agan gven by (4). For a more general class of utlty functons, we show that n certan cases the power-based aucton can acheve an ε-socally optmal allocaton, whch maxmzes the total utlty of the ε-system. Theorem 2. Assume for each {1,..., M}, U (θ, γ ) satsfes Assumpton 1 and U (θ, γ ) U (θ, γ ) (γ + B) > 2, (7) for any γ [0, P/n 0 ]. Then there exsts a prce π pε such that the system s stable and the NE acheves ε-socal optmalty for any ε (0, 1). Condton (7) guarantees that U (θ, γ (p )) s concave n p, where γ (p ) s gven n (2). Ths condton wll be satsfed f the bandwdth s large enough for many utlty functons, some of whch are shown n Table 1. Table 1. Condton (7) for varous utlty functons U (θ, γ) (7) s true for any γ [0, P/n 0] f θ ln (γ) B > P/n 0 θ ln (1 + γ) B > P/n ( ) θγ α 2 (α (0, 1)) B > 1 P/n 1 α 0 1 e θγ B > 2/θ 4 Large System Analyss In ths secton we consder the asymptotc behavor as P, B, M and β go to nfnty, whle keepng P/M, P/B, M/B and β/m fxed. We assume that each user s utlty parameter θ s ndependently chosen accordng to a contnuous probablty densty f (θ) over [ θ, θ ], where 0 θ < θ <. The expected value of θ s E [θ]. Proposton 4. For the SINR-based aucton wth logarthmc utlty, a unque NE exsts n the lmtng system f and only f π s > E [θ] (n 0 + P/B) M P. (8) In ths case, the power and SINR allocatons at the NE are weghted max-mn far wth weghts {θ } 1 M, and user pays θ. Otherwse, no NE exsts. The proof s gven n the Appendx. The system usage effcency at the NE s η = E[θ](n0+P/B) π s P/M. As η 1, the prce π s converges to the rght-hand sde of (8), whch s proportonal to the system load M/P. Ths concdes wth the congeston prcng scheme proposed n [15], where the equlbrum prce reflects the congeston degree of the system. At the NE of the lmtng system, all users receve the same fxed nose plus nterference level (n 0 + P/B). Ths s because each user only gets a neglgble proporton of the total power. Ths makes the SINR-based and power-based auctons equvalent f π s = (n 0 + P/B) π p. In the lmtng system, we defne the socally optmal soluton to be the allocaton that maxmzes the average utlty per user, nstead of the users total utlty, whch s nfnte n ths case.

6 Assumpton 2. The utlty functon U (θ, γ) s asymptotcally sublnear wth respect to γ,.e., lm γ 1 U (θ, γ) = 0. γ Theorem 3. In the lmtng system, f U (θ, γ) satsfes Assumpton 1 and 2, then both the SINR- and power-based auctons can acheve ε-socal optmalty for any ε (0, 1). A sketch of the proof s gven n the appendx. Assumpton 2 s vald for common utlty functons, e.g. θ ln (γ), θ ln (1 + γ), θγ α (α (0, 1)), and any upper-bounded utlty. Under ths assumpton, even f a fnte number of users are allocated non-neglgble proportons of the total power, ther contrbutons to the average utlty becomes neglgble as the number of users ncreases. Because of ths, at the socally optmal soluton every user s allocated a fnte amount of power, and so faces the same nterference level (n 0 + P/B). 5 Myopc Bd Updatng Algorthm In ths secton, we consder how users update ther bds to reach the NE. We use the SINR-based aucton wth logarthmc utltes as an example. User can calculate the sum of other bds gven only P and hs P p own power p, by b = b p. We assume that each user updates the bd usng a myopc algorthm,.e., he submts the best response bd assumng all the other bds are fxed: b (n+1) Ths s smlar to the PUA algorthm used n [17]. = k b (n) (9) Proposton 5. In an SINR-based aucton wth logarthmc utltes, the myopc bd updatng algorthm n (9) globally and geometrcally converges to the unque NE n a stable system f max 1 M k < 1 M 1 or M k < 1. Furthermore, f all users start bddng from zero (the orgn), the bds monotoncally converge to the unque NE. The condtons n Proposton 5 wll be satsfed f the manager announces a hgh enough unt prce. Meanwhle, the prce should be set low enough to acheve a target η. Thus the manager needs to adaptvely search for the rght prce. In our smulatons, we use the followng search method: 1. Intalzaton: Set (π, π) = (0, ), and choose an arbtrary ntal prce π (0) > 0. Also, set a maxmum teraton tme T. 2. For n = 1, 2,..., (a) If the aucton does not converge wthn T teratons, then stop the process. Let π = π (n 1). Moreover, π (n) = 2π (n 1) f π =, otherwse π (n) = (π + π) /2. Restart the aucton. (b) If the aucton converges wthn T teratons at a system effcency η < η, then let π = π (n 1) and π (n) = (π + π) /2. Restart the aucton. (c) If the aucton converges wthn T teratons wth η η, then the search process termnates. Unlke the example n [17], here the sequence of the users bds does not oscllate f users start from the orgn. Ths s because the users best response bds satsfy strategc complementartes roughly, ths means when one user submts a hgher bd, the others want to do the same. Thus f all users start from the lowest bds (the orgnal), the bds monotoncally converge to the unque NE n a stable system. In ths case, takng smaller updatng steps or updatng randomly wth some probablty less than 1 (e.g., the RUA and GUA algorthms n [17]) wll not help convergence. Although we only consder SINR-based auctons wth logarthmc utltes, the myopc bd updatng algorthm also works for the power-based aucton wth logarthmc utltes, as well as some other utlty functons such as U (θ, γ) = θ log (1 + γ). However, we note that n some cases, a target η may not be achevable, due to the non-convexty of the problem.

7 6 Numercal Results In all smulatons shown here, the {θ } 1 M are ndependently and unformly dstrbuted n [1, 100]. Each graph represents one realzaton of the parameters; smlar observatons were obtaned for other realzatons and dfferent dstrbutons of the parameters. Fg. 1 shows a comparson of average utlty per user for the two auctons as well as an upper bound on the socally optmal soluton wth logarthmc utltes. In both auctons, we set the prces so that η s close to 1. From Theorem 2, the power-based aucton acheves socal optmalty for P/ (Bn 0 ) < 0 db. Fg. 1(a) shows that ths s also true for the SINR-based aucton. For P/ (Bn 0 ) > 0 db, the utlty s not concave wth power; n that case, we use a dual formulaton to upper bound the average utlty per user. Note that the two auctons stll acheve a utlty close to the maxmum n ths regme. In Fg. 1(b), we scale the system as n Sect. 4, and choose P/ (Bn 0 ) = 20dB so that the utlty s not concave n power. When M 14, the auctons do not acheve the upper bound on the maxmum average utlty. For large M, the utltes assocated wth both auctons and the socally optmal soluton converge to a constant. For ths example, the asymptotc behavor s accurate wth M 14. Fg. 2 shows the performance of the myopc bd updatng algorthm for users wth logarthmc utlty functons. In Fg. 2(a), users start bddng from the orgn and the bds monotoncally converge to the unque NE. In Fg. 2(b), the performance of the updatng algorthm as the system s scaled s shown. The target system usage effcency η s chosen to be 0.90, 0.95 and 0.98 respectvely. We can see that the number of teratons needed for convergence ncreases wth M and approaches a constant when M s large (.e., M > 20). Ths shows that the algorthm scales well wth the system sze. The fgure also shows that the number of teratons needed for convergence ncreases wth η, mplyng that fast convergence and hgh system usage effcency are generally conflctng objectves Average Utlty Per User: U tot /M SINR based Aucton Power based Aucton Socal Optmal (upperbound) Average Utlty Per User: U tot /M SINR based Aucton Power based Aucton Total Utlty Maxmzaton (upperbound) Power/Background Nose: P/Bn 0 (db) (a) Number of users: M (b) Fgure 1. Average utlty for the two auctons and the maxmum achevable utlty for the logarthmc utlty functon: (a) fnte system wth (P, M, n 0) = ( 10 2, 10, 1 ) for dfferent B; (b) system wth (P, B, n 0) = ( 10 4 M, 10 2 M, 1 ) for dfferent M. 7 Concluson We presented two aucton mechansms (SINR-based and power-based) for sharng spectrum among a group of users subject to a constrant on the nterference temperature at collocated recevers. When combned wth logarthmc utltes, the SINR-based aucton leads to a weghted max-mn far SINR allocaton. The power-based aucton maxmzes the total utlty for a large enough bandwdth. Both aucton mechansms are shown to acheve socal optmalty n a large system lmt where bandwdth and power are ncreased n fxed proporton. We also gave suffcent condtons for global convergence of a myopc bd updatng algorthm, and dscussed the convergence speed both analytcally and numercally. Ths work s prelmnary n that we only consder the nterference temperature at a sngle pont and assume that all recevers are collocated. Relaxng these assumptons s a possble drecton for future

8 Users Bds Number of teratons η * = 0.90 η * = 0.95 η * = Number of Iteratons (a) Number of users M (b) Fgure 2. Performance of the myopc bd updatng algorthm wth logarthmc utlty functons: (a) bds for each user over tme for a fnte system wth (P, B, M, n 0, β) = ( 10 2, 10 3, 10, 1, 1 ) and η = 0.95; (b) Number of teratons requred for a system wth (P, B, n 0) = ( 10 4 M, 10 2 M, 1 ) for dfferent values of M and target η. research. We are also consderng a Commons spectrum usage model, where there s no nterference temperature constrant and each user s constraned only through a techncal standard (e.g., whch mposes a constrant on transmsson power). The problem then s how to avod the tragedy of commons. Another extenson s to consder a dynamc envronment, where the number of actve users vares wth tme. References 1. Spectrum polcy task force report. Federal Communcatons Commsson, US (2002) 2. Mas-Colell, A., Whnston, M., Green, J.: Mcroeconomc Theory. Oxford Unversty Press (1995) 3. Sunnevag, K.J.: Aucton desgn for the allocaton of emsson permts. workng paper (2001) 4. Wang, J.J.D., Zender, J.F.: Auctonng dvsble goods. Economc Theory (2002) Back, K., Zender, J.F.: Auctons of dvsble goods: On the ratonale for the treasury experment. Revew of Fnancal Studes 6 (1993) Johar, R., Tstskls, J.N.: Network resource allocaton and a congeston game. to appear n Mathematcs of Operatons Research (2003) 7. Maheswaran, R., Basar, T.: Nash equlbrum and decentralzed negotaton n auctonng dvsble resources. Group Decson and Negotaton, to appear (2003) 8. Tjdeman, R.: The charman assgnment problem. Dscrete Mathematcs 32 (1980) Waldspurger, C.A., Wehl, W.E.: Strde schedulng: Determnstc proportonal share resource management. Techcal Memorandum MIT/LCS/TM-528, Laboratory for CS, MIT (1995) 10. Dubey, P.: Ineffcency of Nash equlbra. Mathematcs of Operatons Research 11 (1986) D. Famolar, N. B. Mandayam, D.G., Shah, V. In: A New Framework for Power Control n Wreless Data Networks: Games, Utlty and Prcng. Kluwer Academc Publshers (1999) Saraydar, C., Mandayam, N.B., Goodman, D.J.: Effcent power control va prcng n wreless data networks. IEEE Trans. on Communcatons 50 (2002) Alpcan, T., Basar, T., Srkant, R., Altman, E.: CDMA uplnk power control as a noncooperatve game. Wreless Networks 8 (2002) Shroff, N., Xao, M., Chong, E.: Utlty based power control n cellular rado systems. In: Proceedng Inforcom 2001, Anchorage, USA (2001) 15. Hekknen, T.M.: On congeston prcng n a wreless network. Wreless Network 8 (2002) Kelly, F.P., Maulloo, A., Tan, D.: Rate control for communcaton networks: Shadow prces, proportonal farness and stablty. Journal of Operatons Research Socety 49 (1998) Alpcan, T., Basar, T.: Dstrbuted algorthms for nash equlbra of flow control games. Annals of Dynamc Games 7 (2003) to appear. Appendx Proof of Theorem 1 Proof. Frst assume β > 0, thus b = β + M j b j > 0. Usng (2), user s surplus functon can be wrtten as: ( ) b P S (b ; b ) = θ log π s b P. (10) P P (b + b ) n 0 + b B (b + b ) n 0 + b B

9 Notce that b = 0 cannot be the best response bd, snce t leads to a surplus of regardless of b. Dfferentatng (10) wth respect to b yelds S (b ; b ) b = (θ (Bn 0 + P ) b + B (θ n 0 π s P ) b ) (Bn 0 + P ) b b ((Bn 0 + P ) b + Bn 0 b ) 2. (11) Snce b > 0 and b > 0, the sgn of (11) only depends on the sgn of the expresson θ (Bn 0 + P ) b + B (θ n 0 π s P ) b, (12) whch s monotonc n b. Settng (12) equal to 0 and solvng for b yelds b = k b, (13) where k s gven n (3). For k > 0, t can be shown that (13) s the global maxmum of (10), and so s user s best response bd. Alternatvely, f k < 0, then user s best response bd s b =, and there s no NE for the system. If the system has an NE {b } 1 M, t must satsfy the followng set of lnear equatons: b = k b = k b j + β b, and b > 0 for 1 M. (14) j=1 Solvng ths usng (2), we get the unque soluton gven n (4) f and only f M k 1+k < 1. Otherwse, these equatons have no soluton, and so no NE can exst. If β = 0, then (14) can be smplfed as ( ) ( M ) k 1 b = 0, and b = k b > 0 for 1 M. (15) 1 + k 1 + k There are an nfnte number of solutons to (15) f and only f k > 0 for all and M agan, f ths s not the case, then there are no solutons to (15) and so no NE exsts. Proof of Proposton 3 k 1+k = 1. Once Proof. User s surplus functon under a power-based, user-dependent prcng scheme s: ( ) p S (p ) = θ log α n 0 + p p. (16) B As M, each user wll maxmze hs surplus assumng p s fxed, yeldng { θ p = arg max S (p ) = α, α θ P. (17) p [0,P ] P, α < θ P The revenue collected by the manager s then Proof of Proposton 4 α p θ α = α θ. (18) Proof. We obtan (8) by takng the lmt of the condtons n Theorem (1), under the assumed scalng. Let Lm denote lm P,B,M wth P/B, P/M, β/m fxed. Thus, Lm k 1 + k = Lm θ (P/B + n 0 ) P (π s + θ /B) = 1 M M Lm Mθ (P/B + n 0 ) P π s = P/B + n 0 E [θ]. (19) P/Mπ s The frst equalty s from the defnton of k n (3). The second equalty s due to the fact that B. The thrd equalty s because of the weak law of large numbers. Condton (8) then follows drectly. The weghted max-mn far SINR allocaton and payments stay unchanged durng the lmtng process. Snce every user faces the same nose plus nterference n 0 + P/B at the NE, then p = γ (n 0 + P/B) for all. Ths leads to a weghted max-mn far power allocaton at the NE.

10 Proof of Theorem 3 Proof. Due to space consderatons we only gve a sketch of the complete proof. In the lmtng system, the maxmum average utlty per user s the soluton to: [ ( )] Max E p (θ) θ U θ, (20) p(θ) 0 n 0 + (P p (θ)) /B subject to: E θ [p (θ)] = P M The objectve s the average utlty per user n the lmtng system and the constrant corresponds to the total receved power constrant. In both cases we have used the law of large numbers to express these n terms of expectatons over θ. The optmzaton s over all power allocatons, p (θ), whch can be vewed as functons from [θ, θ] to the nonnegatve real numbers. Let U avg (P/M, B/M, n 0 ) denote the soluton to (20) for gven values of P/M, B/M, and n 0. We frst prove the followng lemma: Lemma 1. There exsts a power allocaton p (θ) that solves (20), whch s fnte everywhere,.e., p (θ) lm P P = 0, for all θ [ θ, θ ]. (21) Ths lemma mples that each user receves a neglgble fracton of the total power as the system scales. An outlne of the proof follows. If the lemma s not true, then some user must be allocated nfnte power as the system scales. The key dea s to show that because the utlty s sublnear, ths user wll contrbute a neglgble amount to the average utlty. Thus we can reallocate the user s power among the remanng users so that (21) s satsfed. Ths reallocaton can only ncrease the average utlty, whch gves a contradcton, provng the lemma. Ths lemma ensures that at a soluton to (20), each user faces the same nterference plus nose n 0 +P/B. Ths makes (20) a concave optmzaton problem. By usng calculus of varatons, we can solve for p (θ) n closed form, as well as the correspondng postve Lagrange multpler λ for the average power constrant. Lettng π p = λ or π s = (n 0 + P/B) λ, results n the same power allocaton at the NE for the power- and SINR-based auctons, respectvely.