Near-Optimal Power Control in Wireless Networks: A Potential Game Approach

Transcription

1 Near-Optial Power Control in Wireless Networks: A Potential Gae Approach Utku Ozan Candogan, Ishai Menache, Asuan Ozdaglar and Pablo A. Parrilo Laboratory for Inforation and Decision Systes Massachusetts Institute of Technology Cabridge, MA, 09 Eails: {candogan, ishai, asuan, parrilo}@it.edu Abstract We study power control in a ulti-cell CDMA wireless syste whereby self-interested users share a coon spectru and interfere with each other. Our objective is to design a power control schee that achieves a (near) optial power allocation with respect to any predeterined network objective (such as the axiization of su-rate, or soe fairness criterion). To obtain this, we introduce the potentialgae approach that relies on approxiating the underlying noncooperative gae with a close potential gae, for which prices that induce an optial power allocation can be derived. We use the proxiity of the original gae with the approxiate gae to establish through Lyapunov-based analysis that natural userupdate schees (applied to the original gae) converge within a neighborhood of the desired operating point, thereby inducing near-optial perforance in a dynaical sense. Additionally, we deonstrate through siulations that the actual perforance can in practice be very close to optial, even when the approxiation is inaccurate. As a concrete exaple, we focus on the surate objective, and evaluate our approach both theoretically and epirically. I. INTRODUCTION In contrast to wireline network architectures which can often provide quality of service (QoS) guarantees to end-users by strict division of the network resources, the shared nature of the wireless doain inherently iplies that the perforance of each obile depends on the resources allocated to others. In code division ultiple access (CDMA) systes, the transission power of each obile translates into interference noise for the other obiles and thus degrades their perforance in ters of the obtained data rates. Due to this utual effect and in view of the scarcity of the power resource itself, power control has inarguably becoe a fundaental proble in wireless networks research. The power control proble, even when forulated as a centralized optiization proble with full inforation, is a fairly coplex proble. For exaple, in general-topology CDMA network with ultiple transitters and receivers, each transitter affects each of the receivers in a different anner, therefore it is not a priori clear how to assign the transission powers in a syste-efficient anner. Specifically, basic syste objectives, such as the su-rate, turn out to be difficult-to-solve optiization probles (see, e.g., [] [] and references therein). An additional concern within the power allocation fraework is the possible selfish behavior of obiles, who ay autonoously control their transission power to satisfy their own interests. For exaple, each obile ay plausibly be interested in adjusting its power allocation in order to axiize its individual data-rate (throughput), while sustaining a physical-layer andated power constraint. Naturally, gaetheoretic tools have been widely applied over the last decade to study such copetitive situations in wireless networks (see [4] and [5] for recent surveys). The agenda of bulk of the research in this area includes the study of the conditions for existence and uniqueness of a Nash equilibriu, and analysis of the stability properties of greedy power-updating schees (see, e.g., [4], [6] [8]). In this paper, we consider the power allocation proble fro the viewpoint of a central planner. The planner wishes to ipose a certain power-dependent objective in the network, by properly pricing excessive power usage. As a concrete doain, we consider a ulti-cell CDMA syste with ultiple transitters (henceforth referred to as obiles or users ), each associated with a (possibly different) base-station. The obiles, which share a coon spectru and interfere with each other, are interested in axiizing their net utility (throughput inus onetary costs) subject to individual power constraints. Our objective is to provide a general distributed power control schee that would achieve (or approxiately achieve) any underlying syste objective, despite the selfishness of the obiles. We accoplish this using a novel potential-gae approach, which entails approxiating the original gae with a potential gae that has a (additively) separable structure in the individual power allocations. This enables design of a siple pricing schee that induces the equilibriu of the potential gae to coincide with the optial power allocation of any underlying syste objective. Moreover, since natural user-update schees converge to a Nash equilibriu for potential gaes, the closeness of the two gaes allows us to establish near-optial perforance for user dynaics applied to the original gae. We apply the potential-gae approach to the CDMA doain, by showing that the power gae can be approxiated by a potential gae, with arbitrarily good accuracy as the signal to interference-plus-noise ratio (SINR) increases (e.g., when the interference due to other obiles is negligible). We show that best-response dynaics applied to the original gae converges within a neighborhood of the optial operating point, where

2 the size of the neighborhood depends on the SINR. This shows that this approach can be used for network regulation under any SINR regie with explicit perforance guarantees. We suppleent the theory with experiental results, which deonstrate that the obtained perforance can in practice be very close to optial, even when operating at a relatively low- SINR regie. Related work. There has been uch work in the literature on pricing in counication networks in general, and wireless networks in particular. However, to the best of our knowledge, there is no general fraework that tackles the proble of achieving (near) optial perforance for any given underlying syste objective. One branch of this literature focuses on the profit axiization objective where service providers price usage of network resources to axiize their profits (see, e.g., [9] []). Another branch assues that prices are set by a social planner to regulate selfish user behavior and drive the syste to a ore desirable operating point (e.g., [7], [8], [] and references therein). Paper structure. The rest of the paper is organized as follows. The odel is presented in Section II. We then proceed in Section III to present the potential-gae approach. Section IV provides perforance-loss bounds for the obile interaction under a general SINR regie, and exaines through siulations the actual perforance of the pricing schee. We also briefly describe in this section how to distribute the price generation process. As a concrete exaple, we focus in Section V on the su-rate objective, and evaluate the resulting perforance both theoretically and epirically. We conclude in Section VI. II. THE MODEL A. Preliinaries We consider a set of obiles M = {,...,M} that share the sae wireless spectru. Each obile Mtransits to a pre-assigned base station (we allow for ultiple obiles transitting to the sae base station). Denote by p =(p,...,p M ) the power allocation of the obiles (we shall also refer to p as the operating point of the network). The basic perforance easure that deterines the obiles rates is their SINR. The SINR of user is given by h p SINR (p) = N 0 + k h, () kp k where h k is the gain between user k s transitter and user s base station, and N 0 is the noise power. Note that in CDMA systes, the signals of other users are often treated as interfering noise (i.e., there is no interference cancellation at the receiver), hence the user rate directly depends on its SINR, as we elaborate below. Over the tie-period of interest, we assue that the channel gains are fixed (i.e., fading effects take place at a uch slower tie-scale). Let r (p) be the rate (throughput) of user, as deterined by the power allocation p =(p,...,p M ). Then, r (p) =log(+γsinr (p)), () where γ>0 is the spreading gain of the CDMA syste. The rate function r ( ) can be interpreted as being proportional to the Shannon capacity of user, while we ake the siplifying assuption that the noise plus the interference of all other users constitute an independent Gaussian noise. It is assued that each user can adjust its transission power p within a bounded range 0 <P p P. () The upper bound P represents a constraint on the axial usage of the battery, while the lower bound P > 0 is the lowest power level at which the user s transitter can operate. In practice, a transitter would have a finite nuber of power levels in a bounded range, however we assue for siplicity that any power in the above range can be sustained. B. Utilities and Equilibriu A basic odelling assuption in our work is that users are selfish and adjust their power in a self interested anner. As a result, the individual power usage has to be regulated in soe way. To that end, we consider in this paper the following userbased linear pricing schee: User pays c onetary units per-power unit. The prices c are set in accordance with a global network objective (see Section II-C). Note that the overall payent of user is given by c p. The user objective is to axiize a net rate-utility, which captures a tradeoff between the obtained rate and the onetary cost, given by u (p) =r (p) ζ c p, (4) where ζ > 0 is a user-specific rate vs. oney tradeoff coefficient. The individual user optiization proble, fixing other users power allocation p = (p,...,p,p +,...,p M ), can be forulated as ax u ( p, p ), (5) p P where P = {p P p P }. We refer to P = P P M as the joint feasible strategy space, and to P = P P P + P M as the joint feasible strategy space of all users but the -th one. We forally denote a gae instance by G = M, {u }, {P } and refer to this gae as the power gae. ANash equilibriu (NE) of G is a feasible power allocation p P fro which no user has an incentive to unilaterally deviate, naely u (p) u ( p, p ) (6) for every p P. The existence of a Nash equilibriu follows in view of the fact that the underlying gae is a concave gae [] (i.e., u ( ) is concave in p and the joint strategy space P is convex). In general, a user could decide not to use its power at all. We exclude this option here, by iplicitly assuing that each user ust sustain a nonzero lower bound on its rate, which is guaranteed to be satisfied by transitting at a power of at least P > 0.

3 In this paper we also consider operating points which are approxiately Nash equilibria. To forally address such operating points, we use the concept of ɛ-(nash) equilibriu. An operating point p is an ɛ-equilibriu of the gae G if for every q P and M u (p, p ) u (q, p ) ɛ. (7) C. Syste Utility Assue that a central planner wishes to ipose soe perforance objective over the network. Generally, the objective relates to the transission powers eployed by the users, hence can alternatively be posed as an optiization proble ax U 0(p), (8) p P where U 0 ( ) is the syste utility-function. As a concrete exaple, we will consider in Section V the su-rate objective, given by U 0 (p) = r (p), where r ( ) is defined in (). For the analysis in this paper, we shall assue that the central planner is equipped with the required inforation to solve (8) (i.e., the knowledge of all channel gains and the feasible power region), and actually is able to solve this optiization proble. We denote the optial solution of (8) by p, and refer to it henceforth as the desired operating point. In the sequel, we consider the price setting proble faced by the central planner, who is interested in inducing the optial utility value U 0 (p ). III. THE POTENTIAL-GAME APPROXIMATION As an interediate step in our analysis, we consider in this section a noncooperative gae with odified utilities where ũ (p) = r (p) ζ c p, (9) r (p) =log(γsinr (p)). (0) We refer to this gae as the potentialized gae and denote it by G = M, {ũ }, {P }. When the spreading gain γ satisfies γ (or alternatively h h k for all k ), we say that users operate in high-sinr regie. Note that under this regie, the odified rate forula r (p) r (p) serves as a good approxiation for the true rate, and thus ũ (p) u (p). A. Properties of the Potentialized Gae In this subsection, we obtain soe basic properties of the potentialized gae G. Specifically, we show that a Nash equilibriu point for the gae always exists and is unique. Furtherore, we establish that the gae G is a potential gae. A Nash equilibriu for the potentialized gae is defined as in (6), with u (p) replaced by ũ (p). Noting that ũ (p) reains (strictly) concave in p and that the feasible strategy space reains P, the existence of a NE is guaranteed []. We proceed to show that G belongs to the class of potential gaes. A gae G = M, {ũ }, {P } is said to be a potential gae if there exists a function φ : P R satisfying φ(p, p ) φ(q, p )=ũ (p, p ) ũ (q, p ), () for every M, p,q P, p P. Proposition : The gae G is a potential gae. The corresponding potential function is given by φ(p) = log(p ) ζ c p. () Proof: This follows using the characterization of potential gaes in [4], i.e., = ũ(p), M. The uniqueness of the equilibriu now follows by exploiting the potential forula (). Proposition : The potentialized gae G has a unique Nash equilibriu. Proof: Note that the potential function () is strictly concave and continuously differentiable. It is shown in [5] that if the potential function is concave and continuously differentiable and if the users strategy spaces are convex (intervals in our case), then the set of pure strategy Nash equilibria coincides with the set of axiizers of the potential function. The potential function () is strictly concave over a convex strategy-space, hence the axiizer is unique, iplying that the NE is unique. B. Assigning Prices Our interest in this subsection is in deriving prices c = (c,...,c M ) for the potentialized gae G such that the unique equilibriu of G will coincide with the desired operating point p (recall that the prices affect the utilities of the gae). As entioned earlier, we do not consider here how p is obtained and assue it has been calculated by a central planner; we show below that equipped with p P, the central planner can set the prices c in a siple way. Theore : Let p be the desired operating point. Then the prices c are given by c =(ζ p ), M. () Proof: We show that when c = c for every user, then p is the unique equilibriu of the potentialized gae. Since G is a potential gae, the axiu of its potential φ is a Nash equilibriu. We next show that p is a axiu of the potential. Using {c } as the prices, the partial derivative of the potential is given by = ζ p ζ p = p p. Setting p = p, we thus have φ(p ) =0for all M. Recalling that φ is concave, it follows that p is a global axiu of the potential; hence p Pis an equilibriu of G, which is unique by Proposition. Adopting the price vector c (given in ()), the desired operating point p is generally not an equilibriu of the power gae G. Nonetheless, due to the relation between the

4 4 gaes G and G, eploying c as the per-user prices induces near-optial perforance in G, in the sense that natural gae dynaics converges within a neighborhood of p. This property is foralized in the next section. IV. NEAR-OPTIMAL DYNAMICS In this section, we analyze the dynaical properties of the gae G, for which the per-user prices are set according to (). Our ain results herein are Theores 4 5, which establish that best-response dynaics for G converges within a neighborhood of the desired operating point p, and consequently induce near-optial perforance in ters of the syste utility U 0. The section is organized as follows. We describe best-response dynaics in Section IV-A and analyze their properties in Section IV-B. We then provide in Section IV-C soe nuerical exaples to highlight several aspects of the dynaics. We conclude this section by a brief discussion of the results and soe practical aspects of our ethod. The technical proofs for this section can be found in the Appendix. A. Best-Response Dynaics A natural class of dynaics in ultiuser noncooperative systes is the so-called best-response dynaics, in which each player updates its strategy to axiize its utility, given the strategies of other players. In our specific context, let β : P p denote the best response apping for the th user, which satisfies β (p ) = argax u (p, p ). (4) p P Note that best response aps are in general set-valued, however in our setting β (p ) is single-valued due to strict concavity of each user s utility in its strategy. We assue that users update their power allocation in accordance with their best-response. Specifically, we assue the following update rule: p p + α (β (p ) p ) for all M, where α>0 is a fixed step-size. Assuing that users update their power allocation frequently enough and for sall α, the above update rule ay be approxiated by the differential equation ṗ = β (p ) p for all M. (5) This continuous-tie dynaics is siilar to continuous tie fictitious play dynaics and gradient-play dynaics (see, e.g., [6] and [7]). We henceforth refer to (5) as the best-response (BR) dynaics of our gae. We note that if the users were to play the potentialized gae, this dynaics would converge to p. This observation can be easily shown through a Lyapunov analysis using the potential function of G (e.g., by adapting the analysis in [8] to BR dynaics). Yet, our interest is in studying the dynaical properties of the power gae G, which is the subject of the next subsection. B. Convergence Analysis We study in this subsection the properties of best-response dynaics (5). Before proceeding with the analysis, we require additional notations and definitions. Since our solution ethod copares the outcoes of the power gae G to those of the potentialized gae G, we need to define the user s best-response in the latter. Because G is a potential gae, it follows by definition (see ()) that the corresponding best-response β (p ) of any user can be obtained by axiizing the potential function φ given the other users strategies. Thus, β (p ) = argax φ(p, p ). (6) p P Note that β (p ) is also a single-valued function (by the strict concavity of the potential function). We show below that the BR dynaics (5) operates in a neighborhood of p. To foralize this property, we introduce the notion of unifor convergence (see [9] for related concepts). Let p t be the operating point at tie t. We say that the dynaics converges uniforly to a set S if there exists soe T (0, ) such that for any initial operating point p 0 P, p t S for every t T. In our context, the sets of interest relate to the equilibriu of the potentialized gae. Specifically, for any given ɛ, denote by Ĩɛ the set of ɛ-equilibria of G, naely Ĩ ɛ = { p ũ (p, p ) ũ (q, p ) ɛ (7) for every q P and M }. Our first result establishes that the dynaics (5) converges uniforly to a set Ĩɛ, where ɛ is explicitly characterized by the gae paraeters. Let P SINR = h N 0 + k h kp k be the inial SINR of user. Then, Lea : The best-response dynaics (5) in G converges uniforly to Ĩɛ (i.e., the set of ɛ-equilibria of G), where ɛ satisfies ɛ. (8) γ SINR The proof of the lea follows fro a Lyapunov-based analysis, where the Lyapunov function used is related to the potential function φ of the potentialized gae G (see Appendix for the proof). Observe that ɛ above is inversely proportional to both γ and SINR. This for of dependence is expected fro the rate forula (), as for large values of these ters we have ũ (p) u (p). We note that the above lea still does not provide an answer to how far the set of ɛ-equilibria of G is fro the desired operating point p. However, it is used below to establish that any point in the set Ĩɛ (to which the BR dynaics converges) is in a neighborhood of p. It also follows that the BR dynaics converges to a set of ɛ-equilibriu of the power gae G. This result is oitted here due to lack of space, and can be found in the accopanying technical report for this paper [0].

5 5 Theore 4: Let Ĩɛ be given by (7), where ɛ satisfies (8). Then p p P ɛ for every p Ĩ ɛ and every M. Under soothness assuptions on the syste utility U 0,a sall ɛ leads to near-optial perforance in ters of syste utility. This is stated in the next theore (the proof of which iediately follows fro Theore 4, see [0] for details). Theore 5: Let Ĩɛ be given by (7), where ɛ satisfies (8). (i) Assue that U 0 is a Lipschitz continuous function, with a Lipschitz constant given by L. Then U 0 (p ) U 0 ( p) ɛl P (9) for every p Ĩɛ. (ii) Assue that U 0 is a continuously differentiable function such that U0 L, M. Then U 0 (p ) U 0 ( p) ɛ P L, (0) for every p Ĩɛ. The expression U 0 (p ) U 0 ( p) can be regarded as a perforance-loss easure. Theore 5 iplies that the bound on the perforance-loss decreases with ɛ. This is expected, since by Theore 4, a sall value of ɛ iplies that BR dynaics converges to a sall neighborhood of p ; hence, the dynaics operates in the proxiity of the desired operating point. On the other hand, the bound increases with L (or L ), as the difference between p and p is translated to a difference in the associated values of U 0 (which is scaled proportionally to L or L ). We ephasize that Theores 4 5 hold without requiring any assuptions on the SINRs of the users. Consequently, the perforance bounds we obtain are valid for any choice of the syste paraeters {h k }, {P }, {P },γ. C. Nuerical Exaples Our objective in this subsection is to validate the actual perforance of the suggested pricing schee through basic experients. Specifically, we are interested in exaining how close we get in practice to the desired operating point, as a function of the accuracy of the potential-gae approxiation. As discussed in Section III (and also verified through the theoretical bounds of the preceding section), the approxiation becoes better for larger spreading gain γ, or alternatively when the self-gain coefficients h are uch larger than the cross-gain coefficients h k and the noise power N 0. For siplicity of ipleentation, we execute the siulations below for different values of γ, rather than significantly odifying the ratio between the self-gain coefficients and the cross-gain coefficients. Typical values of the spreading gain in CDMA systes ay range between 5 00 [7], []. In our experients, we focus on a lower subset of this range, as when the spreading gain is above 00, the actual perforance is indistinguishable fro the optial one. We now describe the setup used for the experients. We consider a network with three users. For all siulations, we assue that the desired operating point is p =[5, 5, 5] and that the prices are set as in Theore. Note that we actually do not specify here the underlying syste utility, as our ain concern in this set of siulations is to observe how close to p best-response dynaics eventually converges. Since the gain coefficients and N 0 can be scaled without changing the SINR, we noralize N 0 to. The self-gain coefficients h are chosen uniforly at rando (fro the interval [, 4]), and so do the cross-gain coefficients h k (fro the interval [0, ]). We consider three different values of γ, {5, 0, 50}. We assue that P =, P = 0 for each player M. The dynaics are initialized at the operating point p 0 =[,, ]. We next exaine the evolution of the operating point in tie fro two different angles. Figure shows the tie evolution of the operating point for different values of γ, starting fro p 0 and ending in a neighborhood of p (depending on the value of γ). As expected, the dynaics tends to converge closer to p as γ increases. Figure depicts the L -nor distance of p t fro p. We observe that larger γ s lead to shorter distances fro p, not only as a final outcoe, but also at any point in tie. Note further that all curves are onotonously decreasing, i.e., the dynaics tend to get consistently closer to the desired operating point, which is a desired property. An additional observation which is worth noting is that the trajectories of the dynaics see to converge to a point (rather than to a larger set). This was not guaranteed by our theoretical results. An interesting future research direction is to exaine whether this convergence behavior is theoretically guaranteed. Overall, the qualitative behavior reported above atches our theoretical results in Section IV (e.g., the distance fro p is inversely proportional to γ). We ephasize that the actual deviations fro p are uch saller than the theoretical guarantees. For exaple, the bounds in Theore 4 can be an order of agnitude larger than the ones obtained in our experients. This gap is quite expected, as our bounds are general, independent of the desired operating point and the actual syste-utility. It reains a future research direction to tighten the bounds for specific cases of interest, e.g., by considering concrete syste utilities and restricting the paraeter space of the proble. D. Discussion We briefly discuss here soe consequences of our results, and also highlight soe practical aspects. We have deonstrated in this section, both theoretically and epirically, that we can apply the so-called potential-gae approach in order to enforce the network operating point to be close to a desired one. The appeal of the schee lies in its generality, in the sense that any syste objective can be (nearly) satisfied despite the self-interested nature of the underlying users. The focus of this paper is on the gae-theoretic analysis of a wireless network, where the prices are assued to be set properly by a central network authority. Such central authority thus has to be equipped with full inforation on the paraeters

6 6 p p p γ=50 4 γ=0 γ=5 Fig.. The evolution of the power levels under best response dynaics. The starting point is p 0 =[,, ], the desired operating point is p =[5, 5, 5]. p t p * γ=50 γ=0 γ= Tie Fig.. The distance p t p between the current and desired power allocations, under best-response dynaics. of the proble (such as the power constraints and the gain coefficients). In practice, however, such inforation ay not be available centrally. A plausible way of dealing with this issue, is to set the prices theselves in a distributed anner, as we highlight below. Assue that each base-station has coplete knowledge of the paraeters that affect the perforance of its associated users (such as the cross-gain coefficients). Assue further that the base stations can counicate aong theselves (e.g., through wired-based connections). Consider a syste utility that is separable in, i.e., of the for U 0 (p) = U 0,(p), where U 0, ( ) is the syste-utility coponent associated with user (the su-rate objective is an exaple of such utility with U 0, (p) =r (p)). Under this utility structure, one ay consider the following two stage procedure for distributed price generation. At the first stage, the base stations exchange private inforation and jointly agree on a desired operating point (e.g., by using one of the distributed ethods described in []). Consequently, each base-station sets the price for its associated obile according 5 to (). This two-stage procedure generates the prices c, provided that the axiization of U 0 (p) can be solved in a distributed anner (e.g., when U 0, ( ) are concave functions). Otherwise (e.g., for non-concave syste utilities) distributed price setting reains an open issue, since distributed optiization ethods lack the tools to deal with such cases. A final coent relates to possible inaccuracies in channel gain inforation. We have assued throughout that the inforation about the channel gains is perfect, i.e., coincides with the true gains of the underlying network. Due to technological liitations, however, the estiated gains (i.e., the gains used for price setting) ight differ fro the true gains that deterine the users throughputs (e.g., as a consequence of inforation quantization effects). In our fraework, these differences translate to having different gains for the original gae (the true gains) and the potentialized gae (the estiated gains). Nonetheless, our analysis ethodology can be extended to accoodate such inaccuracies. This direction will be foralized in future versions of the current work. V. THE SUM-RATE OBJECTIVE We consider in this section the natural syste objective of axiizing the su rate in the network. The su-rate objective can be forulated as axiization of the following utility U 0 (p) = r (p). () We next apply Theore 5 to obtain explicit bounds on the perforance of BR dynaics. Theore 6: Let p be the operating point that axiizes (), and let Ĩɛ be the set of ɛ-equilibria to which BR dynaics converges (where ɛ is given by (8)). Then U 0 (p ) U 0 ( p) ɛ(m ) P () P for every p Ĩɛ. Proof: (outline) The proof follows by bounding the partial derivatives of (); we show in Lea (given in the appendix) that U0 M P. We then apply Theore 5(ii), which iediately leads to (). We now exaine through siulations the actual (su-rate) perforance of our schee. Specifically, we are interested in both the teporal perforance (i.e., the evolution of the surate easure () in tie), as well as the effect of γ on the overall deviation (in ters of the su-rate) fro the desired operating point. We consider a network of ten users. We set N 0 =, the cross-gain coefficients h k and the self-gain coefficients h are chosen uniforly at rando (fro the intervals [0, ] and [9, ] respectively). The power constraints are identical for all players, P =0, P =0for all M. We consider different values of γ in the range γ [, 00]. For each siulation instance, we solve for the desired operating point p (the axiizer of ()) and set the prices according to

7 7 (). We note that the underlying optiization proble is nonconvex and therefore (approxiately) solved nuerically by ultiple executions of an optiization solver, each initialized at different starting points. Figure deonstrates the evolution of the su-rate U 0 in tie for a typical siulation run (with γ = 0). We initialize the dynaics at the inial-power operating point [P,P...P 0 ]. We observe that the su-rate in the syste onotonously increases, obtaining a four-fold iproveent in the su-rate. Fro a practical perspective, this is an appealing property, since users, albeit selfish, keep increasing the systeutility, i.e., there is no degradation of perforance at any point in tie. We conclude our experients by exaining the outcoe of the best-response dynaics as a function of γ. As a concrete perforance easure, we are interested in the (percentage) deviation of the obtained su-rate fro its optial value. For agivent this easure is given by 00 U 0(p ) U 0 (p t ) U 0 (p. () ) Since the dynaics is guaranteed to converge in finite tie to a set, the long-run average deviation can be estiated by averaging () over t > Tfor large T (i.e., after the dynaics is confined in a sall neighborhood of the desired operating point). Figure 4 depicts the (estiated long-run) average deviation as a function of γ. Two properties of the graph should be ephasized. First, the average deviation decreases with γ, as expected. More iportantly, we see that even for sall values of γ, the average deviation is quite sall (around %). This indicates that despite setting the prices without considering the true utilities, the potential gae approach leads to prices which induce near-optial perforance. U 0 ( p * ) U 0 ( p t ) Tie Fig.. The change in su-rate as a function of tie for γ = 0. The optial su-rate is arked by a dashed line for reference. VI. CONCLUDING REMARKS This paper has considered the power control proble in CDMA wireless networks with self-interested wireless users. % Deviation fro Optial Su Rate γ Fig. 4. The Effect of γ on perforance loss. This graph depicts the (percentage) deviation of the obtained su-rate fro its optial value, as a function of γ. We have introduced the potential-gae approach for distributed power allocation, which (approxiately) enforces any power-dependent syste-objective. This approach involves approxiating the power gae by a close potential gae, for which target prices can be derived. By exploiting the relation between the power gae and its approxiation, the sae prices induce near-optial perforance in the underlying syste. Applying the potential-gae approach for the CDMA wireless doain, we have established through Lyapunovbased analysis that best-response dynaics converges within a neighborhood of the desired operating point. We deonstrated through siulations that the actual perforance could in practice be very close to optial, even when operating at a low-sinr regie. The scope of our work ay be extended in several respects. As indicated in Section IV-D, the distributed ipleentation of pricing is possible, however we have not pursued in this paper any specific distributed ethod. The design and incorporation of distributed optiization schees into our fraework reains an iportant future research direction. In the networkpricing context, a challenging direction is to consider the case where the syste is restricted to setting a user-independent price for power usage (which could be easier to ipleent in soe configurations). It is of interest to exaine the possible perforance degradation due to this sipler pricing schee. On a higher level, our pricing schee is ade possible because the underlying user utilities for a gae that is close to a potential gae. In general noncooperative gaes, however, it is not apparent how to identify such a potential gae with desirable properties. In an ongoing work [], we develop a systeatic procedure for choosing an appropriate potential gae for any given gae. We believe that the general paradig of identifying near-potential gaes can be applied to other networking applications, thereby iproving the controllability of networked systes with selfish users.

8 8 REFERENCES [] Z. Luo and S. Zhang, Dynaic spectru anageent: Coplexity and duality, IEEE Journal of Selected Topics in Signal Processing, vol., no., pp. 57 7, 008. [] S. Hayashi and Z. Luo, Spectru anageent for interference-liited ultiuser counication systes, IEEE Transactions on Inforation Theory, vol. 55, no., pp. 5 75, 009. [] G. Scutari, D. Paloar, and S. Barbarossa, Asynchronous iterative water-filling for gaussian frequency-selective interference channels, IEEE Transactions on Inforation Theory, vol. 54, no. 7, pp , 008. [4] F. Meshkati, H. Poor, and S. Schwartz, Energy-efficient resource allocation in wireless networks, IEEE Signal Processing Magazine, vol. 4, no., pp , 007. [5] M. Chiang, P. Hande, T. Lan, and C. Tan, Power control in wireless cellular networks, in Foundations and Trends in Networking, Volue, Issue 4, 008, pp [6] E. Altan and Z. Altan, S-odular gaes and power control in wireless networks, IEEE Transactions on Autoatic Control, vol. 48, no. 5, pp , 00. [7] C. Saraydar, N. Mandaya, and D. Goodan, Efficient power control via pricing in wireless data networks, IEEE Transactions on Counications, vol. 50, no., pp. 9 0, 00. [8] T. Alpcan, T. Basar, R. Srikant, and E. Altan, CDMA uplink power control as a noncooperative gae, Wireless Networks, vol. 8, no. 6, pp , 00. [9] T. Basar and R. Srikant, Revenue-axiizing pricing and capacity expansion in a any-users regie, in in Proc. IEEE Infoco, 00, pp. 7. [0] L. He and J. Walrand, Pricing and revenue sharing strategies for internet service providers, IEEE Journal on Selected Areas in Counications, vol. 4, no. 5, pp , 006. [] D. Aceoglu and A. Ozdaglar, Copetition in parallel-serial networks, IEEE Journal on Selected Areas in Counications, vol. 5, no. 6, pp. 80 9, 007. [] A. Ozdaglar and R. Srikant, Incentives and pricing in counication networks, in Algorithic Gae Theory. Cabridge University Press, 007. [] J. Rosen, Existence and uniqueness of equilibriu points for concave n-person gaes, Econoetrica, vol., no., pp , 965. [4] D. Monderer and L. Shapley, Potential Gaes, Gaes and Econoic Behavior, vol. 4, no., pp. 4 4, 996. [5] A. Neyan, Correlated equilibriu and potential gaes, International Journal of Gae Theory, vol. 6, no., pp. 7, 997. [6] D. Fudenberg and D. Levine, The theory of learning in gaes. The MIT Press, 998. [7] J. Shaa and G. Arslan, Dynaic fictitious play, dynaic gradient play, and distributed convergence to Nash equilibria, IEEE Transactions on Autoatic Control, vol. 50, no., pp. 7, 005. [8] J. Hofbauer and W. Sandhol, On the global convergence of stochastic fictitious play, Econoetrica, pp , 00. [9] H. Khalil, Nonlinear systes. Prentice Hall, 00. [0] O. Candogan, I. Menache, A. Ozdaglar, and P. Parrilo, Near-optial power control in wireless networks - a potential gae approach, LIDS, MIT, Tech. Rep., available fro candogan/ publications/networkpricing.pdf. [] L. Andrew, Measureent-based band allocation in ultiband CDMA, IEEE Transactions on Wireless Counications, vol. 4, no., pp , 005. [] A. Nedic and A. Ozdaglar, Cooperative distributed ulti-agent optiization, in Convex Optiization in Signal Processing and Counications. Cabridge University Press, 009. [] O. Candogan, I. Menache, A. Ozdaglar, and P. Parrilo, Near-Potential Gaes, (In preparation). ACKNOWLEDGMENT This research was funded in part by National Science Foundation grants DMI and ECCS-069, by the DARPA ITMANET progra, and by a Marie Curie International Fellowship within the 7th European Counity Fraework Prograe. APPENDIX Throughout this section, we use the notation λ ζ c for all M. Proof of Lea : Let φ ax p P φ(p) and φ in p P φ(p), and define the function V = φ φ 0. The idea behind the proof is to use V as a Lyapunov function for the analysis of the BR dynaics in G. Consider the tie derivative of V, it is easy to see that V (p) = ṗ = (β (p ) p ) = ( ) β (p ) p + ( β (p ) p β ) (p ). (4) Note that by the strict concavity of the potential function it follows that ( β (p ) p ) φ( β (p ), p ) φ(p, p ) =ũ ( β (p ), p ) ũ (p, p ). (5) We next obtain a lower bound on the ter (β (p ) β ) (p ). Considering the bounds on the power investent of players, any p satisfies P p P. In view of (), λ satisfies P P, λ M. (6) Hence, it follows that = λ p. (7) P P For a given fixed p the unconstrained axiu of u with respect to p (denoted by q ) satisfies u(q,p ) = 0, where u (q, p ) = + Hence, solving for q we obtain, γh N 0+ P k h kp k q = λ N 0 γh γ γh q λ. N P 0+ k h kp k k h k h p k. It follows fro (6) that q P ; using the concavity of u, it can be seen that the best response of player satisfies β (p )=ax{p,q }. (8) Siilarly, it can be shown that for a fixed p, the unconstrained axiu of ũ with respect to p (denoted by q ) satisfies ũ( q,p ) = φ( q,p ) =0. Using the partial derivatives of the potential function, it follows that

9 9 q = λ. By (6) it follows that the unconstrained axiu q always satisfies the power constraints, hence β (p )=. (9) λ For every p P and M, (8) (9) iply that β (p ) β (p ), and furtherore β (p ) β (p ) N 0 γh + γ N 0 γh + γ k where ξ N0 h + k follows fro (7) and (0) that ( β (p ) p β (p )) k h k h p k h k h P k = ξ γ, (0) h k h P k for every M. It thus ( ) ξ P P γ. By (4) and (5) it can be readily seen that V satisfies V (p) ũ ( β (p ), p ) ũ (p, p ) ( ) ξ P P γ ξ γp () Therefore, () iplies that V (p) unless ũ ( β (p ), p ) ũ (p, p ) ξ. γp Note that ũ ( β (p ), p ) ũ (p, p ) 0 for every p and M, because β (p ) is the axiizer of ũ when the users other than are using p. Thus, V (p) ξ, unless γp ũ k ( β k (p k ), p k ) ũ k (p k, p k ) ξ γp () for all k M. Hence, with V as a Lyapunov function, we ay apply a standard Lyapunov analysis arguent to conclude that the dynaics in (5) converges to a set such that () holds for all k M. Note that () iplies that no player can iprove ξ γp its utility by ore than by odifying its action. Thus, the dynaics converges to the set of ɛ-equilibria (Ĩɛ) of the gae G where ɛ ξ γp. By the definitions of ξ and SINR it follows that P ξ = SINR, hence ɛ γ SINR. The unifor convergence follows by noting that 0 V φ + φ. Since the potential function has bounded partial derivatives and the joint strategy space is copact the latter ter is ξ γp bounded by soe V 4. Since V if the operating point is not in Ĩɛ, the convergence tie to Ĩɛ, 4 Specifically, it can be shown that V P log P + log P + λ P using the properties of the potential function. which we denote by T, satisfies T P regardless ξ P of the initial operating point of the dynaics. Thus, the BR dynaics, converge uniforly to Ĩɛ. Proof of Theore 4: The key idea behind the proof is to translate a deviation in the potential function value to a deviation in the strategy space P, by using properties of the partial derivatives of the potential. Since p Ĩɛ, it follows that φ(p, p ) φ( p, p ) ɛ, or equivalently (log p λ p ) (log p λ p ) ɛ, () for every M. Let f : P R be a function such that f (p )=(logp λ p ). Using second order expansion, it follows that f ( p )=f (p )+( p p ) f(p ) γv + (p p ) f(p + α( p p )) p for soe α [0, ]. Note that f(p ) = φ(p ) =0, because p is the desired operating point and all the partial derivatives of the potential vanish at this power allocation. Observing that f(p ) p = p, the previous equation can be rewritten as f (p ) f ( p )= (p p ) (p + α( p p )), or equivalently, (p + α( p p )) (f (p ) f ( p )) = (p p ). Using () and the fact that 0 <p, p P, the previous equation iplies that ɛp (p p ). We therefore conclude that P ɛ p p. Lea : The su-rate function () satisfies U0 M P for every M. Proof: Differentiating U 0 with respect to p we obtain U 0 (p) = + Thus it follows that and U 0 (p) U 0 (p) γh N 0+ P k h kp k γh p N 0+ P k h kp k + + l l γh N 0+ P k h kp k + l γh ll h l p l (N 0+ P k l h klp k) γh ll p l. N P 0+ k l h klp k γh p N P p 0+ k h P kp k γh ll h l p l (N 0+ P k l h klp k) γh ll p l N P 0+ k l h klp k h l for all p P. Therefore, U 0 M ( N 0 + ) M k l h klp k P P