Auction-Based Power Allocation for Many-to-One Cooperative Wireless Networks

Transcription

1 Auction-Based Power Allocation for Many-to-One Cooperative Wireless Networks Mohammed W. Baidas and Allen B. MacKenzie Virginia Tech, Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA 246, USA baidas@ieee.org, mackenab@vt.edu Abstract In this paper, a distributed efficient power allocation game-theoretic framework in wireless ad-hoc networks is proposed where multiple source nodes communicate with a single destination node via a relay node. The power allocation among the source nodes is formulated as an alternative ascending-clock auction A-ACA and achieved using a distributed algorithm that converges in a finite number of clocks and is proven to enforce truthful power demands at every clock and maximize the social welfare. Analytical and numerical results are presented to verify the efficient power allocation, truthtelling and social welfare maximization properties of the proposed A- ACA. It is concluded that the proposed A-ACA lends itself to practical implementation in wireless ad-hoc networks. Index Terms Amplify-and-forward AF, auction, cooperation, many-to-one, network coding, power allocation I. INTRODUCTION In decentralized and fully distributed ad-hoc wireless networks without a single authority and with network users acting as independent entities, rational users selfishly aim at maximizing their utility and use of resources. Moreover, selfish users might not have any incentive to share their resources if they cannot gain some reward in return. In other cases, several users could compete in order to consume transmission resources e.g. bandwidth and power from a particular user who might be welling to assist in relaying their data transmissions towards a destination, however, at a price. Since rational users are rather selfish, they may overstate or otherwise not truthfully report their resource demand if doing so can improve their utility. In other scenarios, revealing such proprietary information may have an adverse strategic long-term impact. Hence, a key problem in wireless ad-hoc networks is how distributively and efficiently allocate transmission resources among competing selfish users. Overall, the modeling of resource allocation and repeated interaction of network users is rooted in game and auction theories. An important thrust of recent research works deal with user selfishness and cooperation from a game-theoretic perspective. For instance, in [], a Stackelberg game is proposed for multinode relay selection and power control when a single source node communicates with a single destination. In another line of work, Huang et al. propose two distributed auction mechanisms that achieve a unique Nash Equilibrium are proposed for relay selection and power allocation, namely, the SNR auction and the power auction [2]. It is shown that the SNR auction offers a flexible tradeoff between fairness and efficiency, while the power auction achieves efficient power allocation by maximizing the total rate increase. In [3], a second-price auction is proposed for fair allocation of a wireless fading channel for different channel state distributions. In this paper, the alternative ascending-clock auction A-ACA is proposed for efficient power allocation in many-to-one cooperative wireless networks, where multiple source nodes communicate with a single destination node via a relay node [4] [5]. In particular, a distributed algorithm is proposed to achieve the A-ACA based efficient power allocation algorithm which is proven to enforce truthful power demands among network source nodes and maximize the social welfare i.e. the sum of utilities of the source nodes and the relay. It is envisioned that the presented work will pave the way for a practical distributed and efficient power allocation in manyto-one cooperative communications in wireless ad-hoc networks. In the remainder of this paper, the network model is presented in Section II. The multiuser game-theoretic framework and the proposed alternative ascending-clock auction are presented in Sections III and IV, respectively. Numerical results of the proposed ascending-clock auction based power allocation are presented in Section V. Finally, the conclusions are drawn in Section VI. II. NETWORK MODEL Consider a wireless network consisting of N source nodes N 2, denoted as S, S 2,..., S N. The N nodes are assumed to have their own data symbols x, x 2,..., x N, respectively, and each node aims at communicating its data symbol to a common destination node D. In addition, assume that there is a relay node R that is welling to share its transmission power P R to forward source nodes data symbols to the destination. In this network shown in Fig. for N = 2, each node is equipped with a single antenna and the relay node s cooperative transmission follows the amplify-andforward AF protocol [6]. The channels between any two nodes are modeled as narrow-band Rayleigh fading channels with additive white Gaussian noise AWGN. Let h j,i denote a generic channel coefficient representing the channel between any two nodes j and i, then h j,i CN, σ 2 j,i, where σ2 j,i = d ν j,i is the channel gain with d j,i and ν being the distance between the two nodes and the path-loss exponent, respectively. The communication between the source nodes and the destination node is performed over a total of N + time-slots and is split into two phases, namely the broadcasting phase BP of N time-slots and the cooperation phase CP of a single time-slot. A. Broadcasting Phase In the broadcasting phase, each source node S j for j {, 2,..., N} is assigned a time-slot T j in which it broadcasts its own data symbol x j to the rest of the network. The received signal y j,r t at the relay node R in time-slot T j is expressed as y j,r = P Bj h j,r x j + n j,r, while the received signal at the destination node D is expressed as y j,d = P Bj h j,d x j + n j,d, 2

2 Fig.. Many-to-One Cooperative Network - Broadcasting and Cooperation Phases - N = 2 Nodes where P Bj is the broadcasting transmit power at node S j and n j,r and n j,d are the zero-mean additive white Gaussian noise AWGN processes with variance N, at the relay and destination nodes R and D, respectively. Upon completion of the broadcasting phase, relay and destination nodes R and D will have received a set of N signals {y j,r} N j= and {y j,d } N j=, respectively, comprising symbols {x j } N j= of the N source nodes. B. Cooperation Phase The cooperation phase involves two operations: signal transmission via the relay node, and 2 multinode signal detection, which are discussed in the following subsections, respectively. Signal Transmission: In the cooperation phase, relay node R in its assigned time-slot T N+ forms a linear network code based on its received symbols {y m,r } N, during the broadcasting phase and transmits it to destination node D. For source separation of each transmitted symbol of the different source nodes at the destination, each received signal y j,r is spread using a signature waveform, c j t. It is assumed that the relay and destination nodes know the signature waveforms of all the source nodes. The cross-correlation of c j t and c i t is ρ j,i = c j t, c i t /T s T s c j tc i tdt for j i with ρ j,j =, T s being the symbol duration and denoting complex conjugation. The resulting signal transmitted by the relay, X t is written as X t = β m,ry m,rc mt, 3 where c m t is the signature waveform associated with source node S m and β m,r is a scaling factor defined as [6] P Cm β m,r =. 4 P Bm h m,r 2 + N where P Cm is the cooperative transmit power of symbol x m at the relay node R. The received signal at destination node D during time-slot T N+ is given by Y r,d t = h r,d X t + n d t, 5 where n d t is the AWGN at node D. Upon substitution of, 3 and 4 into 5, the received signal can be expressed as Y r,d t = α m,r,d x m c m t + n d t, 6 where α m,r,d = β m,r PBm h m,r h r,d and n d t is the equivalent noise term, defined as n d t = n d t + h r,d N β m,r n m,r c m t. 7 2 Multinode Signal Detection: Upon receiving signal Y r,d t from the relay node R, a multinode signal detection is performed by the destination D to extract each symbol x j, for j {, 2,..., N}. This is achieved by passing the received signal Y r,d t through a matched filter bank MFB of N branches, matched to the corresponding nodes signature waveforms c j t, yielding Y j,r,d = Y r,d t, c j t = α m,r,d x m ρ m,j + n j,d. 8 Recall that ρ m,j is the correlation coefficient between c m t and c j t. The output of the MFB can be put in a vector form as where Y r,d = RAx + n n n d, 9 Y r,d = [Y,r,d,..., Y j,r,d,..., Y N,r,d ] T, x = [x,..., x j,..., x N ] T, n n n d = [ n,d,..., n j,d,..., n N,d ] T CN, N BR, and R, A, and B are N N matrices with R being defined as ρ,j ρ,n R = ρ j, ρ j,n, ρ N, ρ N,j while the diagonal matrices A and B are and A = diag [α,r,d,..., α j,r,d,..., α N,r,d ], 3 [ B = diag β2,r,d,..., β j,r,d, 2..., β ] N,r,d 2, 4 with β j,r,d 2 being defined as β j,r 2 = βj,r h 2 r,d 2 +. The received signal vector Y r,d can then be decorrelated assuming matrix R is invertible as ỸỸỸ r,d = R Y r,d = Ax + ñññ d, where ñññ d = R n n n d and ñññ d CN, N R B. Thus, the decorrelated received signal Ỹ j,r,d corresponding to symbol x j is obtained as Ỹ j,r,d = β j,r P Bj h j,r h r,d x j + ñ j,d, 5 where ñ j,d CN, N r j βj,r 2 h r,d 2 + and r j is the j th diagonal element of matrix R. Without loss of generality, it is assumed that ρ j,i = ρ for all j i and thus [7] + N 2ρ r j = + N 2ρ N ρ r 2 N. 6 Upon the completion of the broadcasting and cooperation phases, destination node D will have received two signals of each symbol x j for j {, 2,..., N}; a direct signal from the source node S j in the broadcasting phase and a relayed signal in the cooperation phase. The detection of symbol x j, denoted as x j, is achieved through maximal-ratio-combining MRC of the signals received in the broadcasting and cooperation phases. Thus, the instantaneous SNR at the output of the MRC of symbol x j is given by γ j = γj,d B +γc j,d, where γb j,d is the SNR due to the broadcast direct transmission from the source to the destination and is defined as

3 R j,d = N + log 2 + P B j h j,d 2 P Bj P Cj h j,r 2 h r,d 2 +, j {, 2,..., N}. 8 N N r N P Bj h j,r 2 + P Cj h r,d 2 + N [ ] P Cj ξ = max, min Ω 2 j,r,d 2Ω j,r,d + Υ2 j,r,d + 4ηΩ j,r,dυ j,r,d Ω j,r,d + Ω j,r,d + 2Υ j,r,d, P R, j {, 2,..., N}. 25 ξ γj,d B = P B j h j,d 2 /N, while γj,d C is the SNR due to the cooperative transmission via relay node R and is defined as γj,d C P Bj P Cj h j,r 2 h r,d 2 = N r N P Bj h j,r 2 + P Cj h r,d 2 + N. 7 Since a single data symbols is exchanged between every source node S j, for j {, 2,..., N} and the destination once every N + time-slots, the achievable rate in bits per time-slot can be determined using 8 top of page. III. MULTI-USER GAME-THEORETIC FRAMEWORK This section presents a game-theoretic framework for the cooperative relay power allocation among the N source nodes. A. Source Node Utility Function The utility function of each source node is based on the improvement in the transmission rate achieved by relay node s cooperative transmission. Accordingly, the utility function of source node S j for cooperative data transmission to the destination D via the relay node R is U S j P Cj = R j,d P Cj ξp Cj, 9 where ξ is the price per unit of power charged by the relay node to forward data symbols to the destination, and R j,d P Cj is the improvement in the transmission rate due to the cooperative relaying which is given by where and R j,d P Cj = N + log 2 Ω j,r,d = + PC Ω j j,r,d, 2 P Cj + Υ j,r,d P Bj h j,r 2 N r N P Bj h j,d 2 + N, 2 Υ j,r,d = P Bj h j,r 2 + N / h r,d Thus, R j,d P Cj defines the cooperation gain to node S j in terms of an improvement in the transmission rate to the destination node D, when relay node R cooperates. Clearly, R j,d P Cj is a monotonically increasing function of P Cj. Each source node aims at maximizing its utility subject to the total transmit power P R available at the relay R. Thus, each source node s cooperative power demand problem can be modeled as max Uj S P Cj = R j,d P Cj ξp Cj, P Cj s.t. P Cj P R, j {, 2,..., N}. 23 Clearly, the utility function U S j P C j is concave in P Cj, and taking the derivative of U S j P C j with respect to P Cj, yields U S j P Cj P Cj = R j,dp Cj P Cj ξ =. 24 By using the identity log 2 x = lnx/ ln 2, defining η = N+ ln 2 and substituting R j,d P Cj in 2 into 24, the utility function U S j P C j is maximized at P Cj ξ which is expressed in 25. B. Relay Node Utility Function The relay node s utility function is based on selling its cooperative transmit power P R to the source nodes to forward their data symbols to the destination. In this case, the relay s utility is defined as the total payment it receives by selling its transmit power P R to the source nodes minus its own cost of cooperation ζ per unit power i.e. for processing, transmitting and receiving. Thus, the relay s utility function is given by U R P R = ϑ r P R ζp R, 26 with ϑ r P R = N j= ϑ jpc j being the total payment the relay receives from the N source nodes for transmitting their data symbols, ϑ j PC j is the payment source node S j makes when it is assigned cooperative transmit power PC j such that P R N j= P C j as will be formally established in the following section. It should be noted that ζpc j is the cooperation cost due to the transmission of source node S j s data symbol. IV. ALTERNATIVE ASCENDING-CLOCK AUCTION The idea of pricing as a distributed control mechanism aims at encouraging autonomous and independent network users to make rational decisions that result in a social benefit for the entire network. Towards this end, a distributed algorithm based on the alternative ascending-clock auction A-ACA is proposed for efficient power allocation [4] [5]. In particular, the relay auctioneer announces a price, the N source nodes bidders report back their cooperative power demanded at that price and power is allocated to source nodes at the current price whenever they are clinched. The relay then raises the announced price and the process repeats until the total power demand meets the available power supply in which case all the relay s transmit power P R is allocated. In the A-ACA algorithm, at each time τ =,,..., the relay announces a price ξ τ to the N source nodes. In order for the relay to cover the cooperation cost per unit power ζ, it initially sets a reserve price of ξ = ζ and announces it to the source nodes. Based on the announced price ξ τ, each source node S j responds with a bid in the form of an optimal power demand P Cj ξ τ to the relay. After receiving all the demands at each clock, the relay compares the total demanded power P C T otal ξτ = N j= P C j ξ τ and compares it with the total available power P R. If the total demand exceeds the supply i.e. P C T otal ξτ > P R, the auction

4 proceeds to time τ + and the associated price is increased to ξ τ+ = ξ τ + µ = ζ + µτ + µ, where µ is an appropriate step size. The relay then calculates the cumulative vector of quantities C j ξ τ, clinched by source node S j at prices up to ξ τ, defined as C j ξ τ = max, P R P Ci ξ τ, j {, 2,..., N}. i=,i j 27 On the other hand, if the supply meets the total demand i.e. PT C otal ξτ P R, the auction concludes and the current time is denoted T. However, as the price goes up, the demanded cooperative power by each source node decreases and it is possible, for a certain increase in price, that the supply is not covered at the final price ξ T i.e. PT C otal ξt < P R. In turn, a proportional rationing rule is applied and the power allocation is achieved according to [4][8] C jξ T = P Cj ξ T + P Cj ξ T P Cj ξ T N j= P C j ξ T N j= P C j ξ T P R P Ci ξ T, i= 28 with N j= C jξ T = P R. Hence, each source node S j is assigned its demanded cooperative transmit power as PC j = C j ξ T. Also, the payment from source node S j to the relay is written as T ϑ j PC j = C j ξ ξ + ξ τ C j ξ τ C j ξ τ, 29 τ= where the total payment the relay node receives for allocating its power P R is ϑ r P R = N j= ϑ jpc j. It is easily verified that T PC j = C j ξ + C j ξ τ C j ξ τ. 3 τ= Also, by substituting ξ τ = ζ + µτ into 29, the payment of source node S j can be rewritten as ϑ j P C j = ζp C j + BP C j, 3 where BP C j = T µτc j ξ τ C j ξ τ 32 τ= is the surplus the relay obtains from node S j upon allocating cooperative power PC j. The proposed distributed A-ACA algorithm is summarized in Table I. The main properties of the proposed A-ACA algorithm are briefly discussed in the following subsections. A. Convergence Theorem Convergence: The A-ACA algorithm concludes in a finite number of clocks. Proof: See Appendix.A. B. Truth-Telling Theorem 2 Truth-Telling: Reporting optimal cooperative power demand truthfully at every clock in the alternative ascendingclock auction is a mutually best response for all source nodes. Proof: See Appendix.B. In the proposed distributed A-ACA algorithm, each participant has full incentive to truthfully reveal its true power demand and Relay initializes clock index at τ = and step size to µ > and then announces initial price of ξ = ζ. 2 Each source node S j computes and submits its optimal power demand P Cj ξ. 3 Relay sums up all demands PT C otal ξ = N j= P C j ξ. 4 IF PT C otal ξ P R, 5 the auction concludes and time is denoted T. 6 ELSE, 7 set ξ τ+ = ξ τ + µ and τ = τ +. 8 Price ξ τ is announced to source nodes. 9 Each source node S j computes and submits its optimal demand P Cj ξ τ. Relay sums up all demands PT C otal ξτ = N j= P C j ξ τ. IF PT C otal ξτ > P R, 2 compute C j ξ τ = max, P R N i=,i j P C i ξ τ and go to step 7. 3 ELSE, 4 the auction concludes and time is denoted T. 5 END. 6 END. 7 Compute C j ξ T and assign PC j = C j ξ T to source node S j, which makes a payment of ϑ j PC j to the relay node. TABLE I ALTERNATIVE ASCENDING-CLOCK AUCTION ALGORITHM this is because the price each source node pays depends solely on opposing nodes bids and thus need not report their private information i.e. also preserves privacy. Thus, the proposed A- ACA algorithm enforces truth-telling and the best strategy of each source node is to truthfully report its power demand at every clock. C. Social Welfare Maximization Theorem 3 Maximization of Social Welfare: The alternative ascending-clock auction based power allocation P C, P C 2,..., P C N maximizes the social welfare. Proof: See Appendix.C. The proposed A-ACA algorithm maximizes the sum of the source nodes and the relay node utilities when the relay node fully sells out its cooperative transmit power P R. D. Simplicity and Applicability The key characteristic of the proposed A-ACA algorithm is the relative simplicity of the auction process during the iterative clock phase and the final power demand allocation. In order words, the computational complexity on both the N source nodes and the relay is low. Each source nodes needs to calculate its optimal power demand P Cj ξ τ based on the announced price ξ τ while ascertaining that it is within available power P R and then submit a bid. On the other hand, the assessment of excess demand and calculation of the cumulative clinch are relatively trivial. Another important key characteristic is the fact that the relay s aim during the clock phase is to drive the excess demand out of the system by raising the price without ending up with excess power supply. This in turn implies the auction process achieves all gains from trade and is allocatively efficient [9]. Based on all the above discussed properties, it is concluded that the proposed A-ACA algorithm lends itself to practical ad-hoc network implementation.

5 Utilities.5.5 Source Node S Relay Node R Sum of Utilites a Utilities Vs. Relay Power Fig. 2. Cooperative Relay Network - Simulation Scenario V. NUMERICAL RESULTS To validate the proposed A-ACA algorithm, a wireless network with three source nodes S, S 2 and S 3 and a relay R see Fig. 2 is simulated. The network topology illustrates a scenario where the distance between any source node S j for j = {, 2, 3} and the destination is equal i.e. d,d = d 2,d = d 3,d. Also, the relay is closer to the destination than to node S i.e. d,r > d r,d ; while it is closer to node S 2 than to the destination i.e. d 2,r < d r,d. Moreover, the relay is at an equal distance from the destination and node S 3 i.e. d 3,r = d r,d. The simulations assume a path-loss exponent of ν = 3, non-orthogonal signature waveforms with a cross-correlation of ρ =.5, reserve price of ζ = 5, and source broadcasting transmit power P Bj = 2 dbw, j = {, 2, 3}. It is evident from Fig. 3a that with the increase in the available relay power P R, the utility of each source node increases, with node S 2 having the highest utility while node S having the lowest. This is attributed to the location of node S 2 with the relay being closer to it than to the destination and hence the effect of path-loss and channel noise is less severe. Thus, using the relay is significantly beneficial to node S 2 which is translated into a higher cooperative power demand and allocation as evident from Fig. 3b. The lowest utility corresponding to node S is justified by a converse argument to that of node S 2. Interesting to observe in Fig. 3a is the utility of the relay which peaks at P R = 2 dbw and then starts to degrade. This is because when P R is high enough i.e. P R > 2 dbw, there is abundant cooperative power for each of the source nodes i.e. the total power demanded by the source nodes is less than the supply relay power P R. Hence, the relay does not get to raise the price so much in the auction to extract higher revenue i.e. higher payments but instead sells most of the power early in the auction at a relatively lower price, which is seen in the form of lower payments in Fig. 3c. Overall, the sum of utilities keeps on increasing with the increase in available relay power P R. In Fig. 4, the utilities of the source nodes and the relay are evaluated at P R = 2 dbw when source node S falsely reports a demand of PC ξ τ = max [, min κp C ξ τ, P R ], τ =,,..., T with κ being the demand factor. On the other hand, nodes S 2 and S 3 report their truthful demands P C2 ξ τ and P C3 ξ τ, respectively, τ =,,..., T. It is clear that as κ increases, the utility of S improves until it reaches its maximum value at κ = i.e. truthful demand, beyond which the utility degrades. Hence, the proposed A-ACA algorithm enforces truthtelling and each source node must report its true/optimal power demand to maximize their utility. Fig. 3. Fig. 4. Payments Allocated Cooperative Power dbw Relay Power P dbw R Source Node S b Allocated Cooperative Power Vs. Relay Power Relay Power P R dbw Source Node S Sum of Payments c Payments Vs. Relay Power Relay Power P dbw R Simulation Results: Utilities, Allocated Power and Payment Utilities Utilities Vs. Relay Power Source Node S Relay Node R Sum of Utilites Demand Factor κ Simulation Results: Truthful Demand Verification - P R = 2 dbw It is also interesting to notice that with the increase in the demand factor κ, the utilities of nodes S 2 and S 3 decrease and this is due to the increased overall demand on power P R which in turn improves the relay node s utility due to the higher revenue/payments. VI. CONCLUSIONS In this paper, an alternative ascending-clock auction is proposed for efficient cooperative relay power allocation to geographically distributed source nodes in an ad-hoc wireless network. The proposed auction is applied via a distributed algorithm that is proven to enforce truth-telling and also maximize the sum of utilities. In conclusion, the proposed alternative ascending-clock auction lends itself to practical implementation in wireless ad-hoc networks. A. Convergence VII. APPENDIX I From 25, it is clear that P Cj ξ τ is non-increasing in ξ τ and hence P Cj ξ τ P Cj ξ τ+ with equality occurring when P Cj ξ τ+ = P Cj ξ τ = or P Cj ξ τ+ = P Cj ξ τ = P R, τ. Since for a sufficiently large τ, P Cj ξ τ+ < P Cj ξ τ < P R, then there exists a finite number T such that N j= P C j ξ T P R and thus the auction concludes at clock T.

6 B. Truth-Telling Assuming all the other source nodes truthfully report their power demands, let Σ jt = {P Cj ξ,..., P Cj ξ T ; C jξ,..., C jξ T ; T } 33 be the auction profile when source node S j when it reports its power demand truthfully on every clock τ. Moreover, let Σ j T = {P Cj ξ,..., P Cj ξ τ,..., P Cj ξ T ; C jξ,..., C jξ τ,..., C j ξ T ; T } 34 be the auction profile of source node S j when it falsely demands power on clock τ = τ for τ T while nodes S i, i {, 2,..., N} and i j truthfully report their power demands. T Also, let PCj ξ be the power allocated to node S j at the end of the auction when it falsely demands power. If P Cj ξ τ P Cj ξ τ, then P T Cj ξ P Cj ξ T, which implies that T T. From 9 and 29, it is clear that Uj S Σ jt Uj S Σ j T = R j,d P Cj ξ T T R j,d PCj ξ ϑ j P Cj ξ T T + ϑ j PCj ξ. 35 It can be shown that ϑ j P Cj ξ T T ϑ j PCj ξ ξ T C j ξ T ξ T Cj ξ T = ξ T P Cj ξ T ξ T T PCj ξ. Hence, Uj S Σ j T Uj S Σ j T R j,d P Cj ξ T ξ T P Cj ξ T T R j,d PCj ξ + ξ T T PCj ξ. 36 If P Cj ξ τ P Cj ξ τ, then P T Cj ξ P Cj ξ T, and thus T T T. Similarly, it can be shown that ϑ j PCj ξ ϑ j P Cj ξ T ξ T Cj ξ T ξ T C j ξ T = ξ T T PCj ξ ξ T P Cj ξ T. Thus U S j Σ jt U S j Σ j T R j,d P Cj ξ T ξ T P Cj ξ T T R j,d PCj ξ + ξ T T PCj ξ. 37 From 36 and 37, falsely demanding power at least once yields a lower utility than demanding power truthfully at every clock τ. Thus, the best strategy is to truthfully demand power at each clock τ for τ T. C. Social Welfare Maximization The proof aims to show that the A-ACA based power allocation P C, P C 2,..., P C N solves the following optimization problem: max P Cj s.t. R j,d P Cj j= j= P Cj P R P Cj P R, j =, 2,..., N, 38 which is convex, since R j is concave in P Cj and the constraint set is convex. Thus, the Lagrangian problem is formulated as [] N LP Cj, λ, ω j, υ j = R j,d P Cj + λ P Cj P R j= j= j= + ω j P Cj P R υ j P Cj, j= and the Karush-Kuhn-Tucker KKT conditions are given by: ηω j,r,d Υ j,r,d Ω j,r,d + P Cj + Υ j,r,d P Cj + Υ j,r,d + λ + ω j υ j = N λ P Cj P R =, j= 39 ω j P Cj P R =, j {, 2,..., N}, υ j P Cj =, j {, 2,..., N}, P Cj P R, j {, 2,..., N}, P Cj P R, j= 4 where η = N+ ln 2 while λ, ω j, υ j, j {, 2,..., N} are the dual variables associated with the power constraint and transmit power positivity. The solution to the optimization problem is known as water-filling [], in the form of P Cj λ in 25, j {, 2,..., N}, where λ satisfies N j= P C j λ = P R. Thus, the outcome of the A-ACA algorithm, PC, PC 2,..., PC N is the solution to the optimization problem in 38 that maximizes the sum of of the source nodes and the relay node utilities when the relay node fully sells out its cooperative transmit power P R. REFERENCES [] B. Wang, Z. Han, and K. J. R. Liu, Distributed relay selection and power control for multiuser cooperative communication networks using stackelberg game, IEEE Trans. on Mobile Computing, vol. 8, pp , Jul. 29. [2] J. Huang, Z. Han, M. Chiang, and H. V. Poor, Auction-based resource allocation for cooperative communications, IEEE JSAC, Special Issue on Game Theory, vol. 26, pp , 28. [3] J. Sun, E. Modiano, and L. Zheng, Wireless channel allocation using an auction algorithm, IEEE Journal on Selected Areas in Communications, vol. 24, pp , May 26. [4] L. M. Ausubel, An efficient ascending-bid auction for multiple objects, Amer. Econ. Rev., vol. 94, pp , 24. [5] V. Krishna, Auction Theory. Academic Press, 22. [6] K. J. R. Liu, A. K. Sadek, W. Su, and A. Kwasinski, Cooperative Communications and Networking. Cambridge University Press, 28. [7] M. W. Baidas, H.-Q. Lai, and K. J. R. Liu, Many-to-many communications via space-time network coding, Proc. of IEEE Wireless Communcations and Networking Conference, pp. 6, April 2. [8] Y. Saez, D. Quintana, P. Isasi, and A. Mochon, Effects of a rationing rule on the ausubel auction: A genetic algorithm implementation, Computational Intelligence, vol. 23, pp , 27. [9] T. K. Forde and L. E. Doyle, Combinatorial clock auction for OFDMA-based cognitive wireless networks, Proc. of 3rd Int. Symp. on Wireless Pervasive Computing ISWPC, pp , May 28. [] S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press 23. [] T. Cover and J. Thomas, Elements of Information Theory. John Wiley Inc., 99.