- Research Article
- Open Access
Pricing in Noncooperative Interference Channels for Improved Energy Efficiency
© Zhijiat Chong et al. 2010
- Received: 30 October 2009
- Accepted: 14 June 2010
- Published: 7 July 2010
We consider noncooperative energy-efficient resource allocation in the interference channel. Energy efficiency is achieved when each system pays a price proportional to its allocated transmit power. In noncooperative game-theoretic notation, the power allocation chosen by the systems corresponds to the Nash equilibrium. We study the existence and characterize the uniqueness of this equilibrium. Afterwards, pricing to achieve energy-efficiency is examined. We introduce an arbitrator who determines the prices that satisfy minimum QoS requirements and minimize total power consumption. This energy-efficient assignment problem is formulated and solved. We compare our setting to that without pricing with regard to energy-efficiency by simulation. It is observed that pricing in this distributed setting achieves higher energy-efficiency in different interference regimes.
- Utility Function
- Nash Equilibrium
- Power Allocation
- Channel State Information
- Spectrum Sharing
Power management and energy-efficient communication is an important topic in future mobile communications and computing systems. Currently, 0.14% of the carbon emissions are contributed by the mobile telecommunications industry . In order to improve the situation, we study new algorithms at physical and multiple-access layers. This includes resource allocation and power allocation. A common mobile communication scenario is where several communication system pairs utilize the same frequencies and are within interference range from one another. This setting is modeled by the interference channel (IFC). The transmitter-receiver pairs could belong to different operators and these are not necessarily connected. Therefore, noncooperative operation of the systems is assumed.
In a noncooperative scenario without pricing, systems transmit at highest possible powers to maximize their data rates. Transmitting at high powers, however, is detrimental to other users, because it induces interference which reduces their data rates. In such settings, spectrum sharing might lead to suboptimal operating points or equilibria . The case of distributed resource allocation and the conflicts in noncooperative spectrum sharing are best analyzed using noncooperative game theory (e.g., for CDMA uplink in  and usage of auction mechanisms in ). An overview of power control using game theory is presented in . Moreover, analysis of noncooperative and cooperative settings using game theory are performed in .
Studies have shown that the point of equilibrium in a noncooperative game is inefficient but can be improved by introducing a linear pricing . Linear pricing means that each system has to pay an amount proportional to its transmit power. This encourages transmission at lower powers, which reduces the amount of interference and at the same time leads to a Pareto improvement in the users' payoffs. Pricing in multiple-access channels has also been investigated with respect to energy-efficiency in . Studies in an economic framework demonstrates other advantages of proper implementation of pricing, for example, it provides incentives to service providers to upgrade their resources  or increase revenue .
In , the energy-efficiency of point-to-point communication systems is improved by sophisticated adaptation strategies. A coding theoretic approach is proposed in  where "green codes" for energy-efficient short-range communications are developed. Recent proposals define a utility function which incorporates the cost of transmission, for example, the price of spending power is considered in a binary variable in  and as an inverse factor in .
A similar utility function as in this paper is proposed in  for single-antenna systems and used to characterize the Nash equilibrium for the noncooperative power control game. Later in , the approach is extended to multiple-antenna channels in a related noncooperative game-theoretic setting. In , distributed pricing is introduced for power control and beamforming design to improve sum rate.
Different from previous works, we apply linear pricing to improve the energy-efficiency of an IFC with noncooperative selfish links to enable distributed implementation. Our objectives also include global stability and fairness. Compared to the work in , we do not assume that the channel states are chosen such that a unique global stable Nash equilibrium (NE) exists. Instead, we constrain the prices such that uniqueness and global stability follows. We derive the largest set of prices in which both the uniqueness of the NE and concurrent transmission are guaranteed, which is then utilized as a constraint in the optimization problem. The contribution is the derivation of the optimal pricing for transmit power minimization under minimum utility requirements and spectrum sharing constraints. If the utility requirements are feasible (Section 4.4), we derive a closed-form expression for the optimal prices (Proposition 6). Another relevant case is to minimize transmit powers such that rate requirements and global stability as well as fairness are achieved. These optimal power allocation and prices are presented in Section 5.3 and feasibility is checked in Proposition 8.
This paper is organized as follows. In Section 2, the system, channel, and the game models are presented. The game described is then studied in Section 3. Based on uniqueness analysis of the Nash equilibrium, we formulate and solve the energy-efficient optimization problem with minimum utility requirements and with minimum rate requirements constraints in Sections 4 and 5, respectively. In Section 6, simulations comparing the setting with and without pricing are presented. Section 7 concludes this paper.
2.1. System Model
The described competing links belong to different operators or wireless service providers. We assume that there exists an entity which can control the operators indirectly by rules or by changing their utility functions. We could think of this entity as a national or international regulatory body. In contrast to common long-term regulation, the utility function here changes on a smaller time-scale. The role of the arbitrator which represents this authority is discussed in Section 2.3.
A similar model is presented in the context of cognitive radios in , where the primary user decides on the prices which the secondary users have to pay for their transmission. The choice of the prices is not only for interference control but also for revenue maximization. The model in  involves multiple entities, that is, the primary users, who determine the prices imposed on secondary users to limit their aggregate and per-carrier interference in a distributed fashion.
2.2. Channel Model
where , , is the transmit power of . We assume no power constraint on the transmitters, that is, . It is shown later that the maximum power that would be utilized is nevertheless bounded due to a pricing factor.
2.3. Game Model
and similarly for player 2.
which is achieved when the counter transmitter allocates no power, that is, . Thus, the strategy region of player could be confined to .
The authority that can control the elements of the game is assumed to determine the power prices, and . It receives either utility or rate demands from the users and checks if they are feasible. If they are, it calculates the prices and informs the system pairs about the prices imposed on them. The links will have to pay costs proportional to their transmit power, that is, and (Figure 1). In game-theoretic notation, this entity is called the arbitrator . The arbitrator is not a player in the game and chooses the equilibrium that meets certain criteria. In our case, these criteria would be fairness, energy-efficiency, and minimum utility requirements or minimum rate requirements. We assume that the arbitrator also has complete game information.
In contrast to the case in which a central controller decides on the power of the users, the arbitrator imposes prices such that the users voluntarily set their powers. Thereby, the arbitrator indirectly determines the power allocation. In this paper, we study short-term price adaptation based on perfect CSI where prices depend on the instantaneous channel state. Long-term price adaptation based on partial CSI can also be implemented but is not considered here but left for future work.
In this section, we study the game described in Section 2.3. This is done by investigating the existence of pure strategy NEs and characterizing the conditions for uniqueness.
3.1. Existence of Nash Equilibrium
There exists a pure strategy NE in a game if the following two conditions are satisfied . First, the strategy spaces of the players should be nonempty compact convex subsets of an Euclidean space. Second, the utility functions of the players should be continuous in the strategies of all players and quasiconcave in the strategy of the corresponding player.
This condition is satisfied for player 1 and similarly for player 2. Next, we analyze the number of NEs that exist and state the related conditions.
3.2. Uniqueness of Nash Equilibrium
In this section, we study the conditions that lead to a unique NE. Under these conditions and considering only the case where the spectrum is simultaneously utilized by the two systems, we prove that the best response dynamics are globally convergent. Under these conditions, the noncooperative systems are guaranteed to operate in the NE if they iteratively apply their best response strategies.
The proof is given in Appendix .
Following the conditions in (8a) and (8b), we can easily characterize the sufficient conditions for the existence of a unique NE.
There exists a unique NE if .
If the conditions in (8a) and (8b) are fulfilled simultaneously, both transmitters would be transmitting at the same time. We denote this case as the concurrent transmission case. Next, we consider only this case since it is the fair case where both systems operate simultaneously. The other cases in which a unique NE exists correspond to one transmitter allocating maximum transmit power and the other not transmitting. The concurrent transmission case satisfies , which is the sufficient condition for the existence of a unique NE given in Corollary 2.
In the concurrent transmission case, the transmitters operate in the unique NE which is a fixed point of the best response function. In order to reach the NE, the best response dynamics must globally converge.
The best response dynamics globally converge to the NE in the concurrent transmission case, that is, when (8a) and (8b) hold simultaneously.
The proof is given in Appendix .
In comparison to the IFC without pricing, the sufficient conditions for global convergence of the best response dynamics are identical. The reason for that is, however, not obvious. The linear pricing in our utility function leads to a translation of the best response function but as well changes the interference conditions where concurrent transmission takes place. This is seen in the conditions in (8a) and (8b) where the bounds depend on the prices. Therefore, proving the sufficient conditions for global convergence of the best response dynamics is necessary in our case.
3.3. Admissible Power Prices
All prices achieve NEs in the concurrent transmission case. In the case that , the set is, however, empty, that is, there exists no power prices that achieve the concurrent transmission case. This happens since the upper bound on would be less than the lower bound for any , that is, . Another observation is that the set is convex only in the case if and both hold. This corresponds to the weak interference case. In the case if one CCC is larger than one, but still the condition holds, the set is not convex.
where and . Note that the can be omitted because the concurrent transmission implies that the power allocation of both systems are nonzero.
In this section, we investigate how the power prices are chosen such that energy-efficiency as well as minimum utility requirements are satisfied.
4.1. Optimization Problem
The arbitrator decides on the power prices such that the outcome satisfies the following conditions.
(C1)The best response dynamics globally converge to the unique NE.
(C2)Spectrum sharing (concurrent transmission) is ensured so that it is fair for all users.
(C3)Users transmit at the lowest powers possible satisfying minimum utility requirement , to promote efficient energy usage.
The function in (12) is convex in only in the weak interference channel case, that is, , Similarly, the constraint set is also only convex in the weak interference channel case. Thus, the problem in (11a), (11b) and (11c) is in general not a convex optimization problem. However, a closed-form solution is possible, which will be shown in Section 4.3. Before that, we will investigate some interesting properties of the inverse power prices which will facilitate the proof of the solution.
4.2. Analysis in Inverse Price Space
We see that by inserting any between and , (18) is always positive if . This implies the following. There is always an increase in as are increased along a line with slope that takes any value between and .
Definition 4 (Dominating vector by inclination ).
A vector is said to dominate a vector by an inclination of if is nonnegative and .
For a region where all dominate by an inclination of , where , . This also means that is the point with the least sum power for this region.
Setting and in (19) and (20) would return the upper and the lower bounds as in (15a) and (15b), making . As ( resp.) is increased, the slope of the upper (lower) bound decreases (increases). The point of intersection of these two curves is where both utility requirements are fulfilled with equality, as indicated by in Figure 2. The region forms an open triangle which is found within . This implies that is a subset of ( ).
From the properties we have considered above, it is quite intuitive to conclude that the solution to problem (11a), (11b) and (11c) is the pair that corresponds to , where the utility requirements (11b) are fulfilled with equality.
These expressions are found by calculating and when (19) equals (20) and then inverting them.
The constraint (11b) is satisfied in . Furthermore, for (11c) to hold, must be a subset of . This is only fulfilled if the slopes of the upper and lower bounds of are within and . Otherwise, they would cross or , making contain regions outside . Because of this property, Corollary 5 holds. Therefore, for any and that yields a nonempty set , the intersection of (19) and (20) yields the inverse power prices with the least sum power in region , which correspond to .
4.4. Feasible Minimum Utility Requirements
We assume that the arbitrator supports reasonable requirements such that . Given minimum utility requirements, and , the arbitrator should be able to determine if this pair is feasible, that is, whether there exists a power pricing pair that leads to a unique NE that fulfills these requirements simultaneously. They are infeasible if all pricing pairs lead to either nonunique NE or a unique NE whose utility tuple does not fulfill the utility requirements.
A minimum utility requirement chosen under the conditions above is feasible if and only if the optimal power prices calculated in (22a) and (22b) are in the admissible power prices set given in (9), that is, .
The proof is given in Appendix .
Therefore, according to Proposition 7, the arbitrator checks if in order to determine the feasibility of the minimum utility requirements.
In Section 6, we give numerical simulations on energy-efficiency comparing the noncooperative setting with pricing and that without pricing as well as the cooperative setting with pricing. Before that, we analyze the case with minimum rate requirements in the next section.
In contrast to the previous section, we now investigate how the power prices are chosen such that energy-efficiency as well as minimum rate requirements are satisfied.
5.1. Optimization Problem
The arbitrator decides on the power prices such that the outcome satisfies the same conditions as in Section 4.1 with a modification in (C3), which we state as following.
(C3)Users transmit at the lowest powers possible satisfying minimum rate requirement .
where is defined as in (12). Before we come to the solution, we present some analysis that will simplify its derivation.
5.2. Analysis and Feasibility
Unlike in the previous section, where both power allocation and prices have a direct influence on whether the utility requirements are fulfilled, only the power allocation has a direct influence on the fulfillment of the rate requirements. Therefore, we take a different approach by first determining the power allocation that fulfills the rate requirements and simultaneously minimizes the total power, and then calculate the optimal power prices that lead the users to this NE.
However, may be negative. For given rate requirements and channel coefficients, we can verify if there exists a feasible unique power vector (i.e., , , where the inequality is componentwise) that fulfills the rate requirements using the following proposition.
The rate vector is feasible if and only if .
According to Theorem A.51 in , for any , there exists a unique vector if and only if . , which is the spectral radius, where are the eigenvalues of the matrix . is calculated as . This implies that the requirements are feasible if and only if .
The power allocation that minimizes with rate requirements is given by in (27), which fulfills the requirements with equality.
This implies that for any , .
is only valid if . Therefore, to ensure that both feasibility and the uniqueness of the NE are simultaneously fulfilled, has to be satisfied.
Suppose , for example, . There are some values of , for example, , which are feasible but there are no corresponding prices that lead the players to a unique NE that fulfills the requirements with equality. This scenario corresponds to strong interference . Therefore, one solution could be to consider another decoding strategy which is more complex and leads to a different achievable rate expression, which has a different game model.
The prices that solve (23a), (23b) and (23c) are given by as in (32a) and (32b), provided that , which ensures the feasibility of the solution and the constraint (23c), guaranteeing the uniqueness of the NE. The corresponding NE strategy is , which fulfills (23b) with equality. Using Proposition 9, we can conclude that this power allocation also fulfills (23a).
Here, we present numerical simulations on energy-efficiency comparing the noncooperative setting with pricing and that without pricing as well as the cooperative setting with pricing with minimum utility requirements.
As expected, the NE in the utility region, which is calculated by inserting into (2), is found exactly at the utility requirements, independently of the CCC values . The NE is very close to the Pareto boundaries, indicating that it is indeed very close to being a Pareto-efficient operating point for various CCCs. By increasing simultaneously, the utility region is expanded in that the intersections at the and axes increase. The region is also observed to change from being convex to being nonconvex as the product becomes larger. The reason for this is that prices are reduced so that systems can reach the utility requirements at higher CCCs. Lower prices mean that the maximum utility of a system is higher, which is achieved when the other system pair does not transmit. In this case, cooperation among systems is more advantageous than noncooperation in achieving a higher sum utility. Note that for a nonconvex utility region, time-sharing between single-user operating points could be used to improve the utilities. This requires the knowledge of the time-sharing schedule at the transmitters and can be considered in future work.
With regard to the optimal prices, which is shown in the legend of Figure 3, we observe that the system with the smaller CCC has to pay less than the one with the larger. However, if both systems have large CCCs, both pay less. We regard this pricing scheme as fair. On the one hand, the system that causes more interference to the other is charged with a higher price; on the other hand, if both systems suffer from high interference from each other, both are encouraged to transmit more power by means of price reduction so that the utility requirements are met.
where is the transmission rate, as in (1), of system and the corresponding power allocation. A similar function is used to measure energy-efficiency for ad hoc MIMO links in . Figure 4 shows a comparison between energy-efficiency in the following settings.
(S1)The NE achieved with pricing.
(S2)The NE achieved without pricing. The power allocation is upper bounded by as in (6) for a fair comparison.
(S3)Both systems cooperatively choose their strategies to achieve the highest sum utility, that is, . The power allocation here is also upper bounded by for a fair comparison.
The operating point for the cooperative case was determined by numerically finding the power allocation that yields the highest sum utility. The reason for maximizing the sum utility instead of the energy-efficiency in (35) is that the former leads to zero transmit powers.
The systems are to cooperate to maximize energy-efficiency, the result is where both transmit powers are zero.
We see that in the noncooperative case, pricing improves the energy-efficiency significantly. The amount of improvement increases as the CCCs increase. The results with cooperation prove to be superior when the CCCs are large, whereas for low CCCs, noncooperation with pricing yields better energy-efficiency. One might expect the outcome of cooperation to be always superior to that of noncooperation. This is not true here because in the case of cooperation, the sum utility is maximized instead of . In our scenario, systems are only interested in maximizing their sum utility but not energy-efficiency when cooperating.
In this work, we consider two communication system pairs that operate in a distributed manner in the same spectral band. In order to improve the system energy-efficiency, we employ linear pricing to the utility of the systems. Following that, we study the setting from a noncooperative game-theoretic perspective, that is, we analyze the existence and uniqueness of the Nash equilibrium. Based on the assumption that there exists an arbitrator that chooses the power prices, we considered the problem of minimizing the sum transmit power with the constraint of satisfying minimum utility requirements and minimum rate requirements, respectively. We derived analytical solutions for the optimal power prices that solve these problems. Simulation results show that the noncooperative operating points with pricing are always more energy-efficient than those without pricing. A further extension of this work is to consider the case with more than two users. This is much more involved because there is no closed-form characterization of the prices that induce a globally stable NE. However, sufficient conditions for a unique NE can be used to define the set for users. For this case, similar programming problems as in (11a), (11b) and (11c) and (23a), (23b) and (23c) should be solved.
A. Proof of Proposition 1
The analysis for the uniqueness of the NE in a game can be done by studying the reaction curves of the players. Here, we give a simple and geometric derivation.
Conditions for the number of NEs.
Number of NEs
Infinitely many NEs
B. Proof of Proposition 3
where is the outcome at the th iteration and is the identity matrix. The algorithm is globally convergent if the spectral radius of is less than one [31, Proposition ]. The condition, , satisfies this requirement. Since the concurrent transmission case satisfies , the best response dynamics are then globally convergent.
C. Proof of Proposition 7
where denotes , which is the utility tuple at the unique NE corresponding to . All points in achieve concurrent transmission. That is, for as explained above in Section 3.3. As we see, any is feasible by definition.
From Proposition 1, we see that the only region that leads to a unique NE with concurrent transmission is when conditions (8a) and (8b) are satisfied, provided that . Since is the equivalent formulation of the price region where these conditions are satisfied, any would either lead to nonunique NE or to a unique NE without concurrent transmission. The tuple that leads to a unique NE without concurrent transmission never satisfies the utility requirements because for any is never chosen as a requirement. Therefore, any will lead to infeasible utilities.
Hence, any price tuple leads to feasible utilities at NE since all tuples map to , and any leads to infeasible utilities.
The authors thank Prof. Jens Zander for fruitful discussions. This work is supported in part by the Deutsche Forschungsgemeinschaft (DFG) under Grant Jo 801/4-1 and by the German Federal State of Saxony in the excellence cluster Cool Silicon in the framework of the project Cool Cellular under Grant no. 14056/2367. Part of this work  has been presented at the 5th International Conference on Cognitive Radio Oriented Wireless Networks and Communications (CrownCom) 2010.
- Sustainable energy use in mobile communications Ericsson White Paper; 2007.Google Scholar
- Etkin R, Parekh A, Tse D: Spectrum sharing for unlicensed bands. IEEE Journal on Selected Areas in Communications 2007, 25(3):517-528.View ArticleGoogle Scholar
- Alpcan T, Basar T, Srikant R, Altman E: CDMA uplink power control as a noncooperative game. Wireless Networks 2002, 8(6):659-670. 10.1023/A:1020375225649View ArticleMATHGoogle Scholar
- Huang J, Berry RA, Honig ML: Auction-based spectrum sharing. Mobile Networks and Applications 2006, 11(3):405-418. 10.1007/s11036-006-5192-yView ArticleGoogle Scholar
- Chiang M, Hande P, Lan T, Tan CW: Power control in wireless cellular networks. Foundations and Trends in Networking 2007, 2(4):381-533. 10.1561/1300000009View ArticleGoogle Scholar
- Leshem A, Zehavi E: Game theory and the frequency selective interference channel. IEEE Signal Processing Magazine 2009, 26(5):28-40.View ArticleGoogle Scholar
- Saraydar CU, Mandayam NB, Goodman DJ: Efficient power control via pricing in wireless data networks. IEEE Transactions on Communications 2002, 50(2):291-303. 10.1109/26.983324View ArticleGoogle Scholar
- Jorswieck E, Boche H, Naik S: Energy-aware utility regions: multiple access pareto boundary. IEEE Transactions on Wireless Communications 2010, 9(7):2216-2226.View ArticleGoogle Scholar
- Sengupta S, Chatterjee M, Ganguly S: An economic framework for spectrum allocation and service pricing with competitive wireless service providers. Proceedings of the 2nd IEEE International Symposium on New Frontiers in Dynamic Spectrum Access Networks (DySPAN '07), April 2007, Dublin, Ireland 89-98.View ArticleGoogle Scholar
- Gajic V, Huang J, Rimoldi B: On competing wireless service providers. submitted to Computer Research Repository, http://arxiv.org/abs/0903.1972 submitted to Computer Research Repository,
- Bougard B, Lenoir G, Dejonghe A, Der Perre LV, Catthoor F, Dehaene W: SmartMIMO: an energy-aware adaptive MIMO-OFDM radio link control for next-generation wireless local area networks. EURASIP Journal on Wireless Communications and Networking 2007, 2007:-15.Google Scholar
- Grover P, Sahai A: Green codes: energy-efficient short-range communication. Proceedings of the IEEE International Symposium on Information Theory (ISIT '08), July 2008, Toronto, Canada 1178-1182.Google Scholar
- Acemoglu D, Ozdaglar A, Srikant R: The marginal user principle for resource allocation in wireless networks. Proceedings of the 43rd IEEE Conference on Decision and Control (CDC '04), December 2004, Nassau, Bahamas 1544-1549.Google Scholar
- Meshkati F, Chiang M, Vincent Poor H, Schwartz SC: A game-theoretic approach to energy-efficient power control in multicarrier CDMA Systems. IEEE Journal on Selected Areas in Communications 2006, 24(6):1115-1129.View ArticleGoogle Scholar
- Gunturi S, Paganini F: Game theoretic approach to power control in cellular CDMA. Proceedings of the IEEE 58th Vehicular Technology Conference (VTC '03), October 2003, Orlando, Fla, USA 2362-2366.Google Scholar
- Duplicy J, Louveaux J, Vandendorpe L: Utility-based MIMO uplink beamforming. Proceedings of the IEEE Sensor Array and Multichannel Signal Processing Workshop (SAMSP '06), July 2006, Waltham, Mass, USA 254-257.Google Scholar
- Schmidt DA, Shi C, Berry RA, Honig ML, Utschick W: Distributed resource allocation schemes. IEEE Signal Processing Magazine 2009, 26(5):53-63.View ArticleGoogle Scholar
- Razaviyayn M, Morin Y, Luo Z-Q: A Stackelberg game approach to distributed spectrum management. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP '10), 2010, Dallas, Tex, USA 3006-3009.Google Scholar
- Pang J-S, Scutari G, Palomar DP, Facchinei F: Design of cognitive radio systems under temperature-interference constraints: a variational inequality approach. IEEE Transactions on Signal Processing 2010, 58(6):3251-3271.MathSciNetView ArticleGoogle Scholar
- Carleial AB: Interference channels. IEEE Transactions on Information Theory 1978, 24(1):60-70. 10.1109/TIT.1978.1055812MathSciNetView ArticleMATHGoogle Scholar
- Larsson EG, Jorswieck EA: Competition versus cooperation on the MISO interference channel. IEEE Journal on Selected Areas in Communications 2008, 26(7):1059-1069.View ArticleGoogle Scholar
- Myerson RB: An Introduction to Game Theory. Northwestern University, Center for Mathematical Studies in Economics and Management Science, Evanston, Ill, USA; September 1984. Discussion Papers 623MATHGoogle Scholar
- Rosen JB: Existence and uniqueness of equilibrium points for concave n-person games. Econometrica 1965, 33(3):520-534. 10.2307/1911749MathSciNetView ArticleMATHGoogle Scholar
- Boyd S, Vandenberghe L: Convex Optimization. Cambridge University Press, Cambridge, UK; 2004.View ArticleMATHGoogle Scholar
- Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE: On the Lambert W function. Advances in Computational Mathematics 1996, 5(4):329-359.MathSciNetView ArticleMATHGoogle Scholar
- Stanczak S, Boche H, Wiczanowski M: Fundamentals of Resource Allocation in Wireless Networks, W. Utschick, H. Boche, and R. Mathar Eds.. Springer, Berlin, Germany; 2009.MATHGoogle Scholar
- Carleial AB: A case where interference does not reduce capacity. IEEE Transactions on Information Theory 1975, 21(5):569-570. 10.1109/TIT.1975.1055432MathSciNetView ArticleMATHGoogle Scholar
- Myerson RB: Game Theory: Analysis of Conflict. Harvard University Press, Cambridge, Mass, USA; 1997.MATHGoogle Scholar
- Chao L, Dandekar KR: Power management in MIMO ad hoc networks: a game-theoretic approach. IEEE Transactions on Wireless Communications 2007, 6(4):1164-1170.View ArticleGoogle Scholar
- Basar T, Olsder GJ: Dynamic Noncooperative Game Theory, Classics in Applied Mathematics. Volume 23. SIAM, Philadelphia, Pa, USA; 1998.View ArticleGoogle Scholar
- Bertsekas DP, Tsitsiklis JN: Parallel and Distributed Computation: Numerical Methods. Prentice-Hall, Upper Saddle River, NJ, USA; 1998.MATHGoogle Scholar
- Mochaourab R, Chong Z, Jorswieck E: Pricing in noncooperative interference channels for improved energy efficiency. Proceedings of the 5th International Conference on Cognitive Radio Oriented Wireless Networks and Communications, 2010, Cannes, FranceGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.