Nash Bargaining in Beamforming Games with Quantized CSI in Two-user Interference Channels

Transcription

1 Nash Bargaining in Beamforming Games with Quantized CSI in Two-user Interference Channels Jung Hoon Lee and Huaiyu Dai Department of Electrical and Computer Engineering, North Carolina State University, Raleigh, NC, USA Abstract In this paper, we consider a beamforming game of the transmitters in a two-user multiple-input single-output interference channel using limited feedback and investigate how each transmitter should find a strategy from the quantized channel state information CSI. In the beamforming game, each transmitter a player tries to maximize the achievable rate a payoff function via a proper beamforming strategy. In our case, each transmitter s beamforming strategy is represented by a linear combining factor between the maximum-ratio transmission MRT and the zero-forcing ZF beamforming vectors, which is shown to be a Pareto optimal achieving strategy. With the perfect CSI, each transmitter can know the exact achievable rate region, and hence can find the beamforming strategy corresponding to any point in the achievable rate region. With limited feedback, however, the transmitters can only conjecture the achievable rate region from the quantized CSI, so their optimal strategies may not be optimal anymore. Considering the quantized CSI at the transmitter, we first find the Nash equilibrium in a noncooperative game. Then, in a cooperative Nash bargaining game, we find a Nash bargaining solution and test its validity. Finally, we propose three bargaining solutions that improve the validity of the cooperation or the average Nash product. Our proposed bargaining solutions utilize the codebook structure; instead of each quantized channel itself, its Voronoi region is considered. Index Terms Beamforming, Nash bargaining, limited feedback, multiple-input single-output, interference channels I. INTRODUCTION Game theory is a mathematical branch to analyze strategic decision makings of multiple players when their interests conflict [1. Game theory is initially used in economics, but now widely adopted in many research fields including political science, biology, and computer science. There have been many efforts to analyze the wireless communication systems with a game theoretic approach when the multiple transmitters operate in the same spectrum band. In this case, three components of a game players, their payoffs or utilities, and a set of strategies generally can be modeled with transmitters, their achievable rates, and a set of transmission strategies, while there can be various strategy sets for different scenarios. Many problems have been modeled with the game theory for wireless communication systems including a beamforming design [2 [4, a resource allocation [5, [6, and security [7. The performance of a wireless system can be improved by adapting to the channel conditions, which requires the channel state information at the transmitter CSIT. Contrary to time division duplexing systems, where a transmitter can directly estimate the channel to the receiver thanks to the reciprocity between the uplink and the downlink channels, many practical systems are frequency division duplexing systems whose uplink and downlink channels are independent of each other. In such systems, the transmitters are generally aided by their receivers; the quantized CSI at each receiver is conveyed to the corresponding transmitter via the feedback link. There have been many studies on increasing the spectral efficiencies using the quantized CSIT for various system models including pointto-point multiple-input multiple-output MIMO channels [8 and multi-user MIMO channels [9, [10. When only the quantized CSI is available at transmitters, their games will be based on the imprecise information. With quantized CSI, each transmitter or each player cannot know the exact achievable rate or the payoff function. Moreover, the strategies derived from the quantized CSI may not be valid. Although the quantized CSIT is quite general in wireless communication systems, the transmitters games based on the quantized CSI are not fully investigated. In this context, it is important to see how the quantized CSI affects a game among the transmitters and how the transmitters can modify their strategies from the quantized CSI. In this paper, we study a beamforming game of the transmitters in two-user multiple-input single-output MISO interference channels ICs when the transmitters utilize the quantized CSIT. Each receiver sends B-bit feedback for each channel direction to its belonging transmitter via a limited feedback link, and the transmitters share the quantized information with each other. For beamforming, each transmitter linearly combines the maximum ratio transmission MRT and the zero-forcing ZF beamforming vectors as it was shown to be Pareto optimal in a perfect CSIT case [11. Thus, the linear combining factor for beamforming, which is a scalar value, becomes each transmitter s strategy. In the same system model, the beamforming game with the perfect CSIT was investigated in [2. Also, the beamforming game with the partial but perfect CSI was investigated in [3, [4; at each transmitter, [3 assumed perfect knowledge of channel covariance matrices, and [4 assumed perfect knowledge of the channels of partial links. Contrary to these papers, we consider the beamforming game based on the quantized CSI and focus on the improvement of a strategy using the codebook structure.

2 TX 1 Quantized CSI Sharing TX 2 Quantized CSI Quantized CSI h 11 h 22 Fig. 1. System model. h 21 h 12 RX 1 RX 2 Contributions: With the quantized CSIT, we firstly find a Nash equilibrium in a non-cooperative game and a Nash bargaining solution in a non-cooperative game. Next, we show that the Nash bargaining solutions based on the quantized CSI may not be valid 1 because of the quantization error. In this case, it is obvious that the validity depends on the precision of the quantized CSI, i.e., the feedback size. Then, we propose three modified bargaining solutions that improve the valid probability or Nash product. Our proposed solutions utilize the structure of the quantization codebook; the Voronoi regions of a codebook are used to derive the solutions instead of their representatives i.e., codewords. A. System Model II. PROBLEM FORMULATION Our system model is depicted in Fig. 1. We consider a two-user MISO IC, where each of two transmitters having M antennas supports its own single-antenna receiver. For integers i, j {1, 2} such that i j, the received signal at the receiver i denoted by y i C 1 1 is given by y i = h ii x i + h ji x j + n i, 1 where h ij C M 1 is the vector channel from the ith transmitter to the jth receiver, x i C M 1 is the transmitted signal at the ith transmitter, and n i C 1 1 is complex Gaussian noise at the ith receiver with zero mean and unit variance, i.e., n i CN0, 1. Thus, in 1, the terms h ii x i and h ji x j are the desired and the interference signals at the receiver i, respectively. We consider a linear beamforming at each transmitter; the transmitted signal at the ith transmitter, i.e., x i, is constructed by x i = v i s i, where v i C M 1 is the transmitter i s beamforming vector such that v i 2 =1.Also,s i C 1 1 is the transmitted symbol for its own receiver such that E[ s i 2 =P i, where P i is the transmitter i s transmit power such that P i P max with P max the maximum transmit power. 1 In this paper, the bargaining solution is said to be valid if it increases all transmitters achievable rates comparing to the non-cooperative case. B. CSI at the Transmitter Since perfect CSIT is not available in many practical systems, we consider a feedback link with limited capacity between each transmitter and its own receiver. We assume that each receiver can perfectly estimate the desired channel as well as the interfering channel. To facilitate our discussion and without loss of generality, we assume that only the feedback budget for each channel directional information CDI is limited to B bits, while each channel gain can be perfectly delivered to the transmitter. This assumption is based on the fact that the acquisition of a channel gain which is a single scalar value is much easier than that of a channel direction which is a complex vector in limited feedback systems [12. Extension to include the effect of the quantized channel gain feedback is straightforward and left to future work. For all i, j {1, 2}, the receiver j quantizes the channel direction of h ij into ĥij as follows: ĥ ij q ij h ij = arg max w hij 2, 2 w W ij where denotes the normalization of a vector, i.e., hij = h ij / h ij. Also, q ij : C M 1 W ij is a B-bit vector quantizer using a codebook W ij C M 1 that consists of 2 B unit-norm vectors. In this case, the channel direction h ij can be expressed by h ij = 1 Z ij ĥ ij + Z ij e ij, where Z ij [0, 1 is the CDI quantization error defined as Z ij sin 2 [ h ij, ĥij, and e ij C M 1 is a unit-norm vector such that e ij ĥij. Thus, after feedback, the transmitter j has the information of h 1j ĥ1j and h 2j ĥ2j, and then with information sharing, each transmitter can know the quantized CSI of all links, which are h ij ĥij for all i, j {1, 2}. C. Beamforming Design at Each Transmitter In a two-user MISO IC, it was already shown in [11 that any point on the Pareto boundary of the achievable rate region can be obtained by a beamforming vector that is a linear combination of MRT and ZF beamforming vectors. Thus, we focus our attention to this class of beamforming schemes for both transmitters. With perfect CSIT, the first transmitter s beamforming vector takes the following form: v1η 1 = 1 η 1v1 MRT + η 1 v1 ZF 3 1 η 1 v1 MRT + η 1 v1 ZF, where η 1 [0, 1 is a linear combining factor for the first transmitter, and v1 MRT C M 1 is the ideal MRT beamforming vector given by v1 MRT h 11.Also,v1 ZF C M 1 is the ZF beamforming vector given by v1 ZF I M

3 h 12 h 12 h / IM 11 h 12 h 12 h 11. Similarly, we can represent the second transmitter s beamforming vector as v2η 2 = 1 η 2v2 MRT + η 2 v2 ZF 4 1 η 2 v2 MRT + η 2 v2 ZF, where η 2 [0, 1 is a linear combining factor for the second transmitter, and the vectors v2 MRT C M 1 and v2 ZF C M 1 are ideal MRT and ZF beamforming vectors for the second transmitter, respectively. With quantized CSI, however, the transmitters cannot find the accurate MRT and ZF beamforming vectors i.e., 3 and 4 because of the CDI quantization errors. For i, j {1, 2} such that i j, the transmitter i s MRT and ZF beamforming vectors obtained from the quantized CSI are given by v q-mrt i ĥ ii and v q-zf i / I M ĥijĥ ijĥii IM ĥijĥ ijĥii,respectively. Therefore, the transmitter i s beamforming vector based on the quantized CSI becomes 1 η iv q-mrt i 1 η i v q-mrt i + η i v q-zf i + η i v q-zf v q i η i= i. 5 D. Achievable Rate and Beamforming Game Obviously, the achievable rate at each receiver depends on the beamforming vectors of both transmitters. We can interpret the linear combining factor η 1 [0, 1 as the first transmitter s beamforming strategy and η 2 [0, 1 as the second transmitter s beamforming strategy. With perfect CSIT, for i, j {1, 2} such that i j, the transmitter i s achievable rate corresponding to the beamforming strategy profile η 1,η 2 becomes R i η 1,η 2,P 1,P 2 log 2 1+ P i h ii v i η i 2 1+P j h ji v j η j 2. 6 With imperfect CSIT, on the other hand, the corresponding rate becomes R i η 1,η 2,P 1,P 2 log 2 1+ P i h ii vq i η i 2 1+P j h ji vq j η. 7 j 2 In this paper, we consider the beamforming game between the two transmitters when they have only quantized CSI, and investigate how they can modify their strategy via cooperation. III. THREE ACHIEVABLE RATE REGIONS In this section, we define the three achievable rate regions in the limited feedback scenarios, which are 1 achievable rate region with perfect CSIT, 2 achievable rate region with imperfect CSIT, and 3 conjectured achievable rate region with imperfect CSIT. 2 2 As in [11, time sharing transmission and joint decoding at the receivers are not considered in this work for simplicity, so achievable rate regions presented below may not be convex. A. Achievable Rate regions with perfect and imperfect CSIT With perfect CSI, the transmitters exactly know the achievable rate pair for any given beamforming strategy profile and power allocations so that can determine the accurate achievable rate region. For a given set of the channels {h ij i, j {1, 2}}, the achievable rate region obtained from the perfect CSI denoted by R is defined as R { R 1 η 1,η 2,P 1,P 2, R 2η 1,η 2,P 1,P 2 }, 8 P 1,P 2 [0,P where R i η 1,η 2,P 1,P 2 is defined in 6. When the transmitters exploit the quantized CSI for beamforming designs, however, not all points in 8 are achievable because each transmitter s beamforming vector is calculated from the quantized information. The achievable rate region with the quantized CSIT denoted by R is given by R { R1 η 1,η 2,P 1,P 2, R 2 η 1,η 2,P 1,P 2 }, 9 P 1,P 2 [0,P where R i η 1,η 2,P 1,P 2 is defined in 7. B. Conjectured achievable rate region at the transmitters with imperfect CSIT Although the achievable rate regions with perfect CSIT and imperfect CSIT are defined in 8 and 9, respectively, calculations of them still require the perfect knowledge of {h 11, h 12, h 21, h 22 }. Thus, the transmitters with only the quantized CSI actually cannot determine 8 and 9. Instead, the transmitters can conjecture the achievable rate region by regarding the quantized CSIT as the perfect one. For all i, j {1, 2}, each transmitter can substitute the quantized channel h ij ĥij for the perfect one h ij = h ij h ij in 7 to find the achievable rates corresponding to each beamforming strategy profile η 1,η 2. Let ˆR be the conjectured achievable rate region at each transmitter using the quantized CSI in place of the perfect CSI. Then, it is defined as ˆR { ˆR1 η 1,η 2,P 1,P 2, ˆR 2 η 1,η 2,P 1,P 2 }, 10 P 1,P 2 [0,P where for all i, j {1, 2} such that i j, the term ˆR i η 1,η 2,P 1,P 2 is the conjectured achievable rate of the transmitter i given by ˆR i η 1,η 2,P 1,P 2 log 2 1+ P i h ii 2 ĥ ii vq i η i 2 1+P j h ji 2 ĥ ji vq j η. 11 j 2 Note that as the feedback size increases i.e., B, three achievable rate regions converge, i.e., lim R = lim ˆR = B B R.

4 IV. NON-COOPERATIVE COMPETITIVE GAME First, we consider a non-cooperative competitive game between the transmitters, where each transmitter becomes a player, and the transmitter i s strategy and utility function are given by η i and 11, respectively. In a non-cooperative game, the transmitters are rational but selfish, so each transmitter tries to maximize its achievable rate ignoring the other transmitter. Generally, the solution of the non-cooperative game is obtained at a Nash equilibrium whose definition is as follows: Definition 1 Nash equilibrium. A Nash equilibrium is a strategy profile at which no player has an incentive to deviate when the other players strategies are given. In our case, the strategy profile η1 NE such that η1 NE [0, 1 with the power allocation P1 NE,P2 NE such that P1 NE,P2 NE [0,P max is a Nash equilibrium if it satisfies the following two conditions: 1 ˆR1 η1 NE,P1 NE,P2 NE ˆR 1 η 1,P 1,P2 NE for all η 1 [0, 1 and P 1 [0,P max. 2 ˆR2 η1 NE,P1 NE,P2 NE ˆR 2 η1 NE,η 2,P1 NE,P 2 for all η 2 [0, 1 and P 2 [0,P max. The Nash equilibrium is obtained in the next theorem. Theorem 1. For the above game, there exists a unique purestrategy Nash equilibrium, where the corresponding strategy profile is η1 NE =0, 0 with full power transmission, i.e., P 1 = P 2 = P max, invariant with the quantization bits, i.e., B. Proof. Since our system model is symmetric, it is enough to show that η1 NE =0and P 1 = P, i.e., ˆR1 0,η 2,P,P 2 ˆR 1 η 1,η 2,P 1,P 2 for all η 1,η 2 [0, 1 and for all P 1,P 2 [0,P. With the quantized CSI, the transmitters believe their information is real regardless of the feedback sizes, and hence use the conjectured achievable rate 11. We recall 11: ˆR 1 η 1,η 2,P 1,P 2 log 2 1+ P 1 h 11 2 ĥ 11 vq 1 η P 2 h 21 2 ĥ 21 vq 2 η. 2 2 In the numerator in the logarithm function, we have P 1 ĥ 11 vq 1 η 1 2 P, where the equality holds when P 1 = P and v q 1 η 1=ĥ11= v q-mrt 1, which corresponds to η 1 =0. Thus, for an arbitrary given strategy of the second transmitter, i.e., for an arbitrary given η 2 [0, 1 and P 2 [0,P, it is easy to show that ˆR 1 0,η 2,P,P 2 ˆR 1 η 1,η 2,P 1,P 2 η 1 [0,1 P 1 [0,P, which implies that η1 NE =0. Similarly, we can prove that η2 NE = 0. Thus, we obtain the unique Nash equilibrium η1 NE = 0,0. V. COOPERATIVE NASH BARGAINING GAME In this section, we consider a cooperative game with the same setting of the non-cooperative game. It is obvious that each transmitter can achieve a higher rate via cooperation. From the bargaining, the transmitters can reach a Nash bargaining strategy η1 NB,η2 NB that results in rate increase in both. 3 Since the transmitters already know that at least they can achieve the Nash equilibrium via non-cooperation, if the achievable rate at a transmitter with bargaining is smaller than what is obtained at the Nash equilibrium, the cooperation bargaining fails, and the transmitters return to the non-cooperative Nash equilibrium, termed the disagreement point. It was shown in [2 that the Nash bargaining point is on the Pareto boundary of the achievable rate region and obtained with full power transmission at each transmitter, i.e., P 1 = P 2 = P max. For notational simplicity, we will use Rη 1,η 2 in place of Rη 1,η 2,P max,p max in the following discussion. A. Nash Bargaining solution from perfect CSIT With the perfect CSIT, each transmitter knows the achievable rate region R given in 8. Let η1 NB,η2 NB be the strategy profile for the Nash bargaining solution obtained from perfect CSIT. Then, the transmitter can find the optimal strategy profile η1 NB,η2 NB as follows: 2 [ η1 NB,η2 NB arg max R i η 1,η 2 R i η1 NE, η 1,η 2 i=1 where η1 NE =0, 0 as shown in Theorem 1. B. Nash Bargaining solution from quantized CSIT itself and its validity When each transmitter only has the quantized CSI, the Nash bargaining solution should be found in the conjectured achievable rate region ˆR defined in 10. Denoting the Nash bargaining solution found with quantized CSI by ˆη 1 NB, ˆη 2 NB, we obtain 2 [ ˆη 1 NB, ˆη 2 NB arg max ˆRi η 1,η 2 ˆR i η1 NE, 12 η 1,η 2 i=1 where η1 NE =0, 0 as shown in Theorem 1. Although each transmitter can find the Nash bargaining solution from the conjectured rate region ˆR, it may not be valid. Obviously, the solution found in 12 ensures that ˆR i ˆη 1 NB, ˆη 2 NB ˆR i η1 NE i {1, 2}, 13 but the transmitter i s real achievable rate is R i ˆη 1 NB, ˆη 2 NB, not ˆR i ˆη 1 NB, ˆη 2 NB. Therefore, the Nash bargaining solution obtained from the quantized CSI does not guarantee both transmitters rate increments, i.e., the solution 12 does not guarantee that R i ˆη 1 NB, ˆη 2 NB R i η1 NE i {1, 2}. 14 Thus, exploiting the quantized CSI may decrease the incentive for cooperation, and we need modified bargaining solutions for limited environment. In the next subsection, we endeavor to improve the Nash bargaining solutions given in The product of all players excess utilities via cooperation is generally referred to as the Nash product, and the Nash bargaining solution maximizes it.

5 C. Modified Bargaining Solutions from Quantized CSI We can modify the Nash bargaining solution given in 12 by considering the effect of the quantization errors. Although the transmitters do not know the exact channels, they can estimate the channel space containing the exact channels. With a B-bit CDI codebook, we have 2 B Voronoi regions, each of which is represented by a codeword. From the quantization procedure i.e., 2, each transmitter knows the Voronoi regions H ij C M 1 for all i, j {1, 2} such that h ij H ij, which is given by H ij = { h C M 1 h = hij, q ij h ij =ĥij}. We propose three modified bargaining solutions based on the Voronoi regions of the quantized channels. 1 Bargaining solution based on averaged achievable rate region Solution 1: The first modified bargaining solution considers the average quantization effect, so each transmitter utilizes the achievable rate region averaged over the Voronoi regions that contain the quantized channels given by R { R1 η 1,η 2, R 2 η 1,η 2 η 1,η 2 [0, 1 }, 15 where [ R 1 η 1,η 2 E h R1 η 1,η 2 h11 H 11, h 21 H [ R 2 η 1,η 2 E h R2 η 1,η 2 h 22 H 22, h 12 H Thus, each transmitter finds the strategy profile denoted by η NB1 1, η NB1 2 from the averaged rate region as follows: η NB1 1, η NB1 2 arg max η 1,η 2 2 [ Ri η 1,η 2 R i η1 NE. 18 i=1 2 Bargaining solution of maximum average Nash product Solution 2: For an arbitrary strategy profile η 1,η 2, the Nash product is defined by 2 R i η 1,η 2 R i η1 NE. 19 i=1 The second modified bargaining solution denoted by η NB2 1, η NB2 2 maximizes the average Nash product over all quantizations as follows: 2 η NB2 1, η NB2 2 arg max h[ R i η 1,η 2 R i η1 η 1,η 2 E NE i=1 h ij H ij i,j {1, 2} Bargaining solution of maximum valid probability Solution 3: The third modified bargaining solution maximizes the valid probability of the cooperation. An arbitrary bargaining solution η1 NB,η2 NB is valid if the solution increases both transmitters achievable rates, i.e., if R i η1 NB,η2 NB R i η1 NE for Valid probability Nash Bargaining solution with quantized CSIT Modified bargaining solution1 Modified bargaining solution2 Modified bargaining solution Feedback size for each channel direction bit Fig. 2. Valid probabilities of various bargaining solutions according to the feedback sizes when K =4and P =10dB. all i {1, 2}. The third modified bargaining solution denoted by η NB3 2 is obtained as follows: [ η NB3 2 = arg max Pr R 1 η 1,η 2 R 1 η1 NE and η 1,η 2 R 2 η 1,η 2 R 2 η NE 1,η NE 2 h ij H ij i,j {1, 2}. 21 In practice, each modified solution can be numerically found. Once a quantized channel is given, each transmitter can randomly pick multiple channel samples in the corresponding Voronoi region for averaging purpose. In our case, since the channel directions are isotropic uniformly distributed in C M 1, they should also be uniformly chosen in the Voronoi regions. Remark 1 Proportional Fairness. Replacing the achievable rates at the Nash equilibrium with zero in 12, 18, and 20, each bargaining solution maximizes the product of achievable rate, which is an objective of proportional fairness. Thus, our bargaining solutions can also modify the beamforming strategy for proportional fairness. VI. SIMULATION RESULTS In this section, we compare the Nash bargaining solution based on the quantized CSI defined in 12, and the modified bargaining solutions defined in 18, 20, and 21. In all simulations, each transmitter has four antennas i.e., M =4, and the maximum transmit power i.e., P max normalized with respect to the background noise is fixed to 10dB. Also, each CDI codebook is modeled with a random vector codebook [13, where all codewords are independent and isotropic random unit-norm vectors in C M 1. In Fig. 2, we compare the valid probabilities 4 of the Nash bargaining solution based on the quantized CSIT itself, 4 The valid probability of each bargaining solution is defined as Pr [ R i η1 NB,ηNB 2 R iη1 NE,ηNE 2 i {1, 2}, where η1 NB,ηNB 2 {ˆη 1 NB, ˆηNB 2, ηnb1 1, η NB1 2, η NB2 1, η NB2 2, η NB3 2 }

6 Average Nash product Nash Bargaining solution with quantized CSIT Modified bargaining solution1 Modified bargaining solution2 Modified bargaining solution Feedback size for each channel direction bit Fig. 3. Average Nash products according to the feedback sizes when K =4 and P =10dB. The average Nash product with perfect CSIT is i.e., ˆη 1 NB, ˆη 2 NB, and the modified bargaining solutions, i.e., η NB1 1, η NB1 2, η NB2 1, η NB2 2, and η NB3 2, according to the feedback sizes. As described in Section III-B, the conjectured rate region with the quantized CSI approaches the rate region with the perfect CSI as the feedback size i.e., B increases. Thus, the bargaining solutions based on the quantized CSI yield the rate increments at both transmitters with higher probability as the feedback size increases. In Fig. 2, we can observe that our proposed bargaining solutions improve the validity of cooperation for any feedback size. In Fig. 3, we compare the average Nash products 5 of various bargaining solutions with respect to the feedback sizes, while the average Nash product with perfect CSIT is We can observe that the proposed bargaining solutions based on the average rate region i.e., Solution 1 and average Nash product i.e., Solution 2 increase the average Nash product for all feedback sizes. However, the validity maximizing bargaining solution, which yields the highest valid probability in Fig. 2, shows negligible Nash product increment. This phenomenon reveals that maximization of the bargaining validity makes both transmitters conservative to change their strategies from the Nash equilibrium. product and the valid probability can be improved by utilizing the Voronoi regions of the codebook rather than the quantized CSI itself the representatives of the codebook. ACKNOWLEDGMENT This work was supported in part by the US National Science Foundation under Grants ECCS , CNS , ECCS and EARS REFERENCES [1 R. B. Myerson, Game Theory: Analysis of Conflict. Cambridge, MA: Harvard Univ. Press, [2 E. G. Larsson and E. A. Jorswieck, Competition versus cooperation on the MISO interference channel, IEEE J. Sel. Areas Commun., vol. 26, no. 7, pp , Sep [3 J. Lindblom, E. Karipidis, and E. G. Larson, Selfishness and altruism on the MISO interference channel: The case of partial transmitter CSI, IEEE Commun. Letters, vol. 13, no. 9, pp , Sep [4 Z. K. M. Ho and D. Gesbert, Balancing egoism and altruism on interference channel: The MIMO case, in Proc. IEEE Int. Conf. Commun. ICC 2010, May [5 G. P. Zhang, K. Yang, and H. H. Chen, Resource allocation for wireless cooperative networks: A unified cooperative bargaining game theoretic framework, IEEE Wireless Commun., vol. 19, no. 2, pp , Apr [6 Q. Cao, Y. Jing, and H. V. Zhao, Power allocation in multi-user wireless relay networks through bargaining, IEEE Trans. Wireless Commun., vol. 12, no. 6, pp , June [7 A. Mukherjee and A. L. Swindlehurst, Jamming games in the MIMO wiretap channel with an active eavesdropper, IEEE Trans. Signal Process., vol. 61, no. 1, pp , Jan [8 K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, On beamforming with finite rate feedback in multiple-antenna systems, IEEE Trans. Inf. Theory, vol. 49, no. 10, pp , Oct [9 N. Jindal, MIMO broadcast channels with finite-rate feedback, IEEE Trans. Inf. Theory, vol. 52, no. 11, pp , Sep [10 J. H. Lee and W. Choi, Optimal feedback rate sharing strategy in zeroforcing MIMO broadcast channels, IEEE Trans. Wireless Commun., vol. 12, no. 6, pp , June [11 E. A. Jorswieck, E. G. Larsson, and D. Danev, Complete characterization of the Pareto boundary for the MISO interference channel, IEEE Trans. Signal Process., vol. 56, no. 10, pp , Oct [12 T. Yoo, N. Jindal, and A. Goldsmith, Multi-antenna downlink channels with limited feedback and user selection, IEEE J. Select. Areas Commun., vol. 25, no. 7, pp , Sep [13 C. K. Au-Yeung and D. J. Love, On the performance of random vector quantization limited feedback beamforming in a MISO system, IEEE Trans. Wireless Commun., vol. 6, no. 2, pp , Feb VII. CONCLUSION In this paper, we studied the beamforming game in a twouser MISO IC with limited feedback. Considering the quantized CSI at the transmitters, we first found Nash equilibrium in the non-cooperative game. Then, in the cooperative Nash bargaining game, we found a Nash bargaining solution and showed that the strategies based on the quantized CSI may not be valid because of the quantization errors. We proposed three modified bargaining solutions and showed that in the cooperation game with quantized CSI, both of the Nash 5 The average Nash product of each scheme is defined as E h [ 2 i=1 Ri η1 NB,ηNB 2 R iη1 NE,ηNE 2, where η1 NB,ηNB 2 {ˆη 1 NB, ˆηNB 2, ηnb1 1, η NB1 2, η NB2 1, η NB2 2, η NB3 2 }