Table Of Content

1 Balancing Egoism and Altruism on Interference Channel: The MIMO Case Zuleita K. M. Ho and David Gesbert Eurecom 2229 Route des Creˆtes, 06560 Sophia Antipolis, France hokm, gesbert @eurecom.fr { } Abstract—This paper considers the so-called Multiple-Input- interference effects on the one hand, and altruistically mini- Multiple-Output interference channel (MIMO-IC) which has mizing the harm they cause to other non-intended receivers 0 relevance in applications such as multi-cell coordination in on the other hand. An important result in this area was the 1 cellular networks as well as spectrum sharing in cognitive characterization of all so-called Pareto optimal beamforming 0 radio networks among others. We address the design of pre- 2 coding (i.e. beamforming) vectors at each sender with the aim solutions for the two-cell case in the form of positive linear of striking a compromise between beamforming gain at the combinationsof the purely selfish and purely altruistic beam- n intended receiver (Egoism) and the mitigation of interference forming solutions [1], [2]. Unfortunately, how or whether at a created towards other receivers (Altruism). Combining egoistic J all this analysis can be extended to the context of MIMO-IC and altruistic beamforming has been shown previously to be 6 (i.e. where receivers have themselves multiple antennas and instrumental to optimizing the rates in a Multiple-Input-Single- 2 Output(MISO)interferencechannel(i.e.wherereceivershaveno interference canceling capability) remains an open question. interference canceling capability) [1], [2]. Here we explore these Coordination on the MIMO-IC has emerged as a very ] T game-theoretic concepts in the more general context of MIMO popular topic recently, with several important contributions channels and use the framework of Bayesian games [3] which I shedding light on rate-scaling optimal precoding strategies s. adlrlaowwspuasratloleldserwiviteh(ismempoi-r)tdainsttriebxuistteidngprweocorkdinognttehcehnMiqIMueOs.-WICe, based on so-called interference alignment [7], [8] and rate- c including rate-optimizing and interference-alignment precoding maximizing precoding strategies [5], [12], to cite just a few. [ techniques,andshowhowsuchtechniquesmaybere-interpreted In this paper, our contributions are as follows: 1 through a common prism based on balancing egoistic and • We re-visit the problem of precoding on the MIMO-IC v altruistic beamforming. Our analysis and simulations attest the throughtheprismofgame-theoreticegoisticandaltruistic 9 improvements in terms of complexity and performance. beamforming methods. For doing so, we derive analyt- 8 6 ically the equilibria for so-called egoistic and altruistic 4 I. INTRODUCTION bayesian games [3]. 1. The mitigation of interference in multi-point to multi-point • We derive a game-theoretic interpretation of previous 0 radiosystemsisofutmostimportanceincontextssuchascog- workaimedatmaximizingthesum-rateovertheMIMO- 0 nitiveradioandmulti-cellMIMOsystemswith fullfrequency IC, such as [12]. 1 reuse.WemodelanetworkofNc interferingradiolinkswhere • We propose a new simplified precodingtechnique aimed : v eachlinkconsistsofasendertryingtocommunicatemessages atsumratemaximization,basedonbalancingtheegoistic i to a unique receiver in spite of the interference arising from and the altruistic behavior at each transmitter, where the X or created towards other links. Recently, the attention of the balancingweightsarederivedfromstatisticalparameters. r a research community was drawn to the so-called coordinated • Weshowthatouralgorithmexhibitsthesameoptimalrate transmission methodswhere interferenceeffectsare mitigated scaling(whenSNRgrows)asshownbyrecentinteresting or even exploited in exchange for an additional overhead in iterative interference-alignment based methods [7], [8]. exchangingdata symbols and channel state information(CSI) At finite SNR, we show improvements in terms of sum between the transmitters. rate,especiallyinthecaseofasymmetricnetworkswhere In a scenario where the back-haul network cannot support interference-alignment methods are unable to properly a complete sharing of data symbols across all transmitters, weigh the contributions on the different interfering links the channel is a so-called interference-channel whereby the to the sum rate. senders can resort to a milder form of coordination that does not require joint encoding of data packets. Coordination over A. Notations the interference channel may take place over one or several domains characterizing the transmission parameters of each The lower case bold face letter represents a vector whereas sender such as the choice of power levels [4], beamforming the upper case bold face letter represents a matrix. (.)H vectors [2], [5]–[8], assigned subcarriers in OFDMA [9], represents the complex conjugate transpose. I is the identity scheduling [10], [11] etc to cite a few. matrix. V(max)(A) (resp. V(min)(A)) is the eigenvectorcor- Recently an interesting framework for beamforming-based respondingto the largest(resp. smallest) eigenvalueof A. B E coordination was proposed for the MISO case by which the is the expectation operator over the statistics of the random transmitters (e.g. the base stations) seek to strike a compro- variable B. S B define a set of elements in S excluding the mise between selfishly serving their users while ignoring the elements in B.\ 2 II. SYSTEMMODEL Tx1 Tx2 Tx3 Tx4 Tx5 TxNc We study a wireless network of N cells, where a subset of N N transmitters will form a coordination cluster (i.e. willbce≤coordinatedacross)andareespeciallyconsidered.The H11 H12 H13H14 H15 transmitters could be the base stations (BS) in the cellular M1 downlink. Each transmitter is equipped with N antennas and t the receivers (e.g. mobile stations) with N antennas. In each r of the N cells, an orthogonal multiple access scheme is c assumed, hence each transmitter (Tx) communicates with a B Nc unique receiver (Rx) at a time. Transmitters are not allowed H or able to exchange user message information, giving rise to H1Nc H2NcH3NcH4NcH5Nc NcNc an interference channel over which we seek some form of beamforming-based coordination. The channel from Tx i to Rx j Hji Nr×Nt is given by: Rx1 Rx2 Rx3Rx4 Rx5 RxNc ∈C Hji =√αjiH¯ji (1) Fig. 1. Limited channel knowledge model for an example of transmitter, hereTXNc,indicated bydottedlines,andanexampleofreceiver, hereRX Each element in channel matrix H¯ is an independent 1,indicated bysolidlines. ji identicallydistributedcomplexGaussianrandomvariablewith zeromeanandunitvarianceandα denotestheslow-varying ji shadowing and pathloss attenuation. ThesetofCSI locallyavailable(resp.notavailable)atTxi itnheTsehreveceterraiavlnesimmbeiptaobmretfaaonmrtmfcoiornmngtirnvibegucvttoeiocrntoosrfdoRefaxTlixingiisiswvwitihi∈∈coCCoNNrdrt×i×n11a.taiAnodns {kbHnyoBkwli}n(kr(,elr=esps1p.....BNu⊥icn)k\nisBowdi.enfi)SniaemtdiRlbaxyrl:iyB,biyd=eMfin{He(rjteihs}epj.=sM1e,t.⊥..o,)Nfcbcy;h:aBMn⊥ine==ls i i i on the interference channel, we assume linear precoding H ; M⊥ = H M . (beamforming)[1],[2],[7],[11].Withthenoisevarianceσi2 at { Aidj}djit=io1n,..a.,lNrceceiveir fee{dbakclk}:k,Bl=e1c.a..uNsce\locial CSI is insuf- Rx i and transmit power P, the received signal-to-noise ratio ficient to exploit all the degrees of freedom of the MIMO- of Rx i is IC [7], some additional limited feedback will be considered vHH w 2P where indicated, in the form of feedback of the beamforming γ = | i ii i| . (2) i Nc vHH w 2P +σ2 vectors vi used at the receiver. Of course, in the case of j6=i| i ij j| i reciprocalchannels,thefeedbackrequirementcanbereplaced P by a channelestimation step based on uplink pilot sequences. A. Receiver design Additionally, it will be classically assumed that the receivers The receivers are assumed to employ maximum SINR areabletoestimatethecovariancematrixoftheirinterference (Max-SINR) beamforming throughout the paper so as to signal based on transmitted pilot sequences. also maximize their rates [13]. The receive beamformer is classically given by: III. BAYESIAN GAMESON INTERFERENCE CHANNEL C −1H w Bayesiangamesarea classofgamesinwhichplayersmust v = Ri ii i (3) i C −1H w optimize their strategy based on incomplete state information Ri ii i | | [3] and hence are particularly well suited to distributed opti- where C is the covariance matrix of received interference Ri mization problems. Below we provide a few useful definition and noise at Rx i and C = H w wHHHP +σ2I. Ri j6=i ij j j ij i for this framework in the context of the MIMO-IC. Importantly,thenoisewillinPpracticecapturethermalnoise A Bayesian game is defined as the following, effects but also any interference originating from the rest of the network, i.e. coming from transmitters located beyond G=<N,Ω,<Ai,ui,B⊥i >i∈N> (4) the coordination cluster. Thus, depending on path loss and where isthesetofplayersinthegame,herereferstotheset shadowingeffects, the σ2 may be quite differentfrom each N other [14]. { i} ochfatnranneslmstiattteerss{C1,N.r.×.,NNtc}Nc..Ωiisitshethseetaoctfioanllspeotssoifblpelagyloerbail, A here refers to a(cid:8)ll choice o(cid:9)f beamformingvectorswi such that B. Limited Channel knowledge the power constraint is fulfilled w 2 1. u : Ω R i i i | | ≤ ×A → To allow for overhead reduction and a better scalability istheutilityfunctionofplayeri.Inthenextsectionwedefine of multi-cell coordination techniques when the number of egoisticandaltruisticutilities. B⊥ isthemissing channelstate i coordinated links N is large, we seek solutions which can information at player i. c operatebasedonlimited,preferablylocal,CSI.Althoughthere Definition 1: A strategy of player i, here refers to beam- may exist various ranges and definitions of local CSI, we forming design, w : B is a deterministic choice of i i i assume the devices (Tx and Rx alike) are able to gain direct action given information B→oAf player i. i knowledge of those channelcoefficients directly connected to Definition 2: A strategy profile W∗ =(w∗,w∗ ) achieves i −i them, as illustrated in Fig. 1. theBayesianEquilibriumifw∗ isthebestresponseofplayeri, i 3 hereoptimaltransmitbeamformerofplayeri,givenstrategies where A will denote the altruistic equilibrium matrix for ji w∗ for all other players and is characterized by Tx i towards Rx j, defined by A = HHv vHH . The −i ji ji j j ji Note th∀ait, wini∗tu=itivaerlgy,mtahxeEpBl⊥iay(cid:8)eur’is(wstir,awte−∗gyi,Bisi,oBp⊥iti)m(cid:9)ized (b5y) correPsprooonfd:ingTrehceeivaeltrruisisvtici =ut|iCCliRR−−tyii11HHciiiiawwnii|.be rewritten as - averaging over the distribution of all missing channel state j6=i|vjHHjiwi|2 = − j6=iwiHAjiwi. Since the vj are information. kPnown from feedback, thPe optimal wi is the least dominant In the following sections, we derive the equilibria for so- eigenvector of the matrix j6=iAji. called egoistic and altruistic bayesian games respectively. P These equilibria contribute to extreme strategies which do V. SUMRATEMAXIMIZATION WITH RECEIVE not perform optimally in terms of the overall network perfor- BEAMFORMER FEEDBACK mance, yet can be exploited as components of more general From the results above, it can be seen that balancing beamforming-basedcoordination techniques. altruism and egoism for player i can be done by trading-off betweenthedominanteigenvectorsoftheegoisticequilibrium IV. BAYESIAN GAMES WITHRECEIVER BEAMFORMER E andnegativealtruistic equilibrium A (j =i) matrices. i ji FEEDBACK Interestingly, it can be shown that{sum}rate6 maximizing WeassumethateachTxhasthelocalchannelstateinforma- precodingfortheMIMO-ICdoesexactlythat.Thuswehereby tionandtheaddedknowledgeofreceivebeamformersthrough brieflyre-visitrate-maximizationapproachessuchas[12]with a feedback channel. Under this assumption, we analyze the this perspective. Egoistic and Altruistic beamforming solutions. Denote the sum rate by R¯ = Nc R where R = i=1 i i log 1+ |viHHiiwi|2P . P A. Egoistic Bayesian Game 2(cid:18) PNj6=ci|viHHijwj|2P+σi2(cid:19) Lemma 1: The transmit beamforming vector which max- Given receive beamformers as a common knowledge, the imizes the sum rate R¯ is given by the following dominant best response strategy of Tx i which maximizes the utility eigenvector problem, function, its own SINR, Nc u (w ,w ,B ,B⊥)= |viHHiiwi|2P (6) Ei+ λojiptAjiwi =µmaxwi (10) i i −i i i Nj6=ci|viHHijwj|2P +σi2  Xj6=i  is the following: P where real values λopt, µ are defined in the proof. ji max Theorem 1: The best-response strategy of Tx i in the ego- Proof: see appendix VIII-A istic Bayesian game is Note that the balancing between altruism and egoism in sum ratemaximizationisdoneusingasimplelinearcombinationof wEgo =V(max)(E ) (7) i i the altruistic and egoistic equilibriummatrices. The balancing whereEi willdenotethe egoisticequilibriummatrix forTxi, parameters, {λojipt}, coincide with the pricing parameters in- given by E = HHv vHH and the corresponding receiver vokedintheiterativealgorithmproposedin[12].Clearly,these i ii i i ii is given by v = CRi−1HiiwiEgo parameters plays a key role, however their computation is a i |CRi−1HiiwiEgo| function of the global channel state information. Instead we Proof: The knowledge of receive beamformers seek belowa suboptimalegoism-altruismbalancingtechnique decorrelates the maximization problem. The which only requires statistical channel information, while maximization problem can be written as wEgo = i exhibiting the right performance scaling. argmax 1 wHE w . |wi|≤1EBi⊥(cid:26)PNj6=ci|viHHijwj|2P+σi2(cid:27) i i i The egoistic-optimal transmit beamformer is the dominant A. Egoism-altruism balancing algorithm: DBA-RF eigenvector wEgo =V(max)(E ). i i We are proposing the following distributed beamforming algorithm with receiver feedback (DBA-RF), to compute the B. Altruistic Bayesian Game transmit beamformers The altruistic utility at Tx i is defined here in the sense of Nc minimizing the expectation of the sum of interference power wi =VmaxEi+ λjiAji. (11) towards other Rx’s. Xj6=i   u (w ,w ,B ,B⊥)= vHH w 2 (8) DBA-RFiteratesbetweentransmitandreceivebeamformers i i −i i i − | j ji i| Xj6=i in a way similar to recent interference-alignmentbased meth- odssuchase.g.[7],[8].Howeverhere,interferencealignment Theorem 2: Thebest-responsestrategyofTxi inthealtru- is not a design criterion. In [7], an improved interference istic Bayesian game is given by: alignment technique based on alternately maximizing the wAlt =V(min)( A ) (9) SINR at both sides is proposed. In contrast, the Max-SINR i ji Xj6=i criterion is only used at the receiver side. This distinction is 4 important as it dramatically changes performance in certain 22 situations (see Section VI). One important aspect of the algorithm above is whether 20 SR−Max DBA−RF it fully exploits the degree of freedom of the interference 18 Max−SINR channelasshownper[7],i.e.whetheritachievestheso-called Alt−Min 16 interference alignment in high SNR regime. The following z) H theorem answers this question positively. c/14 e s Definition 3: Interference is aligned when the following s/ bit12 equations are satisfied at the same time [7]: e ( mrat10 Definition 4: DevfiiHneHtihjwe jse=t o0f b∀ei,ajm6=foriming vectors s(o1lu2-) Su 8 tions in downlink(respectivelyuplink) interferencealignment 6 to be [7] 4 IADL= (13) 2 (w1,...,wNc):XNc HikwkwkHHHik is low rank, ∀i 10 P (dB1W5), [Nc,Nt,Nr]=[3,22,02], ∆ SNR=0dB, 2S5IR=10 30 k6=i IAUL =  Fig. 2. Sum rate comparison in multi links systems with [Nc,Nt,Nr] = [3,2,2] with increasing SNR. DBA-RF achieves close to optimal perfor- Nc mance. (v1,...,vNc):XHHkivkvkHHki is low rank, ∀i. k6=i   Thus, for all (w ,...,w ) DL, there exist receive the Max-SINR method [7], the alternated-minimization (Alt- beamformers v ,ii=1,...N,Nc ∈sucIhAthat (12) is satisfied. Min) methodfor interferencealignement[8] and the sum rate i c optimization method (SR-Max) [12]. The SR-Max method is Note that the uplink alignment solutions are defined for a by construction optimal but is more complex and requires virtualuplinkhavingthesamefrequencyandonlyappearhere as technical concept helping with the proof. extra sharing or feedback of pricing information among the transmitters. Also, Max-SINR method in [7] does not aim to Theorem 3: Assume the downlink interference alignment null out interference, but maximize receive SINR instead. set is non empty (IA is feasible). Denote average SNR of link i by γ = Pαii. Let λ = 1+γi−1γ , then in the large User located in the cells follow a uniform distribution. To i σi2 ji −1+γj−1 j ensure a fair comparison, all the algorithms in comparisons SNR regime, P , any transmit beamforming vector in areinitializedtothesamesolutionandhavethesamestopping DL is a conve→rge∞nce(stable) point of DBA-RF. condition.We performsumratecomparisonsinbothsymmet- IA Proof: We provide here a sketch of the proof. For full ric channels and asymmetric channels where links undergo details, please refer to [15].To provethatIA is a convergence different levels of out-of-cluster noise. Define the Signal to pointofDBA-RF,wewouldprovethatonceDBA-RFachieves Interference ratio of link i to be SIR = αii . The SIR interference alignment, DBA-RF will not deviate from the i PNj6=ciαij is assumed to be 10dB for all links, unless otherwise stated. solution. DenotethedifferenceinSNRbetweentwolinksinasymmetric Assumed interference alignment is reached and let (wIA,...,wIA) DL and (vIA,...,vIA) UL. channels by ∆SNR. 1 Nc ∈ IA 1 Nc ∈ IA Let QDL = Nc H wIAwIA,HHH and QUL = i k6=i ik k k ik i A. Symmetric Channels Nc HHvIAvIAP,HH . k6=i ki k k ki Fig. 2 illustrates the sum rate comparison of DBA-RF with PAt the transmitters: In high SNR regime, λji becomes Max-SINR, Alt-Min and SR-Max in a system of 3 links and negative infinity and DBA-RF gives w = Vmin(QUL) (11). i i eachTxandRxhave2antennas.Sinceinterferencealignment By (13), QUL is low rank and thus w is in the null space i i is feasible in this case, the sum rate performance of SR-Max of QUL. In directconsequences,the conditionsof IA (12) are i and Max-SINR increase linearly with SNR. DBA-RF achieves satisfied. Thus, (w ,...,w ) DL. 1 Nc ∈IA sumrateperformancewiththesamescalingasMax-SINRand At the receivers: The receive beamformer is defined as SR-Max. v = argmaxviHHiiwiwiHHHiivi. Since QDL is low rank, the opitimal v would mviHaQkeDiLthvei denominator izero and thus, v is InFig.3,weshowthesumrateinasystemof5linkswhere i i each Tx and Rx are equipped wtih 2 antennas. Note that in in the null space of QDL. Hence, v UL. Since bothw and vi stays withini ∈IDAL and UL, IA is this case interference alignment is infeasible. The sum rate i i performancessaturate at high SNR regime. DBA-RF achieves IA IA a convergencepoint of DBA-RF in high SNR. close to optimal performance in spite of the unfeasibility of interference alignment. VI. SIMULATION RESULTS B. Asymmetric Channels In this section, we investigate the sum rate performance of Intheasymmetricsystem,somelinksundergounevenlevels DBA-RF in comparison with several related methods, namely of noise. In Fig. 4, we compare the sum rate performance in 5 on balancing the egoistic and the altruistic behavior at each 35 transmitterwiththeaimofmaximizingthesumrate.Weobtain SR−Max DBA−RF an iterative beamforming algorithm which exhibits the same 30 Max−SINR optimalratescaling(whenSNRgrows)shownbyrecentitera- Alt−Min tiveinterference-alignmentbasedmethods.By simultaneously 25 z) equilibrating egoistic and altruistic solutions for all links, we H c/ are able to obtain close to optimum performancein situations se20 s/ with both symmetric and asymmetric link quality levels. bit ate (15 mr VIII. APPENDIX u S 10 A. Proof of Lemma 1 Define the largrangian of the sum rate maximization prob- 5 lem to be (w ,µ)=R¯ µ (wHw 1). The neccessary condition oLf lairgrangian−∂max(wi,µ)i=−0 gives: ∂ R + 01 0 P (dB1W5), [N,N,N]=[5,22,02], ∆ SNR=0dB, 2S5IR=10 30 Nc ∂ R = µ w∂w.iHWLithielementary matr∂iwxiHcalicu- c t r j6=i ∂wH j max i i Plus, ∂ R = P E w and ∂ R = [F5i,g2.,32.]wSituhminrcarteeascionmgpSaNriRso.nDBinA-mRuFltaiclhiniekvsessycslotesmestowoiptthim[Nalcp,eNrfto,rmNarn]c=e. ∂w|viHjHHijjwj|2PPNj=c1|viHHijwj|2PP+σi2 i i A w∂w.iH j −PNk=c1|vjHHjkwk|2P+σj2PNk6=cj|vjHHjkwk|2P+σj2 j i Thus, λopt is a function of all channel states ji information and beamformer feedback, λopt = ji 24 SR−Max |vjHHjjwj|2P PNj=c1|viHHijwj|2P+σi2. 22 DBA−RF −PNk=c1|vjHHjkwk|2P+σj2PNk6=cj|vjHHjkwk|2P+σj2 Max−SINR Alt−Min 20 REFERENCES z)18 [1] E.A.JorswieckandE.G.Larsson,“Completecharacterizationofpareto H c/ boundary for the miso interference channel,” IEEE Transactions on e s/s16 SignalProcessing,vol.56,no.10,pp.5292–5296, October2008. bit [2] K. M. Ho and D. Gesbert, “Spectrum sharing in multiple antenna e (14 channels: Adistributed cooperative gametheoretic approach,” inProc. at mr IEEE International Symposium on Personal, Indoor, Mobile Radio Su12 Communications (PIMRC),Cannes, 15-18September2008. [3] R.B. Myerson, Gametheory: analysis ofconflict, Harvard University 10 Press,1997. [4] S.G.Kiani, D.Gesbert, A.Gjendemsjø, and G.E.Øien, “Distributed 8 power allocation for interfering wireless links based on channel information partitioning,” IEEETransactions onWireless Communications, 6 2008, toappearinJune2008. 10 12P (dBW14), [N,1N6,N]=[138,2,2], 2∆0 SNR=2220dB, 2S4IR=[102,610,0.12]8 30 [5] S. Ye and R. S. Blum, “Optimized signaling for MIMO interference c t r systems with feedback,” IEEE International Transactions on Signal Processing,vol.51,no.11,Nov,2003. Fig. 4. Sum rate performance for asymmetric channel of 3 links system. [6] AdamJ.TenenbaumandRavirajS.Adve, “Linearprocessingandsum DBA-RFoutperforms mostalgorithms bybalancing EgoismandAltruism. throughput in the multiuser MIMO downlink,” IEEE Transactions on Wireless Communications, vol.8,no.5,pp.2652–2661, May2009. [7] K.S.Gomadam,V.R.Cadambe,andS.A.Jafar, “Approachingtheca- pacityofwirelessnetworksthroughdistributedinterferencealignment,” a 3 links system where SNR of link 1 and link 2 is larger inIEEEGlobecom, 2008. [8] S. W.Peters and R. W. Heath, “Interference alignment via alternation than that of link 3 by ∆SNR = 20dB. The SIR’s of the minimization,” inIEEEICASSP,2009. linksare[10,10,0.1]respectively.Thuslink3notonlysuffers [9] V. L.Prasad and N.X.D. Wang, “Coordinated scheduling and power from strong noise, but also a strong interference channel. allocationindownlinkmulticellOFDMAnetworks,”IEEETransactions onVehicularTechnology, vol.58,no.6,July2009. The asymmetry penalizes the Max-SINR and interference [10] S. G. Kiani and D. Gesbert, “Optimal and distributed scheduling alignmentmethodsbecausetheyare unableto properlyweigh for multicell capacity maximization,” IEEE Transactions on Wireless thecontributionsoftheweakerlinkinthesumrate.TheMax- Communications, vol.7,no.1,January2008. [11] W.ChoiandJ.Andrews, “Thecapacity gainfromintercell scheduling SINR strategy turns out to make link 3 very egoistic in this inmultianetnnasystems,”inIEEETrans.WirelessCommun.,Feb.2008. example,whileitsproperbehaviorshouldbealtruistic.Incon- [12] C.X.Shi,D.A.Schmidt,R.A.Berry,M.L.Honig,andW.Utschick, trast, DBA-RF exploitsuseful statistical information,allowing “Distributed interference pricing for the MIMO interference channel,” inIEEEICC,2009. weakerlinktoallocatetheirspatialdegreesoffreedomwisely [13] A.Paulraj, R.Nabar, andD.Gore, Introduction tospace-time wireless towardstrongerlinksandviceversa,towardabettersumrate. communications, Cambridge University Press,2003. [14] AndreaF.Molisch, Wireless Communications, IEEE,2005. [15] K.M.HoandD.Gesbert, “Balancing egoism andaltruism onMIMO VII. CONCLUSION interference channel,” IEEE Journal on Selected Areas in Communi- cations, 2009, submitted to issue on Cooperative Communications in We derive the equilibria for the egoistic and altruistic MIMOCellular Networks. bayesian games. We suggest a precoding technique based