Exchange Economy in Two-User Multiple-Input Single-Output Interference Channels

1 Exchange Economy in T wo-User Multi ple-Input Single-Output Interfere nce Channels Rami Mochaoura b, Student Member , IEEE, and Eduard Jorswieck, Senior Me mber , IEEE Abstract —W e study the conflict between two links in a multiple-in p ut single-output in terference channel. This settin g is strictly competitiv e and can be related to perfectly competitive market models. In such models, general equili b rium theory is used to determine equili brium measures that are Pareto optimal. First, we consider th e link s to be consumers that can trade goods with in themselves. Th e goods in our setting correspond to beamf orming v ectors. W e utilize the confl ict representation of the consumers in the Edgeworth box, a graphical tool that depi cts the allocation of th e goods for the two consumers, to prov ide closed-fo rm solution to all Pareto opti mal outcomes. Aft erwards, we model the situation between the links as a competitive market which ad ditionally defines p rices fo r the goods. The equilibri u m in this economy is called W alrasian and corr esponds t o the prices that equ ate the demand to the supply of goods. W e calculate the unique W alrasian equilibriu m an d p ropose a coor dination process that is realized by an arbitrator which distributes th e W alrasian prices to the consu mers. The consumers then calculate in a decentralized manner th eir optimal demand corr esponding to beamf orming ve ctors that achiev e the W alrasian equilibrium. This outcome is Par eto optimal and dominates the noncooperativ e outcome of the system s. Thu s, based on the game theoretic model and solution concep t , an algorithm fo r a distributed implementation of the beamf orming problem in multiple-input single-output interference channels is prov ided. I . I N T R O D U C T I O N T wo tra nsmitter-recei ver p airs u tilize the same spectral band simultaneou sly . Each transmitter is eq uipped with N antennas and each r eceiv er with a single anten na. This setting cor re- sponds to the multiple-input single-output (MISO) interference channel (IFC) [2]. The systems’ per forman ce in such a setting is degraded by mutual interferen ce, and their noncoo perative operation is g enerally not efficient [3]. T herefo re, coordination between the links is n eeded in or der to imp rove their join t outcome. Generally , of interest is to d evise coor dination mechanisms in which the o perating po int of the link s is Pareto optimal. A c  2013 IEEE. Personal use of this material is permitte d. Permission from IEEE must be obtained for all other uses, in any current or future media, includi ng reprinting/r epublishi ng this material for advert ising or promotional purposes, creating ne w coll ecti ve works, for resale or redistribu tion to serv ers or lists, or reuse of any copyrighted component of this work in other works. Part of this work has been performed in the frame work of the European research project SAPHYRE, which is partly funded by the European Union under its FP7 ICT Objecti ve 1.1 - The Netwo rk of the Future. This work is also supporte d in part by the Deutsche Forsc hungsgemein schaft (DFG) under grant J o 801/4-1. The authors are with the D epartmen t of Electrical E ngineeri ng and Infor- mation T echnology , Dresden Univ ersity of T echnology , 01062 Dresden, Ger- many . E-mail: { Rami.Mochaourab ,Eduard.Jorswieck } @t u-dresden.de. Phone: +49-351-46 332239. Fax: +49-351-4633 7236. Part of this work has been present ed at IEEE Internati onal Confere nce on Communicat ions, W orkshop on Game Theory and Resource Allocati on for 4G, Kyoto, Japan, June 5–9, 2011 [1]. Pareto o ptimal point is an achiev a ble utility tuple from wh ich it is impossible to incre ase the perform ance of o ne link without degrading the per forman ce of another . Consequen tly , P areto optimality ensures efficient exploitation of the wireless chann el resources. For this purpose, there has been sev eral w ork on characterizin g the set of beamfo rming vectors th at are relevant for Pareto optim al oper ation in the MISO IFC [ 4]–[8]. Next, we will discu ss these appro aches. A. Characterization of P a r eto Optimal P o ints Designing a Pareto optimal m echanism r equires finding the joint beamfo rming vector s u sed at the transmitters that lead to the Pareto op timal point. The set of feasib le beamf orming vectors fo r e ach transmitter is an N -dimension al co mplex ball where N is the n umber of used antenna s. The imp ortance of characterizin g the set of beam formin g vectors necessary for the link s’ Pareto optimal operation is twofo ld. First, th e set of relev an t bea mformin g vectors to consider for finding a Pareto optimal poin t is reduced to a relatively small subset of all feasible beam formin g vectors. Second, the ch aracterized set of efficient beamfor ming vector s is parameterized by a number of scalars which can even reduce the complexity for ind icating the req uired b eamform ing vectors. In [4], the efficient beamf orming vectors are parameterized by K ( K − 1) co mplex-valued para meters, where K is the number of links. For the special two-user ca se, th e efficient beamfor ming vectors a re proven to be a linear comb ination of maximum ratio transm ission an d zero for cing tran smission. Thus, tw o real-valued param eters are required each between zero and o ne to characterize all Pareto op timal op erating points. The extension to a real-valued parametrization fo r the general K - user case is condu cted in [5]–[7] where K ( K − 1) real-valued parameter s are required to achie ve all P a reto optimal points. Recen tly in [ 8], parame trization of th e efficient beamfor ming vector is provided in the mu lti-cell MISO setting with gene ral linear tran smit power con straint at the transmit- ters. For the case of MISO IFC and total power con straint at the transmitter , the number of required parameter s is 2 K − 1 . In this work, we provide a single real-valued parame triza- tion o f the beam formin g vectors that are necessary and suf- ficient to achieve all Pareto optimal po ints. This result is gained whe n we mod el the tw o-user MISO IFC as a pure exc hange e conomy [9]. The links are consumers and th ey possess goods which corr espond to beamfo rming vectors. In a pure exchange econ omy , the consum ers can trade their good s within themselves to improve th eir u tility . The utility function of the consumers in o ur c ase is the signal to interf erence plus noise r atio (SI NR) which is f ormulated in terms of the goo ds. 2 Utilizing the E dgeworth box [9], which is a graphica l tool that depicts the pr eferences of the consumer s over the distribution of th e go ods, we provide a closed-f orm solution to all Pareto optimal points of the SINR region . A subset of all Pareto optimal points satisfy that both links jointly achieve higher utility than at the no ncoop erative point. These poin ts are called exc hange equ ilibria and are related to th e solution con cept, the core, from coa litional g ame theor y . B. Coor d ination to Achieve P a r eto Op timal P oints All of th e m entioned efforts to par ameterize th e efficient beamfor ming vectors in the MI SO are valuable for design- ing efficient low complexity distributed resource allo cation schemes su ch as in [ 10]–[12]. I n [10] and [ 11], the real-valued parametriza tion for the two-user case fr om [4] is utilized and bargainin g algorithm s are prop osed to impr ove the join t perfor mance o f th e systems from the non cooper ativ e state. An extension to these works is made in [12] where a strategic bargaining process is prop osed and proven to terminate at a Pareto optimal outcome . Also based on a stra tegic bargaining approa ch, a co ordina tion mechanism is prop osed in [13] for the two-user MISO IFC where the Han-Kobayashi scheme is applied. In the K -user MISO IFC, a low comp lexity o ne- shot coor dination mechanism is given in [14], where each transmitter independen tly m aximizes its virtual SINR. For the two-user c ase, th e propo sed mechanism is proven to achieve a Pareto optimal solution. In this paper, we propose a co ordinatio n mech anism be- tween two MISO interfering links w hich is Pareto optimal and achieves for each link a u tility higher than at the nonco- operation po int. Our an alysis is based on relatin g the MISO IFC setting to a co mpetitive market [9]. T o the best of our knowledge, this is the first time the beamf orming prob lem in the MISO IFC is r elated to and analyzed using com petitive market models. In a competitive ma rket, as propo sed by L. W alras [15], [16], there exists a populatio n in which each individual po ssesses a n amoun t of divisible goo ds. Th e worth of th ese g oods makes up the budget of each individual. Each individual h as a u tility functio n which reveals his dem and on co nsuming go ods. Moreover, each individual would use the r ev enue from selling all his goods to buy amou nts of goods such that his u tility is m aximized. This e conomic mo del is comp etitiv e because each consu mer seeks to maxim ize his pro fit indep endent of wh at the other consumer s demand. W alras investigated if ther e exists prices for the goods suc h that th e ma rket has neither shor tage no r sur plus. The existence of suc h prices, called W alrasian prices, was later explo red by Arrow [1 7]. The prices in this e conom y are usually assumed to be fixed and not determine d by the consumers. It is assumed that the market or an auc tioneer acts as an ar bitrator to determine th e W alrasian pric es. The competitive market mod el h as fo und a fe w app lications for resour ce allocation in commu nication n etworks. In [18], the W alrasian equilibrium is formulated as a linear comp le- mentarity p roblem for a mu lti-link multi- carrier setting . A decentralized price-adju stment pro cess is p roposed where the users send their po wer allocation s in ea ch iter ation to the spectrum m anager which adjusts the p rices to achieve the equilibriu m. In [16], competitive spectrum market is consid- ered wher e the u sers, sharing a commo n fr equency band, can purcha se their tran smit power sub ject to budget constraints. An ag ent, refer red to as the mar ket, determine s the un it prices of the power spectra. Existence of the equilibrium is proven and c ondition s for its un iqueness ar e p rovided. In [19], the comp etitiv e equilibrium is used f or simultan eous bitrate allocation for m ultiple vid eo streams and the Edg ew orth b ox [15] is u sed to illustrate the con flict betwe en the streams. In the co ntext of co gnitive radio, spectrum trad ing is successfully modeled by eco nomic models and ma rket-equilibriu m, and competitive and coop erative pricing schemes are developed in [20]. Moreover , in [21], hierarchical sp ectrum sharing is modeled as a n interrelated market. The pricing mechanism for the bandwidth allocations between the systems equates the supply to the deman d. In o ur case, the links ar e the consum ers and th e parameters of the beam formin g vector s are the good s the co nsumers possess. W e fo rmulate the con sumers’ dema nd f unction s an d calculate the W alra sian prices which equate the dem and to the supply o f each good. T o achieve the Pareto optimal W alrasian equilibriu m, th e arb itrator coo rdinates the tra nsmission of the links. W e co nsider two cases for th e coo rdination mechan ism. Assuming the arbitrator h as full knowledge of the setting, he can calculate the W alrasian pr ices and f orward these to the links. The links in depend ently c alculate their beam formin g vectors acco rding to their demand function. Assumin g th e arbitrator has limited kn owledge of the setting, we prop ose a price ad justment pro cess, also refer red to as t ˆ atonneme nt, to reach the W alrasian prices. In e ach iteration, th e links send th eir de mands to the arb itrator which updates th e prices accordin g to th e excess demand of each goo d. Outline: The o utline of the paper is as follows. The system and chann el model, as well as the definition of the SINR region an d the beamfor ming vectors that are r elev an t f or P a reto optimal op eration are g iv en in Section II. In Sectio n III, we examine a pure exch ange economy between the lin ks. W e model the parametrization of efficient beam formin g vectors as goo ds and the lin ks as con sumers which trade th ese go ods within themselves. W e characterize all Pareto optim al points in closed form an d define the eq uilibria which cor respond to the co re of a coalition b etween the links. In Section IV, we consider a comp etitiv e market mode l and assume th at the goods ar e b ought by the co nsumers at prices d etermined b y a n arbitrator . Th e equilibr ium of this mar ket model is d etermined, and we provide two co ordin ation mech anisms to achiev e it. In Section V, we illustrate the re sults of this paper b efore we conclud e in Section VI. Notations: Colu mn vectors an d matrices are given in low- ercase and upp ercase b oldface letter s, respectively . k a k is the Euclidean norm o f a ∈ C N . | b | is the absolute v alue o f b ∈ C . sign ( a ) denotes th e sign of a ∈ R . ( · ) T and ( · ) H denote transp ose and Hermitian transpose, respectively . The orthog onal projector onto the column space of Z is Π Z := Z ( Z H Z ) − 1 Z H . The orth ogon al projector onto th e o rthog onal compleme nt of the co lumn space of Z is Π ⊥ Z := I − Π Z , where I is an identity matrix. C N (0 , A ) denote s a circularly- 3 symmetric Ga ussian c omplex rando m vector with covariance matrix A . Thr ough out the p aper, the subscrip ts k , ℓ are from the set { 1 , 2 } . I I . P R E L I M I NA R I E S A. System and Chan nel Mod el The qu asi-static block flat-fading c hannel vector f rom tran s- mitter k to rec eiv er ℓ is d enoted b y h kℓ ∈ C N . W e assume that transmission co nsists of scalar co ding f ollowed by beam- forming . The beamform ing vector used by transmitter k is w k ∈ C N . The matched-filtered, sym bol-sampled com plex baseband d ata re ceiv ed at receiver k is 1 y k = h H kk w k s k + h H ℓk w ℓ s ℓ + n k , k 6 = ℓ, (1) where s k ∼ C N (0 , 1) is the symbol transmitted by transmitter k , and n k ∼ C N (0 , σ 2 ) is additive Gaussian noise. Each transmitter has a total po wer constraint o f P := 1 such that k w k k 2 ≤ 1 . W e d efine the signal to noise ratio (SNR) as 1 /σ 2 . The tr ansmitters ar e assumed to have perfect local ch annel state i nform ation (CSI), i.e., each tra nsmitter has perfect knowledge o f the chan nel vectors only between itself and the two rec eiv ers. Fur ther info rmation at the transmitters requir ed for the co ordinatio n mechanism is discu ssed later in Section IV -C. W e assume there exists an arbitrator who coord inates th e operation of the transmitters. Th e arb itrator could be any central controller which is con nected to both links. Ge nerally , the practical iden tification of the arb itrator depen ds on the sce- nario. For example, in hierarch ical networks in which se veral tiers of networks op erate in the same area it is p ossible that higher network tiers b enefit fro m coordin ating the operation of the networks in lower tiers such as in the mod el used in [2 1]. Moreover , the arbitrator can be the base station of a m acrocell which can coord inate the transmission of smaller micro cells in its coverage range [22]. Th e macrocell b ase station is usually connected to the micro cell base stations via a high capacity link which enables the exchange of channel information re- quired f or the coordination process. The applicability of o ur system mode l in a cognitive radio n etwork is not suitable if the transmitters are restricted to take into accou nt the interference lev els they are allowed to induce at primary receivers. Our setting is suitable f or cognitive ne twork settings, in which the users dy namically adapt th eir tran smissions accordin g to the en vironm ent these users exist in . A cogn itiv e transmitter can choose with wh om it can c oopera te and exchange inf ormation to improve its utility . B. SINR Region and Efficient T ransmission The signal to interferenc e p lus noise ratio (SINR) at receiver k is φ k ( w 1 , w 2 ) = | h H kk w k | 2 | h H ℓk w ℓ | 2 + σ 2 , k 6 = ℓ. (2) 1 Throughout the paper , the subscripts k , ℓ are from the set { 1 , 2 } . This results in the a chiev a ble rate 2 log 2 (1 + φ k ( w 1 , w 2 )) fo r link k when sing le user dec oding is perf ormed at the rece iv ers. The SINR r e g io n is the set of all achiev able SINR tuples defined as Φ :=  ( φ 1 ( w 1 , w 2 ) , φ 2 ( w 1 , w 2 )) : k w k k 2 ≤ 1  . (3) In th e SINR region, tuples ca n be ranked acco rding to their Pareto ef ficien cy . An SINR tup le ( φ ′ 1 , φ ′ 2 ) ∈ Φ is P ar eto superior to ( φ 1 , φ 2 ) ∈ Φ if ( φ ′ 1 , φ ′ 2 ) ≥ ( φ 1 , φ 2 ) , wh ere the inequality is compo nentwise and strict for at least one compon ent. The transition from ( φ 1 , φ 2 ) to ( φ ′ 1 , φ ′ 2 ) is ca lled a P ar eto impr ovement . Situ ations wh ere Pareto improvements are not po ssible ar e c alled P a r eto o ptimal . Th ese p oints constitute the P ar eto bound ary of the SINR region. Forma lly , the set of Pareto o ptimal points of Φ are defined as [23, p. 18] P (Φ) := { x ∈ Φ : there is n o y ∈ Φ with y ≥ x , y 6 = x } , (4) where the ineq uality in (4) is compon entwise. For the two-user MISO IFC, th e set o f beam forming vectors that are relev ant fo r Pareto op timal opera tion are par ameter- ized by a single real-valued parameter λ k ∈ [0 , 1] f or each transmitter k 6 = ℓ as [ 4, Cor ollary 1] w k ( λ k ) = p λ k Π h kℓ h kk k Π h kℓ h kk k + p 1 − λ k Π ⊥ h kℓ h kk k Π ⊥ h kℓ h kk k . (5) This par ametrization is valuable for designing efficient low complexity distributed resource allocation sch emes [12]. The set o f b eamfor ming vector in ( 5) includes maximu m ratio transmission (MR T) ( λ MRT k = k Π h kℓ h kk k 2 / k h kk k 2 ) and zero forcing tra nsmission (ZF) ( λ ZF k = 0 ). Accord ing to [4, Coro l- lary 2 ], it suffices that the parameter s λ k only be fro m the set [0 , λ MRT k ] fo r Pareto o ptimal op eration. Note that a transmitter k has to kn ow the chann el vectors h kk and h kℓ , k 6 = ℓ, perf ectly in o rder to calculate the beamfo rming vectors in (5). Sinc e we are inter ested in tran smissions that lead to Pareto optim al outcomes, we will confin e the strategy set o f each transmitter to the set in (5) an d formu late the SINR expr ession in (2) in terms of the parameters λ k . F or this purpose, we first formulate the power g ains at the receivers. Lemma 1: Th e power gain s at the receivers in terms o f th e parametriza tion in (5) are | h H kk w k ( λ k ) | 2 = ( p λ k g k + p (1 − λ k ) ˇ g k ) 2 , (6) | h H kℓ w k ( λ k ) | 2 = λ k g kℓ , k 6 = ℓ , (7) where λ k ∈ [0 , λ MRT k ] and g k := k Π h kℓ h kk k 2 , ˇ g k := k Π ⊥ h kℓ h kk k 2 , g kℓ := k h kℓ k 2 , where k 6 = ℓ . Pr oof: The proof is provid ed in Appendix A. The SINR o f lin k k can be rewritten using Lemm a 1 in terms of th e par ameters in (5) as φ k ( λ 1 , λ 2 ) =  √ λ k g k + p (1 − λ k ) ˇ g k  2 σ 2 + λ ℓ g ℓk , ℓ 6 = k . (8) 2 W e represent the preference of a link over the used beamforming vecto rs with the SINR utility function in (2 ). The results in this paper also hold for any SINR based utilit y function which is strictly increasin g with SINR such as the achie vable rate function. 4 Notice in (8) th at the interfer ence term λ ℓ g ℓk scales linearly with λ ℓ . With th is respect, the par ameter λ ℓ can be interpr eted as a scaling of in terference at th e cou nter receiver . A redu ction in λ ℓ increases the SINR of link k f or fixed λ k . Assuming that the links are not cooperative, their ope ration point can be predicted usin g non coope rativ e game theor y . The o utcome is a solution of a strategic gam e [ 24, Section 2 .1] betwe en the links. C. Game in S trate gic F orm In [3], the outcome of a strategic game between the links is studied. Th e g ame in strategic fo rm con sists of the set o f p lay- ers, { 1 , 2 } , correspo nding to the two links. The p ure strategies of play er k are the real- valued parameters λ k ∈ [0 , λ MRT k ] in (5). The utility f unction of player k is lo g 2 (1 + φ k ( λ 1 , λ 2 )) , where φ k ( λ 1 , λ 2 ) is given in ( 8). The ou tcome of this strategic gam e is the same also wh en the utility function is cho sen to be φ k ( λ 1 , λ 2 ) . This is d ue to the fact that the preference relation of the players wh ich is repre sented thro ugh the u tility fu nction is inv ar iant to positi ve monotonic transforms [9, Theor em 1.2]. In the above d escribed game, a player always ch ooses the MR T strategy indepen dent of the choice of th e other p layer [3], i.e., MR T is a d ominan t strategy for each player . Hence , the unique N ash equilibrium is ( λ MRT 1 , λ MRT 2 ) . T he extension of the two-p layer strategic gam e described ab ove to the K -player case is straightfo rward. The Nash equ ilibrium correspo nds to the strategy profile in which each p layer cho oses MR T . The outcome in Nash equilibrium is g enerally n ot Pareto op timal. In or der to achieve Pareto imp rovements fr om the Nash equilibriu m, coordina tion between the play ers is requir ed. I I I . E Q U I L I B R I A I N E X C H A N G E E C O N O M Y A. Exchange Economy Mod el In this section , we will use a p ure exch ange econ omy model [9, Chapter 5.1] to deter mine eq uilibria which lie on the Pareto boundar y of the SINR r egion in (3). This model assumes that there exists a set of consu mers wh ich voluntarily exchange goods they possess to increase their payoff. The set of consu mers { 1 , 2 } cor respond s to the two links in our setting. The goods c orrespon d to the parametr ization of the beamfor ming vectors in (5). That is, ther e are two goo ds and λ 1 will stand f or good 1 an d λ 2 for good 2 . The c onsumers are initially endowed with amounts o f these go ods. W e will assume that th e links start th e trade in Nash equ ilibrium. Thus, consumer k is initially end owed with λ MRT k from his g ood and nothing fro m the go od of the o ther con sumer . Specifically , we define ( λ MRT 1 , 0) and (0 , λ MRT 2 ) as th e endowmen ts of co nsumers 1 and 2 , re spectiv ely . Since du ring exchange each co nsumer will possess differ- ent amou nts from both av ailable go ods, we introd uce new variables that indicate these. When con sumer k trades an amount of h is g ood k to consumer ℓ 6 = k , this am ount will be represented by x ( ℓ ) k ≤ λ MRT k . The amoun t left for co nsumer k from h is good is x ( k ) k = λ MRT k − x ( ℓ ) k . In c onnection to the parametriza tion in (5), we define the amoun ts of p ossessed goods as x ( k ) k = λ k , x ( k ) ℓ = λ MRT ℓ − λ ℓ , ℓ 6 = k . (9) O 1 λ MRT 2 λ MRT 1 x (1) 2 x (1) 1 x ′ (1) 2 x ′ (1) 1 I 1 ( x (1) 1 , φ ′ 1 ) (a) Consumer 1. O 2 x ′ (2) 1 λ MRT 2 x (2) 2 x ′ (2) 2 λ MRT 1 x (2) 1 I 2 ( x (2) 2 , φ ′ 2 ) (b) Consumer 2. Fig. 1. Preference representat ion of the consumers. I 1 and I 2 are indif ference curve s of consumer 1 and 2 respect i vel y . If consumer k gives x ( ℓ ) k to the other consumer, this means that tran smitter k uses the beamform ing vector in (5) which correspo nds to λ MRT k − x ( ℓ ) k . Hen ce, if x ( ℓ ) k increases, tran smitter k r educes the interference at recei ver ℓ by using a beamfor m- ing vector nearer to Z F . T he utility fun ction of a con sumer represents his preference over the goods. W e use the SINR in (8) as the utility fun ction of the consum er which we rewrite in term s o f the go ods as φ k ( x ( k ) 1 , x ( k ) 2 ) =  q x ( k ) k g k + q (1 − x ( k ) k ) ˇ g k  2 σ 2 + λ MRT ℓ g ℓk − x ( k ) ℓ g ℓk , (10) where we sub stituted λ k = x ( k ) k and λ ℓ = λ MRT ℓ − x ( k ) ℓ , ℓ 6 = k , from (9). Theor em 1: φ k ( x ( k ) 1 , x ( k ) 2 ) in (10) is continuous, strong ly increasing, a nd strictly quasico ncave on [0 , λ MRT 1 ] × [0 , λ MRT 2 ] . Pr oof: The proof is provid ed in Appendix B. The p referenc e of co nsumers 1 and 2 over the goods is plotted in Fig. 1 (a) and Fig. 1(b ), respectively . For consu mer 1 ( analogo usly consumer 2 ), O 1 is th e o rigin o f th e co ordinate system which has x (1) 1 , the amoun t from good 1 , at the x- axis and x (1) 2 , the a mount from good 2 , at the y- axis. I k is the indiffer ence curve of con sumer k wh ich represents the pairs ( x ( k ) 1 , x ( k ) 2 ) such that the consumer achieves the same payoff as with ( x ′ ( k ) 1 , x ′ ( k ) 2 ) , i.e., φ k ( x ( k ) 1 , x ( k ) 2 ) = φ ′ k := φ k ( x ′ ( k ) 1 , x ′ ( k ) 2 ) . The dark region above I k , correspond s to the pair s ( x ( k ) 1 , x ( k ) 2 ) whe re the co nsumer ach iev es higher payoff th an at the ind ifference curve. The region below I k correspo nds to less pa yoff fo r consumer k . Accor ding to the proper ties of the utility function in Theore m 1, the indif ference curves, w hich co rrespon d to the boun daries of the le vel sets of φ k ( x ( k ) 1 , x ( k ) 2 ) , are conv ex. This result is required later for the proof of Theorem 3 to obtain a unique solution to the co nsumer dem and pro blem in (25). M oreover , T heorem 1 pr oves the existence o f at least o ne W alrasian equilib rium which is con sidered in Section IV. Next, we provid e two alternativ e fo rmulation s fo r the ind if- ference curves. Both formulations are required to determin e special alloca tions in the Edgeworth box. Pr opo sition 1: The indifference c urves I k ( x ( k ) ℓ as a f unc- 5 tion o f x ( k ) k ), are calcu lated for given fixed payoffs φ ′ k as I 1 ( x (1) 1 , φ ′ 1 ) = λ MRT 2 + σ 2 g 21 −  q x (1) 1 g 1 + q (1 − x (1) 1 ) ˇ g 1  2 φ ′ 1 g 21 , (11) I 2 ( x (2) 2 , φ ′ 2 ) = λ MRT 1 + σ 2 g 12 −  q x (2) 2 g 2 + q (1 − x (2) 2 ) ˇ g 2  2 φ ′ 2 g 12 . (12) Pr oof: Th e indifference curve I k for a given utility φ ′ k satisfies φ ′ k =  q x ( k ) k g k + q (1 − x ( k ) k ) ˇ g k  2 σ 2 + λ MRT ℓ g ℓk − x ( k ) ℓ g ℓk , ℓ 6 = k . (13) Exchang ing the LHS and the d enomina tor at the RHS o f (13) we get σ 2 + λ MRT ℓ g ℓk − x ( k ) ℓ g ℓk =  q x ( k ) k g k + q (1 − x ( k ) k ) ˇ g k  2 φ ′ k , (14) Solving fo r x ( k ) ℓ , we g et the expression s in (11) and (12). Note tha t Propo sition 1 cha racterizes a family of ind ifference curves. Each indifference curve has a doma in and ran ge which depend s on the fixed SINR value φ ′ k . Th us, for selected fixed SI NRs, the indifference cur ves sho uld be r estricted to take values in the feasible p arameter set f rom (5), i.e., I 1 ( x (1) 1 , φ ′ 1 ) ∈ [0 , λ MRT 2 ] and I 2 ( x (2) 2 , φ ′ 2 ) ∈ [0 , λ MRT 1 ] . Th e indifference curves can b e alterna tiv ely f ormulated to obtain x ( k ) k as a fu nction o f x ( k ) ℓ . Pr opo sition 2: The indifference curves ˜ I k ( x ( k ) k as a fun c- tion o f x ( k ) ℓ ), ar e calculated for giv en fixed p ayoffs φ ′ k as [ 12, Proposition 1 ] ˜ I 1 ( x (1) 2 , φ ′ 1 ) = f   λ MRT 1 , φ ′ 1 φ 1  λ MRT 1 , λ MRT 2 − x (1) 2    , (15) ˜ I 2 ( x (2) 1 , φ ′ 2 ) = f   λ MRT 2 , φ ′ 2 φ 2  λ MRT 1 − x (2) 1 , λ MRT 2    , (16) where f ( a, b ) := ( √ ab − p (1 − a )(1 − b )) 2 . Similarly , the v alues of the indifference cu rves in Proposition 2 have to be in the f easible parameter set su ch that ˜ I 1 ( x (1) 2 , φ ′ 1 ) ∈ [0 , λ MRT 1 ] a nd ˜ I 2 ( x (2) 1 , φ ′ 2 ) ∈ [0 , λ MRT 2 ] . B. Edgeworth Box The E dgeworth box [ 25], [ 9, Chapter 5], illustra ted in Fig. 2, is a graph ical re presentation that is usefu l for the analysis of an exchange eco nomy . The box is con structed by jo ining Fig. 1(a) and Fig. 1(b). Th us, the Edg ew orth box has two points o f or igin, O 1 and O 2 , c orrespon ding to consumer 1 and 2 , respectively . T he initial e ndowments of the c onsumer s define th e size o f the box . The width o f the box is thu s λ MRT 1 , and the heigh t is λ MRT 2 . The po ssession exc hange lens O 1 O 2 I 1 I 2 x ′ (2) 2 x ′ (1) 1 x ′ (2) 1 x ′ (1) 2 contrac t curve x (2) 1 x (1) 2 x (1) 1 x (2) 2 Fig. 2. An illustrat ion of an Edgew orth box. vectors ( x ′ (1) 1 , x ′ (1) 2 ) and ( x ′ (2) 1 , x ′ (2) 2 ) make up the allocatio n (( x ′ (1) 1 , x ′ (1) 2 ) , ( x ′ (2) 1 , x ′ (2) 2 )) in the b ox. Every po int in the box denote s an allocation, i. e., an assignm ent of a possession vector to eac h con sumer . Th e consumer s’ prefer ences in the Edgeworth box can be revealed accordin g to their indifference curves. Th e dark region in Fig. 2 is called th e e xchange lens and contains all allocatio ns that are Pareto imp rovements to the o utcome in (( x ′ (1) 1 , x ′ (1) 2 ) , ( x ′ (2) 1 , x ′ (2) 2 )) . The locu s of all Pareto optimal points in the Edg ew orth box is called the contract cu rve [25]. On these p oints, the ind ifference c urves are tan gent, and are characterize d by the following co ndition 3 [25, p. 2 1]: ∂ φ 1  x (1) 1 , x (1) 2  ∂ x (1) 1 ∂ φ 2  x (2) 1 , x (2) 2  ∂ x (2) 2 = ∂ φ 2  x (2) 1 , x (2) 2  ∂ x (2) 1 ∂ φ 1  x (1) 1 , x (1) 2  ∂ x (1) 2 . (17) The conve xity of the con sumers’ indifference curves implies that these can only b e tan gent at a single p oint. Thu s, the condition in (17) is necessary and sufficient for an allocation to be on the contract curve. Theor em 2: The co ntract curve cc : [0 , λ MRT 2 ] → [0 , λ MRT 1 ] ( x (1) 1 as a func tion of x (2) 2 ) is the solu tion of the following cubic equation 4 a h x (1) 1 i 3 + b h x (1) 1 i 2 + c h x (1) 1 i + d = 0 , (1 8) where a = − ( g 1 + ˇ g 1 )( C − g 12 ) 2 , d = g 1 σ 4 , (19) b = ( C − g 12 )  2 ˇ g 1 ( C + σ 2 ) + g 1 (2 σ 2 + C − g 12 )  , (20) c = − ˇ g 1 ( C + σ 2 ) 2 + σ 2 g 1 (2 g 12 − 2 C − σ 2 ) , (21) 3 In multipl e consumer settings, the condition provide d by Edge worth [25 ] should hold for e very consumer pair . 4 This result is independe ntly obtai ned in [26]. 6 O 1 O 2 I 1 I 2 contrac t curve x (2) 1 x (1) 2 x (1) 1 x (2) 2 core b endo wm ents Fig. 3. An illustrat ion of the alloc ations in the core. and C is a fun ction of x (2) 2 giv en as C =  q x (2) 2 g 2 + q (1 − x (2) 2 ) ˇ g 2   q g 2 x (2) 2 − q ˇ g 2 1 − x (2) 2   σ 2 g 21 + λ MRT 2 − x (1) 2  . (22) The r oot o f in terest in (18) lies in [0 , λ MRT 1 ] a nd satisfies sign  σ 2 /g 12 + x (1) 1 − C x (1) 1  = sign  σ 2 /g 12 + x (1) 1 + C (1 − x (1) 1 )  . (23) Pr oof: The proof is provided in [ 1, Ap pendix A]. According to Edgeworth [25], the outcome of an exchange between the co nsumers must lie on the co ntract cu rve. The solution co ncept by Edgeworth is related to that of co alitional games called the core [27] which d efines equilibria in our exchange economy . The situation between the two links can be represented as a coalitional gam e without transfera ble payoff [24, Cha pter 13.5]. In our case, the core of this g ame [24, Definition 26 8.3] is the set o f all allo cations in the Edg ew o rth box in which no p layer can achieve h igher payo ffs without cooper ating with the other play er . In Fig. 3, the co re is illustrated as the set o f allo cations on the contract c urve which is boun ded b y the indifference curves cor respond ing to th e initial endowments. Th at is, the co re a llocations correspo nd to all Pareto optimal points which do minate the Nash equ ilibrium in the SI NR re gion. W ith the initial en dowments corr espondin g to the Nash equilibr ium ( λ MRT 1 , λ MRT 2 ) , the indifference curves can be calculated fr om Proposition 2 or Proposition 1. The bound s for the co re, as illustrated in Fig. 3, can b e calculated as the inter section p oints o f the indifference cu rves starting at the e ndowment allocation an d the co ntract cu rve ch aracterized in T heorem 2. Later in Sectio n V, the bou nds fo r th e co re will be used to deter mine the Kalai-Smor odinsky solu tion from axiomatic ba rgaining theory . I V . W A L R A S I A N E Q U I L I B R I U M I N E X C H A N G E E C O N O M Y In the preceding sectio n, we ha ve determin ed the Pareto optimal eq uilibria in o ur p ure excha nge econo my . These equilibria can be achieved requiring th e lin ks to negotiate or bargain ( as for instance is proposed in [12]). Next, we will consider decentralized operation of the lin ks and include the arbitrator to coor dinate tran smission of the links. A. Competitive Market Model In a co mpetitive market, the co nsumers buy quantities of goods and also sell go ods they possess such that they maxi- mize their profit. Eac h good ha s a price and ev ery consume r takes the prices as g i ven. Th e p rices of the good s are not determined by con sumers, but arbitrated by markets. In ou r case, the arb itrator deter mines the price s of the goods. Let p k denote the unit price of goo d k . In order to be able to buy g oods, each consum er k is endowed with a budget λ MRT k p k which is the worth of his initial amounts of goods 5 . The b u dget set o f co nsumer k is the set of bundles of go ods he can af ford to po ssess defined as B k := n ( x ( k ) 1 , x ( k ) 2 ) ∈ R 2 + : x ( k ) 1 p 1 + x ( k ) 2 p 2 ≤ λ MRT k p k o . (24) The budget set of consumer 1 is illustrated by the grey area in Fig. 4. Th e bound ary of the budget set is a line wh ich connects the points ( λ MRT 1 , 0) and (0 , λ MRT 1 p 1 /p 2 ) . Thus, the bo undar y ha s a slope of − p 1 /p 2 . For the consumers, th e p rices of the goo ds are m easures fo r their qualitative v aluation. If p 1 is grea ter than p 2 , th en good 1 has more value than good 2 . Given th e prices p 1 and p 2 , con sumer 1 d emands the amoun ts of g oods x (1) 1 and x (1) 2 such that these m aximize his utility in (10). Thus, consumer k so lves the f ollowing problem : maximize φ k  x ( k ) 1 , x ( k ) 2  subject to p 1 x ( k ) 1 + p 2 x ( k ) 2 ≤ λ MRT k p k . (25) In the above con sumer problem, the objective functio n is the SINR of link k and the constraint is de fined by th e budget set of c onsumer k in (24). The physical interpretatio n of the budget set c onstraint can be related to an inter ference constraint. Con sidering consu mer 1 , the con straint in (25) c an be re formula ted to x (1) 1 ≤ λ MRT 1 − p 2 p 1 x (1) 2 , (26) where, as m entioned bef ore, x (1) 1 = λ 1 ∈ [0 , λ MRT 1 ] is the scaling of in terferenc e transmitter 1 produ ces at receiver 2 . Analogou sly , x (1) 2 = λ MRT 2 − λ 2 is the scaling for interferen ce reduction from transmitter 2 a t receiver 1 . Hence, the con- straint in ( 26) dictates th e tradeoff between the amount o f interferen ce transmitter 1 can gener ate at receiv er 2 and the amount of interfer ence recei ver 1 is to tolerate. The p rices p 1 and p 2 can be inter preted as p arameters to contr ol the fairness between th e links by regulating th e amoun t of inter ference th e links ge nerate on each other . 5 This case corresponds to the Arrow-Debreu market model [16 ]. 7 λ MRT 2 O 1 λ MRT 1 x (1) 2 x (1) 1 B 1 p 1 p 2 λ MRT 1 x ∗ (1) 2 x ∗ (1) 1 increa sing utility indif ference curves (budget set) Fig. 4. An illustrat ion of the budge t set of consumer 1 . Theor em 3: The uniqu e solution to th e problem in (25) is x ∗ (1) 1 ( p 1 , p 2 ) = 1 1 + ˇ g 1 g 1  1 + g 21 p 1 p 2 σ 2 + λ MRT 2 g 21 − λ MRT 1 g 21 p 1 p 2  2 , (27) x ∗ (1) 2 ( p 1 , p 2 ) = p 1 p 2  λ MRT 1 − x ∗ (1) 1  , (28) for consumer 1 , an d x ∗ (2) 2 ( p 1 , p 2 ) = 1 1 + ˇ g 2 g 2  1 + g 12 p 2 p 1 σ 2 + λ MRT 1 g 12 − λ MRT 2 g 12 p 2 p 1  2 , (29) x ∗ (2) 1 ( p 1 , p 2 ) = p 2 p 1  λ MRT 2 − x ∗ (2) 2  , (30) for co nsumer 2 , w here ˇ g k , g k , g kℓ are define d in Lemm a 1. The f easible pr ices ratio are in the range : β := λ MRT 2 g 12 σ 2 + λ MRT 1 g 12 ≤ p 1 p 2 ≤ β := σ 2 + λ MRT 2 g 21 λ MRT 1 g 21 . (31) Pr oof: The proof is provided in Ap pendix C. Theorem 3 characterizes the demand functio ns o f ea ch con- sumer . In econo mic theory , these function s are called M arshal- lian demand fu nctions [ 9] or W alrasian dema nd fu nctions [ 28]. Note that each consumer calculates h is demands independen tly without knowing the other consumer ’ s demand s. From Theo- rem 3, con sumer 1 (a nalogou sly co nsumer 2) needs to know the constan ts g 1 , ˇ g 1 , and g 21 . Th e measure σ 2 + λ MRT 2 g 21 in (10) is the n oise plu s interference p ower in Nash eq uilibrium . This measur e is repo rted from re ceiv er 1 to its transmitter at Nash equ ilibrium which is the initial state of the links b efore coordin ation takes pla ce. The d emand fu nctions of the consumers in Theo rem 3 are homo genou s of d egree zero [9, Definition A2 .2] with the prices p 1 and p 2 . That is, the demand of consume r 1 for good 1 (an alogou sly consumer 2 for good 2 ) satisfies x ∗ (1) 1 ( tp 1 , tp 2 ) = x ∗ (1) 1 ( p 1 , p 2 ) for t > 0 . Hence, given only a prices ratio ¯ p 1 / ¯ p 2 , we can calculate a p rices p air as p 1 = ¯ p 1 / ¯ p 2 and p 2 = 1 which lead s to the same d emand as with ¯ p 1 and ¯ p 2 . W ith this respect, a consume r need only know the p rice ratio p 1 /p 2 from the ar bitrator to calculate his demand s. In Fig. 4, the d emand of consumer 1 is illustrated as the point O 1 O 2 I 1 I 2 contrac t curve slope − p ∗ 1 p ∗ 2 B 1 B 2 (budge t set) (budge t set) x ∗ (1) 1 x ∗ (1) 2 x ∗ (2) 1 x ∗ (2) 2 x (2) 1 x (1) 2 x (1) 1 x (2) 2 Fig. 5. An illustrat ion of an Edgew orth box. I 1 and I 2 are indif ference curves of consumer 1 and 2 respecti vely . The line with slope - p ∗ 1 /p ∗ 2 separat es the budge t s ets of the consumers in W alrasia n equilibri um. where the co rrespond ing indifference curve is tangen t to the bound ary of the budget set. The next r esult provides a significant prope rty tha t the goods in our setting possess. Later in Sectio n IV -B and Section IV -C, this pr operty is required to prove the uniquen ess of the W alrasian equilibrium and also to guaran tee the global conv ergence of the price adjustment pr ocess. Lemma 2: Th e goods in our setting are gr oss substitutes , i.e., increasing the price of o ne good increases the demand of the oth er g ood. Pr oof: De creasing the r atio p 1 /p 2 can be interp reted as decreasing p 1 or in creasing p 2 . Consider the aggr egate excess demand o f go od 1 defin ed as z 1 ( p 1 , p 2 ) = x ∗ (1) 1 ( p 1 , p 2 ) + x ∗ (2) 1 ( p 1 , p 2 ) − λ MRT 1 , (32) where x ∗ (1) 1 ( p 1 , p 2 ) and x ∗ (2) 1 ( p 1 , p 2 ) are the demand fun ctions of go od 1 in (2 7) and (3 0) from Theor em 3. If p 1 /p 2 decreases, then x ∗ (1) 1 ( p 1 , p 2 ) increa ses. If p 1 /p 2 decreases, then x ∗ (2) 1 ( p 1 , p 2 ) also incre ases since p 2 /p 1 increases and x ∗ (2) 2 ( p 1 , p 2 ) decreases. Th us, the ag gregate excess de mand of good 1 in (32) increases if p 1 /p 2 decreases. The analysis is an alogou s for th e secon d go od. If each consum er is to d emand amou nts of g oods without considerin g the demands of the oth er consumer, then it is importan t th at the con sumers’ deman ds equal the co nsumers’ supply of g oods. Prices which fulfill this req uiremen t are called W a lrasian and are calculated next. B. W alrasian Equ ilibrium In a W alrasian equilib rium, the dema nd equa ls the supp ly of each good [9, Definition 5.5 ]. Accord ing to the properties of the utility function in Th eorem 1, ther e exists at least one W alrasian equilibrium [9, Th eorem 5.5] . T he W alrasian price s ( p ∗ 1 , p ∗ 2 ) th at lead to a W alr asian equilibriu m satisfy x ∗ (1) 1 ( p 1 , p 2 ) + x ∗ (2) 1 ( p 1 , p 2 ) = λ MRT 1 , (33) and x ∗ (1) 2 ( p 1 , p 2 ) + x ∗ (2) 2 ( p 1 , p 2 ) = λ MRT 2 . (34) 8 In o ur setting in which only two g oods exist, W alras’ law [9, Chapter 5 .2] provides the property th at if the deman d equals the supp ly o f one good, then the demand w ould equ al the supply of the oth er go od. Hence, in order to calculate th e W alrasian price s, it is sufficient to conside r only one of the condition s in (33) an d (3 4). Theor em 4: The ratio of the W alrasian p rices is the u nique root of a  p 1 p 2  5 + b  p 1 p 2  4 + c  p 1 p 2  3 + d  p 1 p 2  2 + e  p 1 p 2  + f = 0 , (3 5) that satisfies th e condition in (3 1). The co nstant co efficients are a = T 1 T 2 2 T 3 , b = − 2 T 3 T 2 ( T 2 S 2 + T 1 S 1 ) , c = 2 T 4 T 2 S 3 + 4 S 1 S 2 T 2 T 3 + T 1 S 4 T 3 , d = − 2 S 4 S 2 T 3 − 4 T 1 T 2 S 2 S 3 − S 1 T 4 S 3 , e = 2 S 3 S 2 ( T 2 S 2 + T 1 S 1 ) , f = − S 1 S 2 2 S 3 , where T 1 = ( g 1 − ˇ g 1 ) / ( g 1 + ˇ g 1 ) , T 2 = λ MRT 1 + σ 2 /g 12 , T 3 = (1 − λ MRT 1 ) λ MRT 1 , T 4 =  ˇ g 2 1 − ˇ g 1 g 1 + g 2 1  / ( g 1 + ˇ g 1 ) 2 , S 1 = ( g 2 − ˇ g 2 ) / ( g 2 + ˇ g 2 ) , S 2 = λ MRT 2 + σ 2 /g 21 , S 3 = (1 − λ MRT 2 ) λ MRT 2 , S 4 =  ˇ g 2 2 − ˇ g 2 g 2 + g 2 2  / ( g 2 + ˇ g 2 ) 2 , and ˇ g k , g k , g kℓ are d efined in Lemm a 1. Pr oof: Su bstituting (2 7) and (30) in ( 33) and collecting p 1 /p 2 we get the expression in (35). Th e cond ition in (3 1) states th e set of feasible p rices suc h that the demands of the consumer s are feasible. At least one price pair is in this set since a W alrasian equ ilibrium al ways exists in o ur setting. In ad dition, having the property that the g oods are gro ss substitutes in Lem ma 2, im plies that the W alrasian equ ilibrium in ou r setting is u nique [ 28, Proposition 17.F .3]. Note that th e roots in (35) can b e easily ca lculated using a Newton method. And du e to th e uniq ueness o f the W alrasian prices, o nly one root satisfies th e con dition in (3 1). According to the First W elfare Th eorem [9, Theorem 5.7] , the W alrasian equ ilibrium is Pareto o ptimal. Moreover, link ing to the results in the pre vious section, the W alrasian equilibriu m lies in th e core [9, Theorem 5 .6]. In other word s, the W al- rasian equilibr ium dominates the Na sh equ ilibrium outcome. In Fig. 5, the allo cation in W alra sian eq uilibrium which correspo nds to the W alrasian p rices r atio p ∗ 1 /p ∗ 2 is illustrated in th e Edgeworth box. It is the p oint on th e co ntract curve which intersects the line tha t passes through the endowment point (Nash e quilibrium ) with slope − p ∗ 1 /p ∗ 2 (with respec t to the co ordinate system of consumer 1 ). Th e grey area in Fig. 5 is the budget set of consumer 1 as descr ibed in Fig. 4. The white ar ea in the Ed gew orth b ox is the budget set of co nsumer 2 . Acco rding to the axis transform ation in constructing the Edgeworth box , the b ound aries of the consu mers’ budge t sets coincide. The ind ifference curves of the con sumers are tange nt to this line an d also tangen t to o ne another which illustrates the Pareto optimality of th e W alrasian equ ilibrium. T ABLE I R E Q U I R E D I N F O R M AT I O N AT T H E A R B I T R ATO R A N D T R A N S M I T T E R S T O I M P L E M E N T T H E W A L R A S S I A N E Q U I L I B R I U M I N O N E - S H O T . Information Arbitrat or h 11 , h 12 , h 21 , h 22 , σ 2 Tra nsmitter 1 h 11 , h 12 , σ 2 + λ MRT 2 k h 21 k 2 , k h 21 k 2 Tra nsmitter 2 h 22 , h 21 , σ 2 + λ MRT 1 k h 12 k 2 , k h 12 k 2 T ABLE II R E Q U I R E D I N F O R M AT I O N A T T H E A R B I T R ATO R A N D T R A N S M I T T E R S F O R T H E P R I C E A D J U S T M E N T P R O C E S S . Information Arbitrat or k h 21 k 2 , k h 12 k 2 , λ MRT 1 , λ MRT 2 , σ 2 Tra nsmitter 1 h 11 , h 12 , σ 2 + λ MRT 2 k h 21 k 2 , k h 21 k 2 Tra nsmitter 2 h 22 , h 21 , σ 2 + λ MRT 1 k h 12 k 2 , k h 12 k 2 C. Coor din a tion Mechanism In this section, we pr ovide two coo rdination mechanisms which require different amou nt of informa tion at the arb itrator . If the arbitrator h as full knowledge o f all parameters o f the setting, then he can calculate the W alrasian prices fr om Theo - rem 4 and for ward these to th e transmitters. The transmitter s calculate their deman ds f rom Th eorem 3 and choose the beamfor ming vecto rs accor dingly . This mechanism that uses the resu lts in Theor em 3 and Theo rem 4 leads directly to the W alrasian equilibriu m. In T able I, the req uired inf ormation at the arbitrator and the transmitters to implemen t th is one- shot mechanism are listed. W e assume that each transmitter forwards the chann el info rmation it has to th e arbitrator . Note that eac h tra nsmitter k initially kn ows the ch annel vectors h kk and h kℓ , k 6 = ℓ , which are required to calculate the ef ficient beamfor ming vectors in (5). Also, transmitter k kn ows th e sum σ 2 + λ MRT ℓ k h ℓk k 2 , k 6 = ℓ, since this is the n oise plus interferen ce in Nash eq uilibrium for warded throu gh feedba ck from th e intended receiver . The arbitrator, w hich no w has full knowledge o f all chann els, can then forward the missing informa tion on th e chan nel g ain k h ℓk k 2 to a transmitter k . If the arb itrator has limited inform ation about the setting, we cou ld still a chieve the W alrasian price s thro ugh an iter ativ e price adjustment process. For fixed arbitrary initial prices, the transm itters can calculate their d emands and f orward these to the arb itrator . The arbitr ator e xploits the dem and informa tion to update the prices of the good s. Specifically , the ar bitrator would increase the price of the goo d which has higher deman d than its supp ly . Due to the proper ties of the go ods in Lemma 2, this price adju stment pr ocess, also called t ˆ atonnement, is globally convergent to the W alrasian prices giv en in Theo rem 4 [29]. The price adjustment process requires the information listed in T ab le II to b e a vailable at the a rbitrator and the tr ansmitters. In contrast to T able I , th e arbitrator requires aside from the noise power σ 2 only the c ross channel gains k h 21 k 2 , k h 12 k 2 and the p arameters λ MRT 1 , λ MRT 2 from the transm itters. Th is in formation is req uired on ly at the beginning of the price adjustment p rocess in o rder to calcu late the bo unds fo r the fe asible pr ices β and β giv en in (31). In Algorithm 1, th e price adjustment proc ess is described . This process is essentially a bisection method wh ich finds the roots of th e excess d emand fu nction described in the p roof of 9 Algorithm 1: Distributed pr ice adju stment proc ess. Input : x (1) 1 , x (2) 1 , x (1) 2 , x (2) 2 1 initialize: accuracy ǫ , n = 0 , β (0) = β , β (0) = β in (3 1), p (0) 1 p (0) 2 = β (0) 2 + β (0) 2 ; 2 while β ( n ) − β ( n ) > ǫ do 3 receive dem ands x (1) 1 , x (2) 1 , x (1) 2 , x (2) 2 ; 4 n = n + 1 ; 5 if x (1) 1 + x (2) 1 > λ MRT 1 then 6 β ( n ) = p ( n − 1) 1 p ( n − 1) 2 , β ( n ) = β ( n − 1) ; 7 p ( n ) 1 p ( n ) 2 = β ( n ) + β ( n ) 2 ; 8 else 9 β ( n ) = β ( n − 1) , β ( n ) = p ( n − 1) 1 p ( n − 1) 2 ; 10 p ( n ) 1 p ( n ) 2 = β ( n ) + β ( n ) 2 ; Output : p ( n ) 1 /p ( n ) 2 Lemma 2. The accuracy measu re cond itioning th e terminatio n of the algorithm is de fined as ǫ . The terms β and β are the lower and up per b ound s o n the pr ice ratio given in (3 1), respectively . Th e pr ices r atio is initialized to the middle value of these bou nds and forwarded to the links. The link s send their deman ds calculated fr om Theo rem 3 to the arbitrator . If the deman d o f good 1 is greater than its supply , then the arbitrator in creases the ratio of the p rices to half the distance to the up per b ound β . Thus, the pr ice of good 1 relati ve to the pr ice of goo d 2 in creases. T he lower bo und on the p rices ratio β is upd ated to the pr ice ratio of the previous iter ation. If the dem and of go od 1 is less than its supply , the price ratio is decr emented ha lf the distance to th e lower boun d β . The upp er bound β is set to the p rices ratio of th e previous iteration. T he algorithm terminates when th e distance between the updated upp er and lo we r bo unds on the p rices ratio is below an accu racy measure ǫ . 1 2 3 4 5 6 7 8 9 10 0.5 1 1.5 2 2.5 3 iteration prices ratio (p 1 /p 2 ) upper bound on prices ratio lower bound on prices ratio Walrasian prices ratio Fig. 6. Conv ergence of the price ratio in the price adjustment process to the W alrasia n price ratio. In Fig. 6 , the p rices ratio in the price adjustment process is marked with a cro ss and is shown to converge af ter a few iterations to th e W alrasian pr ices ra tio f rom T heorem 4. The dashed lines co rrespon d to the upper and lower bounds in −0.2 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 x 1 (1) x 2 (1) SNR = 0 dB SNR = −5 dB joint ZF SNR = 5 dB SNR = 10 dB SNR = −10 dB joint MRT Fig. 7. Course of the contract curve in the Edge worth box for dif ferent SNR v alues. (31). V . D I S C U S S I O N A N D I L L U S T R A T I O N S In Fig. 7, the contract curve charac terized in Theorem 2 is plotted f or different SNR v alues. The numb er of anten nas at the tran smitters is two and we generate indep enden t in stanta- neous channels h kℓ identically distrib uted as C N (0 , I ) . The contract curve is calculated by taking 10 3 samples of x (2) 2 unifor mly spaced in (0 , λ MRT 2 ) to o btain values of x (1) 1 . The course o f th e co ntract cu rve for 1 0 dB SNR is near to the edge of the Edgeworth box where joint ZF is marked. Th is means that Pareto optimal allocations require either transmitter to choose beamf orming vectors near to ZF . For decrea sing SNR, the co ntract curve moves away fro m the ZF edge. For low SNR, the contract curve is then close to the ed ge with joint MR T . These observations confor m with th e a nalysis in [30] where Pareto optimal maxim um sum utility transmission is stud ied in low and high SNR r egimes. 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.05 0.1 0.15 0.2 0.25 0.3 x 1 (1) x 2 (1) contract curve indifference curve (consumer 1) Walrasian equilibrium indifference curve (consumer 2) Fig. 8. Edgew orth box which depict s the all ocatio n for the W alra sian prices. In Fig. 8, an E dgeworth box is plotted for a sample channel realization with two tran smit antennas at bo th transmitters. For the p rices calculated fro m T heorem 4 we obtain the W alrasian equilibriu m allocation on the contract curve wh ere the cor- respond ing indifference curves are tang ent. The indifference curves are obtain ed from Proposition 1 . The line p assing throug h W alrasian eq uilibrium alloca tion defines th e budget sets of th e con sumers as is illustrated in Fig. 5. 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0 0.5 1 1.5 φ 2 (x (1) 1 ,x (1) 2 ) φ 2 (x (2) 1 ,x (2) 2 ) MRT ZF maximum sum SINR Pareto boundary core allocations Walrasian equilibrium Kalai−Smorodinsky solution Nash bargaining solution virtual SINR coordination weak Pareto boundary Fig. 9. SINR region of a two-user MISO IFC with SNR = 0 dB and tw o antenna s at the transmitt ers. In Fig . 9, the SINR region is plo tted. The points lying inside the SINR r egion correspon d to the beamform ing v ectors characterized in ( 5), where a sub set o f these points are Pareto optimal. The Pareto boundary co rrespon ds to the allocations on the contract cu rve calculated in Theorem 2. Th e weak Pareto bound ary consists of weak P areto optimal poin ts in which the link s can not strictly increase their utility simultaneo usly . Formally , the weak Pareto op timal po ints of the SINR region Φ a re de fined a s [2 3, p. 14 ] W (Φ) := { x ∈ Φ : there is no y ∈ Φ with y > x } , (36) where the inequality in (3 6) is compon entwise. Pareto o ptimal points P (Φ) in (4) define a stron ger optimality f or a utility tuple than weak Pareto optim al p oints. A weak Pareto optimal point is n ot necessarily Pareto optimal. But all Pareto optimal points a re also weak Pareto op timal, i.e . P (Φ) ⊆ W (Φ) . The c ore allocations a re all Pareto optimal points that dominate th e N ash equilibrium (join t MR T). Assuming the links are rationa l, only allocations in the core can be of inter est for the links. In o ther words, the link s will not coo perate if one link would achieve lower pa yoff th an at the Nash equilibrium. The W alrasian equilibrium from Theorem 4 always lies in th e core. In Fig. 9, we also p lot the m aximum sum SINR which is obtained by g rid search over the allocations on the Pareto bound ary . The virtual SINR co ordinatio n po int correspo nds to the coo rdination mechanism in [14], where th e m inimum mean square e rror (MM SE) tran smit beam formin g vector s w MMSE k = [ σ 2 I + h kℓ h H kℓ ] − 1 h kk k [ σ 2 I + h kℓ h H kℓ ] − 1 h kk k , k 6 = ℓ, (37) are proven to ach iev e a P areto optim al po int. These beam - forming vectors re quire only local channel state informatio n at the tran smitters which is an app ealing prope rty in terms of the low overhead in info rmation exch ange betwee n the link s. The virtual SINR coord ination and the m aximum sum SINR points do not necessarily lie in the core . Hence, these p oints are no t suitable fo r distributed implem entation between th e rational links. In Fig. 9, two solutions from ax iomatic b argaining the ory , namely the Nash bargainin g solution (NBS) and the Kalai- Smorodin sky (KS) solutio n are plo tted. These solu tions lie in th e co re and differ by the ax ioms th at d efine them . Th e interested reader is ref erred to [23] fo r a comp rehensive theory on axio matic bargaining . Accord ing to simulations, these two solutio ns ar e n ot far fro m ea ch o ther . The prop erties that the W alr asian equilibrium and the NBS or KS solu tion have in comm on is that they are Pareto optimal an d lie in the core, i.e., each user achiev es h igher utility than at the Nash equilibrium. The d ifference be tween th e solutions is th e fairness aspects in allocating the Pareto o ptimal utilities to the players. The curren t advantage in the W alrasian equilibr ium over NBS and KS solution is that it can b e char acterized in closed-for m u sing T heorem 3 and Theo rem 4. I n ad dition, we devise a coord ination m echanism to im plement the W alrasian equilibriu m in Sectio n IV - C. Next, we will describe how the NBS and KS solutions are obtained. The NBS [24, Chap ter 15] is the solution of the following problem : maximize ( φ 1 − φ NE 1 )( φ 2 − φ NE 2 ) subject to ( φ 1 , φ 2 ) ∈ Φ , (38) where φ NE k := φ k ( λ MRT 1 , λ MRT 2 ) is th e SINR in Nash equ ilibrium and Φ is the SINR region in (3). Note th at the NBS is define d for conv ex utility regions o nly , and the SINR region Φ in our case is not n ecessarily conve x as is sho wn in Fig. 9. Howe ver, solving the optim ization problem in (3 8) by grid search over 10 3 generated Pareto optimal points from Th eorem 2 gives a single solution which we plot in Fig. 9. The KS solution is the solu tion o f the fo llowing p roblem [ 31]: maximize min  φ 1 − φ NE 1 φ CORE 1 − φ NE 1 , φ 2 − φ NE 2 φ CORE 2 − φ NE 2  subject to ( φ 1 , φ 2 ) ∈ Φ , (39) where φ CORE 1 (analogo usly φ CORE 2 ) is the solution of the following problem : maximize φ 1 subject to ( φ 1 , φ NE 2 ) ∈ Φ . (40) The two Pareto o ptimal po ints ( φ CORE 1 , φ NE 2 ) and ( φ NE 2 , φ CORE 2 ) are the bou nds to the co re and are marked with circles on the Pareto b ound ary in Fig. 9. These boun ds, as discussed in Section III-B, can be calculated in th e Edgeworth bo x as the intersection of the co ntract curve and the in difference curves correspo nding to the Nash equilibr ium. The KS solution which solves the p roblem in (39) usin g th e co re bou nds is then fou nd by g rid sear ch over the generated Pareto op timal p oints from Theorem 2. A. Difficulties in th e Ex ten sion to K -User MI S O IF C While the too ls in the p aper can be applied to general K con sumer and M good s economy as can be found in [9], [ 28], the application to the beamf orming pro blem in the MISO IFC can cu rrently be do ne o nly for the two-user case. This is mainly because of the structure of the parametrizatio n av ailab le for the efficient beamform ing vectors in th e general case. 11 Using the param etrization in (5) for two-users, we hav e chosen in Section III the amoun t of good 1 for consumer 1 as x (1) 1 = λ 1 and the amount of good 1 for consumer 2 as x (2) 1 = λ MRT 1 − λ 1 . W ith this relatio n between the parameters and the g oods and due to the structu re o f th e expression in (5), th e SINR in (10) for link 1 depen ds on ly on x (1) 1 and x (1) 2 which are the am ounts f rom g ood 1 and g ood 2 fo r co nsumer 1 . T his method of d efining the goods in terms of the parameters does not carry o n for th e K -user MISO IFC case. W e illustrate this drawback b ased on an example in the 3 -user case. The parametriza tion for th e bea mformin g vectors are [7] w 1 ( λ 11 , λ 12 , λ 13 ) = v max  λ 11 h 11 h H 11 − λ 12 h 12 h H 12 − λ 13 h 13 h H 13  , (41) w 2 ( λ 21 , λ 22 , λ 23 ) = v max  − λ 21 h 21 h H 21 + λ 22 h 22 h H 22 − λ 23 h 23 h H 23  , (42) w 3 ( λ 31 , λ 32 , λ 33 ) = v max  − λ 31 h 31 h H 31 − λ 32 h 32 h H 32 + λ 33 h 33 h H 33  , (43) where v max ( Z ) is the eigen vector that correspo nds to the largest eig en value of Z a nd λ k 1 + λ k 2 + λ k 3 = 1 , k = 1 , 2 , 3 . Note that dif ferent real-valued parameter izations are also pro- vided in [5], [6], [8] which also lead to the same con clusion in terms of the application of the exchange economy model. W e use the p arametrizatio n in [7] in order to h ighlight the usage of th e different parameter s. In (41)-(43), three go ods can be directly disting uished each correspond ing to the parame ters of each transmitter . W e can choose the amount of go od 1 (analogo usly for goo ds 2 and 3 ) to be divided between the three links as x (1) 1 = λ 11 for link 1 , x (2) 1 = λ 12 for link 2 , and x (3) 1 = λ 13 for link 3 . In order to model this setting as an exchange econo my , the utility (SINR) of link k should on ly depend on the a mounts of go ods x ( k ) 1 , x ( k ) 2 , x ( k ) 3 . Howe ver, with the param etrization in (41)-(43), the SINR expression of a link k would depend on all parameter s. Hence, in formulating the de mand of consumer k as is don e in the two-user case in (25), th e solutio n depen ds also on the d emand s of the o ther consumer s. In this case, each consum er cannot find his optimal demand of g oods independen tly witho ut knowing wh at the other consu mers demand . Due to th is fact, it is currently not possible to find th e W alrasian eq uilibrium in the gene ral K - user MI SO I FC case. V I . C O N C L U S I O N S In this work, we model the interac tion betwe en two link s in the M ISO IFC as an exchange eco nomy . Th e lin ks are consid- ered as the consumers and the exchang ed goo ds cor respond to beamforming vectors. Utilizing the conflict repre sentation in the Edgew orth box, all Pareto optimal points could be characterized in clo sed form. The equilibria of the co nsidered exchange eco nomy are related to a solu tion concept from coalitional gam e theor y called th e core. These allocations are Pareto op timal and do minate the Nash equilibr ium of a strategic g ame between the links. W e pro pose a coo rdination mechanism b etween th e lin ks wh ich achieves a Pareto o ptimal outcome in the co re. For th is pur pose, the situation b etween the links is m odeled as a competitive m arket where now each consumer is endowed with a budget and can consume the goods at specific prices. The equilibrium in this economy is called W alrasian and cor respond s to the prices that equate the demand to the sup ply o f good s. T he uniqu e W alrasian prices are calculated and the coord ination mech anism is executed by an arbitrato r that fo rwards the prices to the co nsumers. The consu mers then calculate in a de centralized mann er their optimal d emand c orrespo nding to beam formin g vectors that achieve the W alrasian equilibrium. This o utcome is Pareto optimal and dominates the Nash equilibrium in the SINR region. A P P E N D I X A P R O O F O F L E M M A 1 The direct and interf erence power gain s, | h H kk w k ( λ k ) | 2 and | h H kℓ w k ( λ k ) | 2 , k 6 = ℓ, are ca lculated as functio ns of the parameters λ k by u sing the expression for the beamfor ming vectors in (5). The d irect p ower gain is calculated as: | h H kk w k ( λ k ) | 2 = p λ k h H kk Π h kℓ h kk k Π h kℓ h kk k + p 1 − λ k h H kk Π ⊥ h kℓ h kk k Π ⊥ h kℓ h kk k ! 2 (44) =  p λ k k Π h kℓ h kk k + p 1 − λ k k Π ⊥ h kℓ h kk k  2 . (45) The in terferenc e power is: | h H kℓ w k ( λ k ) | 2 =      p λ k h H kℓ Π h kℓ h kk k Π h kℓ h kk k + p 1 − λ k h H kℓ Π ⊥ h kℓ h kk k Π ⊥ h kℓ h kk k      2 (46) = λ k | h H kℓ Π h kℓ h kk | 2 k Π h kℓ h kk k 2 = λ k k h kℓ k 2 . (47) These expr essions lead to ( 6) an d ( 7) in Le mma 1. A P P E N D I X B P R O O F O F T H E O R E M 1 First, it is easy to see that the SINR expre ssion in (10) is continu ous. Th e SINR φ k ( x ( k ) 1 , x ( k ) 2 ) is strong ly increas- ing with the goo ds x ( k ) 1 and x ( k ) 2 if φ k ( x ′ ( k ) 1 , x ′ ( k ) 2 ) > φ k ( x ( k ) 1 , x ( k ) 2 ) whenever ( x ′ ( k ) 1 , x ′ ( k ) 2 ) 6 = ( x ( k ) 1 , x ( k ) 2 ) and ( x ′ ( k ) 1 , x ′ ( k ) 2 ) ≥ ( x ( k ) 1 , x ( k ) 2 ) [9, Definition A1. 17]. Define the directional d eriv a ti ve of φ k at ( x ( k ) 1 , x ( k ) 2 ) in d irection z as ∇ z φ k  x ( k ) 1 , x ( k ) 2  = lim t → 0 φ k  x ( k ) 1 , x ( k ) 2  + t z  − φ k  x ( k ) 1 , x ( k ) 2  t , (48) Since φ k ( x ( k ) 1 , x ( k ) 2 ) is differentiable, the limit above can be giv en as [9, Chapter A. 2] ∇ z φ k  x ( k ) 1 , x ( k ) 2  = ∇ φ k  x ( k ) 1 , x ( k ) 2  z , (49) 12 where ∇ φ k ( x ( k ) 1 , x ( k ) 2 ) is the grad ient of φ k at ( x ( k ) 1 , x ( k ) 2 ) written as ∇ φ k  x ( k ) 1 , x ( k ) 2  =   ∂ φ k  x ( k ) 1 , x ( k ) 2  ∂ x ( k ) k , ∂ φ k  x ( k ) 1 , x ( k ) 2  ∂ x ( k ) ℓ   , (50) with ℓ 6 = k . Th e directiona l deriv ati ve of φ k ( x ( k ) 1 , x ( k ) 2 ) defines the slope of the tange nt to φ k ( x ( k ) 1 , x ( k ) 2 ) at the point ( x ( k ) 1 , x ( k ) 2 ) in the d irection z . Hence, if the direc tional der iv a- ti ve is positive for z = ( z 1 , z 2 ) T with z 1 and z 2 nonnegative and satisfying k z k = p z 2 1 + z 2 2 = 1 , the n the u tility functio n φ k ( x ( k ) 1 , x ( k ) 2 ) is strongly increasing. Conseq uently , the direc- tional der iv ative in (49) is strictly positiv e if the compon ents of the gradien t ∇ φ k ( x ( k ) 1 , x ( k ) 2 ) are strictly positive. The first compon ent of ∇ φ k ( x ( k ) 1 , x ( k ) 2 ) is ∂ φ k  x ( k ) 1 , x ( k ) 2  ∂ x ( k ) k =  q x ( k ) k g k + q (1 − x ( k ) k ) ˇ g k  q g k x ( k ) k − q ˇ g k 1 − x ( k ) k  σ 2 + λ MRT ℓ g ℓk − x ( k ) ℓ g ℓk . (51) The partial d eriv ative in ( 51) is strictly larger than zero when x ( k ) k < g k / ( ˇ g k + g k ) . Substituting ˇ g k and g k from Lemma 1 we get x ( k ) k < g k ˇ g k + g k = k Π h kℓ h kk k 2 k h kk k 2 = λ MRT k . (52) Since x ( k ) k ∈ [0 , λ MRT k ] , th e partial d eriv ati ve in (51) is strictly larger than zero except fo r x ( k ) k = λ MRT k . The second component of ∇ φ k ( x ( k ) 1 , x ( k ) 2 ) is ∂ φ k  x ( k ) 1 , x ( k ) 2  ∂ x ( k ) ℓ = g ℓk  q x ( k ) k g k + q (1 − x ( k ) k ) ˇ g k  2  σ 2 + λ MRT ℓ g ℓk − x ( k ) ℓ g ℓk  2 , (53) with ℓ 6 = k , which is strictly larger than zero for x ( k ) ℓ ∈ [0 , λ MRT ℓ ] . Hen ce, the directional deriv ative in (4 9) is strictly positive for ( x ( k ) 1 , x ( k ) 2 ) ∈ [0 , λ MRT 1 ] × [0 , λ MRT 2 ] except fo r the case x ( k ) k = λ MRT k and z = (1 , 0) . Since λ MRT k is the upper bound on x ( k ) k , the slop e of the function φ k ( x ( k ) 1 , x ( k ) 2 ) in the direction x ( k ) k as is restricted by the condition z = (1 , 0) is not o f inter est. Next, we will prove that the SINR function is join tly quasiconcave wit h the goo ds. Consider the SINR expression in (10), and define f ( x ( k ) k ) :=  q x ( k ) k g k + q (1 − x ( k ) k ) ˇ g k  2 and g ( x ( k ) ℓ ) := σ 2 + λ MRT ℓ g ℓk − x ( k ) ℓ g ℓk . The function φ k ( x ( k ) 1 , x ( k ) 2 ) = f ( x ( k ) k ) /g ( x ( k ) ℓ ) is strictly quasicon cave if f ( x ( k ) k ) is strictly co ncave and g ( x ( k ) ℓ ) is conv ex [3 2, Proposition 2]. It is clear that g ( x ( k ) ℓ ) is conv ex since the function is linea r in x ( k ) ℓ . In o rder to show that f ( x ( k ) k ) is strictly con cave, we build the second deriv ati ve o f f ( x ( k ) k ) as follows: d 2 f ( x ( k ) k ) d 2 x ( k ) k =  q g k /x ( k ) k − q ˇ g k / (1 − x ( k ) k )  2 −  q x ( k ) k g k + q (1 − x ( k ) k ) ˇ g k  ×   s g k ( x ( k ) k ) 3 + s ˇ g k (1 − x ( k ) k ) 3   (54) = g k x ( k ) k + ˇ g k (1 − x ( k ) k ) − 2 s g k ˇ g k (1 − x ( k ) k )( x ( k ) k ) − g k x ( k ) k − ˇ g k (1 − x ( k ) k ) − v u u t (1 − x ( k ) k ) g k ˇ g k ( x ( k ) k ) 3 − v u u t x ( k ) k g k ˇ g k (1 − x ( k ) k ) 3 (55) = − 2 s g k ˇ g k (1 − x ( k ) k )( x ( k ) k ) − v u u t (1 − x ( k ) k ) g k ˇ g k ( x ( k ) k ) 3 − v u u t x ( k ) k g k ˇ g k (1 − x ( k ) k ) 3 < 0 . (56) The second der iv ative of f ( x ( k ) k ) is strictly less than zero. Thus, f ( x ( k ) k ) is strictly concave. According ly , φ k ( x ( k ) 1 , x ( k ) 2 ) is strictly qu asiconcave. A P P E N D I X C P R O O F O F T H E O R E M 3 Since the function φ k ( x ( k ) 1 , x ( k ) 2 ) is strictly quasiconcave, then this function has a unique max imum. Considerin g c on- sumer 1 (analog ously con sumer 2 ), the Lagrangian function to the con strained op timization p roblem in (25) is L  x (1) 1 , x (1) 2 , µ  = φ 1  x (1) 1 , x (1) 2  + µ  λ MRT 1 p 1 − x (1) 1 p 1 − x (1) 2 p 2  , (57) where µ is a Lagrange mu ltiplier . The Karush–Kuhn– T uc ker (KKT) con ditions for optim ality ar e necessary an d sufficient giv en as: ∂ L  x (1) 1 , x (1) 2 , µ  ∂ x (1) 1 = ∂ φ 1  x (1) 1 , x (1) 2  ∂ x (1) 1 − µp 1 = 0 (58) ∂ L  x (1) 1 , x (1) 2 , µ  ∂ x (1) 2 = ∂ φ 1  x (1) 1 , x (1) 2  ∂ x (1) 2 + µp 2 = 0 (59) ∂ L  x (1) 1 , x (1) 2 , µ  ∂ µ = λ MRT 1 p 1 − x (1) 1 p 1 − x (1) 2 p 2 = 0 (60) According to con ditions (5 8) and (59), we get ∂ φ 1  x (1) 1 , x (1) 2  ∂ x (1) 1 1 p 1 = − ∂ φ 1  x (1) 1 , x (1) 2  ∂ x (1) 2 1 p 2 (61) 13 ⇒  q x (1) 1 g 1 + q (1 − x (1) 1 ) ˇ g 1  √ g 1 q x (1) 1 − √ ˇ g 1 q 1 − x (1) 1 ! σ 2 + λ MRT 2 g 21 − x (1) 2 g 21 =  q x (1) 1 g 1 + q (1 − x (1) 1 ) ˇ g 1  2 g 21  σ 2 + λ MRT 2 g 21 − x (1) 2 g 21  2 p 1 p 2 (62) ⇒ √ g 1 q x (1) 1 − √ ˇ g 1 q 1 − x (1) 1 =  q x (1) 1 g 1 + q (1 − x (1) 1 ) ˇ g 1  g 21 ( σ 2 + λ MRT 2 g 21 − x (1) 2 g 21 ) p 1 p 2 . (63) Substituting x (1) 2 from (60) we get q (1 − x (1) 1 ) g 1 − q x (1) 1 ˇ g 1 =  x (1) 1 q (1 − x (1) 1 ) g 1 + (1 − x (1) 1 ) q x (1) 1 ˇ g 1  ( σ 2 g 21 + λ MRT 2 − λ MRT 1 p 1 p 2 | {z } B + x (1) 1 p 1 p 2 ) p 1 p 2 (64) ⇒ q (1 − x (1) 1 ) g 1 B − q x (1) 1 ˇ g 1 B − x (1) 1 p 1 p 2 q x (1) 1 ˇ g 1 = (1 − x (1) 1 ) q x (1) 1 ˇ g 1 p 1 p 2 (65) ⇒ q x (1) 1 ˇ g 1  B + p 1 p 2  = q (1 − x (1) 1 ) g 1 B (66 ) Squaring bo th sides on the condition that B ≥ 0 we can write x (1) 1 ˇ g 1  B + p 1 p 2  2 = (1 − x (1) 1 ) g 1 B 2 . (67) W e solve for x (1) 1 to get x (1) 1 = 1 + ˇ g 1 g 1  1 + p 1 p 2 B  2 ! − 1 . (68) Substituting B fr om (64) we get the expression in (27). x (1) 2 is calcu lated ac cording to (6 0). R E F E R E N C E S [1] R. Mochaoura b and E. A. Jorswieck, “W alrasia n equilibriu m in two-user multiple -input single-outp ut interference channel, ” in P r oc. ICC , Kyoto, Japan, Jun. 2011, pp. 1–5. [2] S. Vi shwana th and S. 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