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Utilizing the Edgeworth box [9], which is a graphical tool that spectrum manager which adjusts the prices to achieve the depicts the preferences of the consumers over the distribution equilibrium. In [16], competitive spectrum market is consid- of the goods, we provide a closed-form solution to all Pareto ered where the users, sharing a common frequency band, can optimal points of the SINR region. A subset of all Pareto purchase their transmit power subject to budget constraints. optimal points satisfy that both links jointly achieve higher An agent, referred to as the market, determines the unit utility than at the noncooperative point. These points are called prices of the power spectra. Existence of the equilibrium is exchange equilibria and are related to the solution concept, the proven and conditions for its uniqueness are provided. In [19], core, from coalitional game theory. the competitive equilibrium is used for simultaneous bitrate allocation for multiple video streams and the Edgeworth box [15] is used to illustrate the conflict between the streams. In B. Coordination to Achieve Pareto Optimal Points the context of cognitive radio, spectrum trading is successfully All of the mentioned efforts to parameterize the efficient modeled by economic models and market-equilibrium, and beamforming vectors in the MISO are valuable for design- competitive and cooperative pricing schemes are developed ing efficient low complexity distributed resource allocation in [20]. Moreover, in [21], hierarchical spectrum sharing is schemes such as in [10]–[12]. In [10] and [11], the real-valued modeled as an interrelated market. The pricing mechanism parametrization for the two-user case from [4] is utilized for the bandwidth allocations between the systems equates the and bargaining algorithms are proposed to improve the joint supply to the demand. performance of the systems from the noncooperative state. An In our case, the links are the consumers and the parameters extension to these works is made in [12] where a strategic of the beamforming vectors are the goods the consumers bargaining process is proposed and proven to terminate at a possess. We formulate the consumers’ demand functions and Pareto optimal outcome. Also based on a strategic bargaining calculate the Walrasian prices which equate the demand to the approach, a coordination mechanism is proposed in [13] for supply of each good. To achieve the Pareto optimal Walrasian the two-user MISO IFC where the Han-Kobayashi scheme is equilibrium, the arbitrator coordinates the transmission of the applied. In the K-user MISO IFC, a low complexity one- links. We consider two cases for the coordination mechanism. shot coordination mechanism is given in [14], where each Assuming the arbitrator has full knowledge of the setting, he transmitter independently maximizes its virtual SINR. For the can calculate the Walrasian prices and forward these to the two-user case, the proposed mechanism is proven to achieve links. The links independently calculate their beamforming a Pareto optimal solution. vectors according to their demand function. Assuming the In this paper, we propose a coordination mechanism be- arbitrator has limited knowledge of the setting, we propose tween two MISO interfering links which is Pareto optimal a price adjustment process, also referred to as tˆatonnement, and achieves for each link a utility higher than at the nonco- to reach the Walrasian prices. In each iteration, the links operation point. Our analysis is based on relating the MISO send their demands to the arbitrator which updates the prices IFC setting to a competitive market [9]. To the best of our according to the excess demand of each good. knowledge, this is the first time the beamforming problem in Outline: The outline of the paper is as follows. The system the MISO IFC is related to and analyzed using competitive and channel model, as well as the definition of the SINR market models. In a competitive market, as proposed by L. region and the beamforming vectors that are relevant for Pareto Walras [15], [16], there exists a population in which each optimal operation are given in Section II. In Section III, we individual possesses an amount of divisible goods. The worth examine a pure exchange economy between the links. We of these goods makes up the budget of each individual. Each model the parametrization of efficient beamforming vectors individual has a utility function which reveals his demand as goods and the links as consumers which trade these goods on consuming goods. Moreover, each individual would use within themselves. We characterize all Pareto optimal points the revenue from selling all his goods to buy amounts of in closed form and define the equilibria which correspond goods such that his utility is maximized. This economic model to the core of a coalition between the links. In Section IV, is competitive because each consumer seeks to maximize we consider a competitive market model and assume that the his profit independent of what the other consumers demand. goods are bought by the consumers at prices determined by an Walras investigated if there exists prices for the goods such arbitrator. The equilibrium of this market model is determined, that the market has neither shortage nor surplus. The existence and we provide two coordination mechanisms to achieve it. In of such prices, called Walrasian prices, was later explored by Section V, we illustrate the results of this paper before we Arrow [17]. The prices in this economy are usually assumed conclude in Section VI. to be fixed and not determined by the consumers. It is assumed Notations: Column vectors and matrices are given in low- that the market or an auctioneer acts as an arbitrator to ercase and uppercase boldface letters, respectively. a is the determine the Walrasian prices. Euclidean norm of a CN . b is the absolutek valuek of The competitive market model has found a few applications b C. sign(a) denotes∈ the sign| | of a R. ( )T and ( )H for resource allocation in communication networks. In [18], denote∈ transpose and Hermitian transpose,∈ respectively.· T· he the Walrasian equilibrium is formulated as a linear comple- orthogonal projector onto the column space of Z is ΠZ := H 1 H mentarity problem for a multi-link multi-carrier setting. A Z(Z Z)− Z . The orthogonal projector onto the orthogonal Π Π decentralized price-adjustment process is proposed where the complement of the column space of Z is Z⊥ := I Z , users send their power allocations in each iteration to the where I is an identity matrix. (0, A) denotes a circularly-− CN 3

2 symmetric Gaussian complex random vector with covariance This results in the achievable rate log2(1 + φk(w1, w2)) for matrix A. Throughout the paper, the subscripts k,ℓ are from link k when single user decoding is performed at the receivers. the set 1, 2 . The SINR region is the set of all achievable SINR tuples { } defined as

2 II. PRELIMINARIES Φ := (φ (w , w ), φ (w , w )) : wk 1 . (3) 1 1 2 2 1 2 k k ≤ A. System and Channel Model In the SINR region, tuples can be ranked according to their Pareto efficiency. An SINR tuple (φ , φ ) Φ is Pareto The quasi-static block flat-fading channel vector from trans- 1′ 2′ superior to (φ , φ ) Φ if (φ , φ ) (∈φ , φ ), where mitter k to receiver ℓ is denoted by h CN . We assume 1 2 1′ 2′ 1 2 kℓ the inequality is componentwise∈ and strict≥ for at least one that transmission consists of scalar coding∈ followed by beam- component. The transition from (φ , φ ) to (φ , φ ) is called forming. The beamforming vector used by transmitter k is 1 2 1′ 2′ a Pareto improvement. Situations where Pareto improvements w CN . The matched-filtered, symbol-sampled complex k are not possible are called Pareto optimal. These points baseband∈ data received at receiver k is1 constitute the Pareto boundary of the SINR region. Formally, H H the set of Pareto optimal points of Φ are defined as [23, p. 18] yk = h wksk + h wℓsℓ + nk, k = ℓ, (1) kk ℓk 6 (Φ) := x Φ: there is no y Φ with y x, y = x , where s (0, 1) is the symbol transmitted by transmitter k P { ∈ ∈ ≥ 6 }(4) k, and n∼ CN (0, σ2) is additive Gaussian noise. Each k where the inequality in (4) is componentwise. transmitter has∼CN a total power constraint of P := 1 such that 2 2 For the two-user MISO IFC, the set of beamforming vectors wk 1. We define the signal to noise ratio (SNR) as 1/σ . k k ≤ that are relevant for Pareto optimal operation are parameter- The transmitters are assumed to have perfect local channel ized by a single real-valued parameter λk [0, 1] for each state information (CSI), i.e., each transmitter has perfect transmitter k = ℓ as [4, Corollary 1] ∈ knowledge of the channel vectors only between itself and the 6 Π Π two receivers. Further information at the transmitters required hkℓ hkk h⊥kℓ hkk wk(λk)= λk + 1 λk . (5) for the coordination mechanism is discussed later in Section Πh hkk − Π h kℓ h⊥kℓ kk IV-C. p k k p k k We assume there exists an arbitrator who coordinates the This parametrization is valuable for designing efficient low operation of the transmitters. The arbitrator could be any complexity distributed resource allocation schemes [12]. The set of beamforming vector in (5) includes maximum ratio central controller which is connected to both links. Generally, 2 2 transmission (MRT) ( MRT h h h ) and zero the practical identification of the arbitrator depends on the sce- λk = Π kℓ kk / kk ZFk k k k nario. For example, in hierarchical networks in which several forcing transmission (ZF) (λk = 0). According to [4, Corol- tiers of networks operate in the same area it is possible that lary 2], it suffices that the parameters λk only be from the set MRT higher network tiers benefit from coordinating the operation of [0, λk ] for Pareto optimal operation. Note that a transmitter k has to know the channel vectors hkk and hkℓ, k = ℓ, perfectly the networks in lower tiers such as in the model used in [21]. 6 Moreover, the arbitrator can be the base station of a macrocell in order to calculate the beamforming vectors in (5). Since which can coordinate the transmission of smaller microcells in we are interested in transmissions that lead to Pareto optimal its coverage range [22]. The macrocell base station is usually outcomes, we will confine the strategy set of each transmitter connected to the microcell base stations via a high capacity to the set in (5) and formulate the SINR expression in (2) in link which enables the exchange of channel information re- terms of the parameters λk. For this purpose, we first formulate quired for the coordination process. The applicability of our the power gains at the receivers. system model in a cognitive radio network is not suitable if the Lemma 1: The power gains at the receivers in terms of the transmitters are restricted to take into account the interference parametrization in (5) are levels they are allowed to induce at primary receivers. Our H 2 2 h wk(λk) = ( λkgk + (1 λk)ˇgk) , (6) | kk | − setting is suitable for cognitive network settings, in which the H 2 h wk(λk) = λkgkℓ, k = ℓ, (7) users dynamically adapt their transmissions according to the | kℓ | p 6p environment these users exist in. A cognitive transmitter can MRT Π 2 where λk [0, λk ] and gk := hkℓ hkk , gˇk := choose with whom it can cooperate and exchange information Π 2∈ 2 k k h⊥kℓ hkk , gkℓ := hkℓ , where k = ℓ. to improve its utility. k Proof:kThe proofk is providedk in Appendix6 A. The SINR of link k can be rewritten using Lemma 1 in terms of the parameters in (5) as B. SINR Region and Efficient Transmission 2 The signal to interference plus noise ratio (SINR) at receiver √λkgk + (1 λk)ˇgk − φk(λ , λ )= , ℓ = k. (8) k is 1 2  σ2 + λ g  6 H 2 p ℓ ℓk hkkwk φk(w1, w2)= | | , k = ℓ. (2) H 2 2 2We represent the preference of a link over the used beamforming vectors h wℓ + σ 6 | ℓk | with the SINR utility function in (2). The results in this paper also hold for any SINR based utility function which is strictly increasing with SINR such 1Throughout the paper, the subscripts k,ℓ are from the set {1, 2}. as the achievable rate function. 4

(1) (2) x2 x Notice in (8) that the interference term λℓgℓk scales linearly (1) ′ 1 (2) ′ MRT I1(x1 ,φ1) MRT I2(x2 ,φ2) with λℓ. With this respect, the parameter λℓ can be interpreted λ2 λ1 as a scaling of interference at the counter receiver. A reduction in λℓ increases the SINR of link k for fixed λk. Assuming that the links are not cooperative, their operation point can be ′(1) x′(2) predicted using noncooperative game theory. The outcome is x 2 1 a solution of a strategic game [24, Section 2.1] between the ′(1) MRT (1) ′(2) MRT (2) O1 λ O2 λ links. x 1 1 x1 x 2 2 x2 (a) Consumer 1. (b) Consumer 2. C. Game in Strategic Form In [3], the outcome of a strategic game between the links is Fig. 1. Preference representation of the consumers. I1 and I2 are indifference curves of consumer 1 and 2 respectively. studied. The game in strategic form consists of the set of play- ers, 1, 2 , corresponding to the two links. The pure strategies { } MRT of player k are the real-valued parameters λk [0, λk ] in (5). ∈ If consumer k gives x(ℓ) to the other consumer, this means The utility function of player k is log2(1+φk(λ1, λ2)), where k φk(λ1, λ2) is given in (8). The outcome of this strategic game that transmitter k uses the beamforming vector in (5) which (ℓ) (ℓ) is the same also when the utility function is chosen to be corresponds to λMRT x . Hence, if x increases, transmitter k − k k φk(λ1, λ2). This is due to the fact that the preference relation k reduces the interference at receiver ℓ by using a beamform- of the players which is represented through the utility function ing vector nearer to ZF. The utility function of a consumer is invariant to positive monotonic transforms [9, Theorem 1.2]. represents his preference over the goods. We use the SINR in In the above described game, a player always chooses the (8) as the utility function of the consumer which we rewrite MRT strategy independent of the choice of the other player in terms of the goods as [3], i.e., MRT is a dominant strategy for each player. Hence, the unique Nash equilibrium is (λMRT, λMRT). The extension of 2 1 2 (k) (k) the two-player strategic game described above to the K-player xk gk + (1 xk )ˇgk (k) (k) − case is straightforward. The Nash equilibrium corresponds to φk(x1 , x2 )= q q  , (10) 2 MRT (k) σ + λ gℓk x gℓk the strategy profile in which each player chooses MRT. The ℓ − ℓ outcome in Nash equilibrium is generally not Pareto optimal.

In order to achieve Pareto improvements from the Nash (k) MRT (k) where we substituted λk = xk and λℓ = λℓ xℓ ,ℓ = k, equilibrium, coordination between the players is required. from (9). − 6 (k) (k) III. EQUILIBRIA IN EXCHANGE ECONOMY Theorem 1: φk(x1 , x2 ) in (10) is continuous, strongly increasing, and strictly quasiconcave on [0, λMRT] [0, λMRT]. A. Exchange Economy Model 1 × 2 In this section, we will use a pure exchange economy Proof: The proof is provided in Appendix B. model [9, Chapter 5.1] to determine equilibria which lie on The preference of consumers 1 and 2 over the goods is the Pareto boundary of the SINR region in (3). This model plotted in Fig. 1(a) and Fig. 1(b), respectively. For consumer assumes that there exists a set of consumers which voluntarily 1 (analogously consumer 2), O1 is the origin of the coordinate (1) exchange goods they possess to increase their payoff. The system which has x1 , the amount from good 1, at the x- set of consumers 1, 2 corresponds to the two links in our axis and x(1), the amount from good 2, at the y-axis. I is { } 2 k setting. The goods correspond to the parametrization of the the indifference curve of consumer k which represents the beamforming vectors in (5). That is, there are two goods and (k) (k) pairs (x1 , x2 ) such that the consumer achieves the same (k) (k) (k) (k) λ1 will stand for good 1 and λ2 for good 2. The consumers payoff as with (x′1 , x′2 ), i.e., φk(x1 , x2 ) = φk′ := are initially endowed with amounts of these goods. We will (k) (k) φk(x′1 , x′2 ). The dark region above Ik, corresponds to assume that the links start the trade in Nash equilibrium. Thus, (k) (k) the pairs (x , x ) where the consumer achieves higher consumer k is initially endowed with λMRT from his good and 1 2 k payoff than at the indifference curve. The region below Ik nothing from the good of the other consumer. Specifically, we corresponds to less payoff for consumer k. According to the define MRT and MRT as the endowments of consumers (λ1 , 0) (0, λ2 ) properties of the utility function in Theorem 1, the indifference 1 and 2, respectively. curves, which correspond to the boundaries of the level sets Since during exchange each consumer will possess differ- k k of φ (x( ), x( )), are convex. This result is required later ent amounts from both available goods, we introduce new k 1 2 for the proof of Theorem 3 to obtain a unique solution to variables that indicate these. When consumer k trades an the consumer demand problem in (25). Moreover, Theorem amount of his good k to consumer ℓ = k, this amount will (ℓ) 6 1 proves the existence of at least one Walrasian equilibrium be represented by x λMRT. The amount left for consumer k ≤ k which is considered in Section IV. k from his good is x(k) = λMRT x(ℓ). In connection to the k k k Next, we provide two alternative formulations for the indif- parametrization in (5), we define− the amounts of possessed ference curves. Both formulations are required to determine goods as special allocations in the Edgeworth box. (k) (k) MRT (k) xk = λk, xℓ = λℓ λℓ, ℓ = k. (9) − 6 Proposition 1: The indifference curves Ik (xℓ as a func- 5

(k) (1) tion of x ), are calculated for given fixed payoffs φ′ as x2 k k ′(2) x O2 2 x(2) 1 (1) (1) 1 2 x1 g1 + (1 x1 )ˇg1 I1 (1) MRT σ − contract curve I1(x1 , φ1′ )= λ2 + q q  , g21 − φ1′ g21 I2 (11) 2 (2) (2) 2 x2 g2 + (1 x2 )ˇg2 (2) MRT σ − I2(x2 , φ2′ )= λ1 + q q  . exchange lens g12 − φ2′ g12 (12) ′(1) ′(2) x 2 x 2 Proof: The indifference curve Ik for a given utility φk′ (1) x1 ′(1) satisfies O1 x 1 2 x(2) (k) (k) 2 x gk + (1 x )ˇgk k − k φk′ = q q  , ℓ = k. (13) Fig. 2. An illustration of an Edgeworth box. 2 MRT (k) σ + λ gℓk x gℓk 6 ℓ − ℓ Exchanging the LHS and the denominator at the RHS of (13) (1) (1) (2) (2) we get vectors (x′1 , x′2 ) and (x′1 , x′2 ) make up the allocation (1) (1) (2) (2) 2 ((x′1 , x′2 ), (x′1 , x′2 )) in the box. Every point in the (k) (k) x gk + (1 x )ˇgk box denotes an allocation, i.e., an assignment of a possession k − k 2 MRT (k) vector to each consumer. The consumers’ preferences in the σ + λℓ gℓk xℓ gℓk = q q  , − φk′ Edgeworth box can be revealed according to their indifference (14) curves. The dark region in Fig. 2 is called the exchange lens Solving for x(k), we get the expressions in (11) and (12). ℓ and contains all allocations that are Pareto improvements to Note that Proposition 1 characterizes a family of indifference the outcome in ((x (1), x (1)), (x (2), x (2))). The locus of all curves. Each indifference curve has a domain and range which ′1 ′2 ′1 ′2 Pareto optimal points in the Edgeworth box is called the depends on the fixed SINR value φ . Thus, for selected k′ contract curve [25]. On these points, the indifference curves fixed SINRs, the indifference curves should be restricted are tangent, and are characterized by the following condition3 to take values in the feasible parameter set from (5), i.e., (1) MRT (2) MRT [25, p. 21]: I1(x1 , φ1′ ) [0, λ2 ] and I2(x2 , φ2′ ) [0, λ1 ]. The indifference curves∈ can be alternatively formulated∈ to obtain (k) (k) (1) (1) (2) (2) ∂φ1 x , x ∂φ2 x , x xk as a function of xℓ . 1 2 1 2 (k) ˜ (1) (2) Proposition 2: The indifference curves Ik (xk as a func-     (k) ∂x1 ∂x2 tion of xℓ ), are calculated for given fixed payoffs φk′ as [12, (2) (2) (1) (1) Proposition 1] ∂φ2 x1 , x2 ∂φ1 x1 , x2 = . (17) ∂x(2)  ∂x(1)  φ 1 2 ˜ (1) MRT 1′ I1(x2 , φ1′ )= f λ1 , , (15)  MRT MRT (1)  The convexity of the consumers’ indifference curves implies φ1 λ1 , λ2 x2 − that these can only be tangent at a single point. Thus, the    φ condition in (17) is necessary and sufficient for an allocation ˜ (2) MRT 2′ I2(x1 , φ2′ )= f λ2 , , (16) to be on the contract curve.  MRT (2) MRT  φ2 λ1 x1 , λ2 MRT MRT − Theorem 2: The contract curve cc : [0, λ2 ] [0, λ1 ]    (1) (2) → where f(a,b) := (√ab (1 a)(1 b))2. (x1 as a function of x2 ) is the solution of the following 4 Similarly, the values of the− indifference− − curves in Proposition 2 cubic equation p ˜ (1) have to be in the feasible parameter set such that I1(x2 , φ1′ ) 3 2 MRT (2) MRT ∈ (1) (1) (1) [0, λ ] and I˜ (x , φ′ ) [0, λ ]. a x + b x + c x + d =0, (18) 1 2 1 2 ∈ 2 1 1 1 h i h i h i B. Edgeworth Box where The Edgeworth box [25], [9, Chapter 5], illustrated in 2 4 (19) Fig. 2, is a graphical representation that is useful for the a = (g1 +ˇg1)(C g12) , d = g1σ , − − 2 2 analysis of an exchange economy. The box is constructed b = (C g12) 2ˇg1(C + σ )+ g1(2σ + C g12) , (20) − − by joining Fig. 1(a) and Fig. 1(b). Thus, the Edgeworth c = gˇ (C + σ2)2 + σ2g (2g 2C σ2), (21) − 1 1 12 − −
box has two points of origin, O1 and O2, corresponding to consumer 1 and 2, respectively. The initial endowments 3In multiple consumer settings, the condition provided by Edgeworth [25] of the consumers define the size of the box. The width of should hold for every consumer pair. MRT MRT 4 the box is thus λ1 , and the height is λ2 . The possession This result is independently obtained in [26]. 6

(1) x2 equilibria can be achieved requiring the links to negotiate or (2) O2 bargain (as for instance is proposed in [12]). Next, we will x1 I1 consider decentralized operation of the links and include the arbitrator to coordinate transmission of the links. I 2 contract curve

core A. Competitive Market Model In a competitive market, the consumers buy quantities of endowments goods and also sell goods they possess such that they maxi- mize their profit. Each good has a price and every consumer

b (1) takes the prices as given. The prices of the goods are not x1 O1 determined by consumers, but arbitrated by markets. In our (2) x2 case, the arbitrator determines the prices of the goods. Let pk denote the unit price of good k. In order to be able to MRT Fig. 3. An illustration of the allocations in the core. buy goods, each consumer k is endowed with a budget λk pk which is the worth of his initial amounts of goods5. The budget set of consumer k is the set of bundles of goods he can afford (2) and C is a function of x2 given as to possess defined as

(2) (2) x g2 + (1 x )ˇg2 (k) (k) 2 (k) (k) MRT 2 2 k := (x , x ) R : x p + x p λ pk . − B 1 2 ∈ + 1 1 2 2 ≤ k C = q q  . (22) (24) g2 gˇ2 σ2 MRT (1) n o (2) (2) + λ x g21 2 2 The budget set of consumer 1 is illustrated by the grey area in x2 − 1 x2 −  −  q q   Fig. 4. The boundary of the budget set is a line which connects The root of interest in (18) lies in [0, λMRT] and satisfies MRT MRT 1 the points (λ1 , 0) and (0, λ1 p1/p2). Thus, the boundary has a slope of p1/p2. For the consumers, the prices of the goods 2 (1) (1) − sign σ /g12 + x Cx are measures for their qualitative valuation. If is greater 1 − 1 p1  2  (1) (1) than p2, then good 1 has more value than good 2. Given the = sign σ /g12 + x1 + C(1 x1 ) . (23) − prices p1 and p2, consumer 1 demands the amounts of goods   (1) (1) Proof: The proof is provided in [1, Appendix A]. x1 and x2 such that these maximize his utility in (10). Thus, According to Edgeworth [25], the outcome of an exchange consumer k solves the following problem: between the consumers must lie on the contract curve. The (k) (k) solution concept by Edgeworth is related to that of coalitional maximize φk x1 , x2 games called the core [27] which defines equilibria in our (25) (k) (k) MRT subject to p x + p x λ pk. exchange economy. The situation between the two links can be 1 1 2 2 ≤ k represented as a coalitional game without transferable payoff In the above consumer problem, the objective function is the [24, Chapter 13.5]. In our case, the core of this game [24, SINR of link k and the constraint is defined by the budget Definition 268.3] is the set of all allocations in the Edgeworth set of consumer k in (24). The physical interpretation of box in which no player can achieve higher payoffs without the budget set constraint can be related to an interference cooperating with the other player. In Fig. 3, the core is constraint. Considering consumer 1, the constraint in (25) can illustrated as the set of allocations on the contract curve which be reformulated to is bounded by the indifference curves corresponding to the initial endowments. That is, the core allocations correspond to (1) MRT p2 (1) x1 λ1 x2 , (26) all Pareto optimal points which dominate the Nash equilibrium ≤ − p1 in the SINR region. With the initial endowments corresponding MRT MRT to the Nash equilibrium (λ , λ ), the indifference curves (1) MRT 1 2 where, as mentioned before, x1 = λ1 [0, λ1 ] is the can be calculated from Proposition 2 or Proposition 1. The scaling of interference transmitter 1 produces∈ at receiver 2. bounds for the core, as illustrated in Fig. 3, can be calculated (1) MRT Analogously, x2 = λ2 λ2 is the scaling for interference as the intersection points of the indifference curves starting at reduction from transmitter−2 at receiver 1. Hence, the con- the endowment allocation and the contract curve characterized straint in (26) dictates the tradeoff between the amount of in Theorem 2. Later in Section V, the bounds for the core will interference transmitter 1 can generate at receiver 2 and the be used to determine the Kalai-Smorodinsky solution from amount of interference receiver 1 is to tolerate. The prices p1 axiomatic bargaining theory. and p2 can be interpreted as parameters to control the fairness between the links by regulating the amount of interference the IV. WALRASIAN EQUILIBRIUM IN EXCHANGE ECONOMY links generate on each other. In the preceding section, we have determined the Pareto optimal equilibria in our pure exchange economy. These 5This case corresponds to the Arrow-Debreu market model [16]. 7

(1) (1) x2 increasing utility x 2 ∗ x (2) MRT (2) 1 O2 λ2 x1 I1 p1 λMRT p2 1 2 (budgetB set) I2 indifference curves ∗(1) ∗(2) x2 x2

∗(1) x2 contract curve 1 slope 1 (budgetB set) B − p ∗ (budget set) p ∗1 2

x(1) ∗ (1) ∗ 1 O1 (1) MRT O1 (1) x λ1 x1 x 1 1 (2) x2 Fig. 4. An illustration of the budget set of consumer 1.

Fig. 5. An illustration of an Edgeworth box. I1 and I2 are indifference curves ∗ ∗ of consumer 1 and 2 respectively. The line with slope -p1/p2 separates the Theorem 3: The unique solution to the problem in (25) is budget sets of the consumers in Walrasian equilibrium.

(1) 1 x1∗ (p1,p2)= , (27) p1 2 g21 where the corresponding indifference curve is tangent to the gˇ1 p2 1+ 1+ 2 MRT MRT p1 g1 σ +λ g21 λ g21 boundary of the budget set. 2 − 1 p2   The next result provides a significant property that the goods (1) p1 MRT (1) x2∗ (p1,p2)= λ1 x1∗ , (28) in our setting possess. Later in Section IV-B and Section p2 −   IV-C, this property is required to prove the uniqueness of for consumer 1, and the Walrasian equilibrium and also to guarantee the global (2) 1 convergence of the price adjustment process. x2∗ (p1,p2)= , (29) p2 2 g12 Lemma 2: The goods in our setting are gross substitutes, gˇ2 p1 1+ 1+ 2 MRT MRT p2 g2 σ +λ g12 λ g12 i.e., increasing the price of one good increases the demand of 1 − 2 p1   the other good. (2) p2 MRT (2) x∗ (p ,p )= λ x∗ , (30) 1 1 2 2 2 Proof: Decreasing the ratio p1/p2 can be interpreted as p1 −   decreasing p1 or increasing p2. Consider the aggregate excess for consumer 2, where gˇk,gk,gkℓ are defined in Lemma 1. demand of good 1 defined as The feasible prices ratio are in the range: (1) (2) MRT MRT 2 MRT z1(p1,p2)= x1∗ (p1,p2)+ x1∗ (p1,p2) λ1 , (32) λ g12 p1 σ + λ g21 − β := 2 β := 2 . (31) 2 MRT MRT (1) (2) σ + λ1 g12 ≤ p2 ≤ λ1 g21 where x1∗ (p1,p2) and x1∗ (p1,p2) are the demand functions Proof: The proof is provided in Appendix C. of good 1 in (27) and (30) from Theorem 3. If p1/p2 (1) Theorem 3 characterizes the demand functions of each con- decreases, then x1∗ (p1,p2) increases. If p1/p2 decreases, (2) sumer. In economic theory, these functions are called Marshal- then x1∗ (p1,p2) also increases since p2/p1 increases and (2) lian demand functions [9] or Walrasian demand functions [28]. x2∗ (p1,p2) decreases. Thus, the aggregate excess demand Note that each consumer calculates his demands independently of good 1 in (32) increases if p1/p2 decreases. The analysis without knowing the other consumer’s demands. From Theo- is analogous for the second good. rem 3, consumer 1 (analogously consumer 2) needs to know If each consumer is to demand amounts of goods without 2 MRT the constants g1, gˇ1, and g21. The measure σ + λ2 g21 in considering the demands of the other consumer, then it is (10) is the noise plus interference power in Nash equilibrium. important that the consumers’ demands equal the consumers’ This measure is reported from receiver 1 to its transmitter at supply of goods. Prices which fulfill this requirement are Nash equilibrium which is the initial state of the links before called Walrasian and are calculated next. coordination takes place. The demand functions of the consumers in Theorem 3 B. Walrasian Equilibrium are homogenous of degree zero [9, Definition A2.2] with In a Walrasian equilibrium, the demand equals the supply the prices p1 and p2. That is, the demand of consumer 1 for good 1 (analogously consumer 2 for good 2) satisfies of each good [9, Definition 5.5]. According to the properties (1) (1) of the utility function in Theorem 1, there exists at least one x∗ (tp ,tp )= x∗ (p ,p ) for t> 0. Hence, given only a 1 1 2 1 1 2 Walrasian equilibrium [9, Theorem 5.5]. The Walrasian prices prices ratio p¯1/p¯2, we can calculate a prices pair as p1 =p ¯1/p¯2 (p1∗,p2∗) that lead to a Walrasian equilibrium satisfy and p2 = 1 which leads to the same demand as with p¯1 and p¯2. With this respect, a consumer need only know the price (1) (2) MRT x1∗ (p1,p2)+ x1∗ (p1,p2)= λ1 , (33) ratio p /p from the arbitrator to calculate his demands. In 1 2 (1) (2) MRT Fig. 4, the demand of consumer 1 is illustrated as the point and x2∗ (p1,p2)+ x2∗ (p1,p2)= λ2 . (34) 8

TABLE I In our setting in which only two goods exist, Walras’ law [9, REQUIRED INFORMATION AT THE ARBITRATOR AND TRANSMITTERS TO Chapter 5.2] provides the property that if the demand equals IMPLEMENT THE WALRASSIANEQUILIBRIUMINONE-SHOT. the supply of one good, then the demand would equal the Information supply of the other good. Hence, in order to calculate the 2 Arbitrator h11, h12, h21, h22, σ Walrasian prices, it is sufficient to consider only one of the h h 2 MRT h 2 h 2 Transmitter 1 11, 12, σ + λ2 k 21k , k 21k h h 2 MRT h 2 h 2 conditions in (33) and (34). Transmitter 2 22, 21, σ + λ1 k 12k , k 12k Theorem 4: The ratio of the Walrasian prices is the unique root of TABLE II REQUIRED INFORMATION AT THE ARBITRATOR AND TRANSMITTERS FOR THEPRICEADJUSTMENTPROCESS. p 5 p 4 p 3 p 2 p a 1 +b 1 +c 1 +d 1 +e 1 +f =0, (35) p2 p2 p2 p2 p2 Information           h 2 h 2 MRT MRT 2 Arbitrator k 21k , k 12k , λ1 , λ2 , σ h h 2 MRT h 2 h 2 that satisfies the condition in (31). The constant coefficients Transmitter 1 11, 12, σ + λ2 k 21k , k 21k h h 2 MRT h 2 h 2 are Transmitter 2 22, 21, σ + λ1 k 12k , k 12k

2 a = T1T2 T 3, b = 2T3T2(T2S2 + T1S1), − C. Coordination Mechanism c =2T4T2S3 +4S1S2T2T3 + T1S4T3,

d = 2S4S2T3 4T1T2S2S3 S1T4S3, In this section, we provide two coordination mechanisms − − − which require different amount of information at the arbitrator. e =2S S (T S + T S ), f = S S2S , 3 2 2 2 1 1 − 1 2 3 If the arbitrator has full knowledge of all parameters of the where setting, then he can calculate the Walrasian prices from Theo- rem 4 and forward these to the transmitters. The transmitters MRT 2 calculate their demands from Theorem 3 and choose the T1 = (g1 gˇ1)/(g1 +ˇg1),T2 = λ1 + σ /g12, − beamforming vectors accordingly. This mechanism that uses T = (1 λMRT)λMRT,T = gˇ2 gˇ g + g2 /(g +ˇg )2, 3 − 1 1 4 1 − 1 1 1 1 1 the results in Theorem 3 and Theorem 4 leads directly to the S = (g gˇ )/(g +ˇg ), S = λMRT + σ2/g , 1 2 − 2 2 2 2 2
21 Walrasian equilibrium. In Table I, the required information MRT MRT 2 2 2 S3 = (1 λ2 )λ2 , S4 = gˇ2 gˇ2g2 + g2 /(g2 +ˇg2) , at the arbitrator and the transmitters to implement this one- − − shot mechanism are listed. We assume that each transmitter
and gˇk,gk,gkℓ are defined in Lemma 1. forwards the channel information it has to the arbitrator. Note Proof: Substituting (27) and (30) in (33) and collecting that each transmitter k initially knows the channel vectors hkk p1/p2 we get the expression in (35). The condition in (31) and hkℓ, k = ℓ, which are required to calculate the efficient states the set of feasible prices such that the demands of the beamforming6 vectors in (5). Also, transmitter k knows the 2 MRT 2 consumers are feasible. At least one price pair is in this set sum σ + λℓ hℓk , k = ℓ, since this is the noise plus since a Walrasian equilibrium always exists in our setting. interference in Nashk k equilibrium6 forwarded through feedback In addition, having the property that the goods are gross from the intended receiver. The arbitrator, which now has substitutes in Lemma 2, implies that the Walrasian equilibrium full knowledge of all channels, can then forward the missing 2 in our setting is unique [28, Proposition 17.F.3]. Note that the information on the channel gain hℓk to a transmitter k . k k roots in (35) can be easily calculated using a Newton method. If the arbitrator has limited information about the setting, And due to the uniqueness of the Walrasian prices, only one we could still achieve the Walrasian prices through an iterative root satisfies the condition in (31). price adjustment process. For fixed arbitrary initial prices, According to the First Welfare Theorem [9, Theorem 5.7], the transmitters can calculate their demands and forward the Walrasian equilibrium is Pareto optimal. Moreover, linking these to the arbitrator. The arbitrator exploits the demand to the results in the previous section, the Walrasian equilibrium information to update the prices of the goods. Specifically, lies in the core [9, Theorem 5.6]. In other words, the Wal- the arbitrator would increase the price of the good which rasian equilibrium dominates the Nash equilibrium outcome. has higher demand than its supply. Due to the properties of In Fig. 5, the allocation in Walrasian equilibrium which the goods in Lemma 2, this price adjustment process, also corresponds to the Walrasian prices ratio p1∗/p2∗ is illustrated called tˆatonnement, is globally convergent to the Walrasian in the Edgeworth box. It is the point on the contract curve prices given in Theorem 4 [29]. The price adjustment process which intersects the line that passes through the endowment requires the information listed in Table II to be available at point (Nash equilibrium) with slope p1∗/p2∗ (with respect to the arbitrator and the transmitters. In contrast to Table I, the the coordinate system of consumer 1).− The grey area in Fig. 5 arbitrator requires aside from the noise power σ2 only the cross is the budget set of consumer 1 as described in Fig. 4. The channel gains h 2, h 2 and the parameters λMRT, λMRT k 21k k 12k 1 2 white area in the Edgeworth box is the budget set of consumer from the transmitters. This information is required only at the 2. According to the axis transformation in constructing the beginning of the price adjustment process in order to calculate Edgeworth box, the boundaries of the consumers’ budget sets the bounds for the feasible prices β and β given in (31). coincide. The indifference curves of the consumers are tangent In Algorithm 1, the price adjustment process is described. to this line and also tangent to one another which illustrates This process is essentially a bisection method which finds the the Pareto optimality of the Walrasian equilibrium. roots of the excess demand function described in the proof of 9

Algorithm 1: Distributed price adjustment process. SNR = 10 dB joint ZF (1) (2) (1) (2) 0.6 Input: x1 , x1 , x2 , x2 SNR = 5 dB (0) (0) 1 initialize: accuracy ǫ, n =0, β = β, β = β in (31), p(0) (0) β(0) 0.4 1 = β + ; (0) 2 2 (1) 2 SNR = 0 dB p2 x (n) (n) 2 while β β >ǫ do − 0.2 (1) (2) (1) (2) SNR = −5 dB 3 receive demands x1 , x1 , x2 , x2 ; 4 n = n +1; 0 joint MRT (1) (2) MRT SNR = −10 dB 5 if x1 + x1 > λ1 then (n−1) (n) p (n) (n 1) −0.2 0 0.2 0.4 0.6 0.8 1 6 1 − β = (n−1) , β = β ; (1) p2 x (n) 1 p(n) β +β(n) 7 1 (n) = 2 ; p2 Fig. 7. Course of the contract curve in the Edgeworth box for different SNR 8 else values. (n) (n−1) (n) (n 1) p1 9 β = β − , β = (n−1) ; p2 (n) p(n) β +β(n) 10 1 (31). (n) = 2 ; p2 (n) (n) Output: p1 /p2 V. DISCUSSION AND ILLUSTRATIONS In Fig. 7, the contract curve characterized in Theorem 2 is plotted for different SNR values. The number of antennas at Lemma 2. The accuracy measure conditioning the termination the transmitters is two and we generate independent instanta- of the algorithm is defined as . The terms and are ǫ β β neous channels hkℓ identically distributed as (0, I). The the lower and upper bounds on the price ratio given in (31), 3 CN (2) contract curve is calculated by taking 10 samples of x2 respectively. The prices ratio is initialized to the middle value (1) uniformly spaced in (0, λMRT) to obtain values of x . The of these bounds and forwarded to the links. The links send 2 1 course of the contract curve for 10 dB SNR is near to the their demands calculated from Theorem 3 to the arbitrator. edge of the Edgeworth box where joint ZF is marked. This If the demand of good 1 is greater than its supply, then the means that Pareto optimal allocations require either transmitter arbitrator increases the ratio of the prices to half the distance to choose beamforming vectors near to ZF. For decreasing to the upper bound β. Thus, the price of good 1 relative to SNR, the contract curve moves away from the ZF edge. For the price of good 2 increases. The lower bound on the prices low SNR, the contract curve is then close to the edge with ratio β is updated to the price ratio of the previous iteration. joint MRT. These observations conform with the analysis in If the demand of good 1 is less than its supply, the price [30] where Pareto optimal maximum sum utility transmission ratio is decremented half the distance to the lower bound β. is studied in low and high SNR regimes. The upper bound β is set to the prices ratio of the previous iteration. The algorithm terminates when the distance between 0.3 the updated upper and lower bounds on the prices ratio is 0.25 contract curve below an accuracy measure ǫ. 0.2 indifference curve Walrasian (consumer 1) 3

(1) 2 equilibrium

x 0.15 upper bound on prices ratio

) 2.5 0.1 2

/p indifference curve 1 2 0.05 (consumer 2) 0 1.5 Walrasian prices ratio 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 x(1) 1 prices ratio (p 1 lower bound on prices ratio Fig. 8. Edgeworth box which depicts the allocation for the Walrasian prices. 0.5 1 2 3 4 5 6 7 8 9 10 iteration In Fig. 8, an Edgeworth box is plotted for a sample channel Fig. 6. Convergence of the price ratio in the price adjustment process to the realization with two transmit antennas at both transmitters. For Walrasian price ratio. the prices calculated from Theorem 4 we obtain the Walrasian equilibrium allocation on the contract curve where the cor- In Fig. 6, the prices ratio in the price adjustment process responding indifference curves are tangent. The indifference is marked with a cross and is shown to converge after a few curves are obtained from Proposition 1. The line passing iterations to the Walrasian prices ratio from Theorem 4. The through Walrasian equilibrium allocation defines the budget dashed lines correspond to the upper and lower bounds in sets of the consumers as is illustrated in Fig. 5. 10

1.5 maximum sum SINR In Fig. 9, two solutions from axiomatic bargaining theory, namely the Nash bargaining solution (NBS) and the Kalai- Pareto boundary Smorodinsky (KS) solution are plotted. These solutions lie in the core and differ by the axioms that define them. The 1

) virtual SINR coordination interested reader is referred to [23] for a comprehensive (2) 2

,x theory on axiomatic bargaining. According to simulations, core allocations (2) 1 these two solutions are not far from each other. The properties (x

2 ZF Walrasian equilibrium φ that the Walrasian equilibrium and the NBS or KS solution 0.5 Nash bargaining solution have in common is that they are Pareto optimal and lie in Kalai−Smorodinsky solution the core, i.e., each user achieves higher utility than at the

weak MRT Nash equilibrium. The difference between the solutions is the Pareto boundary fairness aspects in allocating the Pareto optimal utilities to the 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 players. The current advantage in the Walrasian equilibrium φ (x(1),x(1)) over NBS and KS solution is that it can be characterized in 2 1 2 closed-form using Theorem 3 and Theorem 4. In addition, we Fig. 9. SINR region of a two-user MISO IFC with SNR = 0 dB and two devise a coordination mechanism to implement the Walrasian antennas at the transmitters. equilibrium in Section IV-C. Next, we will describe how the NBS and KS solutions are obtained. The NBS [24, Chapter 15] is the solution of the following problem: In Fig. 9, the SINR region is plotted. The points lying NE NE inside the SINR region correspond to the beamforming vectors maximize (φ1 φ )(φ2 φ ) − 1 − 2 (38) characterized in (5), where a subset of these points are Pareto subject to (φ1, φ2) Φ, optimal. The Pareto boundary corresponds to the allocationson ∈ NE MRT MRT the contract curve calculated in Theorem 2. The weak Pareto where φk := φk(λ1 , λ2 ) is the SINR in Nash equilibrium boundary consists of weak Pareto optimal points in which and Φ is the SINR region in (3). Note that the NBS is defined the links cannot strictly increase their utility simultaneously. for convex utility regions only, and the SINR region Φ in our Formally, the weak Pareto optimal points of the SINR region case is not necessarily convex as is shown in Fig. 9. However, solving the optimization problem in (38) by grid search over Φ are defined as [23, p. 14] 103 generated Pareto optimal points from Theorem 2 gives a (Φ) := x Φ: there is no y Φ with y > x , (36) single solution which we plot in Fig. 9. The KS solution is W { ∈ ∈ } the solution of the following problem [31]: where the inequality in (36) is componentwise. Pareto optimal NE NE φ1 φ1 φ2 φ2 points (Φ) in (4) define a stronger optimality for a utility maximize min − , − φCORE φNE φCORE φNE (39) tuple thanP weak Pareto optimal points. A weak Pareto optimal  1 − 1 2 − 2  subject to (φ , φ ) Φ, point is not necessarily Pareto optimal. But all Pareto optimal 1 2 ∈ points are also weak Pareto optimal, i.e. (Φ) (Φ). where φCORE (analogously φCORE) is the solution of the following P ⊆W 1 2 The core allocations are all Pareto optimal points that problem: dominate the Nash equilibrium (joint MRT). Assuming the maximize φ1 links are rational, only allocations in the core can be of interest (40) subject to (φ , φNE) Φ. for the links. In other words, the links will not cooperate if one 1 2 ∈ link would achieve lower payoff than at the Nash equilibrium. CORE NE NE CORE The two Pareto optimal points (φ1 , φ2 ) and (φ2 , φ2 ) The Walrasian equilibrium from Theorem 4 always lies in the are the bounds to the core and are marked with circles on core. In Fig. 9, we also plot the maximum sum SINR which the Pareto boundary in Fig. 9. These bounds, as discussed in is obtained by grid search over the allocations on the Pareto Section III-B, can be calculated in the Edgeworth box as the boundary. The virtual SINR coordination point corresponds intersection of the contract curve and the indifference curves to the coordination mechanism in [14], where the minimum corresponding to the Nash equilibrium. The KS solution which mean square error (MMSE) transmit beamforming vectors solves the problem in (39) using the core bounds is then found by grid search over the generated Pareto optimal points from [σ2I + h hH ] 1h wMMSE = kℓ kℓ − kk , k = ℓ, (37) Theorem 2. k 2 H 1 [σ I + hkℓh ] hkk 6 k kℓ − k are proven to achieve a Pareto optimal point. These beam- A. Difficulties in the Extension to K-User MISO IFC forming vectors require only local channel state information While the tools in the paper can be applied to general at the transmitters which is an appealing property in terms of K consumer and M goods economy as can be found in the low overhead in information exchange between the links. [9], [28], the application to the beamforming problem in the The virtual SINR coordination and the maximum sum SINR MISO IFC can currently be done only for the two-user case. points do not necessarily lie in the core. Hence, these points This is mainly because of the structure of the parametrization are not suitable for distributed implementation between the available for the efficient beamforming vectors in the general rational links. case. 11

Using the parametrization in (5) for two-users, we have strategic game between the links. We propose a coordination chosen in Section III the amount of good 1 for consumer 1 mechanism between the links which achieves a Pareto optimal (1) as x1 = λ1 and the amount of good 1 for consumer 2 as outcome in the core. For this purpose, the situation between (2) MRT the links is modeled as a competitive market where now each x1 = λ1 λ1. With this relation between the parameters and the goods and− due to the structure of the expression in (5), the consumer is endowed with a budget and can consume the (1) (1) goods at specific prices. The equilibrium in this economy is SINR in (10) for link 1 depends only on x1 and x2 which are the amounts from good 1 and good 2 for consumer 1. This called Walrasian and corresponds to the prices that equate the method of defining the goods in terms of the parameters does demand to the supply of goods. The unique Walrasian prices not carry on for the K-user MISO IFC case. We illustrate are calculated and the coordination mechanism is executed this drawback based on an example in the 3-user case. The by an arbitrator that forwards the prices to the consumers. parametrization for the beamforming vectors are [7] The consumers then calculate in a decentralized manner their optimal demand corresponding to beamforming vectors that w1(λ11, λ12, λ13) achieve the Walrasian equilibrium. This outcome is Pareto H H H optimal and dominates the Nash equilibrium in the SINR = vmax λ h h λ h h λ h h , (41) 11 11 11 − 12 12 12 − 13 13 13 region.  

w2(λ21, λ22, λ23) APPENDIX A H H H PROOF OF LEMMA 1 = vmax λ21h21h + λ22h22h λ23h23h , (42) − 21 22 − 23 H 2 The direct and interference power gains, hkkwk(λk)   H 2 | | and h wk(λk) , k = ℓ, are calculated as functions of the | kℓ | 6 w3(λ31, λ32, λ33) parameters λk by using the expression for the beamforming H H H vectors in (5). The direct power gain is calculated as: = vmax λ31h31h31 λ32h32h32 + λ33h33h33 , (43) − − H 2   hkkwk(λk) where vmax(Z) is the eigenvector that corresponds to the | | 2 H H largest eigenvalue of Z and λk1 + λk2 + λk3 =1, k =1, 2, 3. Π h Π h hkk hkℓ hkk kk h⊥kℓ kk = λk + 1 λk (44) Note that different real-valued parameterizations are also pro- Πh h Π kℓ kk − h⊥ hkk ! vided in [5], [6], [8] which also lead to the same conclusion in k k k kℓ k p p 2 terms of the application of the exchange economy model. We Πh h Π h = λk kℓ kk + 1 λk h⊥kℓ kk . (45) use the parametrization in [7] in order to highlight the usage k k − k k   of the different parameters. In (41)-(43), three goods can be The interferencep power is: p directly distinguished each corresponding to the parameters H 2 h wk(λk) of each transmitter. We can choose the amount of good 1 | kℓ | H Π H Π 2 (analogously for goods 2 and 3) to be divided between the hkℓ hkℓ hkk hkℓ h⊥kℓ hkk (1) (2) = λk + 1 λk (46) three links as x = λ for link 1, x = λ for link 2, Πh Π 1 11 1 12 kℓ hkk − h⊥ hkk (3) k k k kℓ k and x = λ13 for link 3. In order to model this setting as an p H p 1 h Πh h 2 exchange economy, the utility (SINR) of link k should only = λ kℓ kℓ kk = λ h 2. (47) k | Π 2| k kℓ (k) (k) (k) hkℓ hkk k k depend on the amounts of goods x1 , x2 , x3 . However, k k with the parametrization in (41)-(43), the SINR expression of These expressions lead to (6) and (7) in Lemma 1. a link k would depend on all parameters. Hence, in formulating the demand of consumer k as is done in the two-user case in APPENDIX B (25), the solution depends also on the demands of the other PROOF OF THEOREM 1 consumers. In this case, each consumer cannot find his optimal First, it is easy to see that the SINR expression in (10) demand of goods independently without knowing what the (k) (k) is continuous. The SINR φk(x1 , x2 ) is strongly increas- other consumers demand. Due to this fact, it is currently not (k) (k) (k) (k) ing with the goods x1 and x2 if φk(x′1 , x′2 ) > possible to find the Walrasian equilibrium in the general K- (k) (k) (k) (k) (k) (k) φk(x1 , x2 ) whenever (x′1 , x′2 ) = (x1 , x2 ) and user MISO IFC case. (k) (k) (k) (k) 6 (x′1 , x′2 ) (x1 , x2 ) [9, Definition A1.17]. Define the ≥ (k) (k) VI. CONCLUSIONS directional derivative of φk at (x1 , x2 ) in direction z as In this work, we model the interaction between two links in (k) (k) zφk x , x the MISO IFC as an exchange economy. The links are consid- ∇ 1 2 ered as the consumers and the exchanged goods correspond  (k) (k) (k) (k) φk x1 , x2 + tz φk x1 , x2 to beamforming vectors. Utilizing the conflict representation = lim − , (48) t 0   t    in the Edgeworth box, all Pareto optimal points could be → (k) (k) characterized in closed form. The equilibria of the considered Since φk(x1 , x2 ) is differentiable, the limit above can be exchange economy are related to a solution concept from given as [9, Chapter A.2] coalitional game theory called the core. These allocations (k) (k) (k) (k) zφk x , x = φk x , x z, (49) are Pareto optimal and dominate the Nash equilibrium of a ∇ 1 2 ∇ 1 2     12

(k) (k) (k) (k) (k) where φk(x1 , x2 ) is the gradient of φk at (x1 , x2 ) strictly concave, we build the second derivative of f(xk ) as written∇ as follows:

(k) (k) (k) (k) 2 (k) 2 ∂φk x , x ∂φk x , x d f(xk ) (k) (k) (k) (k) 1 2 1 2 (k) = gk/xk gˇk/(1 xk ) φk x1 , x2 = , , 2 − − ∇   (k)   (k)  d xk   ∂xk ∂xℓ q q   (k) (k)  (50) x gk + (1 x )ˇgk (k) (k) − k − k with ℓ = k. The directional derivative of φk(x1 , x2 )   6 k k q q defines the slope of the tangent to φ (x( ), x( )) at the point k 1 2 gk gˇk (k) (k) in the direction z. Hence, if the directional deriva- + (54) (x1 , x2 ) ×  (k) 3 (k) 3  T s(xk ) s(1 xk ) tive is positive for z = (z1,z2) with z1 and z2 nonnegative − 2 2   and satisfying z = z1 + z2 =1, then the utility function (k) (k) k k φk(x1 , x2 ) is strongly increasing. Consequently, the direc- g gˇ g gˇ g p = k + k 2 k k k tional derivative in (49) is strictly positive if the components (k) (k) − (k) (k) − (k) (k) (k) xk (1 xk ) s(1 xk )(xk ) xk of the gradient φk(x1 , x2 ) are strictly positive. The first − − ∇ k k component of ( ) ( ) is (k) (k) φk(x1 , x2 ) gˇ (1 x )gkgˇk x gkgˇk ∇ k − k k (55) − (k) − v (k) 3 − v (k) 3 (k) (k) (1 xk ) u (xk ) u(1 xk ) ∂φk x1 , x2 − u u − t (k)t (k)   gkgˇk (1 xk )gkgˇk ∂xk = 2 − − (1 x(k))(x(k)) − v (x(k))3 (k) (k) gk gˇk s k k u k x gk + (1 x )ˇgk k k − u k k x( ) 1 x( ) t − k − − k (k) = q q  . x gkgˇk 2 MRT (qk) q k σ + λ gℓk x gℓk < 0. (56) ℓ − ℓ − v (k) 3 (51) u(1 xk ) u − t The partial derivative in (51) is strictly larger than zero when (k) The second derivative of f(xk ) is strictly less than zero. (k) (k) (k) (k) x < gk/(ˇgk + gk). Substituting gˇk and gk from Lemma 1 k Thus, f(xk ) is strictly concave. Accordingly, φk(x1 , x2 ) we get is strictly quasiconcave. 2 g Πh h (k) k kℓ kk MRT (52) xk < = k 2 k = λk . APPENDIX C gˇk + gk hkk k k PROOF OF THEOREM 3 (k) Since x [0, λMRT], the partial derivative in (51) is strictly (k) (k) k k Since the function φk(x , x ) is strictly quasiconcave, ∈ (k) MRT 1 2 larger than zero except for xk = λk . The second component then this function has a unique maximum. Considering con- (k) (k) of φk(x , x ) is sumer 1 (analogously consumer 2), the Lagrangian function ∇ 1 2 2 to the constrained optimization problem in (25) is (k) (k) (k) (k) x gk + (1 x )ˇgk (1) (1) (1) (1) ∂φk x1 , x2 k k − x1 , x2 ,µ = φ1 x1 , x2 = gℓk q q 2 , L  (k)  (k) (57) ∂x 2 MRT    MRT  (1) (1) ℓ σ + λℓ gℓk xℓ gℓk + µ λ p x p x p , − 1 1 1 1 2 2 (53) − −     (k) where is a Lagrange multiplier. The Karush–Kuhn–Tucker with ℓ = k, which is strictly larger than zero for xℓ µ MRT 6 ∈ [0, λℓ ]. Hence, the directional derivative in (49) is strictly (KKT) conditions for optimality are necessary and sufficient (k) (k) MRT MRT given as: positive for (x1 , x2 ) [0, λ1 ] [0, λ2 ] except for the (k) MRT ∈ × MRT case xk = λk and z = (1, 0). Since λk is the upper (1) (1) (1) (1) (k) (k) (k) ∂ x1 , x2 ,µ ∂φ1 x1 , x2 bound on xk , the slope of the function φk(x1 , x2 ) in the L = µp =0 (58) (k)  (1)   (1)  − 1 direction xk as is restricted by the condition z = (1, 0) is ∂x1 ∂x1 not of interest. (1) (1) (1) (1) ∂ x1 , x2 ,µ ∂φ1 x1 , x2 Next, we will prove that the SINR function is jointly L = + µp =0 (59)  (1)   (1)  2 quasiconcave with the goods. Consider the SINR expression in ∂x2 ∂x2 2 (k) (k) (k) (1) (1) (10), and define f(x ) := x gk + (1 x )ˇgk ∂ x1 , x2 ,µ k k k L MRT (1) (1) − = λ1 p1 x1 p1 x2 p2 =0 (60) (k) 2 MRTq (k) q   ∂µ  − − and g(xℓ ) := σ + λℓ gℓk xℓ gℓk. The function (k) (k) (k) (k) − φk(x1 , x2 ) = f(xk )/g(xℓ ) is strictly quasiconcave According to conditions (58) and (59), we get (k) (k) if f(xk ) is strictly concave and g(xℓ ) is convex [32, (1) (1) (1) (1) Proposition 2]. It is clear that g(x(k)) is convex since the ∂φ1 x1 , x2 1 ∂φ1 x1 , x2 1 ℓ (61) (k) (k) (1) = (1) function is linear in . In order to show that is   p1 −   p2 xℓ f(xk ) ∂x1 ∂x2 13

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