1 Coalitional Games in MISO Interference Channels: Epsilon-Core and Coalition Structure Stable Set Rami Mochaourab, Member, IEEE, and Eduard Jorswieck, Senior Member, IEEE,

Abstract—The multiple-input single-output interference chan- single-output (MISO) interference channel (IFC) [3]. Next, we nel is considered. Each transmitter is assumed to know the describe existing work on beamforming mechanisms in this channels between itself and all receivers perfectly and the re- setting assuming perfect channel state information (CSI) at ceivers are assumed to treat interference as additive noise. In this setting, noncooperative transmission does not take into account the transmitters and single-user decoding at the receivers. the interference generated at other receivers which generally leads to inefficient performance of the links. To improve this A. Beamforming in the MISO interference channel situation, we study cooperation between the links using coalitional games. The players (links) in a coalition either perform zero Optimal beamforming in the MISO IFC corresponds to forcing transmission or Wiener filter precoding to each other. The rate tuples at which it is not possible to strictly improve ǫ-core is a for coalitional games which takes into the performance of the users jointly. Such points are called account the overhead required in coalition deviation. We provide Pareto optimal. Finding special Pareto optimal operating points ǫ necessary and sufficient conditions for the strong and weak - in the MISO IFC such as the maximum weighted sum-rate, core of our coalitional game not to be empty with zero forcing transmission. Since, the ǫ-core only considers the possibility of geometric mean, and proportional-fair rate points are NP-hard joint cooperation of all links, we study coalitional games in problems [4]. However, finding the max-min Pareto optimal partition form in which several distinct coalitions can form. operating point is polynomial time solvable [4]. In [5], two We propose a polynomial time distributed coalition formation distributed algorithms are proposed to compute the max-min algorithm based on coalition merging and prove that its solution operating point. After exchanging optimization parameters, lies in the coalition structure stable set of our coalition formation game. Simulation results reveal the cooperation gains for different the computational load of the beamforming vectors is carried coalition formation complexities and deviation overhead models. out sequentially at the transmitters. In [6], a monotonic op- timization framework is proposed to find points such as the Index Terms—interference channel; beamforming; coalitional maximum sum-rate operating point in general MISO settings games; epsilon-core; coalition structure stable set with imperfect channel state information at the transmitters. The interested reader is referred to [7] for characterizations of optimal beamforming in MISO settings. I.INTRODUCTION Since optimal beamforming requires high information ex- In multiuser interference networks, interference can be the change between the transmitters (or central controller), low main cause for performance degradation of the systems [2]. complexity and distributed transmission schemes are desirable With the use of multiple antennas at the transmitters, interfer- for practical implementation. When utilizing the reciprocity ence can be managed through cooperative beamforming tech- of the uplink channel in time division duplex (TDD) systems, niques. For this purpose, backhaul connections are necessary each transmitter is able obtain perfect local CSI of the channels in order to exchange information for cooperation between the between itself and all receivers [8]. Cooperative beamforming transmitters [2]. In this work, we consider cooperation between schemes based on local CSI do not require CSI exchange the transmitters at the beamforming level only, i.e., information through the backhaul connections and are hence favorable [2]. concerning the joint choice of beamforming vectors at the One cooperative beamforming scheme which requires local transmitters is exchanged between the transmitters but not the CSI is zero forcing (ZF) transmission. This transmission signals intended to the users. scheme produces no interference at unintended receivers. The system we consider consists of a set of transmitter- Heuristic ZF transmission schemes in multicell settings have receiver pairs which operate in the same spectral band. The been proposed in [9], [10], where the objective is to efficiently transmitters use multiple antennas while the receivers have select a subset of receivers at which interference is to be single antennas. This setting corresponds to the multiple-input nulled. In [9], the transmitters perform ZF to receivers which are mostly affected by interference. In [10], a successive c 2014 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, greedy user selection approach is applied with the objective including reprinting/republishing this material for advertising or promotional of maximizing the system sum-rate. Another beamforming purposes, creating new collective works, for resale or redistribution to servers scheme which requires local CSI is Wiener filter (WF) precod- or lists, or reuse of any copyrighted component of this work in other works. 1 Part of this work was presented at IEEE CAMSAP, San Juan, Puerto Rico, ing [11]. Joint WF precoding in MISO IFC is proposed in [12] December 13-16, 2011 [1]. Rami Mochaourab is with ACCESS Linnaeus as a non-iterative cooperation scheme. For the two-user case, Centre, Signal Processing Department, School of Electrical Engineering, the obtained operating point is proven to be Pareto optimal. KTH Royal Institute of Technology, 100 44 Stockholm, Sweden. Phone: +4687908434. Fax: +4687907260. E-mail: [email protected]. Ed- Furthermore, in [13], the authors study the reciprocity of the uard Jorswieck is with Communications Theory, Communications Lab, TU Dresden, 01062 Dresden, Germany. E-mail: [email protected]. 1Also called minimum mean square error (MMSE) transmit beamforming. 2 uplink and downlink channels in the MISO IFC to formulate restricted to merging of pairs of coalitions only supporting a distributed beamforming scheme in the MISO IFC. In the low complexity implementation. proposed beamforming schemes in [12], [13], all transmitters While in Dhp-stability and the recursive core solution con- cooperate with each other. cepts a set of players can deviate and form new coalitions, Joint cooperation between the links does not necessarily individual based stability [27] restricts only a single player lead to an improvement in the rates of the users compared to leave a coalition and join another. A deviation in which to a noncooperative and noncomplex beamforming method. a user leaves a coalition and joins another if this improves Accordingly, a user may not have the incentive to cooperate his payoff leads to Nash stability [28] and has been used in with all other users. It is then of interest to devise stable [29] in the context of channel sensing and access in cognitive cooperative mechanisms that also determine which links would radio. In [30], individual stability which is a weaker stability cooperate voluntarily with each other. concept than Nash stability is used for coalitional games in the provides appropriate models for designing multiple-input multiple-output (MIMO) interference channel. distributed resource allocation mechanisms. The conflict in A coalition of links cooperate by performing ZF to each other. multiple antenna interference channels is studied using game Individual stability requires additional to Nash stability the theory in [14]. The noncooperative operating point (Nash equi- constraint that the payoffs of the members of the coalition in librium) in the MISO IFC corresponds to joint maximum ratio which the deviator wants to join do not decrease. transmission (MRT). This is found to be generally not efficient [15]. In order to improve the performance of the Nash C. Contributions equilibrium, interference pricing is applied in [16]. Cooperative games in the MISO IFC have been applied We consider coalitional games without transferable utilities in [17], [18], [19]. The Kalai-Smorodinsky solution from [31] among the links. While noncooperative transmission axiomatic bargaining theory is studied in the MISO IFC in [17] corresponds to MRT, we restrict cooperation between a set and an algorithm is provided to reach the solution. In the two- of links to either ZF transmission or WF precoding. In [1], user case, all cooperative solutions, called exchange equilibria, the necessary and sufficient conditions for a nonempty core of are characterized in [18] and a distributed mechanism is the coalitional game with ZF transmission are characterized. In proposed to reach the Walrasian equilibrium in the setting. this work, we provide the necessary and sufficient conditions Using strategic bargaining, an operating point in the set of for nonempty strong and weak ǫ-core [32] of the coalitional exchange equilibria is reached requiring two-bit with ZF beamforming. The ǫ-core generalizes the core between the transmitters in [19]. The approaches in [18], [19] solution concept and includes an overhead for the deviation are however limited to the two-user case. of a coalition. In contrast to the result in [1] which specifies an SNR threshold above which the core is not empty, the ǫ- core is not empty above an SNR threshold and also below B. Applications of Coalitional Games in Partition Form a specific SNR threshold. These thresholds depend on the Coalitional games provide structured methods to determine deviation overhead measure and the user channels. possible cooperation between rational players. A tutorial on the While the strong and weak ǫ-core solution concepts consider application of coalitional games in communication networks the formation of a grand coalition only, we study coalitional can be found in [20]. In these games, a coalition is a set of games in partition form in which several distinct coalitions players which would cooperate to achieve a joint performance can form. In [1], coalition formation based on merging and improvement. In interference networks, the performance of a splitting of coalitions has been applied. In this work, we coalition of players depends on the coalitions formed outside propose a distributed coalition formation algorithm based on the coalition. Appropriate in this context are coalitional games coalition merging only. We propose a coalition deviation rule, in partition form [21] which take into account what the players q-Deviation, to incorporate a parameter q which regulates the achieve given a coalition structure, a partition of the set of complexity for finding deviating coalitions. The of users into disjoint cooperative sets. the coalition formation algorithm is proven to be inside the There are different stability concepts for coalitional games coalition structure stable set [33] of our coalition formation in partition form. In [22], Dhp-stability is proposed which is game. Accordingly, the stability of the obtained partition of the based on deviation rules of coalition merging and splitting. links is ensured. We provide an implementation of the coalition The stability concept is used for games with transferable formation algorithm and show that only two-bit signaling utility in [23] and also applied in [1] in the MISO IFC for between the transmitters is needed. Moreover, we prove that games with nontransferable utility and partition form. The the proposed coalition formation algorithm terminates in poly- recursive core [24] solution concept for coalitional games in nomial time. Simulation results reveal the tradeoff between partition form has been applied in [25], [26]. In [25], the set coalition formation complexity and the obtained performance of cooperating base stations (a coalition) performs interference of the links. In addition, we compare our algorithm regarding alignment. An algorithm is proposed in which the coalitions complexity and performance to the algorithms in [26] and [30]. can arbitrarily merge and split and proven to converge to To the best of our knowledge, the application of the ǫ-core of an element in the recursive core. In [26], MISO channels coalitional games and the coalition structure stable set solution are considered and the set of cooperating transmitters apply concepts are new for resource allocation in wireless networks. network MIMO techniques. Coalition formation in [26] is Note, that the coalition structure stable set solution concept 3 for coalitional games in partition form is different than the B. Noncooperative Operation recursive core [24] solution concept used in [25], [26]. In game theory, games in strategic form describe outcomes Outline: In Section II, we provide the system and channel of a conflict situation between noncooperative entities. A model and also describe the noncooperative state of the links. strategic game is defined by the tuple , , (ui)i∈N , where In Section III, the game in coalitional form is formulated is the set of players (links), ishN theX strategy spacei of and its solution is analyzed. In Section IV, we formulate the N X the players given in (3), and uk is the utility function of game in partition form and specify our coalition formation player k given in (4). In [34], it is shown that maximum ratio mechanism. We also study the complexity of the proposed transmission (MRT), written for a transmitter i as deviation model and provide an implementation of the coali- MRT tion formation algorithm in our considered system. In Section wi = hii/ hii , (5) k k V, we provide simulation results. is a unique dominant strategy. A dominant strategy equilibrium Notations: Column vectors and matrices are given in low- [31, Definition 181.1] of a strategic game is a strategy profile ercase and uppercase boldface letters, respectively. a is the ∗ ∗ k k (w1, ..., wK ) such that for every player i Euclidean norm of a CN . b and denote the absolute ∈ N ∈ | | |S| ∗ value of b C, and the cardinality of a set , respectively. ui(w , w i) ui(wi, w i), (wi, w i) , (6) i − ≥ − ∀ − ∈ X ( )H denotes∈ the Hermitian transpose. The orthogonalS projec- where w := (w ) is the collection of beam- tor· onto the null space of Z is Π⊥ := I Z(ZH Z)−1ZH , −i j j∈N \{i} Z forming vectors of all users other than user i. Hence, each where I is an identity matrix. (x ) −denotes a profile i i∈S transmitter chooses MRT irrespective of the strategy choice with the elements corresponding to the set . The notation of the other transmitters. Consequently, the strategy profile f(x) (g(x)) means that the asymptotic growthS of f(x) in (wMRT, ..., wMRT) is the unique of the strategic x is upper∈ O bounded by g(x). 1 K game between the links. In [15], it is shown that joint MRT is II. PRELIMINARIES near to the Pareto boundary of the achievable rate region in the A. System and Channel Model low SNR regime. In the high SNR regime, joint MRT has poor performance [34] while zero forcing (ZF) transmission is near Consider a K-user MISO IFC and define the set of links as to the Pareto boundary [15]. However, ZF beamforming cannot := 1, ..., K . Each transmitter i is equipped with Ni 2 N { } ≥ be implemented if the links are not cooperative. Therefore, we antennas, and each receiver with a single antenna. The quasi- will study cooperative games between the links. static block flat-fading channel vector from transmitter i to receiver j is denoted by h CNi×1. Each transmitter is ij III. COALITIONAL GAME assumed to have perfect local∈ CSI. The local CSI is gained through uplink training pilot signals [8]. We assume time A. Game in Coalitional Form division duplex (TDD) systems with sufficiently low delay In game theory, cooperative games are described by games between the downlink and uplink time slots such that, using in coalitional form. A game in coalitional form [31, Definition channel reciprocity, the downlink channels are estimated to be 268.2] is defined by the tuple the same as the uplink channels. , ,V, (ui)i , (7) The beamforming vector used by a transmitter i is denoted hN X ∈N i by wi i, where the set i is the strategy space of where is the set of players, is the set of possible joint transmitter∈i Adefined as A actionsN of the players in (3), V assignsX to every coalition (a S Ni×1 2 nonempty subset of ) a set V ( ) , and uk is the utility i := w C : w 1 , (1) N S ⊆ X A { ∈ k k ≤ } of player k given in (4). A coalition is a set of players that where we assumed a total power constraint of one (w.l.o.g.). are willing to cooperate, and V ( ) definesS their joint feasible The basic model for the matched-filtered, symbol-sampled strategies. The game considered inS (7) is in characteristic form complex baseband data received at receiver i is and the mapping V ( ), called the characteristic function, S H H assumes a specific behaviour for the players outside . yi = hii wisi + hji wj sj + ni, (2) S j6=i There exist several models that describe the behavior of the where s (0, 1) is theX symbol transmitted by transmitter players outside [35]. For our model, we adopt the γ-model j S j and n ∼ CN (0, σ2). We assume that all signal and noise from [36] and specify that all players outside a coalition i S variables are∼ CN statistically independent. Throughout, we define do not cooperate, i.e., build single-player coalitions. Later in the SNR as 1/σ2. Section IV, we consider a coalitional game in partition form in A strategy profile is a joint choice of strategies of all which the formation of several coalitions is feasible. Through- transmitters defined as out, we assume that the payoff of a player in a coalition cannot be transferred to other players in the same coalition. (w , ..., wK ) := K . (3) 1 ∈ X A1 ×···×A Thus, we consider games with nontransferable utilities which Given a strategy profile, the achievable rate of link i is is appropriate for our model in which the achievable rate of H 2 one link cannot be utilized at other links. hii wi ui(w1, ..., wK ) = log 1+ | | , (4) A solution of the coalitional game is the core which is a 2 hH w 2 2 j6=i ji j + σ ! set of joint strategies in with which all players want to | | X where we assume single-user decodingP receivers. cooperate in a grand coalition and any deviating coalition 4 cannot guarantee higher utilities to all its members. With this a set of links, then it performs ZF in the direction of the respect, the core strategies are stable. We adopt the following corresponding receivers. Hence, we define the mapping variant of the core [32].2 ZF ZF Definition 1: The weak ǫ-core of a coalitional game is the V ( )= (wi)i∈N : wi = wi→S for i , S { ∈ X MRT ∈ S set of all strategy profiles (xi)i∈N V ( ) for which there is wj = w for j , (8) ∈ N j ∈ N\S} no coalition and (y )i V ( ) such that ui(y , ..., y ) S i ∈N ∈ S 1 K − where wZF is transmitter i’s ZF beamforming vector to the ǫi >ui(x , ..., xK ) with ǫi 0 for all i . i→S 1 ≥ ∈ S links in written as The weak ǫ-core is not empty if there exists no coalition S Π⊥ h whose members achieve higher payoff than in ZF Zi→S ii S ⊂ N wi→S = ⊥ , Zi→S = (hij )j∈S\{i}. (9) the grand coalition taking the additional overhead ǫi in ΠZ hii N k i→S k deviation of each player i . Alternatively, ǫi can be ∈ S Observe that if the number of antennas Ni < , then ZF in considered as a reward which is given to player i in order |S| to motivate him to stay in the grand coalition. However, since (9) is the zero vector, i.e. transmitter i switches its transmission ZF in our model, no external entity is assumed which can give off. Similar to the definition of the strategy profile V ( ) in (8), it is possible to consider different cooperative transS mit such a reward to the users, the interpretation of ǫi as an overhead is more appropriate. Incorporating the overhead in beamforming than ZF in a coalition. the solution concept is appropriate in communication networks According to Definition 1, the weak ǫ-core is not empty if since coalition deviation requires an additional complexity and only if ZF ZF for searching for possible coalitions to cooperate with. This ui(V ( )) ǫi ui(V ( )), i and . (10) overhead is discussed later in Section IV-A in detail. S − ≤ N ∀ ∈ S ∀S ⊂ N The next result provides the conditions under which the weak In Definition 1, the overhead ǫi for deviation of a player i in a coalition is fixed. A stronger notion for the weak ǫ-core ǫ-core of our game is not empty. accounts forS an overhead which depends on the size of the Proposition 1: For ǫi > 0 for all i , the weak ǫ-core ∈ N 2 2 coalition which deviates. is not empty if and only if the noise power satisfies σ σ¯ and σ2 σ2 where ≤ Definition 2: The strong ǫ-core of a coalitional game is the ≥ set of all strategy profiles (xi)i∈N V ( ) for which there is 2 2 2 2 σ¯ := min min σ¯i,S , σ := max max σi,S , (11) ∈ N S⊂N i∈S S⊂N i∈S no coalition and (y )i V ( ) such that ui(y , ..., y ) S i ∈N ∈ S 1 K − ǫi/ >ui(x1, ..., xK ) with ǫi 0 for all i . with   The|S| interpretation of the overhead≥ in the definition∈ S of the , ∆i,S < 0 or Ψi,S 0; strong ǫ-core originates from the original definition in [32] for 2 ∞ ≥ σ¯i,S := −Ψi,S −√∆i,S (12) ǫ , ∆i, 0 and Ψi, < 0; games with transferable utility where the overhead ǫ required ( 2(2 i −1) S ≥ S for the deviating coalition is shared by its members. Hence, and the overhead decreases forS each member of as the size of S 0, ∆i,S < 0 or Ψi,S 0; increases. This is in contrast to the weak ǫ-core where the 2 ≥ (13) S σi,S := −Ψi,S +√∆i,S overhead is constant for each player. The strong and weak ǫ , ∆i, 0 and Ψi, < 0; ( 2(2 i −1) S ≥ S ǫ-core definitions can be regarded as generalizations of the and the used parameters are defined as traditional solution concept of the core for which the overhead 2 ǫi ǫi is set as ǫi =0 for all i . Interestingly, taking into account ∆i, := Ψ 4(2 1)2 CiBi, , (14a) ∈ N S i,S − − S the deviation overhead, the solution set of the coalitional game ǫi Ψi, := (2 (Bi, + Ci) (Bi, + Ai, )), (14b) S S − S S is enlarged, i.e., the core is a subset of the strong ǫ-core [32]. H ZF 2 Ai, := h w , (14c) Also, the strong ǫ-core is a subset of the weak ǫ-core. For S | ii i→S | H MRT 2 H ZF 2 coalitional games in which the core is empty, including the Bi,S := hji wj , Ci := hii wi→N . (14d) overhead in the deviation could lead to stability of the system. j∈N \S | | | | X Next, we will specify the characteristic function V ( ). Proof: The proof is provided in Appendix A. 2 2 While we adopt the γ-model to assume that the players outsideS Proposition 1 implies that for all SNR values 1/σ 1/σ¯ 2 2 ≥ a coalition are noncooperative, in order to define V ( ) we and 1/σ 1/σ it is profitable for all players to jointly S ≤ need to specify the cooperation strategies in a coalition . We perform ZF. In the case where the overhead ǫi = 0 for all consider two simple non-iterative transmission schemesS which players, the conditions under which the core is not empty has can be applied in a distributed manner. These are ZF and WF been given in [1, Proposition 1] and restated below as a special beamforming defined in the next subsections. case of Proposition 1. Corollary 1: For ǫi = 0 for all i , the weak ǫ-core is not empty if and only if σ2 σˆ2 where∈ N B. Coalitional Game with Zero Forcing Beamforming ≤ 2 The transmitters choose MRT if they are not cooperative σˆ := min min Bi,S Ci/(Ai,S Ci) . (15) according to Section II-B. If a transmitter cooperates with S⊂N i∈S { − } Interestingly, in comparison to the core without deviation 2The definition of ǫ-core in [32] is for games with . Here overhead, the weak ǫ-core is not empty above an SNR thresh- we formulate the solution concept for games with nontransferable utility such that the overhead ǫ is different for each player and not transferable to other old and also below an SNR threshold. The weak ǫ-core is players in its coalition. also not empty at low SNR is due to the fact that the noise 5

1.2 20 1 18.2 1 1 conditions for 3 empty strong ε-core 3 15 8 0.8 ǫ 2 2 =

0.6 ··· 10 = 1

ǫ conditions for

y-axis [km] 6 0.4 6 = empty weak ε-core ǫ 5 5 8 8 0.2 5 3.6 4 7 4 7 0 0 0 0.2 0.4 0.6 0.8 1 1.2 -40 -20 0 20 40 51.7 60 x-axis [km] signal-to-noise ratio [dB]

Fig. 1. A setting with 8 links and each transmitter uses 8 antennas. In order Fig. 2. Conditions for empty strong and weak ǫ-cores are plotted in the to include the effect of distances between the links on the received power filled regions for the setting with 8 links in Fig. 1. gains we use the following path loss model: Let dkℓ be the distance between transmitter k and a receiver ℓ in meters and δ be the path loss exponent, we h −δ/2h˜ h˜ h˜ I write the channel vector kℓ = dkℓ kℓ/k kℓk with kℓ ∼ CN (0, ). −δ 2 if and only if ∆i,S < 0 or Ψi,S 0, for all i , , We define the SNR as SNR= dkk /σ and we set δ = 3. ≥ ∈ S S ⊂ N where Ψi,S and ∆i,S are given, respectively, in (14a) and (14b) in Proposition 1. From Proposition 1, in the case where the overhead ǫ is power at low SNR is much larger than the interference. Then, i zero for all i , the core is not empty only above an SNR the performance difference between joint ZF beamforming in threshold. Next,∈ N we show that also in this case, a player does the grand coalition compared to the performance of another not have an incentive to build a coalition with another player beamforming strategy is not large enough to compensate for at low SNR, i.e., the overhead leading to the formation of the grand coalition. ZF ZF The derivation of the conditions for nonempty strong ǫ-core ui(V ( i )) >ui(V ( )), i , , > 1. (16) in Definition 2 is analogous to that in Proposition 1 in which { } S ∀ ∈ S ∀S ⊆ N |S| Notice that V ZF( i ) = (wMRT, ..., wMRT). The conditions for for a player i , the term ǫi is replaced with ǫi/ . It must { } 1 K be noted that in∈ orderS to calculate the conditions for|S| nonempty (16) to hold are given in [1, Proposition 2] and restated here. Proposition 2: Single-player coalitions exist if weak and strong ǫ-core in Proposition 1, an exhaustive search |N| 2 over 2 1 nonempty subsets of must be performed. Ai, B hii Bi, − N 2 2 S i,{i} −k k S (17) σ > σˇ := max max 2 , In Fig. 2, we plot the conditions for empty strong and S⊆N i∈S hii Ai,S weak ǫ-core from Proposition 1 for the setting in Fig. 1. Only  k k −  with A and B defined in (14c) and (14d), respectively. for an overhead strictly larger than zero does a lower SNR i,S i,S threshold exists for nonempty strong and weak ǫ-core. As the overhead increases, the lower threshold 1/σ¯2 increases and the C. Coalitional Game with Wiener Filter Precoding upper threshold 1/σ2 decreases. Consequently, the SNR region In this section, we assume the players cooperate by perform- where all players have an incentive to jointly perform ZF ing WF precoding with the players in their own coalition. For transmission becomes larger. It is shown that if the conditions link i in coalition , transmitter i’s WF precoding is S for the weak ǫ-core not to be empty are satisfied then they are 2 H −1 (Iσ + j i hij hij ) hii also satisfied for the strong ǫ-core. The operation of wireless wWF = ∈S\{ } . (18) i→S 2 H −1 systems is usually in the range between 5 and 20 dB SNR. It (Iσ + Pj∈S\{i} hij hij ) hii can be seen from Fig. 2 that the conditions for the stability of k k In comparison to ZF beamforming in (9), WF precoding is the grand coalition with ZF beamforming requires relatively P interesting when the number of antennas at the transmitters higher overhead measure at the links. is smaller than the number of links in the coalition. WF There is a relation between the result in Proposition 1 and precoding in (18) has interesting behavior for asymptotic SNR the notion of cost of stability [37]. In [37], it is assumed that cases [11]. In the high SNR regime (σ2 0), wWF converges all users have the same ǫ . The cost of stability specifies the i→S i to wZF in (9). In the low SNR regime→ (σ2 ), wWF smallest overhead ǫ such that the weak ǫ-core is not empty. i→S i→S i converges to wMRT in (5). → ∞ Fig. 2 illustrates the cost of stability which corresponds to i The game in coalitional form with WF precoding is the overhead ǫ on the boundary points of the region where , , V WF, (R ) where the mapping V WF which defines the weak ǫ-core is empty. That is, for a fixed SNR value, the k k∈N hNthe strategyX profile accordingi to WF cooperation scheme is boundary point is the smallest overhead value with which the weak ǫ-core is not empty. WF WF V ( )= (wk)k∈N : wk = wk→S for k , In Fig. 2, it is shown that above a certain overhead level, the S { ∈ X MRT ∈ S wℓ = w for ℓ . (19) ǫ-core is nonempty for any SNR value. This overhead level is ℓ ∈ N\S} obtained during the proof of Proposition 1 and stated here. Conditions for nonempty strong and weak ǫ-core of the coali- Corollary 2: The weak ǫ-core is not empty for any σ2 > 0 tional game with WF precoding in terms of SNR thresholds 6 are hard to characterize because the noise power in (18) is Algorithm 1 Coalition formation algorithm. inside the matrix inverse. 1: Input: , (ǫ ,...,ǫK ), q N 1 Next, we will study coalition formation games between 2: Initilize: k =0, = 1 ,..., K C0 {{ } { }} the links and use the WF and ZF beamforming schemes as 3: for k, q do Tq, ⊂CT |T | ≤ cooperative beamforming strategies. 4: k k ; C −→ C +1 5: if k k then C ≺T C +1 IV. COALITION FORMATION 6: k = k +1; In the previous section, we have used the strong and weak 7: Go to Step 2;

ǫ-core as solutions to our coalitional game which only consider 8: Output: k the feasibility of the formation of the grand coalition. In this C section, we enable the formation of several disjoint coalitions to form a coalition structure. A coalition structure is a parti- coalition structure to the next is acceptable if this leads to a tion of , the grand coalition, into a set of disjointC coalitions N L L preferable coalition structure. Afterwards, the stability of the 1, ..., L where j=1 j = and j=1 j = . coalition formation process must be studied. For this purpose, {SLet Sdenote} the set ofS all partitionsN of .S We consider∅ the P S 3T N a stability concept for coalition structures must be specified. following game in partition form [21]: A set of at most q coalitions in some arbitrary coalition T , ,F, (ui)i , (20) structure 0 can merge to form a single coalition. In hN X ∈N i C ∈ P doing so, the coalition structure 0 changes to 1 . We where F : is called the partition function. We C C ∈ P P → X formally define this mechanism as follows. consider two scenarios for player cooperation in a coalition. q,T Definition 3 (q-Deviation): The notation indi- The scenarios correspond to ZF and WF transmissions. Given 0 1 cates that the coalitions in , where C −→and C q a coalition structure , the strategy profile of the players T T ⊂C0 ∈P |T | ≤ C merge to form the coalition = . Then, the coalition according to ZF or WF is defined by S T structure 0 changes to 1 = 0 in the set of coalition bf bf structuresC . C C \TS ∪S F ( ) := (wi)i∈N : wi = wi→Sj for i j , j , C { ∈ X ∈ S S ∈(21) C} The motivationP behind the merging deviation model in Defi- where bf = ZF,WF with wZF and wWF defined in (9) nition 3 is that coalition formations starts from the noncoop- { } i→Sj i→Sj and (18), respectively. Notice that if j = 1 and i j , erative state of single-player coalitions and hence cooperation then wZF = wWF = wMRT. For a|S coalition| structure∈ S , requires merging of coalitions. The deviation complexity in i→Sj i→Sj i C F ZF( ) is a strategy profile in which each player chooses ZF Definition 3 is tunable through the parameter q. The larger q to theC players in his coalition. Similarly, F WF( ) is the strategy is, the more complex it is to search for possible merging coali- profile when WF is applied. In our case, the coalitionC structure tions since the number of possible combinations increases with uniquely determines the associated strategy of each player. q. We study the complexity for this search in Section IV-A. Consequently, the payoff of each player is directly related to Given a coalition structure, we assume that the coalitions the formed coalition structure. can communicate with each other in order to find possible The payoffs of the members of a coalition depend on which performance improvement through deviation. A deviating set coalitions form outside. The effects caused by other coalitions of coalitions according to q-Deviation in Definition 3 can on a specific coalition are called externalities [38]. In our compare the resultingT coalition structure with the previous C1 game, due to the interference coupling between the links, coalition structure by the Pareto dominance relation C0 ≺T externalities exist. These are categorized under negative and specified as follows: positive externalities. In the case of negative externalities, the merging of two coalitions reduces the utility of the other C0 ≺T C1 ⇔ bf bf coalitions. While positive externalities lead to an increase in i : ui(F ( 0)) ǫi ui(F ( 1)), and (22) ∀ ∈ T C − ≤ C the payoff of other coalitions when two coalitions merge. In bf bf j [ : uj (F ( )) ǫi

40 10 preferred to 0 by the players in and the preferences of C T T (K, 2),D (K,K ),W (K,K ) the players in are not considered in the comparison. ) N \ T This choice enables the set of deviating coalitions to T K , q 30 ( ) T K , q 10 decide on their own if they want to merge without consulting ( D ( K , q ) the remaining players. However, note that the strategies of ,W W ( K , q ) q = 4 ) q = 8 q = 32 deviating coalitions do affect the utilities of all players. 20

K , q 10

Based on the deviation rule in Definition 3 and the coalition (

in (22), we formulate a binary relation to compare ,D q = 16 two coalition structures. ) 1010 Definition 4 (q-Dominance): The coalition structure q- K , q q = 8 C1 (

q T dominates 0, written as 1 0, if there exists a set of C C ≫q,TC q = 2 coalitions 0 such that 0 1, and 1 T 0. 0 T ⊂C C −→ C C ≺ C 10 According to the previous definitions, we define the coali- 0 20 40 60 80 100 tion formation game as ( , q), where is the set of all K coalition structures and Pq is≫ the dominanceP relation defined ≫ Fig. 3. Illustration of the measures D(K,q), T (K,q), and W (K,q) defined in Definition 4. The solution of the coalition formation game in (23), (27), and (28) respectively. ( , q) is a set of coalition structures with special stability properties.P ≫ We use the coalition structure stable set as a solution concept for ( , q) adopted from [33, p. 110]. the coalition structure stable set of ( , q), then according P ≫ Definition 5 (CoalitionP ≫ Structure Stable Set): The set of to external stability, there exists a coalition structure ′ in the ′ q C coalition structures is a coalition structure stable set coalition structure stable set such that k. However, if q R⊂P C ≫ C of ( , ) if is both internally and externally stable: this is the case, k would not be a solution of Algorithm 1 C P ≫ R ′ • is internally stable if there do not exist , such since Algorithm 1 terminates when no coalition structure is Rthat q ′, C C ∈ R found that q-dominates the obtained coalition structure. If C ≫ C q • is externally stable if for all there exists k is inside the coalition structure stable set of ( , ) C ′′ P ≫ R′ such that ′ q . C∈P\R and there is a coalition structure in the stable set such C ∈ R C ≫ C that ′′ q , then also wouldC not be a solution of The coalition structure stable set for coalition formation k k AlgorithmC ≫ 1. Accordingly,C theC coalition structure resulting games is a modification of the stable set solution concept k from Algorithm 1 must be in the coalition structureC stable set of coalitional games in characteristic form which has been of ( , q). originally proposed in [40]. Unlike the core, the stable set is P ≫ not necessarily unique. Next, we discuss the complexity for finding a set of coali- tions which merge according to -Deviation in Definition 3 and The coalition structure stable set of ( , q) has at least q compare it to an analogous rule based on coalition splitting. one element which is the grand coalitionP ≫. As q-Deviation (Definition 3) only allows merging of coalitions,N the coalition structure is stable because no deviation is possible. We A. Complexity assume howeverN that the players start their operation in the Our coalition formation algorithm is influenced by the noncooperative state corresponding to the Nash equilibrium following works on coalitional games in partition form [41], (Section II-B). In order to reach a solution in the coalition [33]. In [41] a solution concept called equilibrium binding structure stable set, we provide Algorithm 1.4 Step 4 finds agreements is proposed. The algorithm starts in the grand a q-Deviation (Definition 3) by searching over all coalition coalition, and only splitting operations can occur. The sta- merging possibilities given coalition structure and q, the k bility of a coalition structure is based on finding a sequence maximum number of coalitions that are allowedC to merge. of splitting deviations which achieve higher utilities in the For a possible deviating coalition , the resulting coalition final coalition structure. Moreover, the players are considered structure is compared with theT previous coalition ac- k+1 farsighted, i.e., can anticipate the effects of their actions to the cording toCq-Dominance (Definition 4). If Step 5 is true, then actions of other players. This mechanism is however shown the new coalition structure is adopted and the index k k+1 to be inefficient and has motivated the extension in [33]. In is incremented. If no deviatingC coalitions satisfy Step 5, then [33], a coalition formation algorithm is proposed where the the algorithm terminates. splitting operation proposed in [41] is adapted such that the Proposition 3: Algorithm 1 converges to a coalition struc- coalitions that split can also merge in an arbitrary manner. ture in the coalition structure stable set of ( , q). Despite the increased complexity, Pareto efficiency is achieved Proof: First, the algorithm is guaranteedP ≫ to converge only in special cases. since only merging operations are allowed and the number The reason for choosing coalition deviation based only on of merging steps is finite. We prove by contradiction that the merging of coalitions in our work is twofold: First, since the resulting coalition structure is in the coalition structure stable users start their operation in the noncooperative state of single- set. Let the solution of the algorithm be k. If k is outside C C player coalitions, coalition formation must be able to merge 4Other initializing coalition structures can also be supported by Algorithm coalitions. Second, the splitting operation is far more complex 1 which may lead to different outcomes. than the merging operation, as discussed next. 8

According to Definition 3, the number of possible ways to Algorithm 2 Implementation of Algorithm 1. merge a set of at least two and at most q coalitions from the 1: Input: , (ǫ1,...,ǫK ), q, bf = ZF,WF N { } iter coalition structure is 2: Initialize: k = 0, 0 = 1 ,..., K , n , r = C C {{ } { }} min{q,|C|} min q, 0 , Θ=0 (23) { |C |} D( , q)= |C| . 3: while r 2 and k > 2 do |C| j=2 j ≥ |C |   4: Each user generates lexicographically ordered r- The expression above has noX closed form. For the special case combinations of k: 1,..., |Ck | ; q = , D( , )=2|C| 1 and for q = 2 we get C {T T( r )} |C| |C| 2|C| − |C| − 5: |Ck| D( , 2)=( )/2. The worst case complexity corre- for ℓ = 1 : r do |C| |C| − |C| 6: niter = niter + 1; sponds to single-player coalitions because then the number of  7: Each user in ℓ temporarily generates k from k coalitions in is largest and equal to K. This is the initial T C +1 C coalition structureC which starts Algorithm 1. by merging ℓ; T bf Lemma 1: The growth rate of D( , q) in (23) is bounded 8: Each user i ℓ compares his utility ui(F ( k)) bf∈ T C − q |C| ǫi to ui(F ( k )); by (K ) for fixed q. C +1 OProof: First, observe that D( , q) D(K, q) for 9: Increment the number of utility comparisons: Θ = |C| ≤ |C| ≤ Θ+ ; K. According to the definitions in [42, 24.1.1], and for q K S∈Tℓ |S| ≤ 10: Each user in ℓ sends a message to the other users we can calculate P T in ℓ from the set M1, M2, M3 ; q K q K T { } D(K, q)= = (24) 11: if all messages are M1 or M2 and some M1 then j=2 j j=2 K j 12: Users in ℓ merge to form a single coalition;    −  T XK(K 1) KX(K 1)(K 2) 13: Users in ℓ send M4 to users outside ℓ; = − + − − + T T 2! 3! ··· 14: k = k +1 and r = min q, k ; { |C |} q factors 15: Go to Step 4; K(K 1) (K q + 1) 16: r = r 1; + − ··· − . (25) − q! 17: Output: k, Θ z }| { C Accordingly, D(K, q) (Kq). In comparison to the∈ mergingO rule, the number of combi- nations for splitting a set to k, k , subsets is given by index which is commonly known to all users. Moreover we the Stirling number of theS second kind≤ |S|[43, Theorem 8.2.6]: assume that each user knows the members of each coalition ,i =1,...,L. In Step 4 in Algorithm 2, each user generates 1 k k i S( , k)= ( 1)t (k t)|S|. (26) aS list of r-subsets of the indexes 1,...,L of the coalition |S| k! t=0 − t − structure and these subsets are{ ordered} in lexicographic X   k If we allow, as we did in the merging case, the splitting of order. TheC generation of this list can be done using the S into at least two and at most q subsets, we get the following algorithm provided in [43, Section 4.4]. In this manner, each number of combinations: user has the same ordered list of the subsets of 1,...,L of q { } T ( , q)= S( , k). (27) size r. We assume that all users are synchronized in the sense |S| k=2 |S| that the users consider the same element ℓ of the generated T For q = , the number aboveX corresponds to the th Bell list for a period of time in which negotiation takes place. number minus|S| one.5 In Fig. 3, we compare the complexity|S| of Then, in Step 7 in Algorithm 2, the new coalition structure the merging and splitting operations for increasing number of k+1 is temporarily formed by merging ℓ. In Step 8, each C T users and different values of . It can be observed that the user in ℓ evaluates his utility in the new coalition. In Step 9, K q T splitting operation requires much more searching combinations the number of utility comparisons is incremented according than the merging operation. to the number of users in ℓ. Following Step 8 in which T the utility comparisons are made, in Step 10 each user in ℓ T B. Implementation in the MISO interference channel communicates one of the following messages to the other users in ℓ: In Algorithm 2, we provide an implementation of Algorithm T 1 in the MISO setting. The algorithm is initialized according • (M1) utility improves ← to the Nash equilibrium, i.e., all coalitions are singletons. • (M2) utility is the same ← The term r = min q, 0 is the number of coalitions that • (M3) utility decreases { |C |} ← are allowed to merge. The quantity Θ is initialized to zero • (M4) coalition forms and will aggregate the total number of utility comparisons ← during the algorithm. This measure will be used later in the Note that any of the above four messages can be exchanged simulations in Section V to reveal numerically the complexity between two links requiring two bits of information. In Step of the algorithm. 11, the Pareto dominance relation defined in (22) is examined. We assume that when a coalition structure forms, such If the condition in Step 11 is true then the coalitions in ℓ merge and all users outside ℓ are informed of the new as k = 1,..., L , each coalition is given a unique C {S S } coalitionT structure by exchangingT the message M4. Algorithm 5 th q The |S| Bell number is Pk=0 S(|S|,k) and S(|S|, 0)+S(|S|, 1)) = 1. 2 terminates if the grand coalition is formed or if r, the number 9

bf = ZF 20 bf = ZF 1 bf = WF 1 1 bf = WF maximum average user rate [7] 1 1 14 maximum average user rate [7] Nash equilibrium 1 15 1 Nash equilibrium 1 1 12 1 1 1 10 1 10 2 8 3 3 3 3 6 6 4 7 5 4 8 4 user rate [bits/s/Hz] average user rate [bit/s/Hz] 8 8 2 8 8 8 8 8 8 0 0 8 8 8 1 2 3 4 5 6 7 8 user -20 -10 0 10 20 30 40 50 60 signal-to-noise ratio [dB] Fig. 5. User rates at 25 dB SNR.

Fig. 4. Average rate of the 8 users in the setting in Fig. 1 using Algorithm 2. The number under (above) the curve is the number of coalitions with ZF (WF). low SNR regime, single-player coalitions exist supporting Proposition 2. Note that in the low SNR regime, the outcome of coalitions to merge, is less than two. with joint MRT is efficient [15]. In the mid SNR regime, Algorithm 2 provides a structured method to find a possible coalition formation improves the joint performance of the mergings between coalitions by exploiting the algorithm in links from the Nash equilibrium. Optimal average user rate [43, Section 4.4]. The complexity of Algorithm 2 is however is obtained using the monotonic optimization algorithm from related to the complexity of an exhaustive search with the the supplementary material of [7]. There, the beamforming restriction on the maximum number of coalitions that can space is not restricted to ZF or WF scheme. The average user merge, q. rate in Nash equilibrium saturates in the high SNR regime. Theorem 1: The number of iterations of Algorithm 2 is This is contrary to ZF and WF coalition formation where the bounded as niter (Kq). ∈ O average user rate increases linearly due to the formation of Proof: In the worst case, only two coalitions merge at a the grand coalition. The formation of the grand coalition in time, i.e. with r = 2, and the two coalitions which merge the high SNR regime supports Proposition 1. are according to the last element in the r-combination list Comparing the WF and ZF schemes, it is observed in generated in Step 4 in Algorithm 2. In addition, in worst case Fig. 4 that with WF precoding, larger coalitions form at this described behaviour occurs for every merging occasion lower SNR values than with ZF beamforming. As a result, until the grand coalition forms. Accordingly, the worst case higher average user rate gains are achieved with WF coalition number of iterations of Algorithm 2 is given as: formation than with ZF coalition formation. For example, at K−2 min{q,K−i} K i SNR = 25 dB, the coalition structure with WF precoding is niter W (K, q) := − , (28) ≤ i=0 j=2 j 1, 3, 4, 5, 6, 7, 8 , 2 while the coalition structure with ZF X X   {{beamforming is }1{, 3}}, 2 , 4, 5, 6, 7, 8 . Observe that the =D(K−i,q) in (23) coalition 4, 5, 6{{, 7, 8 }is{ the} { cluster of}} links in the bottom where the first summation in (28)| accounts{z for the maximum} right side{ of Fig. 1. This} set of links form a single coalition of K 1 merging of two coalitions before the grand coalition to reduce the interference between one another. The achieved − forms and the second summation corresponds to the worst user rates at 25 dB SNR are shown in Fig. 5. For user one, case number of iterations within the while loop in Step 3 of it is observed that optimal beamforming reduces his utility Algorithm 2 before exactly two coalitions merge. The binomial compared to the Nash equilibrium. Hence, the maximum sum K−i coefficient j is the worst case number of iterations within rate beamforming strategy is not acceptable for player one and the for loop before the if condition in Step 11 is true consequently the maximum sum rate strategy is not stable for  according to which coalition merging occurs. From Lemma joint cooperation. For both WF and ZF schemes, player two is 1 and observing that i = 0 in (28) admits the largest growth in a single-player coalition. However, his rate in both schemes for the upper bound, we obtain the complexity result. is higher than in Nash equilibrium. This reveals that in this Accordingly, Algorithm 2 has polynomial time complexity case, the formed coalitions outside 2 have positive effects for fixed q. The worst case number of iterations W (K, q) in on player two. { } (28) is illustrated in Fig. 3. In the following figures, we consider 8 users with 8 antennas each and we average the performance over 103 V. NUMERICAL RESULTS random channel realizations. First, in Fig. 6 - Fig. 9 we We first consider the setting with 8 links in Fig. 1. Using compare the performance of WF and ZF coalition formation the coalition formation algorithm in Algorithm 2, the average for different values of q. We also relate and compare our user rate is plotted for increasing SNR in Fig. 4 where we results to the coalition formation algorithms in [26] and [30]. set q = 8 and ǫ = 0 as input to the algorithm. In the Then, in Fig. 11 - Fig. 13 we plot the performance of coalition 10

14 q = 2, bf = ZF, recursive core [26] q = 8, bf = ZF 10 q = 2, bf = WF, recursive core [26] 12 q = 8, bf = WF 9 bf = ZF, individual stability [30] q = 4, bf = ZF bf = WF, individual stability [30] 10 q = 4, bf = WF 8 q = 3, bf = ZF 7 8 q = 3, bf = WF 6 q = 2, bf = ZF 6 5 q = 2, bf = WF q = 3 4 Nash equilibrium 4 bf = ZF average user rate [bit/s/Hz] 3 q = 4 2 average number of coalitions 2 0 bf = WF q = 8 -20 -10 0 10 20 30 40 1

signal-to-noise ratio [dB] -20 -10 0 10 20 30 40 signal-to-noise ratio [dB] Fig. 6. Comparison of average user rates achieved with Algorithm 2 for different values of . The overhead is set to ǫ 0. q = Fig. 8. Comparison of the average number of coalitions obtained by Algorithm 2 for different values of q.

1200

1.2 Θ q = 8, bf = ZF 1000 q = 8, bf = WF 1 q = 4, bf = ZF q = 3, bf = ZF 800 q = 4, bf = WF 0.8 q = 3, bf = WF q = 3, bf = ZF q = 2, bf = ZF, recursive core [26] 600 q = 3, bf = WF 0.6 q = 2, bf = WF, recursive core [26] 400 0.4 bf = ZF, individual stability [30]

average user rate [bit/s/Hz] bf = WF, individual stability [30] 200 0.2

Nash equilibrium number of utility comparisons 0 0 -20 -10 0 10 20 30 40 -20 -10 0 10 20 30 40 signal-to-noise ratio [dB] signal-to-noise ratio [dB]

Fig. 9. Comparison of the average number of utility comparisons Θ required Fig. 7. Comparison of average user rates achieved with Algorithm 2 and by Algorithm 2 for different values of q. coalition formation algorithms from [26] and [30]. The overhead is ǫ = 0.

performance of the links for q = 2 is less than for q = 3 formation for two overhead models defined in (29) and (30). and is higher than for [30, Algorithm 1] which is based on In Fig. 6, the average user rate is plotted for different values individual stability. In Fig. 7, it is shown that the average of q. The parameters q influences the deviation rule as defined performance of individual stability is slightly higher than in in Definition 3 and specifies the largest number of coalitions Nash equilibrium. Note however that the performance of [30, which are allowed to merge in one iteration of Algorithm 2. It Algorithm 1] can be significantly improved for sufficiently is shown that as q increases, higher performance is obtained. large number of antennas at the transmitters. Interestingly, although the number of iterations of Algorithm 2 Fig. 6 and Fig. 7 lead us to the conclusion that in is not restricted, performance loss still occurs for smaller q. multi-antenna interference channels, performance improve- In Fig. 7, we compare our algorithm with the algorithms ment through cooperation with ZF or WF beamforming de- in [26] and [30, Algorithm 1]. Note that the system models pends greatly on the number of users that are allowed to and cooperation models in [26] and [30] are different than deviate and cooperate at a time. ours. In [26] cooperation between a set of base stations is In Fig. 8, the average number of coalitions obtained from based on network MIMO schemes which require exchange Algorithm 2 are plotted and compared to [30, Algorithm 1]. of user data between the transmitter. In [30], a MIMO inter- The average number of coalitions with the ZF beamforming ference channel is considered and cooperation in a coalition scheme is larger than with the WF beamforming scheme, is according to ZF transmission. In the proposed coalition i.e., with WF more coalitions merge than with ZF. At low formation algorithm in [26], the deviation rule is based on SNR, single player coalitions exist with ZF beamforming. merging of two coalitions, i.e., corresponds to 2-Deviation This illustrates the result in Proposition 2. Since, WF beam- in Definition 3, and the comparison relation is according to forming converges to MRT beamforming at low SNR, we 2-Dominance in Definition 4. Hence, the algorithm in [26] observe that in this SNR regime some coalitions form with corresponds to our algorithm with q = 2, and the authors WF beamforming. As q increases, the average number of prove that the resulting coalition structure lies in the recursive coalitions decreases. This explains the performance loss in core of their coalition formation game. In Fig. 7, the average Fig. 6. Both, WF and ZF coalition formation obtain the same 11

12 q = 2, bf = ZF, recursive core [26] ε 4 = 0, bf = ZF 300 q = 2, bf = WF, recursive core [26] i 10 ε = 0, bf = WF Θ bf = ZF, individual stability [30] i 3.5 250 ε in (29), bf = ZF bf = WF, individual stability [30] i 8 ε in (29), bf = WF 3 200 i q = 3 ε in (30), bf = ZF 6 i 2.5 ε in (30), bf = WF 150 i 2 4 Nash equilibrium 100 1.5

average user rate [bit/s/Hz] 2 50 1

number of utility comparisons q = 8 0 5 10 15

0 -20 -10 0 10 20 30 40 -20 -10 0 10 20 30 40 signal-to-noise ratio [dB] signal-to-noise ratio [dB] Fig. 11. Comparison of average user rates of Algorithm 2 with q = 8 for Fig. 10. Comparison of the average number of utility comparisons Θ required the overhead models specified in (29) and (30). by Algorithm 2 for different values of q.

8 ε = 0, bf = ZF i 7 ε = 0, bf = WF i number of coalitions for the same q at high SNR. This is 6 ε in (29), bf = ZF i because WF beamforming converges to ZF beamforming in 5 ε in (29), bf = WF i this SNR regime. Coalition formation with [30, Algorithm 1] 4 ε in (30), bf = ZF leads to high average number of coalitions which means that i 3 ε in (30), bf = WF the number of users that cooperate is small. i 2

In Fig. 9, the average number of utility comparisons during average number of coalitions 1 Algorithm 2 is plotted for different q. Note, that Algorithm 2 -20 -10 0 10 20 30 40 starts in single-player coalitions and the users initially search signal-to-noise ratio [dB] for the largest number of coalitions to merge. The number of utility comparisons Θ during the search is specified in Fig. 12. Comparison of average number of coalitions resulting from Step 8 in Algorithm 2 and corresponds to the total number of Algorithm 2 with q = 8 for the overhead models specified in (29) and (30). utility comparisons the users need to do before the algorithm converges. It is shown in Fig. 9 that the WF scheme requires to specify ǫ for a given coalition structure as follows: generally much lower number of utility comparisons than the C ZF scheme. The reason for this is with the WF scheme more bf bf ǫi = |S| ui(w ,..., w ), i , , (29) coalitions merge than with the ZF scheme according to Fig. 8 1 K ∈ S S∈C |N | which leads to faster convergence rate of Algorithm 2 with 1 bf bf ǫi = ui(w ,..., w ), for all i , (30) WF beamforming than with the ZF scheme. The number of 1 K ∈ N utility comparisons generally decreases for smaller since |N | Θ q with bf = ZF,WF . According to (29) and following the the number of possible deviations decreases. For a specific q, comparison{ relation} in (22), the overhead for a user which it is observed that Θ is not monotonic with SNR. With ZF examines whether to join a coalition increases with the beamforming, the number of utility comparisons increases at size of . In (30), the overhead is assumedS to be fixed. In around 5 dB SNR and afterwards decreases. The reason for S − both overhead models in (29) and (30), we assume that the this behavior is at very low SNR no coalitions merge and hence overhead is proportional to the utility in the grand coalition Algorithm 2 terminates after searching over all merging pos- since it corresponds to the largest overhead needed to examine sibilities of single-player coalitions. At around 5 dB SNR, − whether the grand coalition forms. a small number of coalitions merge and hence Algorithm 2 In Fig. 11 - Fig. 13, we set q =8 and plot the performance requires additional iterations to search for possible merging in of Algorithm 2. In Fig. 11, the average user rate in both ZF the new coalition structure. Since the number of coalitions that and WF schemes is similar with both overhead models in (29) merge is small, the number of comparisons is large. At larger and (30) and no overhead (ǫi =0). With ZF beamforming, it is SNR, the number of coalitions that merge increases and hence shown that higher average user rate improvement is achieved the coalitions are found faster than at smaller SNR. At high for larger overhead. The explanation of this effect is that SNR, the grand coalition is then favorable for all the users with larger overhead more coalitions merge as is shown in and for q = 8 it is possible to build the grand coalition in a Fig. 12. With WF coalition formation, it is shown that with the single iteration of Algorithm 2. This is contrary to the case proposed overhead models, the grand coalition always forms q < 8 which explains the performance advantage of q = 8. in the simulation setup. In Fig. 10, we can see that [30, Algorithm 1] requires a low In Fig. 13, the average number of utility comparisons Θ number of utility comparisons and hence has low complexity. required during Algorithm 2 is plotted. With WF beamform- For the subsequent plots, we define two overhead models ing, Θ is very low which reveals that the WF beamforming 12

TABLE I

Θ 1500 ε = 0, bf = ZF CASE STUDY FOR ANALYSING THE QUADRATIC EQUATION IN (33). i ε = 0, bf = WF i Case I ǫi = 0 ε in (29), bf = ZF 1000 i Case II ǫi > 0 and ∆i,S < 0 2 2 ε in (29), bf = WF Case III ǫi > 0 and ∆i,S ≥ 0 and σ1 + σ2 ≤ 0 i 2 2 Case IV ǫ > 0 and ∆ S ≥ 0 and σ + σ > 0 ε in (32), bf = ZF i i, 1 2 i 500 ε in (30), bf = WF i APPENDIX A number of utility comparisons 0 PROOF OF PROPOSITION 1 -20 -10 0 10 20 30 40 signal-to-noise ratio [dB] Considering an arbitrary player i in an arbitrary coalition , we write the condition in (10) as Fig. 13. Comparison of average number of utility comparisons Θ in S Algorithm 2 with q = 8 for the overhead models specified in (29) and (30). H ZF 2 hii wi→S log 1+ | | ǫi 2 2 H MRT 2 σ + hji wj ! − j∈N \S | | H scheme requires less complexity in coalition formation than P h wZF 2 log 1+ ii i→N , (31) ZF beamforming. Generally, with ZF beamforming and higher 2 | 2 | ≤ σ ! overhead, the number of utility comparisons decrease. This is observed after approximately 2.5 dB SNR. At low SNR, which is equivalent to − the number of utility comparisons are larger including the Ai, Ci S ǫi ǫi (32) overhead compared to no overhead because a small number 1+ 2 2 +2 2 , σ + Bi,S ≤ σ of users merge to form coalitions as is shown in Fig. 12. Consequently, a larger number of searches during Algorithm 2 where the introduced notations Ai,S ,Bi,S , and Ci are given is required. In comparison to no overhead, it can be observed in (14c) and (14d) in Proposition 1. Cross multiplying (32) 2 that the maximum number of utility comparisons shifts to the and solving for σ we get the following condition left on the SNR axis as the overhead increases. f(σ2) 0, σ2 > 0, (33) ≥ 2 ǫi 2 2 ǫi where f(σ ) := (2 1)(σ ) + (2 (Bi,S + Ci) (Bi,S + 2 ǫi − − VI. CONCLUSIONS Ai,S ))σ +2 CiBi,S . In order to analyze (33), a case study is summarized in Table I and illustrated in Fig. 14. We study cooperation in the MISO IFC using coalitional 2 If ǫi =0, then f(σ ) in (33) is a straight line as in Case I games. A transmitter’s noncooperative transmission strategy in Fig. 14. The condition in (33) reduces to is MRT. We consider ZF or WF precoding for cooperative 2 user transmission in a coalition. The necessary and sufficient (Ci Ai,S )σ + CiBi,S 0, (34) − ≥ conditions under which all users profit from joint cooperation 2 CiBi,S 2 σ0 := σ > 0, (35) with ZF are characterized. It is shown that incorporating Ai,S Ci ≥ an overhead in deviation of a coalition enlarges the set of − and corresponds to the shaded region in Fig. 14. conditions under which the users have the incentive to coop- If ǫ > 0, and having (2ǫi 1) > 0 then the quadratic poly- erate compared to the case without considering the overhead. i nomial f(σ2) in (33) describes− a parabola with a minimum Afterwards, we consider coalitional formation games and and opens upwards as illustrated in Cases II-IV in Fig. 14. If propose a distributed coalition formation algorithm based on the minimum of the parabola is strictly larger than zero (Cases coalition merging. The complexity of the algorithm is tuneable II in Fig. 14), then the quadratic equation has no real roots and its implementation requires two-bit signalling between and the discriminant of f(σ2) is negative, i.e., the transmitters. We prove that the algorithm terminates in polynomial time and generates a coalition structure in the ǫi 2 ∆i, := (2 (Bi, + Ci) (Bi, + Ai, )) coalition structure stable set. The performance improvement S S − S S ǫi ǫi 4(2 1)2 CiBi, < 0. (36) of the links through cooperation with simple distributed trans- − − S mission schemes is achieved and depends on the complexity If (36) is satisfied, so is the condition in (33) for any 0 < σ2 < for coalition deviation and user-specific overhead measures. because the entire parabola has strictly positive values. ∞ 2 As future work, we will extend the current work to consider For ∆i,S 0, f(σ ) in (33) has two real roots which multiple antennas at both the transmitters and receivers. More- corresponds to≥ Cases III and IV in Fig. 14. Then condition over, we will generalize to multi cell settings in which multiple (33) holds for receivers are associated with each transmitter. Significant to σ2 σ2 or σ2 σ2 and σ2 > 0, (37) study for these settings is the appropriate design for the ≤ 1 ≥ 2 cooperative precoding schemes used within the coalition. Nev- where σ2 and σ2 are the roots of f(σ2) in (33) given as ertheless, our interest is also to investigate different solution 1 2 concepts for coalitional games in partition form and also to 2 Ψi,S ∆i,S 2 Ψi,S + ∆i,S σ1 = − − , σ2 = − . (38) draw comparisons between them. 2(2ǫi 1) 2(2ǫi 1) −p −p 13

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