Agile Broadcast Services: Addressing the Wireless Spectrum Crunch via Coalitional Game Theory

Transcription

1 Agile Broadcast Services: Addressing the Wireless Spectrum Crunch via Coalitional Game Theory 1 Nikhil Karamchandani, Paolo Minero, and Massimo Franceschetti Dept. of EE, UCLA, Los Angeles, CA 90095, USA and Dept. of ECE, UCSD, La Jolla, CA 92093, USA nikhil@ee.ucla.edu Dept. of EE, University of Notre Dame, Notre Dame, IN 46556, USA pminero@nd.edu Dept. of ECE, UCSD, La Jolla, CA 92093, USA massimo@ece.ucsd.edu Abstract The performance of cooperation strategies for broadcast services sharing a common wireless channel is studied in the framework of coalitional game theory. Two strategies are examined. The first represents an open sharing model where each service provider is allowed to transmit at any time but simultaneous transmissions result in interference. It is shown analytically that in the absence of coordination cost, the grand coalition formed by all providers cooperating to avoid simultaneous transmissions is both sumrate optimal and stable. The second strategy represents a time-division regulated access method where service providers are granted exclusive access to the channel at different times, each having a guaranteed successful transmission opportunity in a round-robin fashion. In the absence of coordination cost, the grand coalition where all providers cooperate by sharing their guaranteed right to access the channel is sum-rate optimal but unstable, in the sense that some group of providers may have an incentive to deviate from the grand coalition. In the presence of coordination cost, a different scenario arises. In both models large coalitions do not form, and simulation results suggest that the open access model for large networks can lead to a regime where performance is considerably limited by interference.

2 2 Index Terms Coalitional games, cooperative communications, multiple-access. I. INTRODUCTION In today s overpopulated wireless networks bandwidth is a precious resource. In the United States, the Federal Communications Commission (FCC) has the responsibility of allocating portions of the spectrum to various industries. A currently preferred method is to auction off bands that then become property of the purchaser. Due to high demand by service providers, this method has the advantage of raising government revenues. However, once allocated, bandwidth usage remains inefficient. One of the problems is the bursty activity of the users that can lead to idle channels and inefficient use of the available resource. This has led the FCC to warn about the nation s technology renaissance possibly turning into a chimera of poor services and budget-busting prices for the end users 1. The presence of hidden resources that are allocated but not fully utilized opens up the possibility of developing network economic models for resource sharing that can lead to design guidelines with enhanced payoffs. In this framework, our contribution is to study cooperation strategies among service providers offering downlink wireless data access. We focus on transmission of delay-sensitive information over a single narrowband frequency channel: different service providers wish to access the channel and broadcast real-time data to a set of co-located users. Simultaneous packet transmissions, however, result in data loss at the receivers due to interference, in which case the transmitted packets are dropped and are not retransmitted. This simple model is suitable, for example, for applications such as realtime multimedia with tolerance to losses, audio-video multicast, non-emergency news-feeds (e.g. advertisement), and event-driven control where the packets are subject to continuous update at the transmitter. We examine two natural ways of accessing the channel. In the first scenario, which we refer to as the uncoordinated access model, the channel is allocated by the authority to a given service 1 According to FCC Chairman Julius Genachowski, The explosive growth in mobile communications is outpacing our ability to keep up. If we don t act to update our spectrum policies for the 21st century, we re going to run into a wall a spectrum crunch that will stifle American innovation and economic growth and cost us the opportunity to lead the world in mobile communications. Source: Cable News Network (CNN), October 2010.

3 3 and all providers can access it at any given time. Providers can cooperate to coordinate access in order to minimize interference and consequent data loss at the receivers. In the second scenario, which we refer to as the coordinated access model, the authority sets permissions to access the channel in time slots, giving to each provider a guaranteed successful transmission opportunity in one time slot, and these access privileges are repeated in a round robin fashion. In this case, providers can cooperate to share time slots, in order to avoid wasteful idle slots due to the bursty nature of their operation. We analyze the performance of cooperation strategies in these two scenarios using coalitional game theory [1] and determine how stable coalitions among providers arise and evolve in response to the potential throughput gains enabled by cooperation. We assume that members of a coalition share their activity states so that a scheduler can ensure that at most one member per coalition transmits at any given time. Hence, to maximize the network throughput there is an incentive in forming the grand coalition composed by all providers. In this case the system is fully scheduled in a centralized fashion and can maximize channel utilization avoiding interfering transmissions and minimizing idle times. On the other hand, from the providers point of view, a larger coalition requires sharing the aggregate throughput with other providers and there may be an incentive to defect from it. The key question is whether there exist rate allocations, satisfying certain fairness criteria such as the individually egalitarian and proportional profit, that are both achievable and able to stabilize the grand coalition. Our results are as follows. In both scenarios there is an incentive from the global throughput perspective to form the grand coalition. However, while in the first scenario the grand coalition can be stabilized by an appropriate design of a central scheduler satisfying some desirable fairness criteria, in the second scenario the grand coalition cannot always be stabilized, as some subset of providers may always benefit by defecting from it, regardless of the fairness criteria adopted by the central scheduler. An intuitive explanation of these results is as follows. Granting a priori privileges to providers for accessing the channel in given time slots restricts the competitive space of the game, lowering the degree of cooperation, in favor of more conservative actions by which the players tend to preserve their baseline, guaranteed, individual throughput. This suggests that when the objective is to maximize the aggregate throughput to the end users, having an open sharing model is preferable to a regulated access mechanism. Next, we investigate the impact of the cost of establishing a coalition. Under both cooperative

4 4 scenarios, as coalitions grow in size the grand coalition does not form in settings where the cost for forming a coalition exceeds the gain due to cooperation. In such cases, the network splits into multiple smaller, stable coalitions. In order to characterize these stable network partitions, we consider a coalition formation algorithm in which service providers start in a state of noncooperation and iteratively take turns in forming coalitions that allow to improve their individual utility until the network reaches a partition where no user has an incentive to make a switch to another coalition. The impact of the underlying system parameters on the evolution of the algorithm is illustrated by means of simulations. What emerges is that the uncoordinated access model is preferable to the coordinated access model for small networks, or when the cost of cooperation is small. However, as the number of service providers increases or the cost increases, so does the number of disjoint coalitions, and the network enters an interference-limited regime in which the uncoordinated access is not anymore the preferred mechanism for spectrum sharing. In this case, the coordinated access model is preferable as it reduces interference among competing service providers. The general guideline from these results is that algorithms with small overhead in forming coalitions, or operating on small networks, are fully-cooperative, can converge to the grand coalition, and have better throughput. On the other hand, algorithms with substantial cost of cooperation or operating on large networks are subject to cooperative inertia and break off into multiple scattered coalitions creating interference limitations. In this case, the coordinated access model achieves better performance by regulating transmissions and decreasing the interference. Next, we wish to place this study 2 in the context of the related literature. The problem of cooperation among wireless service providers has been previously studied under different angles of attack. In a series of papers [3] [5], Aram, Kumar, Singh, and Sarkar considered a network setup where service providers cooperate by serving each other s customers. In this setting, tools from coalitional game theory are used to distribute the aggregate revenue earned by a coalition such that none of its member has an incentive to deviate. Similarly, the problem of cooperation and profit sharing among internet service providers carrying one other s traffic has been investigated in [6]. In [7], wireless service providers are assumed to cooperate by spectrum sharing and concepts from bargaining game theory are used to study the resulting exchange of 2 Part of this work has been published in the proceedings of the 2011 Allerton Conference on Communication, Control, and Computing [2].

5 5 resources. None of these existing works, however, considers the issue of interference among simultaneous downlink transmissions. In this paper, we investigate how service providers can cooperate to mitigate the amount of interference due to the activity of the users. Concepts from coalitional game theory have been previously applied by several authors to study interference in wireless networks. La and Anantharam [8] considered a Gaussian multiple access channel where senders cooperate in jamming the channel and showed that in this adversalial setting there exists a fair rate allocation that stabilizes the grand coalition, where fairness is defined in a specific axiomatic sense. In a related work, Mathur, Sankar, and Mandayam [9] considered a Gaussian interference network and investigated the stability of the grand coalition under different cooperation models. Cooperative games over the Gaussian multiple access channel have been also studied in [10] under different models of cooperations among transmitters. None of these previous works, however, takes into account the random activity of the users accessing the channel, nor the cost of cooperation in the formation of a coalition. In this paper, instead, we assume that service providers access the channel at random times, depending on the availability of data to send, and focus on two simple models for which it is possible to fully characterize the set of achievable communication rates by the members of different coalitions. In addition, we also consider the impact of the cost of cooperation on the performance of the system. The rest of the paper is organized as follows. A review of preliminary concepts in geometry and coalitional game theory is presented in the next section. Sections III and IV contain the analysis for the uncoordinated access model and the coordinated access model, respectively. Section V is devoted to the analysis of the cost of cooperation and to the presentation of simulation results. We discuss some generalizations of the model and draw final remarks in Section VI. Throughout the paper we use the notation in [11] and we denote by N the finite set of the first n integers, i.e., N = {1,2,...,n}. II. BASIC DEFINITIONS A. Submodular functions and polyhedra Let f be a real-valued function defined on subsets of N. We say that f is normalized if f( ) = 0, non-decreasing if S 1 S 2 N implies f(s 1 ) f(s 2 ), and non-increasing if f is non-decreasing. f is submodular if, for S 1,S 2 N, f(s 1 S 2 )+f(s 1 S 2 ) f(s 1 )+f(s 2 ),

6 6 and it is supermodular if f is submodular. If f is normalized, non-decreasing and submodular, we call it a rank function. Given a rank function f defined on N, we define the polymatroid P(f) as the polyhedron { P(f) = r R n + : } r i f(s) for all S N, (1) i S The set of points in P(f) on the hyperplane i N r i = f(n), i.e., { B(f) = r P(f) : } r i = f(n) i N (2) is called the base polytope of P(f). A vector v P(f) is an extreme point of P(f) if it is not a convex combination of two other vectors in P(f). One of the most important properties of polymatroids is that there is an explicit characterization of the extreme points strictly inside the positive orthant. Given a rank function f and any permutation σ = ( ) σ,...,σ n on the set N, let the vector r σ (f) R n be defined as rσ σ 1 (f) = f({σ 1 }) and rσ σ i (f) = f({σ 1,,σ i }) f({σ 1,,σ i 1 }) (3) for i = 2,,n. Lemma 1 ( [12] [14]): Let P(f) be a polymatroid. Then r σ (f) is a vertex of P(f) for every permutation σ. Any vertex strictly inside the positive orthant must be r σ (f) for some permutation σ. Moreover, for any c R n a solution of the linear program maxc r subject to r P(f) is attained at a point r σ (f) for any permutation σ such that c σ1 c σ2 c σn. It follows from the above lemma that the base polytope B(f) is equal to the convex hull of all extreme point vectors {r σ (f)} σ, and that every point in P(f) is dominated (component-wise) by an element of B(f). For this reason, B(f) is often called the dominant face of P(f) [14]. B. Coalitional game theory A cooperative game with transferable utility is a pair (N,v), where v : 2 N R is a function, called characteristic function, such that v( ) = 0. The elements of N are indices for the players of the game and any non-empty subset S N is called a coalition. In particular, N is called the grand coalition. We say that a game (N,v) is a cost game if v(s) measures the cost incurred by the coalition S and is a profit game if v(s) measures the profit of S. The game is superadditive if v is superadditive, convex if v is non-decreasing and supermodular, and concave if v is a rank

7 7 function, i.e., non-decreasing and submodular. In a superadditive profit (subadditive cost) game, the players have an incentive to form the grand coalition. A profit (cost) allocation is a vector x R n such that i N x i = v(n) and specifies a rule for distributing the profit (cost) of the grand coalition among the players. The core of a game consists of all allocations such that no group of players S N has an incentive to leave the grand coalition N. The core of a profit game (N,v) is defined as C(v) = { x R n : x i = v(n), i N } x i v(s) for all S N. (4) i S The core of a cost game (N,v) is defined by (4) with replaced by. The core denotes the set of profit (cost) allocations that assign to each coalition at least (at most) what it can obtain on its own and thus stabilize the grand coalition. Thus, the grand coalition is stable if there is at least one allocation in the core. In many games the core is empty, but whenever it is not empty there exists an allocation called the nucleolus which is always guaranteed to be in the core. The nucleolus minimizes a certain metric of dissatisfaction of the players amongst all possible allocations in the core, for the formal definition we refer the reader to [1]. Let (N,v) be a profit game. The dual game of (N,v) is the cost game (N,v ) defined by the dual characteristic function v (S) = v(n) v(s c ) (5) where S c = N \S. Similarly, a dual profit game can be defined for a cost game. The following proposition, proved in [11], establishes a connection between a cooperative game and its dual. Proposition 1: Let (N,v) be a profit (cost) game and let the dual (N,v ) be a cost (profit) game. Then, (N,v) is a convex (concave) game iff (N,v ) is a concave (convex) game. Furthermore, C(v) = C(v ). For any game (N,v) and any permutation σ on the set N, the vector r σ (v) defined as in (3) is called a marginal worth vector. The convex hull of all marginal vectors is called the Weber set. The Weber set always includes the core of a game, and the two sets coincide if and only if the game is convex, in which case the extreme points of the core are precisely the marginal worth vectors [15]. The Shapley profit allocation of a game (N, v) is the centroid of the marginal

8 8 vectors, i.e., 1 n! r σ (v) (6) σ where the summation is over the set of all permutations of N. Shapley provided an axiomatic characterization of this profit allocation, showing that (6) is the unique vector satisfying a set of four axioms capturing a notion of fairness (cf. [13]). III. UNCOORDINATED ACCESS MODEL Suppose that n service providers wish to broadcast delay-sensitive to a set of co-located subscribers. Let time be divided into slots of unit duration. At the beginning of every slot each service provider has an empty queue of packets and receives a new packet to transmit with probability p i, independently of other providers. When a provider is active, it attempts to broadcast the newly received packet to the subscribers by transmitting over the common channel. Simultaneous transmissions can lead to packet collisions because of the mutual interference over the shared wireless medium. In practice, the severity of a collision depends on many factors including the separation among the wireless devices, the signal strength, the communication schemes etc. As a first order approximation, we adopt the classical collision model for random access communications [16], according to which a packet collision occurs when two or more providers are simultaneously transmitting. Packets are assumed to carry delay-sensitive information that has to be communicated within one slot, so they are discarded by the service provider at the end of each slot in the event of a packet collision. The motivation for such an assumption is that for time-sensitive applications dropping packets is preferable to waiting for delayed packets. Examples of such applications are real-time multimedia with tolerance to losses, such as audio-visual multicast, non-critical news-feeds (i.e. advertisement), and eventdriven control applications where the packets are subject to updates at the transmitter. Since packets are not retransmitted, the packet queues at the end of each slot are assumed to be empty regardless the outcome of a packet transmissions. In this setting, p i denotes the rate of arrival of real-time traffic, i.e., 1 p i is the probability that service provider i does not have any new packet to send to any of the service subscribers. To avoid collisions and hence to enhance the throughput of the network, providers can form coalitions and cooperate by sharing their activity states with other members of the coalition. This

9 9 setting is representative of wireless systems where a common channel is allocated by the authority to a given service and providers cooperate to utilize the common resource in the most efficient way by mitigating the amount of interference generated by simultaneous transmissions. Coalitions operate in fully cooperative mode, so a scheduler determines the active service provider that can access the channel while all other active members of the coalition remain silent. Collisions only occur when providers belonging to different coalitions transmit simultaneously. Formally, for each non-empty coalition S N, a scheduler is a function g S : 2 S S having the properties that (P1) g S (S ) S for all S S and (P2) g S (S ) = iff S =. Associated with any such scheduler g S is a vector (r 1,,r S ) defined for i {1,2,..., S } as r i = A S p j j A l S\A (1 p l ) ( ) 1 {gs (A)={i}} (1 p k ), k S c ] [ = E A 1{gS (A)={i}} (1 p k ), (7) k S c where 1 E denotes the indicator function for the event E and the expectation is taken with respect to the random set of active service providers A S. The above definitions can be interpreted as follows. Suppose that in every slot the scheduler selects a member of S according to the scheduling function g S. As time progresses, the ratio between the number of slots where provider i, i S, is chosen for transmission and there is no collision over the total number of slots converges to r i. Thus, (r 1,,r S ) denotes the vector of average packet transmission rates that can be simultaneously achieved by the providers forming the coalition S by means of the scheduler g S. The (closure of) the union of all rate vectors corresponding to a choice of the scheduling function is denoted as the S-capacity region C S. Formally, C S = conv g S { r R n + : (7) holds for i {1,2,..., S } }. Any element of the S-capacity region is called an achievable rate vector for coalition S. Our first result is to provide the a characterization of C S. To this end, we define a set function m S : 2 S S defined on all subsets of S as m S (B) = (1 i B(1 p i )) k S c (1 p k ) for all B S (8)

10 10 and m S ( ) = 0. In words, m S (B) denotes the probability that at least one element of B is active and that all service providers in S c are inactive. The function m S satisfies the following property. Lemma 2: For every S N, m S is a rank function. Proof: The function m S is normalized because by definition m S ( ) = 0. Also, for any S 1 S 2 S m S (S 1 ) m S (S 2 ) 1 i S 1 (1 p i ) 1 i S 2 (1 p i ) 1, which proves that m S is non-decreasing. Similarly, for any S 1,S 2 N, we have m S (S 1 S 2 )+m S (S 1 S 2 ) m S (S 1 )+m S (S 2 ) = 2 i S 1 S 2 (1 p i ) j S 1 S 2 (1 p j ) 2 i S 1 (1 p i ) j S 2 (1 p j ) = 1 (1 i S 2 \S 1 (1 p i ))( i S 1 S 2 (1 p i ) j S 1 (1 p j )) 2 i S 1 (1 p i ) j S 2 (1 p j ) 1 and thus m S is submodular. The following theorem provides an explicit characterization of the S-capacity region as the polymatroid P(m S ) associated with the rank function m S. Theorem 1: For every S N, the capacity region C S is the polymatroid associated with the rank function m S, i.e., C S = P(m S ). Proof: To begin with, we prove that C S P(m S ). Observe from (7) that for any scheduling function g S and B S, [ r i = E A i B i B 1 {gs (A)={i}} ] k S c (1 p k ) P( B A > 0) k S c (1 p k ) = m S (B) where the inequality follows from (8) and property (P1) of the scheduler, which implies that g S (A) B for all A such that A B =, and it is equality iff g S (A) B for all A such

11 11 that A B. Thus, C S P(m s ). To show the reverse inclusion, we prove that any rate vector on the boundary of P(m S ) is achievable. Since any rate vector in P(m S ) is dominated component-wise by a rate vector on the base polytope B(m S ) defined in (2), it suffices to prove the achievability of the S! extreme points of B(m S ) given in Lemma 1. For any permutation σ on the set S, we will show that the corresponding extreme point r σ is achievable. Let the scheduling function g S (A) select the maximum element from the set of active providers A, according to the ordering σ 1 > > σ S, i.e., g S (A) = σ i iff σ i A and σ j A for all j = 1,,i 1. Then, E A [ 1{gS (A)={σ 1 }}] = pσ1 and thus and for i = 2,, S, [ ] E A 1{gS (A)={σ 1 }} (1 p k ) = m S ({σ 1 }) = rσ σ 1 (m S ) k S c [ ] E A 1{gS (A)={σ i }} (1 p k ) k S c i 1 = p σi (1 p σj ) (1 p k ) k S c j=1 = m S ({σ 1,,σ i }) m S ({σ 1,,σ i 1 }) = r σ σ i (m S ), and hence the extreme point r σ (m S ) is achievable. Remark 1: It follows from the above theorem that the S-sum capacity region, i.e., the set of the rates in C S for which the sum rate is maximized, is equal to the dominant face B(m S ) of P(m S ) and that the corner points of B(m S ) can be achieved by simply scheduling providers based on a predefined priority list. Remark 2: The S-capacity region C S has the same geometric structure as the informationtheoretic capacity region of the multiple access channel [17]. We remark that an interpretation of the capacity region of the multiple access channel via coalitional game theory appears in [18]. In the extreme case of no cooperation the system operates in full anarchy and thes-capacity of the singleton S = {i} is simply the probability that service provider i is the only active service provider in the system. By forming coalitions, providers can reduce the collision probability and thus increase the aggregate sum rate. On the other hand, from the providers point of

12 12 view, forming a coalition means sharing the aggregate sum rate with other competitors. Thus, a coalition will be stable only if no subset of service providers has an incentive to defect. To study the tradeoff between cooperation and competition in the formalism of coalitional game theory, we consider the profit game (N, v) with characteristic function v defined as v(s) = (1 i )) i S(1 p (1 p k ) for all S N (9) k S c and v( ) = 0. This choice of the characteristic function is motivated by the observation that v(s) represents the average sum rate achieved by the members of S when they form a coalition, while assuming adversarial behavior from the providers outside S. In fact, v(s) is the probability that at least one element of S is active while all service providers in S c do not have data to transmit. It turns out that the game (N, v) has many interesting properties. In particular, the next theorem gives a characterization of the core of this game as the N-sum capacity region. Theorem 2: The game (N,v) is a convex game and its core C(v) is equal to the N-sum capacity region, i.e., C(v) = B(m N ). Proof: The proof is based on duality. From Lemma 2, m N is a rank function and thus the cost game (N,m N ) is concave. It then follows from the definition of the core (4) and from (2) that C(m N ) = B(m N ). Next, observe that the profit game (N,v) is the dual of the cost game (N,m N ). In fact from (8), for all S N v(s) = (1 k )) i S(1 p (1 p k ) i S c = m N (N) m N (S c ) = m N(S). Hence, the claim follows from Proposition 1. Remark 3: The core of a game is one of the most important solution concepts in coalitional game theory. There are games for which the core is empty or cannot be characterized [19] and games for which part of the core cannot be achieved [8]. It is thus remarkable that the core of the profit game (N,v) can be fully characterized and that any stabilizing rate allocation can be achieved by appropriate design of the central scheduling function. Remark 4: The core of any convex game is non-empty, as it contains at least the Shapley profit allocation (6). It follows from Theorem 2 that the Shapley profit allocation of the game (N,v)

13 13 is simply the centroid of the N-sum capacity region. Furthermore, since the core is non-empty it also contains the nucleolus profit allocation. Given that the core is non-unique and that any rate allocation in the core is achievable, a natural next step is finding stabilizing allocations which satisfy specific notions of fairness. Potential candidates are the proportional profit allocation, in which rates are assigned proportionally to the data arrival probabilities at each service provider, the individually rational egalitarian profit allocation, whereby the extra profit due to cooperation is divided equally among the members of the coalition, and the envy-free allocation recently introduced by La and Ananthram [8], which is based on an axiomatic notion of fairness named envy-free fairness [8], [20]. Corollary 1: The following hold: 1) Let r be the proportional profit allocation vector defined by r i = p i k N p v(n) (10) k for i N. Then, r is an element of the core C(v) for all p 1,,p n. 2) Let r be the individually rational egalitarian profit allocation vector defined by r i = 1 ( v(n) ) v({j}) +v({i}). (11) n j N for i N. Then, for all n > 2 there exists a tuple (p 1,p 2,,p n ) such that r is not an element of C(v). 3) Let r be the envy-free profit allocation vector defined by ( r i = 1 1 (1 p i ) i i n k=i+1 (1 p k ) ) n r k, (12) k=i+1 for i N. Then, r is an element of C(v) for all p 1 p 2... p n. Example 1: Consider for example a network composed by two service providers that are active with probabilities p 1 and p 2, respectively. In this case, the capacity region C N is equal to the set of non-negative rate pairs (r 1,r 2 ) such that r 1 p 1 r 2 p 2 r 1 +r 2 p 1 +p 2 p 1 p 2.

14 Proportional profit Envy-free Rational egalitarian and Shapley 0.4 r2 0.2 Capacity C N Core C(v) = B(m N) r 1 Fig. 1. Capacity and core for a systems with two providers (p 1 = 0.8 and p 2 = 0.4) in the uncoordinated access model. In this case, the core coincides with the dominant face of the pentagon delimiting the capacity region. The two extreme points of C N strictly inside the positive orthant have coordinates (p 1,p 2 (1 p 1 )) and (p 1 (1 p 2 ),p 2 ), respectively, and can be achieved by a scheduler which always gives priority to provider 1 (provider 2) over provider 2 (provider 1). On the other hand, the core C(v) is equal to the set of non-negative allocations (x 1,x 2 ) such that x 1 p 1 (1 p 2 ) x 2 p 2 (1 p 2 ) x 1 +x 2 = p 1 +p 2 p 1 p 2. It is immediate to verify that C(v) is equal to the base polytope B(m N ), i.e., the dominant face of the pentagon delimiting C N, and hence that the Shapley profit allocation, which is in this case equal to the individually rational egalitarian allocation, is the midpoint of B(m N ). Similarly, it can be checked that the proportional profit allocation (p 1 (1+p 1 p 2 /(p 1 +p 2 )),p 2 (1+p 1 p 2 /(p 1 + p 2 ))) and the envy-free allocation (1 p 1 (1 p 2 ) p 2 (1 p 2 /2),p 2 (1 p 2 /2)) are in B(m N ) and hence are achievable. Fig. 1 illustrates the geometry of the problem in special case where p 1 = 0.8 and p 2 = 0.4. Proof: By Theorem 2, proving that a rate allocation is in the core is equivalent to proving

15 15 that it is an element of the N-sum capacity region B(m N ). By definition of the proportional profit allocation (10), i N r i = v(n). Furthermore, for any S N we have r i = i S p i m N (N) m N (S), i S k N p k where the inequality follows since f(x) = ( i S p i + x)/(1 (1 x) i S (1 p i)) is an increasing function of x (0,1) for any S N. Thus the proportional profit allocation vector is an element of B(m N ). Consider now the individually rational egalitarian profit allocation (11). We claim that if we set p 1 = 1/n and p j = 1/2 for all j 1, then the profit allocation r defined in (11) is not an element of C(v). By contradiction, suppose that for this choice of the parameters r C(v). Then, the inequality i N\{1} r i v(n \{1}) implies that (n 1)v(N)+(n 1)v({n}) < v(n \{1}). Using the definition of the functionsm N and v in (8) and (9) respectively, and after some algebra, the above inequality reduces to p 1 1 ( 1 1 )( 1 2 ) 1 < 1 n 2 n n, which is a contradiction with the assumption that p 1 = 1/n. This establishes that r C(v). Finally, the proof that the envy-free profit allocation is always an element of the core C(v) is reported in Appendix A. In conclusion, in the uncoordinated access model where cooperation is used as a form of collision avoidance, the grand coalition can be stabilized so that no member of a coalition has an incentive to defect. As an example, this can be accomplished by designing a central scheduler that assigns rates according to the proportional profit allocation or the envy-free profit allocation. IV. COORDINATED ACCESS MODEL Consider now a TDMA system, in which each slot is assigned by the authority for transmission to one service provider in a periodically repeated order, so that collisions are always prevented. Provider i is active in each slot with probability p i, independently of other operators in the network, so if it is inactive during one of its allocated slots, no transmission occurs and the shared channel resource remains unused. To prevent this inefficient use of the common resource, providers can cooperate by sharing their respective slots, such that these can be used by any

16 16 active member of the coalition. A scheduler controls the transmission of packets within each coalition, by choosing one provider among the pool of active ones. The formal definition of a scheduling function g S for coalition S is as in Section III, i.e., it is a set function defined on the subsets of S such that properties (P1) and (P2) are satisfied. Associated with any scheduler g S is a vector (r 1,,r S ) defined for every i {1,2,..., S } as r i = S n E [ ] A 1{gS (A)={i}}, where the expectation is taken with respect to the random set of active providers A S. The above definitions can be interpreted as follows. Suppose that in each of the S slots in which a member of the coalition S holds the right to access the channel, the scheduler selects the active provider in S according to the scheduling function g S. As time progresses, the ratio between the number of slots where provider i is chosen for transmission over the total number of slots converges to r i. Thus, (r 1,,r S ) denotes the vector of average packet transmission rates that can be simultaneously achieved by the the members of coalition S via the scheduler g S. In the extreme case of no cooperation the system operates in TDMA mode and the maximum achievable rate by provider i is p i /n. On the other hand, in the opposite extreme where the grand coalition is formed, the system is fully scheduled by a central scheduler and any of the active providers can be chosen to transmit in a slot. Note that this is identical to the case where the grand coalition is established in the uncoordinated access model studied in Section III and hence the (closure of) set of all achievable rate vectors is the N-capacity region C N, as before. From the system perspective, the formation of a grand coalition ensures that all slots can potentially be utilized, however it is not clear a priori if it is possible to design a scheduling function for the grand coalition and divide the aggregate sum rate among the members such that no group has an incentive to deviate. as To answer this question we consider the profit game(n,w) with characteristic function defined w(s) = S n m N(S) for all S N. (13) where m N is as defined in (8). This choice of the characteristic function is motivated by the observation that w(s) denotes the expected sum rate achievable by the providers in coalition S.

17 17 Proposition 2: The profit game (N, w) is superadditive. Proof: For any two disjoint coalitions S 1,S 2 N w(s 1 S 2 ) = S 1 + S 2 m N (S 1 S 2 ) n = S 1 n m N(S 1 S 2 )+ S 2 n m N(S 1 S 2 ) S 1 n m N(S 1 )+ S 2 n m N(S 2 ) = w(s 1 )+w(s 2 ) where the inequality follows from the fact that m N is a non-decreasing function. Since the game is superadditive, providers have an incentive to form the grand coalition N. The next step is to find achievable rate allocations which stabilize the grand coalition, i.e., rate vector that are elements of C(w) B(m N ). The next proposition shows that unlike in the uncoordinated access model considered in Section III, this is not always possible. Proposition 3: The following facts are true for n = 2: 1) (N,w) is a convex game. 2) For every pair (p 1,p 2 ) there exists an achievable stabilizing rate allocation, i.e., C(w) B(m N ). 3) There exists a pair (p 1,p 2 ) such that the Shapley profit allocation (6) and the nucleolus are not in the N-sum capacity region B(m N ). For all n 2, instead, 1) There exists a set of probabilities p 1,,p n for which no rate allocation in the core is achievable, i.e., C(w) B(m N ) =. 2) In the symmetric setting p 1 = p 2 =... = p n = p, there exists an achievable stabilizing rate allocation, i.e., C(w) B(m N ). Example 2: Consider again the setup of Example 1 in which two service providers are active with probabilities p 1 and p 2, respectively. In this case, the core of the profit game w C(w) is equal to the set of non-negative allocations (x 1,x 2 ) such that x 1 p 1 /2 x 2 p 2 /2

18 Proportional profit Envy-free Rational egal. and Shapley r2 0.4 B(m N) Capacity C N Core C(w) r 1 Fig. 2. Capacity and core for a systems with two providers (p 1 = 0.25 and p 2 = 0.75) in the coordinated access model. In this case, the core is only partially overlapping with the dominant face of the pentagon delimiting the capacity region. x 1 +x 2 = p 1 +p 2 p 1 p 2. In general, C(w) partially overlaps with the base polytope B(m N ) of the capacity region C N. For example, Fig. 2 illustrate the boundaries of C N (solid line) and the boundaries of the inequalities describing C(w) (dotted lines) in the special case where p 1 = 1/4 and p 2 = 3/4. It is clear from the figure that part of the core is not achievable because it lies outside B(m N ). In particular, the Shapley profit allocation, which is in this case equal to the individually rational egalitarian allocation, and the envy-free allocation are not achievable for this specific choice of the parameters. Proof: Let n = 2. It is immediate to check that the two-user game is convex. For any achievable stabilizing rate allocation (r 1,r 2 ), inequalities (1) and (4) require that p j /2 r j p j, j N, and r 1 + r 2 = p 1 + p 2 p 1 p 2. These constraints are simultaneously satisfied by both the proportional profit allocation which assigns r j = w(n) p j /(p 1 + p 2 ), j N, and the profit allocation which assigns r j = p j p 1 p 2 /2, j N. Since the game is convex for any (p 1,p 2 ), the Shapley profit allocation and the nucleolus are trivially in the core. However, let (p 1,p 2 ) = (1/4,3/4) (cf. Fig. 2). It is immediate to verify that the Shapley profit allocation is given by (13/32,1/4), the nucleolus is given by (9/32,17/32) (cf. [1]), and these rate allocations

19 19 do not lie in the capacity region as they violate the constrain r 1 p 1 = 1/4. Suppose now that n 2, and let p n = 1 1 2n and p i = 1/n for all i < n. We claim that for this choice of the parameters C(w) B(m N ) =. By contradiction, suppose that there exists a rate vector r which is both in the N-sum capacity region and the core. Then, r satisfies w(s) i S r i m N (S) for all S N with strict equality if S = N. From i N r i = m N (N) and i N\{1} r i w(n \{1}), it follows that r 1 m N (N) w(n \{1}), which combined with r 1 +r n w({1,n}) gives r n w({1,n}) m N (N)+w(N \{1}) = 1 n α(n)(1 p 1), where α(n) = 1 n n+2 (2n n +n 4 2n 3 ) (0,1). Combining the above inequality and r n m N ({n}) yields, p n n1+α(n) 1 1 n, which is a contradiction with the hypothesis that p n = 1 1 > 1 1. Finally, consider the 2n n symmetric case where all providers are active with probability p. In this case, the symmetric profit allocation r 1 = = r n = (1 (1 p) n )/n is both in the core and in the N-sum capacity region, as it can be verified using the fact that 1 (1 p) n n 1 (1 p) S S for every S N. Thus we can conclude that unlike the uncoordinated setting in Section III where cooperation is always cohesive, the randomness of traffic arrival at each service provider plays a crucial role in determining the stability of TDMA networks in which cooperation is used to share transmission slots among service providers. In particular, in asymmetric scenarios where transmission probabilities are different it is not always possible to design a scheduler such that no provider has an incentive to leave the grand coalition. It should be remarked that our cooperative model does not pose any restriction on the way the scheduler manages transmissions within a coalition.

20 20 The fact that the grand coalition need not be stable remains true under more restrictive models of cooperation where time slots cannot be assigned to the coalition constituent members in an arbitrary fashion. V. IMPACT OF COST OF COOPERATION In the previous sections we have studied the benefits of cooperation under the assumption that there is no cost associated with the formation of a coalition. In practice, however, there is a cost involved in establishing cooperation due to the local communication among service providers. Formally, the effect of this overhead cost can be taken into account via a function c : 2 N R + which returns the non-negative cost of forming a coalition S N. For example, assuming that the cost of cooperation increases linearly with the size of the coalition, we let α S for S > 1, c(s) = (14) 0 otherwise, for all S N and for some cost amplification factor α > 0. The choice in (14) is justified by the fact that the cost of cooperation grows proportionally to the number of providers forming a coalition, because to cooperate providers must transmit to a designated coalition leader at least one bit of information indicating their intent to transmit. We assume that in any given time slot an active user can transmit q bits of information if there is no collision, so once we account for the cost of cooperating, the net value of a coalition S in the uncoordinated model becomes ṽ(s) = qv(s) c(s), (15) where v(s) is the characteristic function defined in (9). We can similarly define a coalition game (N, w) for the coordinated access model, where the characteristic function is given by w(s) = qw(s) c(s), (16) and w(s) is defined in (13). Since the cost of cooperation increases with the size of the coalition, (N,ṽ) and (N, w) are not superadditive games in general. Hence the grand coalitions will not always form and the network will typically result in multiple disjoint coalitions.

21 21 Algorithm 1 Coalition formation algorithm Initial network configuration At the beginning, the network partition Π initial = N. Coalition formation algorithm with three stages Stage 1: Neighbor discovery Each provider discovers neighboring coalitions and initial network partition Π initial. Stage 2: Coalition formation Π current = Π initial repeat Given the current network partition Π current, each provider i performs the following steps in its turn. 1) Providerisearches for a better coalitiont Π current than its current coalitions Π current, according to (17), (18). 2) If i finds such a coalition T, i updates its history set h(i) to include S \{i}. i leaves the current coalition S and joins T. Update Π current with (Π current \(S T)) ((S \{i}) (T {i})). until Current network partition Π current converges to a stable partition. Stage 3: Cooperative network operation Providers operate cooperatively according to the stable current network partition Π current. A. Coalition formation algorithm As customary in the coalition formation game literature [21] [24], we study the equilibria for the games (N,ṽ) and (N, w) in the dynamic setting where service providers are allowed to distributedly form coalitions. Given some initial partition of the network, service providers consider deviating according to a set of rules defined by a coalition formation algorithm until a stable equilibrium point is reached.

22 22 We assume that the algorithm starts with all the service providers acting non-cooperatively. In each round, providers take turns to discover other coalitions in the network and exchange relevant information, e.g., identities and activity probabilities. A service provider switches if it finds a coalition which is preferable to the one he is part of. The process is repeated until the network reaches a partition where no player has an incentive to make a switch to another coalition. It is shown in [21] that any sequence of switch operations converges to a stable partition of the network. In addition, since each provider can always return to the non-cooperative state, the process is guaranteed to result in a network partition which is better, or at least equivalent to the non-cooperative setting. Service providers agree to join or leave a coalition based on a notion of preference. Formally, a preference is a binary relation over the set of coalitions. Given i N and any two coalitions S 1,S 2 containing i, S 1 S 2 implies that i prefers being a member of S 1 over S 2. In our setting, we assume that S 1 S 2 if and only if f i (S 1 ) > f i (S 2 ), (17) where f i : 2 N R is a preference function defined as r i (S) if ( r j (S) r j (S \{i}) for all j S \{i} f i (S) = and S / h(i) ) or ( S = 1), otherwise, (18) where r i (S) is the payoff received by service provider i in coalition S, for example according to the proportional profit allocation as in (10), and h(i) is a history set which keeps track of all the coalitions that i joined and left in the past. Given the preference function (17), the steps of the proposed coalition formation algorithm are summarized in Algorithm 1. B. Simulation Results In this section we present some simulation results that illustrate how the stable equilibria for the games (N,ṽ) and (N, w) depend on the number of providers n, the cost factor α, as well as the geometry of the network. We consider a network with n service providers distributed uniformly at random over the [0,1] [0,1] square region. For each provider i, we choose the

23 23 activity probability p i uniformly at random from (0,1). It is assumed that in any time slot an active provider can transmit q = 200 bits if there are no collisions. Service providers start from the non-cooperative state and employ Algorithm 1 to establish cooperation in the network. For the preference function defined in (18), we assume that the net value of a coalition is divided among its members according to the proportion profit allocation in (10). Apart from the cost in (14) associated with each coalition S, we also impose a locality constraint that a coalition is allowed only if all its members are within a threshold distance d (0, 2] from each other. This is reasonable since the cost of establishing cooperation among far-apart players might be prohibitive. All the results presented in this section are averaged over the random positions of the providers as well as their corresponding activity probabilities. Any run of the coalition formation algorithm converges to a stable partition of the network into disjoint coalitions. The total utility of such a network partition is measured as the sum of the net values of the individual coalitions, where the net value of a coalition is as defined in (15), (16). Fig. 3a shows the average network utility as a function of the number of providers n for both the uncoordinated and the coordinated access models. Notice that the former strictly outperforms the latter whenα = 0. This is due to the fact that Algorithm 1 always results in the grand coalition in the case of the uncoordinated access model while it ends with multiple smaller coalitions in the coordinated access case., see Fig. 3b for a plot of the average maximum coalition size as a function of the number of providers. To see why this is the case, recall from Algorithm 1 that a necessary condition for provider i to join coalition S is that none of the members of S should suffer a drop in their payoff when i joins. For the uncoordinated access model, it is easy to see from (13) and (10) that the above condition is satisfied if 1 j S {i} (1 p j ) = m N (S {i}) j S {i} p j (19) which holds true for any choice of activity probabilities. On the other hand, the above condition is satisfied for the coordinated access model only if ( ( ) ) S 1 mn (S) p i j S p ( S +1) 1. (20) j m N (S) where m N is as defined in (8). The above condition is not always true and may prevent the formation of the grand coalition even when α = 0.

24 24 Net utility Net utility (a) α = 0 Coordi nate d Uncoordinated Population size n (c) α = Coordi nate d Uncoordinated Population size n Max coalition size Max coalition size (b) α = 0 Coordi nate d Uncoordinated Population size n (d) α = Coordi nate d Uncoordinated Population size n Fig. 3. Network utility vs number of service providers. Fig. 3c shows that when the cost of establishing cooperation is significant (α = 0.5), the coordinated access model outperforms the uncoordinated access model as the number of providers in the network increases. As shown in Fig. 3d, when α = 0.5 coalitions formed under the uncoordinated access model are smaller in size than those formed under the coordinated access model. To see why this is the case, recall from Algorithm 1 that a necessary condition for provider i to join coalition S is that the net value of the coalition S should increase when i joins. From (15), (16), it is easy see that this condition holds if α q < p i j/ S {i} for the uncoordinated access model and α q < 1 ( 1 (1 ( S +1)p i ) ) j ) n j S(1 p (1 p j ) (21) (22)

25 25 Net utility Net utility (a) Coalition price α N=5 N=10 N=15 (c) N=5 N=10 N= Coalition radius d Net utility Net utility (b) 130 N=5 N=10 N= Coalition price α (d) 60 N=5 N=10 N= Coalition radius d Fig. 4. Network utility vs coalition price α and coalition radius for the uncoordinated model (cases (a) and (c)) as well as for the coordinated model (cases (b) and (d)). for the coordinated access model. Since the coalition formation algorithm starts from the noncooperative state, notice that the right hand side of the condition in (21) decreases exponentially with the number of providers n, whereas the right hand side in (22) only decreases polynomially. It follows that as n increases it is more likely to converge to a network partition with small coalitions under the uncoordinated access model than it is under the coordinated access model, and this results in a lower average utility. Fig. 4a and Fig. 4b show the average network utility achieved versus the cost factor α for the uncoordinated and the coordinated access model. Notice that in the uncoordinated access model the average network utility decreases in α as the number of providers grows. As already noticed, the coalition formation algorithm is likely to converge to a partition with small coalitions as n increases and thus the average network utility drops with n. On the other hand, the decay is less