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Agile Broadcast Services: Addressing the Wireless Spectrum Crunch via Coalitional Game Theory Nikhil Karamchandani, Member, IEEE, Paolo Minero, Member, IEEE, and Massimo Franceschetti, Senior Member, IEEE

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 sum-rate optimal and stable. The second strategy represents an orthogonal access method where service providers are granted exclusive access to a subset of the available channels, each having a guaranteed successful transmission opportunity. 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. Index Terms—Coalitional game theory, cooperative communications, multiple-access.

I. I NTRODUCTION N 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

I

Manuscript received February 18, 2013; revised July 2, 2013; accepted October 16, 2013. The associate editor coordinating the review of this paper and approving it for publication was W. Su. This work was supported by the National Science Foundation and the UCSD Center for Wireless Communications. The material in this work was presented in part at the IEEE Annual Allerton Conference on Communication, Control, and Computing, Sept. 2011. N. Karamchandani is with the Department of Electrical Engineering, University of California Los Angeles, Los Angeles, CA 90095-1594 (e-mail: [email protected]). P. Minero is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 (e-mail: [email protected]). M. Franceschetti is with the Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, CA 92093-0407 (e-mail: [email protected]). Digital Object Identifier 10.1109/TWC.2013.120613.130317

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 users1 . 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 a setup where different service providers share a common wireless channel and wish to broadcast data to a set of users. We analyze performance from the point of view of the typical user in the network. Simultaneous packet transmissions result in data loss at the user 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 real-time multimedia with tolerance to losses, audio-video multicast streaming, newsfeeds, advertisements, 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 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, service providers are granted exclusive access to a set of orthogonal communication channels, each having a guaranteed successful transmission opportunity. In this case, providers can cooperate by sharing their assigned channels, to ensure that no one of them remains unused in the event that the service provider that owns the right to transmit does not have any packet to transmit. 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 1 According to FCC former 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.

c 2014 IEEE 1536-1276/14$31.00 

KARAMCHANDANI et al.: AGILE BROADCAST SERVICES: ADDRESSING THE WIRELESS SPECTRUM CRUNCH VIA COALITIONAL GAME THEORY

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, in the sense of achievable sum rates from the service providers to the typical subscriber, there is an incentive in forming the grand coalition composed of 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, i.e., no subset of providers has an incentive to defect from 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 a 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 of the providers, having an open sharing model is preferable to a regulated access mechanism. The possible instability of the grand coalition in the second scenario implies that the network partition into coalitions may continuously change. This is clearly undesirable since it will require continuous communication and coordination between the providers to ensure that they have updated information about the current network partition and can setup cooperation with the other members of their coalitions. Next, we investigate the impact of the cost of establishing a coalition. Under both cooperative 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 non-cooperation and iteratively take turns in forming coalitions that allow to improve their individual utility until the network reaches a partition where no provider 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

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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 in the context of the related literature. A. Non-cooperative games There is a rich literature on the use of non-cooperate games to model selfish user behavior in medium-access protocols. [2] studies the problem of selfish users in an Aloha network by casting it in a non-cooperative game framework. They analyze the stability of the system by characterizing the Nashequilibria of the game and the transmission probabilities that optimize the throughput. [3], [4], and [5] generalize the above setup to allow for heterogenous users, multi-packet reception capabilities, and partial information respectively. [6], [7] use game theory to study the impact of selfish behavior in more general medium-access control protocols. Our paper instead focuses on the impact of cooperation amongst the nodes in such networks. To highlight the gains of cooperation vis-a-vis the non-cooperative scenario, we will explicitly compare the throughput of the network under the two setups. B. Coalitional games The problem of cooperation among wireless service providers has been previously studied under different angles of attack. In a series of papers [8]–[10], 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 [11]. In [12], wireless service providers are assumed to cooperate by spectrum sharing and concepts from bargaining game theory are used to study the resulting exchange of resources. None of these existing works, however, considers the issue of packet collision due to simultaneous downlink transmissions. In this paper, we investigate how service providers can cooperate to mitigate the amount of interference they cause to one another’s transmissions. Concepts from coalitional game theory have been previously applied by several authors to study interference in wireless networks. La and Anantharam [13] considered a Gaussian multiple access channel where senders cooperate in

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“jamming” the channel and showed that in this adversarial 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 [14] considered a Gaussian interference network and investigated the stability of the grand coalition under different cooperation models. Coalitional games over the Gaussian multiple access channel have been also studied in [15] 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. Sections II and III contain the analysis for the uncoordinated access model and the coordinated access model, respectively. Section IV 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 V. A review of preliminary concepts in submodular functions and coalitional game theory is presented in the appendices A and B respectively. Throughout the paper we use the notation in [16] and we denote by N the finite set of the first n integers, i.e., N = {1, 2, . . . , n}. II. U NCOORDINATED ACCESS M ODEL Suppose that n service providers wish to broadcast delaysensitive information to a set of 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 of size q bits to transmit with probability pi , 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 take the point of view of a typical network subscriber, that is, a user that can be served by any of the available service providers with equal likelihood, and thus assume that the signals broadcast by the active service providers are received by the subscriber at the same signal strength. Moreover, we assume that communication occurs in the high signal-to-noise ratio regime, a valid assumption in dense networks, such that the interference between two or more transmitted packet can severely impact the received signal-to-noise-plus-interference ratio. Therefore, we adopt the classical collision model for random access communications [17], 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 event-driven 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, pi denotes the rate of arrival of real-time traffic, i.e., 1 − pi 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 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 gS : 2S → S ∪ {∅} defined on the subsets of S having the properties that (P1) gS (S  ) ∈ S  for all S  ⊆ S and (P2) gS (S  ) = ∅ iff S  = ∅. Associated with any such scheduler gS is an achievable rate vector (r1 , · · · , r|S| ) defined as follows. For each i ∈ {1, 2, . . . , |S|} ri

   = q · EA 1{gS (A)={i}} · (1 − pk ) =q

  A⊆S j∈A

pj

 l∈S\A

k∈S c

   (1 − pl ) 1{gS (A)={i}} (1 − pk ), k∈S c

(1) where recall that q denotes the number of bits in any packet, 1E 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 definition can be interpreted as follows. Suppose that in every slot the scheduler selects an active member of S according to the scheduling function gS . Then the second term in the above product corresponds to the probability that provider i ∈ S is chosen for transmission in a slot. It is equal to the probability that the set of active providers A in a slot is such that gS (A) = i. The third term represents the probability that in a given slot, no provider outside the coalition S transmits and thus there is no collision. As time progresses, the ratio between the number of slots where provider i is chosen for transmission and there is no collision and the total number of slots converges to the product of the probabilities in the second and the third terms. Since in any such slot the provider i successfully transmits a packet of q

KARAMCHANDANI et al.: AGILE BROADCAST SERVICES: ADDRESSING THE WIRELESS SPECTRUM CRUNCH VIA COALITIONAL GAME THEORY

bits, (r1 , · · · , r|S| ) denotes the vector of average transmission rates (in bits / slot) that can be simultaneously achieved by the providers forming the coalition S by means of the scheduler gS . The closure of the union of all rate vectors corresponding to all choices of the scheduling function is denoted as the S-capacity region CS . Formally,

 r ∈ Rn+ : (1) holds for i ∈ {1, 2, . . . , |S|} . CS = conv gS

(2) Any element of the S-capacity region is called an achievable rate vector for coalition S. Note that the above definition assumes that in each coalition, a central entity schedules the transmissions and does not include the overhead of sharing activity information that this entails. We assume that the number of bits required for sharing such information is negligible compared to the packet size q and hence does not impact the analysis. Section IV provides simulation results for the case when the overhead costs are also included in evaluating the utility of a coalition. Also, while we assume a central scheduler for each coalition, the scheduling function may be implemented in a distributed fashion at the expense of additional overhead costs. In what follows, we will require some basic concepts from the theory of polymatroids. We present all the relevant definitions in Appendix A. Our first result is to provide a characterization of CS . To this end, we define a set function mS : 2S → R defined on all subsets of S as   mS (B) = q·(1− (1−pi )) (1−pk ) for all B ⊆ S (3) k∈S c

i∈B

and mS (∅) = 0. In words, mS (B) denotes the product of the packet size q and the probability that at least one element of B is active and that all service providers in S c are inactive. The function mS satisfies the following property. Lemma 1: For every S ⊆ N , mS is a rank function. Proof: From the definition in Appendix A, mS is a rank function if it is normalized, non-decreasing, and submodular. It is normalized since by definition mS (∅) = 0. Also, for any S1 ⊆ S2 ⊆ S

1 − i∈S1 (1 − pi ) mS (S1 )

= ≤ 1, mS (S2 ) 1 − i∈S2 (1 − pi ) which proves that mS is non-decreasing. Similarly, for any S1 , S2 ⊆ N , we have mS (S1 ∪ S2 ) + mS (S1 ∩ S2 ) mS (S1 ) + mS (S2 )   2 − i∈S1 ∪S2 (1 − pi ) − j∈S1 ∩S2 (1 − pj )   = 2 − i∈S1 (1 − pi ) − j∈S2 (1 − pj ) = 1− ⎛

⎝1 −

⎞   (1 − pi )⎠ (1 − pi ) − (1 − pj )

 i∈S2 \S1

2−



i∈S1 ∩S2

(1 − pi ) −

i∈S1

≤1

and thus mS is submodular.



j∈S2

j∈S1

(1 − pj )

797

The following theorem provides an explicit characterization of the S-capacity region CS in (2) as the polymatroid P (mS ) associated with the rank function mS . Theorem 1: For every S ⊆ N , the S-capacity region CS is the polymatroid associated with the rank function mS , i.e., CS = P (mS ). Proof: To begin with, we prove that CS ⊆ P (mS ). Consider any achievable rate vector (r1 , r2 , . . . , r|S| ) in CS , with associated scheduling function gS . Observe from (1) that for any B ⊆ S,      ri = q · EA 1{gS (A)={i}} (1 − pk ) i∈B

i∈B



≤ q · P (|B ∩ A| > 0)

k∈S c

(1 − pk )

k∈S c

= mS (B) where the expectation in the first equation is taken with respect to the random set of active service providers A ⊆ S, the inequality follows from property (P1) of the scheduler which implies that gS (A) ∈ B for all A such that A∩B = ∅, and the last equality follows from (3). Thus we have CS ⊆ P (mS ). To show the reverse inclusion, we prove that any rate vector on the boundary of P (mS ) is achievable. Since any rate vector in P (mS ) is dominated component-wise by a rate vector in the base polytope B(mS ) defined in (22), it suffices to prove the achievability of the |S|! extreme points of B(mS ) given in Lemma 2 in Appendix A. For any permutation σ on the set S, we will show that the corresponding extreme point rσ (mS ) is achievable. Consider a scheduling function gS such that for any subset S  ⊆ S it selects the maximum element from the set of providers in S  , according to the ordering σ1 >  · · · > σ|S| , i.e., gS (S  ) = σi iff σi ∈ S  and  σj ∈ S for all j = 1, · · · , i − 1. Then it follows that EA 1{gS (A)={σi }} =

i−1 pσi j=1 (1 − pσj ) and thus    (1 − pk ) = mS ({σ1 }) = rσσ1 (mS ) q · EA 1{gS (A)={σ1 }} k∈S c

and for i = 2, · · · , |S|,    q · EA 1{gS (A)={σi }} (1 − pk ) k∈S c

= q · pσi

i−1 

(1 − pσj )

j=1



(1 − pk )

k∈S c

= mS ({σ1 , · · · , σi }) − mS ({σ1 , · · · , σi−1 }) = rσσi (mS ), and hence the extreme point rσ (mS ) is achievable. Remark 1: Let the S-sum capacity region denote the set of the rates in CS for which the sum-rate is maximized. It follows from the above theorem that the S-sum capacity region is equal to the dominant face B(mS ) of P (mS ) and that the corner points of B(mS ) can be achieved by simply scheduling active providers based on a predefined priority list. Note that the maximum achievable sum-rate for a coalition S is given by   (1 − pi )) (1 − pk ). mS (S) = q · (1 − i∈S

k∈S c

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Remark 2: The S-capacity region CS has the same geometric structure as the information-theoretic capacity region of the multiple access channel [18]. We remark that an interpretation of the capacity region of the multiple access channel via coalitional game theory appears in [19]. In the extreme case of no cooperation the system operates in full anarchy and the S-capacity region of the singleton S = {i} is simply the product of the packet size q and 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 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, we look at the problem from the perspective of coalitional game theory, see Section B for notation and preliminaries. We consider the profit game (N, v) with characteristic function v defined as   (1 − pi )) (1 − pk ) for all S ⊆ N (4) v(S) = q · (1 − k∈S c

i∈S

and v(∅) = 0. This choice of the characteristic function is motivated by the observation that v(S) represents the maximum achievable average sum-rate by the members of S when they form a coalition, while assuming adversarial behavior from the providers outside S, i.e., there will be a packet collision whenever at least one provider in each of S and S c is active. In fact, v(S) is the product of the packet size q and the probability that at least one element of S is active while all service providers in S c are inactive. It turns out that the game (N, v) has many interesting properties. To begin with, it has transferable utility since the profit of a coalition represents the average number of bits per slot that each coalition can transmit and can be arbitrarily apportioned between the members of the coalition. The next theorem gives a characterization of the core C(v) of this game, defined in (24), as the N -sum capacity region B(mN ). 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(mN ). Proof: The proof is based on duality. From Lemma 1, mN is a rank function and thus the cost game (N, mN ) is concave. It then follows from the definition of the core in (24) and from the definition of the base polytope in (22) that C(mN ) = B(mN ). Next, observe that the profit game (N, v) is the dual of the cost game (N, mN ). In fact from (3), for all S ⊆ N   (1 − pj )) (1 − pk ) v(S) = q · (1 − j∈S

= mN (N ) − mN (S c )

k∈S c = m∗N (S).

(5)

where m∗N denotes the dual characteristic function of mN . Hence, the claim follows from Proposition 3. Remark 3: Recall that in a coalitional game with transferable utility, the profit of a coalition S can be divided in any manner among its members. To ensure that these allocations

are also achievable, we will restrict attention to only those allocations which are also elements of the S-capacity region. In particular, for S = N the above theorem shows that all allocations in the core are in fact achievable as well. Remark 4: 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 [20] and games for which part of the core cannot be achieved [13]. 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, as described in Remark 1. Remark 5: For any profit game (N, v) and any permutation σ on the set N , the vector rσ (v) defined in (23) 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 [21]. The Shapley profit allocation of a profit game (N, v) is the centroid of the marginal vectors, i.e., 1  σ r (v) (6) n! 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. [22]). An important property of any convex profit game is that the Shapley profit allocation lies in its core. As a corollary, the core of any convex profit game is non-empty. It follows from Theorem 2 that the Shapley profit allocation of the game (N, v) is simply the centroid of the N -sum capacity region B(mN ). Furthermore, since the core is non-empty it also contains the nucleolus profit allocation. Given that the core has multiple rate allocations and that any of these 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 [13], which is based on an axiomatic notion of fairness named envy-free fairness [13], [23]. Corollary 1: The following hold: 1) Let r be the proportional profit allocation vector defined by pi v(N ) (7) ri =  k∈N pk for i ∈ N . Then r is an element of the core C(v) for all p1 , · · · , pn . 2) Let r be the individually rational egalitarian profit allocation vector defined by  1 v({j}) + v({i}). (8) ri = v(N ) − n j∈N

for i ∈ N . Then for all n > 2, there exists a tuple (p1 , p2 , · · · , pn ) such that r is not an element of C(v).

KARAMCHANDANI et al.: AGILE BROADCAST SERVICES: ADDRESSING THE WIRELESS SPECTRUM CRUNCH VIA COALITIONAL GAME THEORY

(n − 1)v(N ) + (n − 1)v({n}) ≥ nv(N \ {1}) + (n − 1)v({1})

 =⇒ (n − 1)mN (N ) + (n − 1) mN (N ) − mN (N \ {n}) ≥ 



n mN (N ) − mN ({1}) + (n − 1) mN (N ) − mN (N \ {1})

 =⇒ n · mN ({1}) + (n − 1) mN (N \ {1}) − mN (N \ {n}) ≥ mN (N )

1 1 q =⇒ = q · n · p1 ≥ mN (N ) = q · 1 − n−1 · 2 2 2n

where the last inequality follows since mN (N \ {1}) < mN (N \ {n}) and using the definition of the function mN in (3) and the values of pi ’s. It is easy to check that the last inequality is not true for all n > 2 and 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 C. Next, we present an example to illustrate the results presented in this section. Example 1: Consider a network composed of two service providers that are active with probabilities p1 and p2 , respectively and let packet size q = 1. Then it follows from Theorem 1 that the capacity region CN = P (mN ) and is equal to the set of non-negative rate pairs (r1 , r2 ) such that r1 ≤ p1 r2 ≤ p2 r1 + r2 ≤ p1 + p2 − p1 p2 . The two extreme points of CN strictly inside the positive orthant have coordinates (p1 , p2 (1 − p1 )) and (p1 (1 − p2 ), p2 ), respectively, and can be achieved by a scheduler which always

0.6 Proportional profit Envy-free Rational egalitarian and Shapley

0.4 r2

3) Let r be the envy-free profit allocation vector defined by   n n   q i 1 − (1 − pi ) (1 − pk ) − rk , ri = i k=i+1 k=i+1 (9) for i ∈ N . Then, r is an element of C(v) for all p1 ≥ p2 ≥ . . . ≥ pn . Proof: By Theorem 2, proving that a rate allocation is in the core is equivalent to proving that it is an element of the N sum capacity region B(m  N ). By definition of the proportional profit allocation (7), i∈N ri = v(N ). Furthermore, from (25) we have v(N ) = mN (N ) and thus for any S ⊂ N   pi ri =  i∈S · mN (N ) ≤ mN (S), k∈N pk i∈S  where the

inequality follows since f (x) = ( i∈S pi +x)/(1− (1 − x) i∈S (1 − pi )) is an increasing function of x ∈ (0, 1) for any S ⊆ N . Thus the proportional profit allocation vector is an element of B(mN ), as defined in (22). Next, consider the individually rational egalitarian profit allocation in (8). We claim that if we set p1 = 1/2n and pj = 1/2 for all j = 1, then the profit allocation r is not an element of C(v). By contradiction, suppose that for this choice of the parameters r ∈ C(v). Then using the inequality  i∈N \{1} ri ≥ v(N \ {1}), (8), and (25) we have

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Capacity CN

0.2

Core C(v) = B(mN )

0

0

0.4

0.8 r1

Fig. 1. Capacity and core for a systems with two providers (q=1, p1 = 0.8, and p2 = 0.4) in the uncoordinated access model. In this case, the core coincides with the dominant face of the pentagon delimiting the capacity region.

gives priority to provider 1 (provider 2) over provider 2 (provider 1). On the other hand, the core C(v) of the profit game (N, v) is equal to the set of non-negative allocations (x1 , x2 ) such that x1 ≥ p1 (1 − p2 ) x2 ≥ p2 (1 − p2 ) x1 + x2 = p1 + p2 − p1 p2 . It follows from Theorem 2 that C(v) is equal to the base polytope B(mN ), i.e., the dominant face of the pentagon delimiting CN , and hence that the Shapley profit allocation, which is in this case equal to the individually rational egalitarian allocation, is the midpoint of B(mN ). It follows from Corollary 1 that the proportional profit allocation (p1 (1 + p1 p2 /(p1 + p2 )), p2 (1 + p1 p2 /(p1 + p2 ))) and the envy-free allocation (1 − p1 (1 − p2 ) − p2 (1 − p2 /2), p2 (1 − p2 /2)) are in the core C(v) and are achievable. Fig. 1 illustrates the geometry of the problem when p1 = 0.8, and p2 = 0.4. 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. III. C OORDINATED ACCESS M ODEL Consider now an orthogonal medium access scheme where each service provider is granted exclusive access to one of many mutually orthogonal channels. This setup occurs, for instance, in radio and television broadcast services and, more generally, in scenarios where service providers transmit in different frequency bands. For simplicity, we consider here a time-division multiple access (TDMA) system, in which each slot is assigned by the authority for transmission to one service

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provider in a periodically repeated order, so that collisions are always prevented. The same analysis can be repeated, however, in the case of other orthogonal medium access schemes, such as frequency division multiple access (FDMA), code division multiple access (CDMA). We assume that provider i is active in each slot with probability pi , independently of other providers in the network, and transmits a packet consisting of q bits. 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 active member of the coalition. We will now define the notion of an achievable rate region for this setup. Many of the definitions are analogous to those in Section II. 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 gS for coalition S is as in Section II, 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 gS is an achievable rate vector (r1 , · · · , r|S| ) defined for every i ∈ {1, 2, . . . , |S|} as   |S| · EA 1{gS (A)={i}} ri = q · n    |S|    · =q· pj (1 − pl ) 1{gS (A)={i}} n A⊆S j∈A

l∈S\A

(10) where recall that q denotes the number of bits in any packet, 1E 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, similar to the explanation in Section II. The second term in the above product denotes the average fraction of slots assigned to the coalition S. In each such slot, a member of the coalition S holds the right to access the channel and the scheduler selects an active provider in S according to the scheduling function gS . Then the third term in the above product corresponds to the probability that provider i ∈ S is chosen for transmission in such a slot. It is equal to the probability that the set of active providers A in a slot is such that gS (A) = i. As time progresses, the ratio between the number of slots where provider i ∈ S is chosen for transmission and the total number of slots converges to the product of the second and the third terms. Since in any such slot the provider i successfully transmits a packet of q bits, (r1 , · · · , r|S| ) denotes the vector of average transmission rates (in bits / slot) that can be simultaneously achieved by the members of coalition S via the scheduler gS . In the extreme case of no cooperation the system operates in TDMA mode and the maximum achievable rate by provider i is q · pi /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 II and hence the closure of the union of all the achievable rate vectors is the N -capacity

region CN , as defined before in (2) in Section II. Similar to Theorem 1 before, CN is equal to the polymatroid P (mN ) and the base polytope B(mN ) represents the set of achievable rate vectors for which the sum-rate is maximized. 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. Note that the above definition does not include the overhead of sharing activity information among the members of a coalition. We assume that the number of bits required for sharing such information is negligible compared to the packet size q and hence is not included in the analysis. Section IV provides simulation results for the case when the overhead costs are also included in evaluating the utility of a coalition. Also, while we assume a central scheduler for each coalition, the scheduling function may be implemented in a distributed fashion at the expense of additional overhead costs. To answer this question we consider the profit game (N, w) with characteristic function defined as |S| · mN (S) for all S ⊆ N. (11) w(S) = n where mN is as defined in (3). This choice of the characteristic function is motivated by the observation that w(S) denotes the maximum achievable average sum-rate by the members of S when they form a coalition. In fact, w(S) is the product of the packet size q, the fraction |S| /n of the slots assigned to coalition S, and the probability that at least one element of S is active in a slot. As in the previous section, this profit game (N, w) has a transferable utility. Next, we derive some interesting properties of this game. Proposition 1: The profit game (N, w) is superadditive. Proof: For any two disjoint coalitions S1 , S2 ⊆ N |S1 | + |S2 | · mN (S1 ∪ S2 ) n |S1 | |S2 | · mN (S1 ∪ S2 ) + · mN (S1 ∪ S2 ) = n n |S1 | |S2 | ≥ · mN (S1 ) + · mN (S2 ) n n = w (S1 ) + w (S2 )

w (S1 ∪ S2 ) =

where the inequality follows from the fact that mN is a nondecreasing 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 vectors that are elements of C(w) ∩ B(mN ). The next proposition shows that unlike in the uncoordinated access model considered in Section II, this is not always possible. Proposition 2: The following facts are true for n = 2: 1) (N, w) is a convex game. 2) For every pair (p1 , p2 ) there exists an achievable stabilizing rate allocation, i.e., C(w) ∩ B(mN ) = ∅. 3) There exists a pair (p1 , p2 ) such that the Shapley profit allocation (6) and the nucleolus are not in the N -sum capacity region B(mN ).

KARAMCHANDANI et al.: AGILE BROADCAST SERVICES: ADDRESSING THE WIRELESS SPECTRUM CRUNCH VIA COALITIONAL GAME THEORY

i∈S

 for all S ⊆ N with strict equality if S = N . From i∈N ri = mN (N ) and i∈N \{1} ri ≥ w(N \ {1}), it follows that r1 ≤ mN (N ) − w(N \ {1}), which combined with r1 + rn ≥ w({1, n}) gives rn ≥ w({1, n}) − mN (N ) + w(N \ {1}) q = − qα(n)(1 − p1 ), n 1 where α(n) = nn+2 (2nn + n4 − 2n3 ) ∈ (0, 1). Combining the above inequality and rn ≤ mN ({n}) yields, pn ≤ 1 −

1 1 1 ≤1− , n 1 + α(n) n

which is a contradiction with the hypothesis that pn = 1 1 − 2n > 1 − n1 . Finally, consider the symmetric case where all providers are active with probability p. In this case, the symmetric profit allocation r1 = · · · = rn = q(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 q

1 − (1 − p)|S| 1 − (1 − p)n ≤q for every S ⊆ N. n |S|

Remark 6: The above result shows that there exist a set of probabilities p1 , · · · , pn for which the intersection between the capacity region and the core is empty. Note that this does not rule out the possibility that the core itself is empty, but in this case stability can never be reached and so the question of achieving a stable point is ill posed.

Proportional profit Envy-free

0.8

Rational egal. and Shapley

r2

For all n > 2, instead, 1) There exists a set of probabilities p1 , · · · , pn for which no rate allocation in the core is achievable, i.e., C(w) ∩ B(mN ) = ∅. 2) In the symmetric setting p1 = p2 = . . . = pn = p, there exists an achievable stabilizing rate allocation, i.e., C(w) ∩ B(mN ) = ∅. Proof: Let n = 2. It is immediate to check that the two-user game is convex. For any achievable stabilizing rate allocation (r1 , r2 ), inequalities (24) and (21) require that qpj /2 ≤ rj ≤ qpj , j ∈ N , and r1 + r2 = q(p1 + p2 − p1 p2 ). These constraints are simultaneously satisfied by both the proportional profit allocation which assigns rj = w(N ) · pj /(p1 + p2 ), j ∈ {1, 2}, and the profit allocation which assigns rj = q(pj − p1 p2 /2), j ∈ {1, 2}. Since the game is convex for any (p1 , p2 ), the Shapley profit allocation and the nucleolus are trivially in the core. However, let q = 1 and let (p1 , p2 ) = (1/4, 3/4) (cf. Fig. 2). Then 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 do not lie in the capacity region as they violate the constraint r1 ≤ p1 = 1/4. Suppose now that n > 2, and let 1 and pi = 1/n for all i < n. We claim that pn = 1 − 2n for this choice of the parameters, C(w) ∩ B(mN ) = ∅. 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  ri ≤ mN (S) w(S) ≤

801

B(mN )

0.4

Capacity CN Core C(w)

0

0

0.4

0.2 r1

Fig. 2. Capacity and core for a systems with two providers (q=1, p1 = 0.25, and p2 = 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.

Next, we present an example to illustrate the results presented in this section. Example 2: Consider again the setup of Example 1 in which two service providers are active with probabilities p1 and p2 , respectively and packet size q = 1. In this case, the core C(w) of the profit game (N, w) is equal to the set of non-negative allocations (x1 , x2 ) such that x1 ≥ p1 /2 x2 ≥ p2 /2 x1 + x2 = p1 + p2 − p1 p2 . In general, C(w) partially overlaps with the base polytope B(mN ) of the capacity region CN . For example, Fig. 2 illustrate the boundaries of CN (solid line) and the boundaries of the inequalities describing C(w) (dotted lines) in the special case where p1 = 1/4 and p2 = 3/4. It is clear from the figure that part of the core is not achievable because it lies outside B(mN ). In particular, the Shapley profit allocation, which in this case is equal to the individually rational egalitarian allocation, and the envy-free allocation are not achievable for this specific choice of the parameters. Thus we can conclude that unlike the uncoordinated setting in Section II where the grand coalition can always be stabilized, the randomness of traffic arrival at each service provider plays a crucial role in determining the stability of orthogonal medium access networks in which cooperation is used to share transmission orthogonal resources 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. The fact that the grand coalition need not be stable remains true under more restrictive models of cooperation where orthogonal channels

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cannot be assigned to the coalition constituent members in an arbitrary fashion. IV. I MPACT OF C OST OF C OOPERATION 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 : 2N → R+ which returns the non-negative cost in bits / slot 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, (12) c(S) = 0 otherwise, for all S ⊆ N and for some cost amplification factor α > 0. The choice in (12) is motivated by the fact that the cost of cooperation grows proportionally to the number of providers forming a coalition. This is because in every slot, each provider must transmit some auxiliary information bits to a designated coalition leader to set up the cooperative protocol. For example, each provider needs to indicate their intent to transmit at the beginning of a slot. The parameter α represents the cost per user per slot and can be scaled to model different scenarios. Recall that in the uncoordinated access model and without any cost of cooperation, the average number of bits / slot that a coalition S can transmit is given by qv(S), where q is the number of bits an active provider can transmit if there is no collision and v(S) is the characteristic function defined in (4). Since c(S) bits are spent each slot in setting up the cooperation, the net value of a coalition S in the uncoordinated model becomes v˜(S) = v(S) − c(S),

(13)

where v(S) is the characteristic function defined in (4). We can similarly define a coalition game (N, w) ˜ for the coordinated access model, where the characteristic function is given by w(S) ˜ = w(S) − c(S),

(14)

and w(S) is defined in (11). Since the cost of cooperation increases with the size of the coalition, (N, v˜) 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. A. Coalition formation algorithm As is customary in the coalition formation game literature [24]–[27], we study the equilibria for the games (N, v˜) 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. We assume that the algorithm starts with all the service providers acting non-cooperatively. In each round, providers

Algorithm 1 Coalition formation algorithm Initial network configuration Let the network partition into coalitions at the beginning, Πinitial = N . Coalition formation algorithm with three stages Stage 1: Neighbor discovery Each provider i discovers neighboring coalitions, i.e., coalitions such that all their members are within a pre-specified threshold distance d from i. d is called the coalition radius. Stage 2: Coalition formation Let the current network partition into coalitions Πcurrent = Πinitial . repeat Given the current network partition Πcurrent , each provider i performs the following steps in its turn. 1) Provider i searches for a better neighboring coalition T ∈ Πcurrent than its current coalition S ∈ Πcurrent , such that T ∪ {i} i S according to (15), (16). 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 Πcurrent with (Πcurrent \ {S, T }) ∪ {(S \ {i}), (T ∪ {i})}. until Current network partition Πcurrent converges to a stable partition, i.e., no provider has an incentive to switch to another coalition. Stage 3: Cooperative network operation Providers operate cooperatively according to the stable current network partition Πcurrent .

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 next paragraph discusses specific conditions which might make a coalition preferable to another. The process is repeated until the network reaches a partition where no provider has an incentive to make a switch to another coalition. It is shown in [24] 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 S1 , S2 containing i, S1 i S2 implies that i prefers being a member of S1 over S2 . In our setting, we assume that S1 i S2 if and only if fi (S1 ) > fi (S2 ), where fi : 2N → R is a preference function defined as

(15)

KARAMCHANDANI et al.: AGILE BROADCAST SERVICES: ADDRESSING THE WIRELESS SPECTRUM CRUNCH VIA COALITIONAL GAME THEORY

(b) α = 0

Max c oal i t i on si z e

210

190

170

where ri (S) is the payoff received by service provider i in coalition S, for example according to the proportional profit allocation as in (7), and h(i) is a history set which keeps track of all the coalitions that i joined and left in the past. In words, the value of a coalition S for provider i is the payoff it receives in the coalition, if none of the other members of the coalition S will be better off without i in the coalition and the coalition has not been visited before during the run of the algorithm, or the coalition just consists of i, i.e., S = {i}. Given the preference function (15), the steps of the proposed coalition formation algorithm are summarized in Algorithm 1.

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B. Simulation Results In this section we present some simulation results that illustrate how the stable equilibria for the games (N, v˜) 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 activity probability pi uniformly at random from (0, 1). This particular choice is just for illustration and the same setup can be used with other distributions of activity probabilities. It is assumed that in any time slot an active provider can transmit q = 200 bits if there are no collisions2 . Service providers start from the non-cooperative state and employ Algorithm 1 to establish cooperation in the network. For the preference function defined in (16), we assume that the net value of a coalition is divided among its members according to the proportion profit allocation in (7). Apart from the cost in (12) 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. Hereafter, we will refer to d as the coalition radius. All the results presented in this section are averaged over the random positions of the providers as well as their corresponding activity probabilities. Given the parameters of the setup, we then use the coalition formation algorithm to simulate the evolution of the coalition structures in the network. In each round, any provider i attempts to discover a neighboring coalition which is preferable to its current coalition. To enable this operation, the provider i and the coalition need to exchange some auxiliary information with each other. For example, the provider i can send the value of its activation probability pi to a designated leader in the neighboring coalition. The coalition leader can calculate the updated payoffs that its members and the provider i will achieve if i was allowed to join the coalition. The members can then decide whether to invite the provider i to join their 2 While we have fixed the value of packet size q here for the simulations, it is easy to see that the performance of the coalition formation algorithm qualitatively only depends on α/q. Thus, the simulation results are also representative of the results for other values of q as well.

Fig. 3.

Network utility vs number of service providers.

coalition. If they decide in the affirmative, payoff information is sent back to provider i, upon which it can decide whether to switch coalitions. In any round, the number of times a provider needs to repeat this process is at most the number of coalitions in the current partition of the network, which is at most the total number of providers n. Thus, the computational complexity for each provider per round is at most O(n). Now, each provider i maintains a history of all the coalitions it has been part of and does not return to a coalition it was part of before and left (except the singleton {i}). Thus, at the end of each round, the network partition in to coalitions is different from any partition observed before during the run of the algorithm. Thus, the number of rounds that an algorithm needs to run is at most the total number of possible partitions of the network, which is finite. 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 (13), (14). 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 when α = 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 and (15), (16) 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 can be easily checked using (4) and (7) that the above condition is satisfied if for any k ∈ S, 

pk j∈S pj

v(S) ≤ 

pk pi +j∈S

pj

v(S ∪ {i})

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which holds true for any choice of activity probabilities. On the other hand, from (11) and (7) the above condition is satisfied for the coordinated access model only if pk p  k w(S) ≤  w(S ∪ {i}) p j j∈S pi +j∈S pj (|S| + 1)mN (S ∪ {i}) |S| mN (S)  ≤ =⇒  p pi + j∈S pj j j∈S  =⇒ pi |S| mN (S) ≤ (|S| + 1)pi · pj · (1 − mN (S))

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Ne t u t i l i ty

   mN (N ) − mN (S c ) mN (N ) − mN ((S ∪ {i})c )   =⇒ ≤ pi + j∈S pj j∈S pj   =⇒ pi mN (N ) − mN (S c )    pj · mN (S c ) − mN ((S ∪ {i})c ) ≤

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j∈S

+  =⇒ pi





|S|

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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)).

pj · mN (S)

j∈S

1 − mN (S) mN (S)

 ≤ 1.

(18)

where mN (S) is as defined in (3). The above condition is not always true and may prevent the formation of the grand coalition even when α = 0. 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 and (15), (16) 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 S. This implies that the net value of the coalition S should not decrease when i joins the coalition. From (13), (14), it is easy see that this condition holds if  α ≤ pi · (1 − pj ) (19) q j ∈S∪{i} /

for the uncoordinated access model and  α 1 ≤ 1 − (1 − (|S| + 1)pi ) (1 − pj ) q n

(20)

j∈S

for the coordinated access model. When the coalition formation algorithm starts, the coalition S that a provider i attempts to join is of small size since all providers start from the non-cooperative state. Then the right hand side of the condition in (19) involves a product of close to n terms, each smaller than one, and thus decreases exponentially with the number of providers n. On the other hand, the right hand side in (20) only decreases polynomially in n. It follows that as n increases, the coalition formation algorithm 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 vs. the cost factor α for the uncoordinated and the coordinated access model. Notice that in the uncoordinated access model the average network utility decreases sharply in α if the number of providers is large. 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 steep for the coordinated access model. This is due to the fact that the average network utility is less sensitive to the coalition sizes in the coordinated access model. Fig. 4c and Fig. 4d plot the average network utility achieved vs. the coalition radius d for the uncoordinated and the coordinated access models. As expected, the network utility increases for both models as d grows, with the stable network partition evolving from one with many small coalitions when d is small to one with few big coalitions when d is large. The gain is more significant in the uncoordinated access model than in the coordinated access model because of the different sensitivities to coalition sizes. The optimal centralized coalition formation algorithm will exhaustively search for the best possible coalition structure among all possibilities. While this will give the highest network utility, the algorithm will become computationally infeasible as the network size grows. The other extreme is the scenario where there is no cooperation amongst the providers, which is computationally very light but also suffers from sub-optimal network utility. Our proposed coalition formation algorithm treads the middle ground between these two extremes. Fig. 5 compares the average network utility achieved using the optimum centralized coalition formation algorithm, our proposed distributed coalition formation algorithm, and non-cooperation, averaged as before over multiple simulation outputs. Fig. 5a and Fig. 5c correspond to the uncoordinated and the coordinated access models respectively, with no cost or coalition radius constraints. In both cases, there is a big

KARAMCHANDANI et al.: AGILE BROADCAST SERVICES: ADDRESSING THE WIRELESS SPECTRUM CRUNCH VIA COALITIONAL GAME THEORY

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Fig. 5. Network utility vs number of providers n under the optimal centralized coalition formation algorithm, the proposed distributed algorithm, and non-cooperation, for the uncoordinated model (cases (a) and (c)) as well as for the coordinated √model (cases (b) and (d)). Cases (a) and (b) correspond to α = 0 and d = 2 while cases (c) and (d) correspond to α = .25 and d = 1.1.

difference between the achieved network utility with and without cooperation. For the uncoordinated access model, the proposed coalition algorithm works as well as the best centralized coalition formation algorithm. On the other hand, for the coordinated access model, there is a gap between the performance of the two algorithms which seems to remain bounded as the number of providers grows. These results are easily explained by the analysis in equations (17) and (18). Fig. 5b and Fig. 5d consider the case with a positive cost factor α and a non-trivial coalition radius d. In this case, there is a gap between the performance of the proposed coalition formation algorithm and the optimal centralized algorithm under the uncoordinated access model. The performance under the coordinated access model remains more or less the same since the average network utility is less sensitive to the coalition sizes in the coordinated access model. V. C ONCLUDING R EMARKS In this paper, we used tools from coalitional game theory to study the stability of coalitions in real-time broadcast services over a common wireless channel. While we focused on a simple setup to illustrate the main ideas, many of the key notions can be generalized. For example, we considered the symmetric scenario where all signals broadcast by the active service providers are received at the same power level at a given receiver and thus we adopted the classic collision channel model where transmissions are successful if only one service provider transmits in a slot. A natural generalization is to consider the case where signals from different providers in the network are received at different power levels, for which we can assume that if the signal to interference ratio is higher than a certain threshold, then the interference problem can be ignored and a transmitted packet can be successfully decoded. To this end, one can follow the approach in [28] and allow the access probabilities pi ’s to depend on the received signal-tonoise-plus-interference ratios. However, the access probability

805

of each provider will also have to include the corresponding activity probability. The right modeling to include both effects is not immediate and is not covered by the setup of this paper. Once the access probabilities are fixed, the analysis proposed in this paper can be repeated to understand in what scenario cooperation among service providers can result in stable coalitions. Furthermore, as in most previous work, we assumed that the utility of a coalition is the average sum-rate achieved by the providers in the coalition while assuming adversarial behavior from the players outside the coalition. The motivation for this assumption is that it leads to a coalitional game in characteristic form [20], where the utility does not depend on the players which are not part of the coalition. A more general model would allow the value of a coalition to depend on the way the service providers outside the coalition are partitioned. This will result in a coalitional game in partition form, which is typically less amenable to analysis than the game in characteristic form. We investigated the impact of the cost of establishing a coalition by studying the performance of a distributed coalition formation algorithm [24]. The impact of the underlying system parameters on the performance of the scheme is illustrated via simulation results and partial analysis. While we have used a linear cost model for illustration, the setup can be naturally generalized to other cost models. Also, we chose to compare the performance of two canonical medium access schemes where service providers either act as completely independent agents (uncoordinated model) or are fully coordinated (orthogonal transmission model). More practical medium access schemes, such as those employing carrier sense or RTS / CTS, are likely to be somewhere in the middle, as real systems always require some level of coordination among nodes to avoid an excessive amount of interference. By studying the two extreme cases we hope to have given some guidelines towards optimal design by emphasizing the tradeoffs between these two cases. Another assumption we make in our setup is that the activity probabilities remain constant. This can be readily addressed by running the coalition formation phase repeatedly, as long as the time required for coalition formation is small compared to the time it takes for the activity probabilities to change. Our study reveals that coalitional game theory can be a useful tool to study cooperative communications, providing insights into the design of real systems. Specifically, it suggests that in networks where the number of providers is small or the cost of cooperation is negligible, from a global throughput perspective an open sharing model is preferable to a regulated access mechanism, since in the latter case players tend to preserve their baseline, guaranteed, individual throughput, while in the former case they are forced to cooperate to achieve the global optimum by forming the grand coalition. On the other hand, as the size of the network or the cost of cooperation increases, there is significant inertia to cooperate and providers break off into multiple scattered coalitions creating interference limitations. In this case, the regulated access mechanism allows a higher degree of cooperation by regulating transmissions and achieves better performance.

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A PPENDIX A S UBMODULAR F UNCTIONS AND P OLYHEDRA

The game is convex if v is supermodular, i.e.,

Let f be a real-valued function defined on subsets of N . We say that f is normalized if f (∅) = 0, non-decreasing if S1 ⊆ S2 ⊆ N implies f (S1 ) ≤ f (S2 ), and non-increasing if −f is non-decreasing. f is submodular if, for S1 , S2 ⊆ N , f (S1 ∪ S2 ) + f (S1 ∩ S2 ) ≤ f (S1 ) + f (S2 ), and it is supermodular if −f is submodular. If f is normalized, nondecreasing 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   ri ≤ f (S) for all S ⊆ N . (21) P (f ) = r ∈ Rn+ : i∈S

 The set of points in P (f ) on the hyperplane i∈N ri = f (N ), i.e.,   B(f ) = r ∈ P (f ) : ri = f (N ) (22)

v(S ∪ T ) + v(S ∩ T ) ≥ v(S) + v(T ) ∀S, T ⊆ N. The game (N, v) is subadditive or concave if the corresponding conditions above are satisfied with “≥" replaced by “≤". In a superadditive profit or a subadditive cost game, the players have an incentive to form the grand coalition. A profit (cost)  allocation is a vector x ∈ Rn such that i∈N xi = 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 payoff allocations such that no group of players S ⊂ N has an incentive to leave the grand coalition N and thus, the grand coalition once formed is stable. The core of a profit game (N, v) is defined as   xi = v(N ), C(v) = x ∈ Rn : i∈N



i∈N

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 a rank   orthant. Given function f and any permutation σ = σ, . . . , σn on the set N , let the vector rσ (f ) ∈ Rn be defined as rσσ1 (f ) = f ({σ1 }) and rσσi (f ) = f ({σ1 , · · · , σi }) − f ({σ1 , · · · , σi−1 })

(23)

for i = 2, · · · , n. Lemma 2 ( [22], [29], [30]): Let P (f ) be a polymatroid. Then rσ (f ) is a vertex of P (f ) for every permutation σ. Any vertex strictly inside the positive orthant must be rσ (f ) for some permutation σ. Moreover, for any c ∈ Rn+ a solution of the linear program max c r subject to r ∈ P (f ) is attained at a point rσ (f ) for any permutation σ such that cσ1 ≥ cσ2 · · · ≥ cσn . It follows from the above lemma that the base polytope B(f ) is equal to the convex hull of all extreme point vectors {rσ (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 ) [30]. A PPENDIX B C OALITIONAL G AME T HEORY: BASIC D EFINITIONS A coalitional game with transferable utility is a pair (N, v), where v : 2N → 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 profit game if v(S) measures the profit earned by a coalition S and is a cost game if v(S) measures the cost incurred by S. The transferable utility property of the game implies that the profit (cost) v(S) of a coalition can be divided in any manner among its members. The game (N, v) is superadditive if the characteristic function v satisfies v(S ∪ T ) ≥ v(S) + v(T ) ∀S, T ⊂ N such that S ∩ T = φ.

xi ≥ v(S) for all S ⊆ N .

(24)

i∈S

The core of a cost game (N, v) is defined by (24) with “≥” replaced by “≤”. In words, 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 a specific allocation called the nucleolus 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]. The dual of a profit game (N, v) is the cost game (N, v ∗ ) defined by the dual characteristic function v ∗ (S) = v(N ) − v(S c )

(25)

where S c = N \S. Similarly, a dual profit game can be defined for a cost game. The following proposition, proved in [16], establishes a connection between a coalitional game and its dual. Proposition 3: 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 ∗ ). A PPENDIX C T HE E NVY-F REE A LLOCATION IS IN THE C ORE Let r(N ) denote the envy-free profit allocation (9) in a system where the set of players is N . We to show that  need (N ) construction r = v(N ), r(N ) ∈ B(mN ). Since by i∈N  it suffices to show that i∈S r(N ) ≤ mN (S) for any strict subset S of N . The proof is by induction over the number of providers. If there is only one provider in the game, the above claim is true. Now suppose that the claim is true for a set of n providers. We will now show that the claim is true for a game with a player set of n + 1 providers and for which p1 ≥ p2 ≥ . . . ≥ pn+1 . To do so, we prove that (N )

ri

(N \{j})

≤ ri

(26)

KARAMCHANDANI et al.: AGILE BROADCAST SERVICES: ADDRESSING THE WIRELESS SPECTRUM CRUNCH VIA COALITIONAL GAME THEORY

for all i ∈ N \ {j}. In fact, by (26) and the induction hypothesis, it follows immediately that  (N )  (N \{j}) ri ≤ ri ≤ mN (S) i∈S

i∈S

for any strict subset S of N , which proves the claim. The rest of the proof is then devoted to showing (26). Let qi = (1 − pi ) for all i ∈ N . We consider three cases: 1) j = n + 1: In this case, it can be easily verified that for any i ∈ {1, 2, . . . , n − 1} (N \{j})

ri

(N )

− ri

(N )

(N \{j})

≥ ri+1

On the other hand, rn+1 =

n+1 1−qn+1 n+1

(N )

− ri+1 .

(27)

n+1 qn+1 − (n + 1)(1 − qn+1 )qnn

where (a) follows since pn ≥ pn+1 . Combining (27) and (28) yields (26). 2) j = 1: Simple algebra shows that for all i ∈ {2, . . . , n}   n+1  i = qi+1 − qii qk k=i+1 n    i−1 ≤ qi+1 − qii−1 qk

−

k=i+1 (N \{j}) ri+1 ),

where the inequality follows from the fact that xi − y i xi−1 − y i−1 ≥ for any 1 ≥ x > y ≥ 0. i i−1 (29) for all i ≥ 2. Thus, (N \{j})

ri

(N )

− ri

(N \{j})

≥ ri+1

(N )

− ri+1 .

(30)

for all i ∈ {2, . . . , n}. On the other hand, (N \{j})

rn+1

(N )

− rn+1 =

n+1 1 − qn+1 1 − qnn − ≥ 0, n+1 n

where the inequality follows from (29). Hence, (26) is proved also in the case j = 1. 3) 1 < j < n + 1: Repeating the same steps as in the analysis of the case j = 1, it is possible to show that the inequality in (26) holds for all i ∈ {j+1, · · · , n+1}. We analyze the cases i = j − 1 and i ≤ j − 1 separately. First, we write (N )

(N )

(N )

(N )

(N )

(N \{j})

rj−1

(N \{j})

− rj+1

(N )

(N )

≥ rj−1 − rj+1

it suffices to assume pj = pj+1 . In this case, we have j−1 j−1 n+1 − qj−1 ) k=j+1 qk (qj+1 (N \{j}) (N \{j}) . rj−1 − rj+1 = j−1

(N \{j})

(rj−1

(N \{j})

(N )

− rj+1

(N )

) − (rj−1 − rj+1 ) j−1 j−1 n+1 jpj+1 (qj+1 − qj−1 ) k=j+1 qk ≥ 0, = j(j − 1)

(N \{j})

≤ (1 − p2n+1 )n ≤ 1

=

(31)

It can be verified that the right hand side in the above equation is a decreasing function of pj in the interval [pj+1 , pj−1 ]. Hence, to show that

ri

n+1 n ≤ qn+1 − (n + 1)(1 − qn+1 )qn+1

(N \{j}) i(ri

n+1 k=j+1 qk

j(j − 1)

(N \{j})

(a)

−



which yields (26) in the case i = j − 1. It can be easily verified that for all i < j − 1,

The inequality above holds because

(N ) ri+1 )

=

j−1 j + (j − 1)qj+1 qjj − jqj qj−1

Subtracting to the left hand side of (31) and after some algebra we obtain

and

n(rn(N \{j}) − rn(N ) )  1  n+1 = 1 − qn+1 − (n + 1)(1 − qn+1 )qnn ≥ 0. n+1 (28)

(N ) i(ri



807

(N )

rj−1 − rj+1 = (rj−1 − rj ) + (rj − rj+1 )

j−1 n+1 j (qj+1 (qjj−1 − qj−1 ) k=j qk − qjj ) n+1 k=j+1 qk + = j−1 j

(N )

− ri

(N \{j})

≥ ri+1

(N )

− ri+1 .

(N )

Since rj−1 − rj−1 ≥ 0, we have that (26) is proved also in the case i < j − 1.

R EFERENCES [1] B. Peleg and P. Sudholter, Introduction to the Theory of Cooperative Games, 2nd ed. Springer, 2007. [2] A. B. MacKenzie and S. B. Wicker, “Selfish users in Aloha: a gametheoretic approach,” in Proc. 2001 IEEE Veh. Technol. Conf., vol. 3, pp. 1354–1357. [3] Y. Jin and G. Kesidis, “Equilibria of a noncooperative game for heterogeneous users of an ALOHA network,” IEEE Commun. Lett., vol. 6, no. 7, pp. 282–284, 2002. [4] V. Krishnamurthy and M. H. Ngo, “A game theoretical approach for transmission strategies in slotted ALOHA networks with multi-packet reception,” in Proc. 2005 IEEE International Conf. Acoustics, Speech, Signal Process., vol. 3, pp. 653–656. [5] E. Altman, R. E. Azouzi, and T. Jiménez, “Slotted Aloha as a game with partial information,” Comput. Netw., vol. 45, no. 6, pp. 701–713, Aug. 2004. Available: http://dx.doi.org/10.1016/j.comnet.2004.02.013 [6] M. Cagalj, S. Ganeriwal, I. Aad, and J.-P. Hubaux, “On selfish behavior in CSMA/CA networks,” in Proc. 2005 IEEE INFOCOM, vol. 4, pp. 2513–2524. [7] L. Jang-Won, T. Ao, H. Jianwei, M. Chiang, and A. Robert, “Reverseengineering MAC: a non-cooperative game model,” IEEE J. Sel. Areas Commun., vol. 25, no. 6, pp. 1135–1147, 2007. [8] A. Aram, C. Singh, S. Sarkar, and A. Kumar, “Cooperative profit sharing in coalition based resource allocation in wireless networks,” in Proc. 2009 IEEE INFOCOM, pp. 2123–2131. [9] S. Sarkar, C. Singh, and A. Kumar, “A coalitional game model for spectrum pooling in wireless data access networks,” in Proc. 2008 Inf. Theory Applications Workshop, pp. 310–319. [10] C. Singh, S. Sarkar, and A. Aram, “Provider-customer coalitional games,” IEEE/ACM Trans. Netw., vol. 19, no. 5, pp. 1528–1542, Oct. 2011. [11] R. Ma, D. M. Chiu, J. Lui, V. Misra, and D. Rubenstein, “Internet economics: the use of Shapley value for ISP settlement,” IEEE/ACM Trans. Netw., vol. 18, no. 3, pp. 775–787, June 2010. [12] P. Lin, J. Jia, Q. Zhang, and M. Hamdi, “Cooperation among wireless service providers: opportunity, challenge, and solution,” IEEE Wireless Commun., vol. 17, no. 4, pp. 55–61, Aug. 2010. [13] R. J. La and V. Anantharam, “A game-theoretic look at the Gaussian multiaccess channel,” in Proc. 2003 DIMACS Workshop Netw. Inf. Theory, pp. 87–106.

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Nikhil Karamchandani (S’05-M’12) received the M.S. degree from the Department of Electrical and Computer Engineering at the University of California at San Diego in 2007, and the Ph.D. degree in the Department of Electrical and Computer Engineering at the University of California at San Diego in 2011. He is currently a postdoctoral research scholar in the Department of Electrical Engineering at the University of California Los Angeles. His research interests are in communication networks and information theory. He received the California Institute for Telecommunications and Information Technology (CalIT2) fellowship in 2005. Paolo Minero (M’11) received the Laurea degree (with highest honors) in electrical engineering from the Politecnico di Torino, Torino, Italy, in 2003, the M.S. degree in electrical engineering from the University of California at Berkeley in 2006, and the Ph.D. degree in electrical engineering from the University of California at San Diego in 2010. He is an Assistant Professor in the Department of Electrical Engineering at the University of Notre Dame. Before joining the University of Notre Dame, he was a postdoctoral scholar at the University of California at San Diego for six months. His research interests are in communication systems theory and include information theory, wireless communication, and control over networks. Dr. Minero received the U.S. Vodafone Fellowship in 2004 and 2005, and the Shannon Memorial Fellowship in 2008. Massimo Franceschetti (M’98–SM’11) is an associate professor in the Department of Electrical and Computer Engineering of University of California at San Diego. He received the Laurea degree, magna cum laude, in Computer Engineering from the University of Naples in 1997, and the M.S. and Ph.D. degrees in Electrical Engineering from the California Institute of Technology in 1999, and 2003. Before joining UCSD, he was a post-doctoral scholar at University of California at Berkeley for two years. Prof. Franceschetti was awarded the C. H. Wilts Prize in 2003 for best doctoral thesis in Electrical Engineering at Caltech; the S. A. Schelkunoff award in 2005 for best paper in the IEEE Transactions on Antennas and Propagation; an NSF CAREER award in 2006, an ONR Young Investigator award in 2007; the IEEE Communications society best tutorial paper award in 2010; and the IEEE Control theory society Ruberti young researcher award in 2012. He has held visiting positions at at the Vrije Universiteit Amsterdam in the Netherlands, the Ecole Polytechnique Federale de Lausanne in Switzerland, and the University of Trento in Italy. He was associate editor for communication networks for 2009-2012 of the IEEE T RANSACTIONS ON I NFORMATION T HEORY, guest editor for two issues of the IEEE J OURNAL ON S ELECTED A REAS IN C OMMUNICATION, and is now serving as associate editor for the IEEE T RANSACTIONS ON C ONTROL OF N ETWORK S YSTEMS for the period 2013-2016.