Cooperation in multi-access networks via coalitional game theory Nikhil Karamchandani
Paolo Minero
Massimo Franceschetti
Department of ECE UCSD La Jolla, CA 92093, USA
[email protected]
Department of EE University of Notre Dame Notre Dame, IN 46556, USA
[email protected]
Department of ECE UCSD La Jolla, CA 92093, USA
[email protected]
Abstract—Cooperation in multiple-access networks is studied via coalitional game theory. Two strategies are examined, both inspired by practical scenarios. The first strategy is representative of a cooperative random access system. Each node randomly alternates between two states, indicating its desire to transmit. Nodes cooperate to avoid interfering transmissions by sharing knowledge of their state with members of the same coalition. A scheduler determines the active user within each coalition that can access the channel. Collisions occur when users belonging to different coalitions transmit simultaneously. In this case, the grand coalition formed by all nodes is both sum-rate optimal and stable, in the sense that nodes do not have any incentive to leave the coalition. The second strategy is representative of a cooperative token based system. Nodes are statically scheduled in a round robin fashion. They cooperate to avoid wasteful idle cycles by sharing their right to access the channel with members of the same coalition. A scheduler selects the node that transmits among the ones within the coalition. In this case, the grand coalition is sum rate optimal but cannot be stabilized because some user may always have some incentive to deviate. Finally, an algorithm for coalition formation is provided for the case in which there is a cost associated to cooperation and the advantage of cooperation in terms of average queuing delay is investigated via a simple example from queuing theory. Index Terms: Multiple-access, coalitional games, cooperative communications.
I. I NTRODUCTION Modern wireless communication systems have been designed and developed to be inherently centralized, as well exemplified by cellular communication systems, where communication among wireless devices occurs through a network of central access points or base stations. However, a paradigm shift is taking place in wireless communications as new decentralized network architectures allow local communications among neighboring nodes. One of the opportunities and challenges that we face in this changing technological landscape is to solve classical resource allocation problems in a distributed and collaborative way. As well stated by Gallager [1], a major technical problem in multi-access communications is how to enable sharing of the spectrum of a channel among multiple users, most of which have nothing to send most of the time. The solution adopted in many of today’s communication systems is to employ either fully independent scheduling, as in Ethernet, where users can independently decide when to access the shared channel, or
fully cooperative scheduling, as in Token Ring or in time division multiple access (TDMA), where users are periodically granted access to the common resource. The main motivation of this work is to study distributed strategies to enhance the throughput of multi-access networks by cooperation between nodes. To this end, we consider two network settings in which cooperation is used to coordinate packet transmission over the wireless medium. The former, which we refer to as dynamic cooperative setting, models an interference-limited system in which cooperation between users is used as a form of collision avoidance, while the latter, referred to as static cooperative setting, is representative of a TDMA-based system where cooperation is used to avoid wasteful idle cycles during which the channel is not utilized, see Fig. 1(a)-1(b). In these settings, we analyze the performance of cooperation strategies in the framework of coalitional game theory [2] to determine how stable coalitions arise and evolve in response to the potential throughput gains enabled by cooperation. More specifically, in the dynamic cooperative model we consider a random access system composed by power-limited users communicating to a common access point in an uncoordinated way, where each user randomly alternates between two modes of operations, an active state in which it attempts communication and a dormant state in which it shuts off to maximize lifetime by preserving energy. Because of interference over the shared medium, transmitted packets are successfully decoded only if there is a single active user in the network, otherwise we say that they result in a collision. To avoid collisions and hence to enhance the throughput of the network, users can form coalitions and cooperate by sharing their activity states with other members of the coalition. A scheduler controls the transmission of packets within each coalition so that only one active user per coalition transmits, see Fig. 1(a). In the extreme case of no cooperation the system operates in full anarchy as in an ALOHA-type of system [3], while in the opposite extreme when all nodes cooperate, the system is fully scheduled by a central authority. Clearly, to maximize the network throughput, there is a incentive in forming the coalition composed by all users, i.e., the grand coalition, in such a way that packets transmissions never result in a collision. On the other hand, from the users’ point of view,
Access Point Access Point
1
5 2 6 1
4
Coalition 3
5 2 6
Coalition 1
4
Coalition 3
Coalition 1
3
3 Coalition 2 Coalition 2 User 1 Active Coal. 1 Active
User 2 Inactive Coal. 2 Inactive
User 3 Inactive
User 4 Active
User 5 Active
User 6 Active
Slot 4
Slot 5
Slot 6
Coal. 3
User 1
User 2
User 3
User 4
User 5
User 6
Active
Coal. 1
Coal. 1
Coal. 2
Coal. 1
Coal. 3
Coal. 3
(a) In any time slot, each coalition having at least one active user schedules a user for transmission.
Slot 1
Slot 2
Slot 3
(b) Each user is assigned a unique slot and a coalition can utilize all the slots assigned to its members.
Fig. 1. An illustration of the dynamic (left) and static (right) cooperative models considered in the paper. Grey nodes denote the active nodes in a slot. The dynamic model can have a collision if in a given time slot more than one coalition has an active node, while the static model never has any collisions.
a larger coalition means sharing the channel with more users. Thus, the grand coalition will be stable only if no collection of users has an incentive to defect. The key question here is whether there are ways to share the payoff of the grand coalition amongst all its members so that it is stable. In order to provide an answer, we first characterize the set of rates that can be simultaneously achieved by all users in different coalitions. Then, we consider the setup where there is no cost associated with the formation of a coalition and study the transferable utility game that sets a coalition’s utility equal to the maximum sum rate of the members of the coalition. We show that for this game the only viable stable coalition structure is the grand coalition, we provide an explicit characterization of the set of rate allocations for which no users have an incentive in leaving the grand coalition, and show that these can all be achieved by appropriate design of a central scheduler. Thus, in this setting users’ cooperation is cohesive and can be a useful tool to improve throughput of real systems, regardless of the duty-cycle of the nodes composing the network. In the static cooperative model, instead, we consider a network where users’ transmissions are coordinated by means of a round-robin scheduler that let users take turns in accessing the channel by assigning one slot per user in a periodically repeated order. If a user is inactive during one of its allocated slots, then no transmission occurs and the channel resource remains unused. To prevent this undesirable situation, users are allowed to cooperate by sharing their respective slots, such that these can be used by any active member of the coalition. A scheduler controls the transmission of packets within each coalition, see Fig. 1(b). In this setup, we study the transferable utility game that sets a coalition’s utility equal to the maximum sum rate of the members of the coalition. We show that while users have an incentive to form the grand coalition, this cannot always be stabilized, so users have an incentive in leaving the grand coalition. These results highlight that the stability of the grand coalition depends on the duty-cycles of the users in the
network, so in these types of networks cooperation may not always be cohesive. Since large coalitions also imply higher cooperation overhead, it is important to study the effect of the cost of cooperation on a coalitional game. To do so, we revisit the two setting described above in the general case where there is a non-zero cost associated with forming a coalition and study the non-transferable utility game that results when the utility of a coalition is given by the sum rate of its members minus the cost of cooperation. Lastly, we investigate the impact of cooperation over the average queuing delay by combining ideas from queuing theory and coalitional game theory. Cooperative game theory has been recently applied to study multi-hop communications [4], interference networks [5]–[7], spectrum sensing [8], [9], and various aspects of communication networks [2], [10]–[12]. Numerous other papers have studied coalition formation games in wireless networks [13]– [18], while connections between cooperative game theory and information theory have been explored in [19], [20]. Finally, cooperative games with cost have received a lot of recent attention, both in game theory [16]–[18] as well as in networking [13]–[15]. Unlike existing works, our attempt here is to use tools from coalitional game theory to quantify the benefits of cooperation in multi-access system. The rest of the paper is organized as follows. A review of preliminary concepts in geometry and cooperative game theory is presented in the next section. Sections III and IV contain the analysis for the dynamic and static cooperative settings, respectively, under the assumption that there is no cost associated with forming a coalition. Section V, instead, is devoted to the analysis of the cost of cooperation, while Section VI analyzes how the average queuing delay is impacted by node cooperation.
II. BASIC D EFINITIONS A. Submodular functions and polyhedra Let N be a finite set and 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, non-decreasing and submodular, we call it a rank function. Given a rank function f defined on N , the associated polymatroid P (f ) is the polyhedron |N |
P (f ) = {R ∈ R+ : R(S) ≤ f (S) for all S ⊆ N }, (1) P where R(S) = i∈S Ri . The set of points in P (f ) on the hyperplane R(N ) = f (N ), i.e., B(f ) = {R ∈ P (f ) : R(N ) = f (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 orthant. If σ is a permutation on the set N , define the vector Rσ by Rσσ1 = f ({σ1 }) and Rσσi = f ({σ1 , · · · , σi }) − f ({σ1 , · · · , σi−1 })
(2)
for i = 2, · · · , |N |. Lemma 1 ( [21]–[23]): Let P (f ) be a polymatroid. Then Rσ is a vertex of P (f ) for every permutation σ. Any vertex strictly inside the positive orthant must be Rσ for some σ. Moreover, for any c ∈ R|N | a solution of the linear program max c′ R subject to R ∈ P (f ) is attained at a point Rσ for any permutation σ such that cσ1 ≥ · · · , ≥ cσ|N | . It follows from the above lemma that the base polytope B(f ) is equal to the convex hull of all extreme point vectors {Rσ }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 ) [23]. B. Cooperative game theory A cooperative game with transferable utility is a pair (N, v), where N is a finite set and v : 2N → R is a function, called characteristic function, such that v(∅) = 0. The elements of N are 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 nondecreasing and supermodular, and concave if v is a rank function, i.e., non-decreasing and submodular. A profit (cost) allocation is a vector x ∈ R|N | such that x(N ) = v(N ). The allocation is efficient if no coalition has an advantage in
seceding the grand coalition. The core of a profit game (N, v) is defined as the set of all efficient allocations: n C(v) = x ∈ R|N | : x(N ) = v(N ), x(S) ≥ v(S) for all S ⊆ N } .
(3)
The core of a cost game (N, v) is defined by (3) with “≥” replaced by “≤”. The core denotes the set of profit (cost) allocations that assigns to each coalition at least (at most) what it can obtain on its own. In many games the core is empty, but when it is not empty there exists a rate allocation called the nucleolus which is always guaranteed to be in the core. For the formal definition of nucleolus we refer the reader to [2]. 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 ) where S c = N \S. Similarly, a dual profit game can be defined for a cost game. The following proposition, proved in [24], establishes a connection between a cooperative game and its dual. Proposition 2: 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 v and any permutation σ on the set N , the marginal worth vector Rσ is defined by Rσσ1 = v({σ1 }) and by (2) with “f ” replaced by “v” for k > 1. The convex hull of all marginal vectors is called the Weber set [25]. 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. The Shapley profit allocation of a game (N, v) is the centroid of the marginal vectors, i.e., 1 X Rσ φ(v) = n! σ∈Π(N )
where Π(N ) is the set of all permutations on N . Shapley provided an axiomatic characterization of this profit allocation, showing that φ(v) is the unique vector satisfying a set of four axioms capturing a notion of fairness (cf. [22]). III. DYNAMIC COOPERATIVE MODEL Consider a finite set of nodes N = {1, 2, . . . , n} sending data to a common access point. Let time be divided into slots of unit duration. In each slot, every sensor i is active with probability pi , independently of other users. Depending on the applications, the parameter pi represents the burstiness of data arrivals in a random access system [3] or the sensor duty-cycle in a sensor network [26]. When a node is active, it attempts to communicate a unit of information to the access point. A packet collision occurs when two or more users are simultaneously transmitting. To avoid collisions and hence to enhance the throughput of the network, users can form coalitions and cooperate by sharing their activity states with other members of the coalition. This setting is representative of
random access systems employing fully decentralized medium access schemes, such as Ethernet, and where users’ cooperation is used as a form of collision avoidance. Coalitions operate in full cooperative mode, so a scheduler determines the active user that can access the channel while all other active members of the coalition remain silent. Collisions only occur when users belonging to different coalitions transmit simultaneously. Formally, for each non-empty coalition S ⊆ N , a scheduler is a function gS : 2S → S such that gS (A) ∈ A for all A ⊆ N and gS (A) = ∅ iff A = ∅. We say that (R1 , · · · , R|S| ) is an achievable rate vector for coalition S if there exists a scheduler gS such that Y (1 − pi ), Ri = EA 1{gS (A)={i}} i∈S c
where the expectation is taken with respect to the random set of active users A ⊆ S. The closure of the set of achievable rate vectors is called the S-capacity region CS . The above definitions can be interpreted as follows. Suppose that in every slot the scheduler selects the active user in S according to the scheduling function gS . As time progresses, the ratio between the number of slots where user i is chosen for transmission and there is no collision over the total number of slots converges to Ri . Thus, Ri denotes the average packet transmission rate, and the S-capacity represents the set of average rates that can be simultaneously achieved by all users in S. Let mS : 2S → S be a set function defined by Y Y mS (A) = (1 − (1 − pi ) for all A ⊂ N (1 − pi )) i∈S c
i∈A
and mS (∅) = 0. In words, mS (A) is the probability that at least one element of S is active and that all users in S c are inactive. Then, Lemma 3: For every S ⊆ N , mS is a rank function. Proof: It can be easily checked that mS is non-decreasing and submodular. The following theorem provides an explicit characterization of the S-capacity region. Theorem 4: For every S ⊆ N , the capacity region CS is the polymatroid defined by mS , i.e., CS = P (mS ). Proof: For the proof of the converse, observe that for any scheduling function gS and B ⊆ S " # X Y 1{g(A)={i}} R(B) = EA (1 − pi ) i∈S c
i∈B
Y (1 − pi ) ≤ EA 1|B∩A|>0 i∈S c
= P (|B ∩ A| > 0)
Y
(1 − pi )
i∈S c
= mS (B)
where the inequality follows from the fact that gS (A) 6∈ B for all A such that A ∩ B = ∅ and it is equality iff gS (A) ∈ B for all A such that A ∩ B 6= ∅. Thus, CS ⊆ P (ms ). To show the reverse inclusion, we prove that any rate vector on the boundary of P (mS ) is achievable. We begin by
observing that from the properties of polymatroids, any rate in P (mS ) is dominated component-wise by a rate vector on the base polytope B(mS ). Thus, it is sufficient to prove the achievability of the |S|! extreme points of B(mS ) given in Lemma 1. Let σ be a permutation on the set S, and let the scheduling function gS (A) select the maximum element of A according to the ordering σ1 > · · · > σ|S| , i.e., g(A) = σi iff σ i ∈ A and σj 6∈ A for all j = 1, · · · , i − 1. Then, EA 1{g(A)={σ1 }} = p1 and thus Y EA 1{g(A)={σ1 }} (1 − pi ) = mS ({σ1 }) = Rσσ1 i∈S c
and for i = 2, · · · , |S|, Y (1 − pi ) EA 1{g(A)={σi }} i∈S c
= p σi
i−1 Y
(1 − pσk )
k=1
Y
(1 − pi )
i∈S c
= mS ({σ1 , · · · , σi }) − mS ({σ1 , · · · , σi−1 }) = Rσσi , hence the extreme point Rσ is achievable. Remark 1: It follows from the above theorem that the S-sum capacity region, i.e., the set of the rates in CS for which the sum rate is maximized, is equal to the dominant face of P (mS ) and that the corner points of B(mS ) can be achieved by simply scheduling users based on a predefined priority list. Remark 2: The S-capacity region CS has the same geometric structure as the information-theoretic capacity region of the multiple access channel [27]. In fact, using techniques similar to those developed in [28] it is possible to show that CS is formally identical to the ergodic information-theoretic capacity of a class of binary multiple-access channels with fading. In the extreme case of no cooperation the system operates in full anarchy and the S-capacity of the singleton S = {i} is simply the probability that user i is the only active user in the system, as in a slotted ALOHA system [3]. By forming coalitions, users can reduce the collision probability and thus increase the aggregate sum rate. On the other hand, from the users’ point of view, forming a coalition means sharing the channel with more users. Thus, a coalition will be stable only if no collection of users has an incentive to defect. To study the tradeoff between cooperation and competition in the formalism of cooperative game theory, we consider the profit game (N, v) with characteristic function v defined as Y Y v(S) = (1 − (1 − pi ) for all S ⊂ N. (4) (1 − pi )) i∈S
i∈S c
Note that v(S) denotes the average sum rate achieved when the users in S form a coalition, while assuming adversarial behavior from the users outside S. It turns out that this game has many interesting properties. In particular, the next theorem gives a characterization of the core of this game as the N -sum capacity, i.e., the maximum achievable sum rate by the grand coalition.
Theorem 5: 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. Consider the cost game (N, mN ). By Lemma 3 mN is a rank function, thus the game is concave and, by definition of core, C(mN ) = B(mN ). Next, observe that the profit game (N, v) is the dual of the cost game (N, mN ). In fact, for all S ⊂ N Y Y v(S) = (1 − (1 − pi ) (1 − pi )) i∈S c c
i∈S
= mN (N ) − mN (S )
= m∗N (S). Hence, the claim follows from Proposition 2. Remark 1: 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 [9] and games for which part of the core cannot be achieved [19]. It is thus remarkable that the core of the game (N, v) can be fully characterized and that any efficient rate allocation can be achieved by appropriate design of the central scheduling function. Remark 2: The core of any convex game is non-empty, as it contains at least the the Shapley profit allocation. It follows from Theorem 5 that Shapley profit allocation of the game (N, v) is simply the centroid of the N -sum capacity region. Given that the core is non-unique and that any rate allocation in the core is achievable, a natural question to ask is finding efficient allocations which satisfy specific notions of fairness. Potential candidates are the proportional profit allocation, in which rates are assigned proportionally to the duty-cycle or burstiness of each user, 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 [19], which is based on an axiomatic notion of fairness named envy-free fairness [19], [29]. Corollary 1: The following holds: 1) Let the proportional profit allocation R be given by pi v(N ) Ri = P k∈N pk for i ∈ N . Then, R is an element of C(v) for all p1 , · · · , pn . 2) Let the individually rational egalitarian profit allocation given by X 1 v(N ) − v({j}) + v({i}). Ri = n j∈N
for i ∈ N . Then, for all n > 2 there exists a tuple (p1 , · · · , pn ) such that R is not an element of C(v). 3) Let the envy-free profit allocation R be given by ! n n X Y 1 i Ri = Rk , 1 − (1 − pi ) (1 − pk ) − i k=i+1
k=i+1
for i ∈ N . Then, R is an element of C(v) for all p1 ≥ p 2 ≥ . . . ≥ pn . Proof: By Theorem 5, proving that a rate allocation is efficient is equivalent to proving that it is an element of B(mN ). By definition of proportional profit allocation, R(N ) = v(N ). For any S ⊂ N , we have P pi · mN (N ) ≤ mN (S), R(S) = P i∈S k∈N pk P where the inequality follows since f (x) = ( i∈S pi + Q x)/(1 − (1 − x) i∈S (1 − pi )) is an increasing function of x ∈ (0, 1) for any S ⊆ N . Thus the proportional payoff rate vector is in the N -sum capacity region B(mN ). Consider now the individually rational egalitarian profit allocation. By construction R(N ) = mN (N ), so if R is achievable it must be in B(mN ) and hence, by Theorem 5, in the core. Let p1 = 1/n and pj = 1/2 for all j 6= 1. We claim that R 6∈ C(v). By contradiction, suppose that R ∈ C(v). Then, R(N \ {1}) ≤ v(N \ {1}) implies that (n − 1)v(N ) + (n − 1)v({n}) < v(N \ {1}). Using the definition of the functions v and mN and after some algebra, the above inequality reduces to −1 2 1 1 1− n < , p1 ≤ 1 − 1 − n 2 n which is a contradiction with the assumption that p1 = 1/n. Finally, the proof that the envy-free profit allocation is an element of C(v) is reported in Appendix A. In conclusion, in the dynamic network setting where cooperation is used as a form of collision avoidance, the grand coalition can be stabilized so that no user 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. S TATIC COOPERATION MODEL Consider now a TDMA system, in which each slot is assigned for transmission to one user in a periodically repeated order, so collisions are always prevented. User i is active in each slot with probability pi , independently of other users, so if it is inactive during one of its allocated slots, no transmission occurs and the channel resource remains unused. To prevent this inefficient use of the common resources, users are allowed to cooperate by sharing their respective slots, such that these can be used by any active member of the coalition. A scheduler controls the transmission of packets within each coalition, by choosing one node among the pool of active nodes. The formal definition of a scheduling function gS for coalition S is as above. In this setting, we say that (R1 , · · · , R|S| ) is an achievable rate vector for coalition S if there exists a scheduling function gS such that Ri =
|S| EA 1{gS (A)={i}} , n
where the expectation is taken with respect to the random set of active users 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 user in S according to the scheduling function gS . As time progresses, the ratio between the number of slots where user i is chosen for transmission over the total number of slots converges to Ri . Thus, Ri denotes the average packet transmission rate. In the extreme case of no cooperation the system operates in TDMA mode and the maximum achievable rate by user i is pi /n, while in the opposite extreme where the grand coalition is formed, the system is fully scheduled by a central authority, exactly as in the dynamic setting discussed above, and the (closure of) the set of all achievable rates is CN , as defined above. 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 how to design a central scheduling function for the grant coalition such that no user has an incentive to deviate. To answer this question we consider the profit game (N, w) with characteristic function defined as |S| mN (S) for all S ⊂ N. (5) w(S) = n Observe that w(S) denotes the maximum achievable sum rate by users in S. Proposition 6: 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 |S2 | |S1 | 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, users have an incentive to form the grand coalition N . However, the next proposition shows that it is not always possible to find a scheduling function that stabilizes the grand coalition. Proposition 7: 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 efficient rate allocation, i.e., C(w) ∩ B(mN ) 6= ∅. 3) There exists a pair (p1 , p2 ) such that the Shapley profit allocation and the nucleolus are not in the sum-capacity region B(mN ). 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 efficient rate allocation, i.e., C(w) ∩ B(mN ) 6= ∅.
Proof: Let n = 2. It is immediate to check that the two-user game is convex. For any achievable efficient rate allocation (R1 , R2 ) inequalities (1) and (3) require that pj /2 ≤ Rj ≤ pj , j ∈ N , and R1 + R2 = p1 + p2 − p1 p2 . These constraints are simultaneously satisfied by both the proportional profit allocation which assigns Rj = pj /p(N )w(N ), j ∈ N , and the profit allocation which assigns Rj = pj − p1 p2 /2, j ∈ N . Since the game is convex for any (p1 , p2 ), the Shapley profit allocation and the nucleolus are trivially in the core. However, let (p1 , p2 ) = (1/4, 3/4). 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. [2]), and these rate allocation do not lie in the capacity region as they violate the constrain R1 ≤ p1 = 1/4. Suppose now that n ≥ 2, and let 1 p1 = 1 − 2n and pi = 1/n for all i 6= 1. We claim that for this choice of the parameters C(w) ∩ B(mN ) = ∅. By contradiction, suppose that there exist a rate vector R which is both in the sum-capacity and the core. Then, R satisfies w(S) ≤ R(S) ≤ m(S) for all S ⊆ N with strict equality if S = N . Now, R(N ) = m(N ) and R(N \ {n}) ≥ w(N \ {n}) give Rn ≤ m(N ) − w(N \ {n}), which combined with R1 + Rn ≥ w({1, 2}) gives R1 ≥ w({1, 2}) − m(N ) + w(N \ {n}) =
1 − α(n)(1 − p1 ), n
1 where α(n) = nn+2 (2nn + n4 − 2n3 ). Combining the above inequality and R1 ≤ m({1}) yields, making use of the fact that 0 ≤ α(n) < 1,
p1 ≤ 1 −
1 1 1 ≤1− , n 1 + α(n) n
1 > which is a contradiction with the hypothesis that p1 = 1− 2n 1 1 − n . Finally, consider the symmetric case where all users are active with probability p. In this case, the symmetric profit allocation R1 = · · · = Rn = (1 − (1 − p)n )/n is both in the core and in the sum-capacity region, as it can be verified using the fact that
1 − (1 − p)n 1 − (1 − p)|S| ≤ for every S ⊆ N. n |S| Thus we can conclude that unlike in the dynamic setting discussed above, where cooperation is always cohesive, the burstiness or duty-cycle of each user plays a crucial role in determining the stability of TDMA networks in which cooperation is used to share transmission slots among users. In particular, in asymmetric scenarios where users have different transmission probabilities it is not always possible to design a central scheduler such that no user has an incentive to leave the grand coalition. V. I MPACT OF COST OF COOPERATION In the previous sections we have studied the benefits of cooperation in two types of networks under the assumption that there is no cost associated with the formation of a
coalition. However, in practice there is a cost involved in establishing cooperation as this require local communication among nodes. In this section, in addition to the gain from cooperation, we also consider the cost for forming a coalition. We use the dynamic cooperative model from Section III for illustration but the same procedure can be repeated, mutatis mutandis, for the static model discussed in Section IV. It is assumed here that in any given time slot an active user can transmit r bits of information if there is no collision. In this model, users in a coalition cooperate with each other when transmitting information to the access point. In every time slot, one of the active users in the coalition is scheduled. To implement such a scheduling scheme, let each coalition choose a “coalition head” from amongst its members. Users in the coalition take turns to transmits a bit corresponding to its activity state to the coalition head, which then decides which active node from the coalition will be scheduled in the slot. This phase requires |S| bits and thus the net value of a coalition is given by +
v˜(S) = (r − |S|)
v(S),
(6)
where (x)+ = max(x, 0) and v(S) is defined in (4). Since the cost of cooperation increases with the size of the coalition, the game is in general not superadditive. For example, note that the value v˜ for any coalition of size greater than r is zero. Hence the grand coalition of users will not always form and typically the network will consist of multiple small coalitions. Our objective is to use tools from game theory to study the coalitions that form in the network. We do so by providing a simple “merge-and-split” algorithm, inspired by recent works [16]–[18], that is guaranteed to converge to a cooperative network which has better performance than in the corresponding non-cooperative setting. A. Coalition formation games Coalition formation has received a lot of attention in the game theory community. Recent advancements in [16]–[18] have developed a mathematical framework for studying the evolution of coalitions in a network. In particular, a simple “merge-and-split” algorithm has been proposed for allowing nodes to make and break coalitions in a distributed fashion. The main ideas of the procedure are as follows: • Firstly, we define an “order” relation ⊲ to compare any two different partitions P = (P1 , P2 , . . . , Pl ) and Q = (Q1 , Q2 , . . . , Qm ) of any subset A of N . We say P ⊲Q if the way A is partitioned according to P is preferable to Q. There can be many different ways to define an order. For example, we order which Plcan have thePutilitarian m gives P ⊲ Q if i=1 v˜(Pi ) > j=1 v˜(Qj ). Or we can define the pareto order which sets P ⊲ Q when at least one user in A improves its payoff under the partition P without hurting any of the other users. • Given this order, two simple rules are defined for the forming and breaking coalitions, referred to as “merge” and “split”. A collection of coalitions P = (P1 , P2 , . . . , Pl ) merge into a single coalition Q =
∪li=1 Pi if Q ⊲ P. Similarly, a coalition R splits into smaller disjoint coalitions S = {S1 , S2 , . . . , Sm } if S ⊲R. The above basic operations have recently been used to devise distributed coalition formation algorithms for various different scenarios [13]–[15]. Inspired by the ideas there, we describe a similar algorithm for our setup below. B. Coalition formation algorithm Assume that the network starts with the non-cooperative setting where each user forms a distinct coalition. The algorithm consists of three basic steps: Discovery, Merge-and-Split, and Cooperative communication. We briefly discuss each of these steps now. • Discovery: In this phase, each coalition tries to identify other coalitions in the network and exchanges information with them which is relevant to coalition formation, for example they can share the identity and the activity probability pi ’s of their members. • Merge-and-Split: Following the collection of information in the discovery phase, each coalition starts pair-wise negotiations with the discovered coalitions to explore the possibility of a merge. Two coalitions merge if it is beneficial for both according to the order relation ⊲, and this process is repeated till no further merge possibilities remain. Following the merge procedure, the coalitions attempt the split operations. An iteration consisting of multiple merge-split cycles is conducted until it terminates. • Cooperative communication: Once the coalitions in the network have fixed, the nodes in each coalition start cooperating so that they can communicate data to the access point. From [17], [18], any arbitrary sequence of the merge and split operations in the second phase finally converges to a final partition. Furthermore, each merge or split operation is performed only if the resulting network partition is better than the current one. Thus, irrespective of the order of the mergesplit operations, it is guaranteed to result in a cooperative network which is better than the non-cooperative setting. Various different notions of stability of the final partition have also been proposed and studied using the idea of a defection function. Also note that once the first two steps of the algorithm have been performed, they only need to be repeated if there is a change in the network state, for example if an existing node departs or a new node arrives, or if the activity probabilities pi ’s change. VI. I MPACT OF COOPERATION ON DELAY In the previous sections, we have evaluated the benefits of cooperation in terms of achievable throughput, but we have neglected another crucial parameter in multi-access communication, the communication delay. A comprehensive study of the tradeoff between throughput and delay goes beyond the purpose of this paper, but in this section we wish to demonstrate, via a simple queueing example, that the delay performance can also be improved through node cooperation.
Consider a set N = {1, 2, . . . , n} of M/M/1 queues whereby each queue receives customers according to a Poisson process with rate λ and the probability distribution of the service time is exponential with mean 1/µ. Let ρ = λ/µ and assume that ρ < 1 so that each queue is stable. The average waiting time for a customer in any queue is then given by [3] ρ W = . µ−λ Now suppose that the set N of queues is divided into disjoint coalitions. The queues in any such coalition merge all their arrivals in a single queue and the customers can be served by any available server in the coalition. Let S ⊆ N be such a coalition and let |S| = s. Thus we have an M/M/s queueing system such that arrivals into the system are drawn from a Poisson process with rate sλ and customers can be served by any of the s servers, each with a service time which is exponentially distributed with mean 1/µ. The average waiting time for any customer in the coalition is then given by [3] ρ · Ps (7) WS = µ−λ where 1 Ps = 1−ρ
s−1 X s!/i! 1+ s−i (sρ) i=0
!−1
.
(8)
We define a non-transferable utility cost game (N, d) such that for any coalition S ⊆ N , v(S) is a singleton set given by d(S) = {D ∈ R|S| : Di = WS for each i ∈ S}.
(9)
Theorem 8: The cost game (N, d) is subaddititve. Proof: From (7) and (9), we need to show that Ps decreases with s. From (8), this is equivalent to showing that Pbs =
s−1 X s!/i! (sρ)s−i i=0
increases with s. We have s X (s + 1)!/i! Pbs+1 = ((s + 1)ρ)s+1−i i=0 =
=
s+1 X (s + 1)!/(s + 1 − i)!
i=1 s+1 X i=1
1 ρi
s X 1 ≥ ρi i=1
=
((s + 1)ρ)i i−1 1 ... 1 − 1− s+1 s+1 i−1 1 ... 1 − 1− s s
s−1 X s!/i! s−i (sρ) i=0
= Pbs .
Thus, the average waiting time is smallest when all the queues in N form a coalition and thus there is an incentive, in terms of average waiting time in queue, to form the grand coalition.
VII. C ONCLUSION We have studied cooperative communication strategies in multiple access networks using a coalitional game approach. We considered two different modes of operation showing an advantage of a dynamic setting compared to a static one. In the dynamic setting users alternate at random between two modes of operation: an active state and a dormant state. Users within the same coalition share knowledge of their active states and a scheduler assigns the right to transmit within a coalition. In this set up, the grand coalition formed by all users is sum rate optimal and is also stable, in the sense that no user has incentive to leave the coalition. The second mode of operation is the static setting. Here users are scheduled for transmission in a static round robin fashion. However, members of the same coalition can share their right to transmit in a given time slot. In this case, the grand coalition although optimal, can be unstable. Since large coalitions also imply higher cooperation overhead, we then considered the effect of the cost of cooperation on the coalitional game. When this is taken into account the game is not anymore superadditive and the grand coalition does not always form. In this case, we have proposed an algorithm for coalition formation that although it might not reach the optimum grand coalition, it is always superior to the non-coalition setting. Finally, we have showed by a simple queueing example that coalition formation is not only advantageous in terms of rate, but also in terms of average waiting time. Our study reveals that coalitional game theory can be a useful tool to study cooperative communications, providing insights into the design of real systems. Protocol overhead due to coordination is still an issue that can lead to sub-optimal solutions in practice, and requires further attention through either simulation or additional analytic tools. A PPENDIX A T HE ENVY- FREE ALLOCATION IS IN THE CORE The proof follows along similar lines as in [19]. Let R(N ) denote the envy-free profit allocations in a system where the set of players is N . We to show that R(N ) ∈ B(mN ). P need (N SincePby construction i∈N R ) = v(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 users. If there is only one user in the game, the above claim is true. Now suppose that the claim is true for a set of n users. We will now show that the claim is true for a game with a player set N of n + 1 users and for which p1 ≥ p2 ≥ . . . ≥ pn+1 . To do so, we prove that (N ) (N \{j}) Ri ≤ Ri (10) for all i ∈ N \ {j}. In fact, by (10) and the induction hypothesis, it follows immediately that X (N ) X (N \{j}) Ri ≤ Ri i∈S
i∈S
=1−
Y
i∈S
(1 − pi )
for all i ∈ {2, . . . , n}. On the other hand,
= mN (S) for any strict subset S of N , which proves the claim. The rest of the proof is then devoted to showing (10). Let qi = (1 − pi ) for all i ∈ N . We consider three cases: 1) j = n + 1: In this case, it can be verified that for any i ∈ {1, 2, . . . , n − 1} (N )
i(Ri
(N )
i − Ri+1 ) = qi+1 − qii
i ≤ qi+1 − qii
=
Y n+1
k=i+1 n Y
qk qk
k=i+1 (N \{j}) (N \{j}) i(Ri − Ri+1 ),
(N \{j})
(N )
− Ri
(N )
(N )
(N \{j})
n+1 1−qn+1
n+1
(N )
− Ri+1 .
(11)
and
n(Rn(N \{j}) − Rn(N ) ) (N )
= Rn+1 − (1 − qn+1 )qnn 1 n+1 = − (n + 1)(1 − qn+1 )qnn 1 − qn+1 n+1 1 1 − (1 − p2n+1 )n ≥ (n + 1) ≥ 0. (12) The first inequality above holds because n+1 qn+1 − (n + 1)(1 − qn+1 )qnn n+1 n ≤ qn+1 − (n + 1)(1 − qn+1 )qn+1
n = qn+1 (qn+1 + (n + 1)(1 − qn+1 )) = (1 − pn+1 )n (1 + npn+1 )
i qi+1
=
−
qii
Y n+1
(N )
(N \{j})
(N )
− Rj+1
(N )
) − (Rj−1 − Rj+1 ) j−1 j−1 Qn+1 jpj+1 (qj+1 − qj−1 ) k=j+1 qk j(j − 1)
which yields (10) in the case i = j − 1. Next, we that for all i < j − 1, (N )
qk
(N )
i − Ri+1 ) = (qi+1 − qii )
k=i+1 n Y i−1 − qii−1 qi+1 qk k=i+1 (N \{j}) (N \{j}) i(Ri − Ri+1 ),
(N )
− Ri
(N )
≥ Rj−1 − Rj+1
it suffices to assume pj = pj+1 . In this case, we have j−1 j−1 Qn+1 − qj−1 ) k=j+1 qk (qj+1 (N \{j}) (N \{j}) . Rj−1 − Rj+1 = j−1
i(Ri
i ≤ (qi+1 − qii )
(N \{j})
≥ Ri+1
(N )
− Ri+1 .
n+1 Y
k=i+1 j−1 Y
qk
qk
k=i+1 (N \{j})
= i(Ri
n+1 Y
(N \{j})
− Ri+1
qk
k=j+1
),
which implies that for all i < j − 1,
xi−1 − y i−1 xi − y i ≥ for any 1 ≥ x > y ≥ 0. (13) i i−1 for all i ≥ 2. Thus, (N \{j})
(N \{j})
− Rj+1
≥ 0,
where the inequalities follows from the fact that
Ri
(N \{j})
Rj−1
=
where (a) follows since pn ≥ pn+1 . Combining (11) and (12) yields (10). 2) j = 1: Simple algebra shows that for all i ∈ {2, . . . , n}
=
(N )
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
(N \{j})
= (1 − p2n+1 )n
≤
(N )
) + (Rj − Rj+1 ) Qn+1 j j−1 j−1 Qn+1 (qj+1 − qjj ) k=j+1 qk (qj − qj−1 ) k=j qk + = j−1 j Q n+1 j j−1 j qj − jqj qj−1 + (j − 1)qj+1 k=j+1 qk = (15) j(j − 1)
(Rj−1
≤ (1 − pn+1 )n (1 + pn+1 )n
−
(N )
Subtracting to the left hand side of (15) and after some algebra we obtain
(a)
(N ) Ri+1 )
(N )
(N )
On the other hand, Rn+1 =
(N ) i(Ri
where the last inequality follows from (13). Hence, (10) 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 (10) holds for all i ∈ {j+1, · · · , n+1}. We analyze the cases i = j − 1 and i ≤ j − 1 separately. First, we write
= (Rj−1 − Rj
≥ Ri+1
n+1 1 − qn+1 1 − qnn − n+1 n ≥ 0,
(N )
− Rn+1 =
Rj−1 − Rj+1
which can be rewritten as Ri
(N \{j})
Rn+1
(14)
(N \{j})
Ri
(N \{j})
(N )
− Ri
(N )
(N \{j})
≥ Ri+1
(N )
− Ri+1 .
Since Rj−1 − Rj−1 ≥ 0, we have that (10) is proved also in the case i < j − 1.
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