PeertoPeer File Sharing Game using Correlated Equilibrium


 Myron Atkins
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1 PeetoPee File Shaing Game using Coelated Equilibium Beibei Wang, Zhu Han, and K. J. Ray Liu Depatment of Electical and Compute Engineeing and Institute fo Systems Reseach, Univesity of Mayland, College Pak, MD 2742, USA Depatment of Electical and Compute Engineeing, Univesity of Houston, Houston, TX 774, USA Abstact Peetopee (P2P) systems have become moe and moe popula in nowadays by poviding decentalized, selfoganizing and fault toleant file shaing sevices. As the selfish uses do not benefit fom poviding fee sevice, they tend to download files fom othe uses and yet not to upload fo the othes, esulting in low system efficiency. In this pape, we popose a coelated equilibiumbased file shaing game to enhance uses pefomance. We fist chaacteize uses utility with thei expected delay. Then, using the coelated equilibium concept, instead of optimizing thei own benefits alone, the uses ae awae of the best esponse fo them to jointly optimize thei stategies togethe and achieve the coelated equilibium. Simulation esults ae pesented to demonstate the efficiency of the poposed scheme. I. INTRODUCTION With the apid development of netwoking and communication technologies, in ecent yeas, peetopee (P2P) file shaing sevices have become moe and moe popula, such as Napste [1], Gnutella [2], KaZaA [3], and BitToent [4]. By combining sophisticated seaching techniques with a lage scale decentalized file stoage, P2P file shaing systems allow uses to download files diectly fom each othe. Theefoe, they offe a decentalized, selfoganizing, scalable, and fault toleant file shaing sevice poviding an effective balancing of stoage and bandwidth esouces [5]. Cuent P2P systems ely on the assumption that uses will altuistically seve files to the community fo fee. Howeve, eseving a cetain upload bandwidth to seve the othes incus a cost to a use. Since uses ae selfish and do not benefit fom poviding fee sevice, many of them tend not to contibute to uploading thei files while leech the othe uses [6]. Hence, it is of geat impotance to motivate use coopeation in P2P systems to make moe files available. In the liteatue, vaious incentive schemes based on gametheoetic modeling have been poposed to incease the popotion of uses that shae files. Titfotat stategy was poposed in BitToent application to pevent use feeiding [7] [9]. The authos of [1] studied the incentives fo coopeation based on epetition and eputation by modeling file shaing as an evolutionay pisone s dilemma. A diffeential sevicebased incentive scheme was poposed in [11] to impove the system s pefomance. The wok in [12] poposed a micopayment incentive mechanism in which uses ean ewads by uploading to othes and pay fo downloading. The authos of [13] modeled the bandwidth allocation in P2P systems as an exchange economy and poposed the popotional esponse dynamics to achieve the maket equilibium. Howeve, most of the existing gametheoetic famewoks of P2P systems assumed that the system uses optimize thei own benefits individually without consideing the othe uses actions, and would stick to noncoopeative stategies even at the high isk of having no use upload files and suffeing poo individual pefomance. In this pape, we popose a coelated equilibiumbased file shaing game to enhance the uses pefomance. We fist model the file shaing as a noncoopeative game, whee we chaacteize the uses utility using the expected delay they expeience in file shaing. Specifically, if no uses contibute to uploading files fo the othes, the seve has to espond to all data equests fom the uses and the uses may expeience a long delay. If some of the uses coopeate in uploading files, all the othe uses can download files fom both the seve and the coopeative uses, so the expected delay can be educed. The noncoopeative uses who fee ide may enjoy a shote delay, but ae also awae of the isk of having an unaffodable delay if no one coopeates. Using the coelated equilibium (CE) concept [15], the selfish but ational uses ae awae of the isk of having no one contibute to uploading files, so that it is the best esponse fo them to jointly optimize thei actions and achieve a highe degee of coopeation at CE with a bette pefomance. Simulation esults also illustates that the expected delay at CE is geatly educed compaed to the delay pefomance at Nash equilibium. The est of the pape is oganized as follows. In Section II, we pesent the system model of P2P file shaing and deive the expected delay using queuing theoy. We model file shaing as a noncoopeative game in Section III, and analyze the poposed game using the coelated equilibium concept in Section IV. Simulation esults ae pesented in Section V. Finally, Section VI concludes the pape. We adopt the following notation: λ and µ ae aival ate and sevice ate, espectively. D is the delay, R is the download ate, is the upload ate, U is the utility, p is the pobability, E is the expectation, S is use set, Ω is the action set, and R is the eget. II. SYSTEM MODEL In this section, we define a geneal model of a P2P file shaing system in which we assume symmetic exchange of data between pees. By symmetic, we mean that the seve has all the equied data of all pees, while some big data file is divided into seveal chunks, each pee owns some of the chunks, and can download the emaining chunks fom
2 2 eithe the seve o the othe pees who would contibute some bandwidth fo uploading data. We begin by modelling a simple twopee (N = 2) file shaing, and then genealize it to multiple pees (N > 2). A. TwoPee File Shaing In geneal, the seve has only a limited bandwidth, so not all the equests fo data chunks fom the pees can be immediately satisfied, and the pees may expeience delay. Thus, in this pape, we chaacteize the pefomance in file shaing by the expected delay a pee expeiences (i.e., mean steadystate time in queue). Without loss of geneality, we assume fo the aival pocess of the chunk equests, the inteaival time is exponentially distibuted with paamete λ 1, and the duation that a data souce (i.e., the seve o a pee) uploads a chunk is µ 1. Then we can model the file sevice pocess as an M/D/1 queue, and the delay function, denoted by f(λ, µ), can be witten as [21] f(λ; µ) = λ 2µ(µ λ). (1) In Figue 1, we show the system model fo twopee file shaing. Fo case I, neithe pee 1 no pee 2 would upload data chunks fo each othe, so they both send thei equests to the seve, and the expected delay fo pee 1 and pee 2 in this case can be epesented as D i = f(r 1 + R 2 ; ), i = 1, 2, whee R 1 and R 2 is the downloading ate (o chunk equest ate) of pee 1 and pee 2, espectively, and is the uploading ate (o sevice ate, uploading bandwidth) of the seve, with stability equiement R 1 + R 2 < 1. As we mentioned ealie, both the pees may expeience a cetain delay due to the limited capacity/bandwidth of the seve. A possibly bette choice fo the pees is to contibute some bandwidth to uploading data so that they can download chunks fom two souces, which may potentially lowe the buden of the seve and educe the delay. This coesponds to case II in Figue 1 whee both pee 1 and pee 2 coopeate. We assume that each pee equests data chunks fom the seve and the othe pee with equal pobability, so the downloading ate fom eithe of them is R i /2. We also assume the total upload bandwidth is equally shaed by the two coopeative pees. Since the queuing pocess at the seve and each pee is assumed to be independent fom each othe, by applying Little s Law [21], we get the expected delay of pee i as D i = 1 [ R 1 + R 2 f( ; ) + f( R i ; 2 )], i = 1, 2, (3) with stability equiement R 1 + R 2 < 2 and R i <. Howeve, pees in p2p file shaing ae selfish and would pefe to download chunks fom the othe pees athe than contibute to upload fo the othes, since eseving uploading 1 We assume thoughout the pape that the stability equiements ae always satisfied when using (1), othewise we set the delay as infinity without using (1). Fig. 1. System model bandwidth incus cost to pees. This case is illustated in case III and IV of Figue 1. In case III, whee pee 1 is coopeative and pee 2 leeches, we have the pee s expected delay as D 1 = f(r 1 + R 2 2 ; ), (4) and D 2 = 1 [ f(r1 + R ; ) + f( R 2 2, )], (5) with stability equiement 2R 1 + R 2 < 2 and R 2 < 2. Similaly fo case IV whee pee 1 leeches and pee 2 is coopeative, we have the expected delay as D 1 = 1 [ R 1 f( R 2; ) + f( R 1 2, )], (6) and D 2 = f( R R 2; ), (7) with stability equiement R 1 + 2R 2 < 2 and R 1 < 2. B. NPee File Shaing (N > 2) In the following, we extend the deivation above to Npee file shaing with N > 2. Assume thee ae in total N pees, denoted by set S. A subset S c S, would eseve thei own bandwidth fo uploading data chunks. Then the emaining noncoopeative pees fom a set S n = S \ S c. We futhe assume a pee diects his/he equests to the seve and all the coopeative pees with equal pobability
3 3 fo simplicity. Hence, if pee i chooses to coopeate (C) in uploading data, he/she can eithe download data fom the seve o the emaining (K 1) othe coopeative pees, and the equest ate of pee i to each of them is essentially R i (1 K)+1 = R i K. On the othe hand, pee j who pefes not to coopeate (N) can choose downloading fom the seve o the K coopeative pees, and his/he equest ate to each data R povide is j K+1. Theefoe, fo the queue at the seve side, the aival ate of data equests is λ s = j S n R j K i S c R i K, (8) and the sevice ate is µ s =. Fo the queue at a coopeative pee, say pee i, the aival ate of equests is λ pi = j S n R j K l S c,l i R l K, (9) and the sevice ate is µ pi = K since the K coopeative pees equally contibute to an upload bandwidth of. Theefoe, fom the assumption that each queue is independent and using Little s Law [21], we obtain the expected delay of a coopeative pee i as Dp c i = 1 [ f(λs ; µ s ) + f(λ pl ; µ pl ) ], if K [1, N], K l S c,l i (1) and the delay of a noncoopeative pee j is D n p j = 1 K + 1 [f(λ s; µ s )+ i S c f(λ pi ; µ pi )], if K [1, N 1], and (11) D n p j = f( i S R i ; µ s ), if K =. (12) III. FILE SHARING GAME AND CORRELATED EQUILIBRIUM As the pees ae geedy, and aims at minimizing thei delay, we can model the file shaing as a game. Let G = {S, (Ω i ) i S, (U i ) i S } be a finite Nplaye 2 game in stategic fom, whee Ω i = {C, N} (C stands fo coopeation and N stands fo noncoopeation) is the stategy space fo playe i, and U i is the utility function fo playe i. Define Ω i as the stategy space fo playe i s opponents, and denote the action fo playe i and his/he opponents as i and i, espectively. In the file shaing game, since the pees ty to minimize thei expected delay, we can define thei utility function as a noninceasing function of the expected delay. Fo simplicity, we use the negative of the expected delay a pee expeiences as his/he utility, i.e., U i = D i. In othe wod, we want to maximize the utility by minimize the delay. Othe utility functions with simila popeties can also be analyzed, e.g. the delayed sensitive utilities fo multimedia applications. 2 We use use, playe and pee intechangeably. Nash equilibium (NE) is a wellknown concept to chaacteize the outcome of a game, which states that at equilibium evey playe will select a utilitymaximizing stategy given the stategies of evey othe playe. Howeve, computing the mixedstategy NE of a Nplaye game in polynomial time is geneally vey difficult, since it equies to solve multiple highode polynomial equations. Nevetheless, coelated equilibium can be computed in polynomial time in essentially all kinds of multiplaye games [2]. Futhe, due to pees selfish natue, NE may not be system efficient as pees only aim at maximizing thei individual utility (i.e., minimizing the delay). Theefoe, we next study the concept of coelated equilibium (CE), which is moe geneal than the NE [15]. The idea of CE is that a stategy pofile is chosen andomly accoding to a cetain distibution. Define p( i, i ) as the joint distibution of uses to pefom a specific action. Given the ecommended stategy, it is to the playes best inteests to confom with this stategy, and the distibution is called the coelated equilibium, i.e., fo all i S, i, i Ω i, and i Ω i, p( i, i )[U i ( i, i ) U i ( i, i )]. (13) i Ω i Inequality (13) means that when the ecommendation to playe i is to choose action i, then choosing action i instead of i cannot esult in a highe expected utility to playe i. In the file shaing game, pees can obtain some infomation fom the seve, such as data equest ates of othe pees, the total shaed bandwidth (), and the seve s uploading ate. This way, they will conside the potential long delay when no use contibutes to seving files, and it is the best esponse fo them to jointly optimize thei actions so as to attain a highe degee of coopeation and educe delay. We note fom (13) that the set of coelated equilibia is nonempty, closed and convex in evey finite game [16]. Moeove, it may include the distibution that is not in the convex hull of the Nash equilibia. In fact, evey Nash equilibium is a point inside the coelated equilibia set, and the NE coesponds to the special case whee p( i, i ) is a poduct of each individual use s pobability fo diffeent actions, i.e., the action of the diffeent playes is independent [15], [16]. Among the multiple CE, which one is the most suitable should be vey caefully consideed in pactical design. In [17] [22], the authos poposed the citeion of coelated optimal, which is the solution that achieves the highest social welfae, i.e., Definition 1: A multistategy all is coelated optimal if it satisfies the following conditions, s.t. p( all ) = ag max p E p (U i ), (14) i S i Ω i p( i, i )[U i ( i, i ) U i ( i, i )], i, i Ω i, and i S. Othe citeion can also be consideed, such as the maxmin
4 4 fainess citeion, defined as p( fai ) = ag max min E p (U i ). (15) p TABLE I TWOPEER GAME (A) UTILITY TABLE (LEFT); (B) STRATEGY NOTATION (RIGHT). (a) C N C (U (1) 1,1,U 1,1 ) (1) (U 1,2,U 1,2 ) N (U (1) 2,1,U 2,1 ) (U (1) 2,2,U 2,2 ) (b) C N C p 1,1 p 1,2 N p 2,1 p 2,2 Oveall, the genealized system model is to employ any system optimization goal with the coelated equilibium constaint and othe physical constaints, so as to achieve highe mutual benefits by coopeating on the joint pobability distibution of uses actions. In this pape, we use sum utility as an example, but othe optimization citeia can be employed in a simila way. IV. ANALYSIS OF THE PROPOSED GAME In this section, we investigate two appoaches fo the poposed game. The fist one is by linea pogamming method, and the second one is distibuted leaning algoithm. A. Linea Pogamming Method We study the popeties of CE using a twopee game as an example. A geneal utility table is shown in Table I, in which U (k) i,j is the utility fo playe k when the joint action pai is in the i th ow and the j th column, and p i,j is defined as the coesponding joint pobability fo that action pai. The linea pogamming poblem (14) to obtain the coelated optimal stategy becomes constained by max {p i,j} 2 j=1 i=1 2 (U (1) i,j + U i,j )p i,j, (16) p 1,1 + p 1,2 + p 2,1 + p 2,2 = 1, (17) p 1,1 (U (1) 1,1 U (1) 2,1 ) p 1,2(U (1) 2,2 U (1) 1,2 ), (18) p 2,1 (U (1) 2,1 U (1) 1,1 ) p 2,2(U (1) 1,2 U (1) 2,2 ), (19) p 1,1 (U 1,1 U 1,2 ) p 2,1(U 2,2 U ), p 1,2 (U 1,2 U 1,1 ) p 2,2(U 2,1 U ), (21) whee inequalities (18)(21) epesent the constaints of CE in (13). Let 3 = {(p 1,1,, p 2,2 ) R 4 + p 1,1 + + p 2,2 = 1} denote the 3dimensional simplex of R 4. Since the above constaints and the objective function ae all linea, simila to [18], we can deive the set of coelated equilibia as follows. Lemma 1: The set of coelated equilibia fo the game defined in Table I is a hexahedon of 3 with five vetices 2,1 2,2 TABLE II FIVE VERTICES OF THE CORRELATED EQUILIBRIA SET p p 1,1 p 2,2 p 2,1 p 1,2 NE1 1 NE2 1 NE3 1 ab b CE1 1 1+a+b b CE2 ab a+b+ab given in Table II, whee 1+a+b b a+b+ab a a 1+a+b a a+b+ab (1) U 1,1 a = U (1) 2,1 U1,1 U (1) 2,2 U and b = U 1,2 (1) 1,2, U 2,2 U (22) 2,1. Because (16) is a linea pogamming poblem, its solution will be one of the five vetices shown in Table II, accoding to the coefficients of p i,j. Specifically, NE1 and NE2 ae the two puestategy NE, and will be the solution to the poblem in (16) if U (1) 1,2 + U (1) 1,2 (o U 2,1 + U 2,1 ) is the lagest among all U (1) i,j + U i,j s. Howeve, it is moe often the case that the highest social welfae is achieved if all playes coopeate, i.e. U (1) 1,1 + U 1,1 is the lagest coefficient in (16), and when no use coopeates the social welfae U (1) 2,2 + U 2,2 is the lowest. Due to the CE constaints, always coopeating can not be achieved among the selfish uses (p 1,1 < 1 in all five vetices). Howeve, since the uses who adopts CE stategy now jointly optimize thei actions by consideing the low utility of mutual noncoopeation and the high utility when both coopeate, the degee of coopeation is inceased. Fo instance, at CE1, p 1,1 is the lagest and p 2,2 is the smallest among the five vetices, so the social welfae at CE1 is the highest. Moeove, since the set of Nash equilibia is fomed by all convex combinations of the thee NE vetices, it is impossible to achieve an even highe social welfae in the NE set than that of CE1. Theefoe, the system pefomance using the coelated optimal stategy will be highe than any stategy in the NE set. Fo multiuse asymmetic case, the linea pogamming can still be employed. The solution can be computational easy due to some fast algoithms such as Simplex algoithm [23]. Howeve, extensive signalling might be necessay to gathe all the infomation. Next, we will popose a distibuted solution. B. Multiuse Distibuted Leaning Algoithm It is shown that egetmatching algoithm [16] can convege to the set of CE. Specifically, fo any two distinct actions i i in Ω i, the eget of use i at time T fo not playing i is R T i ( i, i) := max{q T i ( i, i), }, (23) whee Q T i ( i, i) = 1 T ( U t i ( i, i ) Ui t ( i, i ) ). (24) t T
5 5 TABLE III REGRETMATCHING LEARNING ALGORITHM.9 pob. of coop. vs. bandwidth Initialize abitaily pobability fo taking action of use i, p 1 i ( i), i K. Fo t=2,3,4, Find Q t 1 i ( i, i ) as in (24). 2. Find aveage eget R t 1 i ( i, i ) as in (23). 3. Let i Ω i be the stategy last chosen by use i, i.e. t 1 i = i. Then the pobability distibution of the actions fo the next peiod, p t i is defined as p t i ( i ) = 1 µ Rt 1 i ( i, i ) i i, p t i ( i) = 1 i pt i i ( i ), whee µ is a cetain constant that is sufficiently lage. pobability of coopeation Use 1 Use 2 Use total upload bandwidth Q T i ( i, i ) can be viewed as the aveage exta payoff that use i would have obtained, if it had played action i evey time in the past instead of choosing i, and R T i ( i, i ) can be viewed as a measue of the aveage eget. The pobability p i ( i ) fo use i to take action i is a linea function of the eget. The details of the egetmatching algoithm is shown in Table III. The complexity of the algoithm is O( Ω i ). Fo evey peiod of T, let s define the elative fequency of uses joint action played till T peiods of time as z T () = 1 T #{t T : t = }, (25) total delay Fig Pobability of coopeation at mixedstategy NE. Delay vs. upload bandwidth No coopeation Nash Equil. Coelated Equil. whee #( ) denotes the numbe of times the event inside the backet happens and t is all uses action at time t. It is poved in [16] that z T conveges almost suely to a set of CE in the long un if evey playe adjusts stategy accoding to the algoithm in Table III. Howeve, z T afte convegence may not be the optimal coelated stategy. Actually in geneal, the leaning algoithm in Table III will lean the mixedstategy NE, which is wose than the optimal CE stategy. In ode to obtain the optimal CE stategy by maximizing the objective function in (16), we next deive a new leaning algoithm, which allows negotiation between playes. V. PRELIMINARY SIMULATION RESULTS In ode to evaluate the delay pefomance in the poposed file shaing game, we conducted seveal peliminay simulations. We assume a esidential netwok, whee the downloading ate of each use is set to be R = 4 kb/s, and the sevice ate (uploading ate) of seve is = 125 kb/s. We assume thee pees ae in the P2P system, and vay the shaed upload bandwidth in the inteval of [42, 18] kb/s. In Figue 2, we plot the pobability of pee coopeation at mixedstategy NE. Since the pees downloading ates ae identical, the file shaing game is symmetic in tems of the utility, so the pobability of coopeation is the same fo all thee pees. It can be seen fom Figue 2 that as the shaed upload bandwidth inceases, the pobability of coopeation also inceases. This is because when pees shae a geate total upload bandwidth Fig. 3. Compaison of total delay. upload bandwidth, a highe degee of coopeation can moe geatly educe an individual use s delay. In Figue 3, we compae the total delay of the thee uses unde nocoopeation, mixedstategy NE, and the coelated optimal stategy. We see that when thee is no coopeation, the uses expeience a vey long delay; since the uses only ely on the seve to download files, the delay is a hoizontal line with inceasing. At the mixedstategy NE, the delay deceases apidly as inceases, because the pees shae an inceasing upload bandwidth and become moe coopeative. The CE stategy has the best pefomance, since the uses now jointly optimize thei actions. Moeove, as the delay does not decease too much when gows beyond 9 kb/s, it suffices to shae a total upload bandwidth at aound 9 kb/s, in ode to educe the cost to the uses. In Figue 4, we show the pobability of joint actions of all thee uses at thei coelated optimal stategy. The yaxis epesents the pobability that a subset of the pees choose coopeation, and the indices of the coopeative pees ae listed inside the squae backets beside each subplot. Since the pobability of all pees coopeating and the pobability
6 6 P [2,3] P [1,3] P [3] P [1,2] P [2] P [1] (a) Joint pobabilities when pee 3 coopeates (b) Joint pobabilities when pee 3 does not coopeate Fig. 4. Joint pobability at the coelated optimal stategy (the index inside the squae backets indicates which use contibutes to uploading). of no pee coopeating ae both vey close to zeo, we only show the othe nonzeo pobabilities. We can see that when the shaed upload bandwidth is less than 135 kb/s, since the pobability of only one pee coopeating is aound 1/3, the coelated optimal stategy andomly suggests one use to povide sevice to the othe two. The eason why no moe uses choose coopeation is that the shaed bandwidth is elatively small. Fo instance, if thee uses shae a 9 kb/s upload bandwidth, each of them can only offe 3 kb/s uploading ate, while thei equest ates ae 4 kb/s. Moe uses will choose coopeation when the shaed bandwidth is geate. Fo instance, when is geate than 135 kb/s, two of the thee uses will be andomly picked to upload files, because moe uses can povide bette sevices if they upload data with a geate ate. In the final vesion of this pape, the authos plan to finish the simulation with moe than thee uses. Moeove, the asymmetic case will be investigated and the distibuted solution will be tested. VI. CONCLUSIONS Selfish uses in P2P file shaing always ty to download files fom the othe uses without contibuting thei own content. This may esult in undesiable delay such as the bottleneck at the seve. In ode to impove the pefomance in file shaing, we popose a gametheoetic famewok whee uses jointly optimize thei stategies. With the coelated equilibium concept, uses ae awae of the vey long delay when thee is little coopeation, so it is the best choice fo them to jointly optimize thei stategies and incease the degee of coopeation. Simulation esults show that the delay is much smalle using the coelated optimal stategy than that of Nash equilibium. REFERENCES [1] Napste. [Online]. Available: [2] The Gnutella Potocol Specification, 2. [Online]. Available: clip2.com/gnutellapotocol4.pdf/ [3] Kazaa media desktop, 21. [Online]. Available: [4] Bittoent, 23. [Online]. Available: bitconjue.og/bittoent/ [5] E. K. Lua, J. Cowcoft, M. Pias, R. Shama, and S. Lim, A suvey and compaison of peetopee ovelay netwok schemes, IEEE Communications Suvey and Tutoial, Ma. 24. [6] S. Saoiu, P. Gummadi, and S. Gibble, Measuement study of peetopee file shaing systems, Tech Repot UWCSE162, Univesity of Washington, 21. [7] B. Cohen, Incentives build obustness in BitToent, 1st Wokshop on Economics of PeetoPee Systems, June 23. [8] S. Jun and M. Ahamad, Incentives in BitToent induce fee iding, in Poc. of the 25 ACM SIGCOMM Wokshop on Economics of Peeto Pee Systems, 25. [9] D. Qiu and R. Sikant, Modeling and pefomance analysis of BitToentlike peetopee netwoks, in Poc. of ACM SIGCOMM, pp , 24. [1] K. Lai, M. Feldman, I. Stoica, and J. Chuang. Incentives fo coopeation in peetopee netwoks, in Wokshop on Economics of PeetoPee Systems, 23. [11] C. Buagohain, D. Agawal, and S. Sui, A game theoetic famewok fo incentives in P2P systems, in Poc. of the Intenational Confeence on PeetoPee Computing, pp , Sep. 23. [12] P. Golle, K. LeytonBown, and I. Mionov. Incentives fo shaing in peetopee netwoks, ACM Electonic Commece, Oct. 21. [13] F. Wu and L. Zhang, Popotional esponse dynamics leads to maket equilibium, in Poc. of the 39th ACM symposium on Theoy of computing, pp , 27. [14] M. Osbone and A. Rubenstein, A Couse in Game Thoey, MIT pess, [15] R. J. Aumann, Subjectivity and coelation in andomized stategy, Jounal of Mathematical Economics, vol. 1, no. 1, pp , [16] S. Hat and A. MasColell, A simple adaptive pocedue leading to coelated equilibium, Econometica, vol. 68, no. 5, pp , Sep. 2. [17] E. Altman, N. Bonneau, and M. Debbah, Coelated equilibium in access contol fo wieless communications, Lectue Notes in Compute Science, no. 3976, pp , SpingeVelag, Gemany, 26. [18] A. Amengol, The set of coelated equilibia of 2 2 Games, [Online] Available: [19] B. Shestha, D. Niyato, Z. Han and E. Hossain, Wieless Access in Vehicula Envionments Using Bit Toent and Bagaining, in Poceedings of IEEE Global Communications Confeence, New Oleans, LA, Novembe 28. [2] C. Papadimitiou, Computing coelated equilibia in multiplaye games, [Online] chistos/. [21] L. Kleinock, Queueing Systems, Wiley, [22] Z. Han and K. J. R. Liu, Resouce Allocation fo Wieless Netwoks: Basics, Techniques, and Applications, Cambidge Univesity Pess, 28. [23] S. Boyd and L. Vandenbeghe, Convex Optimization. Cambidge Univesity Pess, Cambidge, UK, 24.
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