Cooperative Game Theory for Distributed Spectrum Sharing

Transcription

1 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2007 proceedngs. Cooperatve Game Theory for Dstrbuted Spectrum Sharng Juan E. Surs, Luz A. DaSlva, Zhu Han, and Allen. Macenze radley Department of Electrcal and Computer Engneerng, Vrgna Tech, lacksburg, VA Electrcal and Computer Engneerng Department, ose State Unversty, ose, ID Abstract There s a need for new spectrum access protocols that are opportunstc, flexble and effcent, yet far. Game theory provdes a framework for analyzng spectrum access, a problem that nvolves complex dstrbuted decsons by ndependent spectrum users. We develop a cooperatve game theory model to analyze a scenaro where nodes n a mult-hop wreless network need to agree on a far allocaton of spectrum. We show that n hgh nterference envronments, the utlty space of the game s non-convex, whch may make some optmal allocatons unachevable wth pure strateges. However, we show that as the number of channels avalable ncreases, the utlty space becomes close to convex and thus optmal allocatons become achevable wth pure strateges. We propose the use of the Nash arganng Soluton and show that t acheves a good compromse between farness and effcency, usng a small number of channels. Fnally, we propose a dstrbuted algorthm for spectrum sharng and show that t acheves allocatons reasonably close to the Nash arganng Soluton. I. INTRODUCTION Opportunstc spectrum access has become a hgh prorty research area, snce the lmted spectrum avalable s neffcently utlzed. We study the problem where nodes n a wreless network try to gan access to bandwdth by compettvely allocatng ther own transmsson power across multple channels. We specfcally study the case where no coordnatng entty exsts n the network, and nodes need to arrve n a dstrbuted fashon at a far and effcent sharng of avalable channels. However, we assume the presence of an entty such as the FCC that can enforce agreements between players, as ths s a requrement for cooperatve games. Game theory studes mathematcal models of conflct and cooperaton among ntellgent and ratonal decson makers. Non-cooperatve game theory has been used to analyze wreless networks n [1] [4]. Iteratve water-fllng [5] and nterference avodance [6] can also be vewed as types of noncooperatve games. In cooperatve game models, players coordnate to acheve a mutually desrable soluton. Cooperatve game theory for analyss of networks and spectrum sharng has been studed n [7] [9]. Dfferent farness defntons have been proposed n the lterature, ncludng proportonal farness [10] and max-mn farness [11]. Our prmary contrbuton s to analyze the utlty space of the spectrum sharng game as the number of channels ncreases and show that the Nash arganng Soluton acheves a far and effcent spectrum allocaton. We begn by showng that n a hgh nterference envronment and a fnte number of channels, the utlty space of the spectrum sharng game s non-convex. Non-convexty can lead to optmal ponts that are mxed strateges, possbly not achevable wth pure strateges 1. We show that by ncreasng the number of channels avalable, the utlty space becomes closer to convex and more optmal ponts can be acheved wth pure strateges. Next, we show that the NS allocaton provdes a reasonable compromse between effcency and farness. We argue what the far spectrum allocaton should be and show that the NS acheves t. In our smulaton results we show that maxmum NS effcency can be acheved even wth a small number of channels. Another contrbuton of our work s to propose an algorthm that can acheve the proposed allocaton n a dstrbuted manner, usng only local nformaton. The algorthm focuses not on mplementng a barganng process between players, but on arrvng at the NS, snce that s the expected outcome of the barganng process. To fnd the NS pont requres the soluton to a non-lnear, non-convex optmzaton problem. The algorthm mplements a dstrbuted approxmaton to the optmzaton problem. In our smulaton results, we show that the algorthm reasonably approxmates the NS allocaton usng only nformaton about nodes wthn approxmately two hops. The remander of the paper s organzed as follows. Secton II gves a bref descrpton of cooperatve game theory and the NS. Secton III dscusses the spectrum sharng problem, ncludng the game model, the utlty space and the NS n the context of the spectrum sharng game. Secton IV presents the dstrbuted algorthm and show that t converges. Secton V provdes our smulaton results. Fnally, secton VI provdes some concludng remarks. II. COOPERATIVE GAME THEORY AND NASH ARGAINING Game theory provdes a set of mathematcal tools that are useful n analyzng complex decson problems wth nteractons between self-nterested decson makers, called players. The basc component of game theory s a game, G = M,A,{u }. M = {1,...N} s the set of players, A s the set of actons for player, A = A 1... A N, and u s the objectve functon, sometmes called utlty functon, whch player wshes to maxmze. For convenence, the set of actons for all players except player s denoted as A = A 1... A 1 A A N. In a cooperatve game, players bargan wth each other before the game s played. If an agreement s reached, players act accordng to the agreement reached, otherwse players act n a non-cooperatve way. Note that the agreements reached must be bndng, so players are not allowed to devate from 1 A mxed strategy s composed of a set of possble actons and a probablty dstrbuton over ths acton space. A pure strategy s a mxed strategy that conssts of a sngle possble acton /07/$ IEEE 5282

2 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2007 proceedngs. what s agreed upon. John Nash wrote n hs semnal paper on cooperatve games [12] that to understand the outcome of a barganng game, we should not focus on tryng to model the barganng process tself, but nstead, we should lst the propertes, or axoms, that we expect the outcome of the barganng process to exhbt. Ths way of analyzng cooperatve games s called axomatc barganng game theory [13]. efore we proceed, we need to ntroduce some termnology. An agreement pont s any acton vector a A that s a possble outcome of the barganng process. A dsagreement pont s an acton vector a A that s expected to be the result of non-cooperatve play gven a falure of the barganng process.e., what wll happen f players cannot come to an agreement. Clearly, the utlty acheved by every player at any agreement pont has to be at least as much as the utlty acheved at the dsagreement pont. A barganng soluton s a map that assgns a soluton to a gven cooperatve game. Followng s the barganng soluton proposed by Nash n [12]. Defnton 1: Nash arganng Soluton NS. LetU = {u a a A} be a convex, closed and upper bounded subset of R N, a 0 be the dsagreement pont, u 0 = u a 0 be the utlty of player acheved at the dsagreement pont, and U 0 = {u U u u 0 } be the set of achevable utltes. Then u = φu, u 0 s a NS f t meets the followng condtons: 1 Indvdual ratonalty IR: u u0. That s, u U 0. 2 Pareto optmalty PO: If there exsts u U 0 such that u u, then u = u,. 3 Invarance to affne transformatons INV: f ψ : R N R N, ψu =u wth u = c u + d,c,d R,c > 0,, then φψu,ψu 0 = ψφu, u 0. 4 Independence of rrelevant alternatves IIA: f u V U and u = φu, u 0 then φv,u 0 =u. 5 Symmetry SYM: f U s symmetrc wth respect to and j, u 0 = u0 j, and u = φu, u 0, then u = u j. Condtons 3-5 are the so called farness axoms. The INV axom assures that the soluton s nvarant f affnely scaled. The IIA axom states that f the doman s reduced to a subset of the doman that contans the NS, then the NS s not changed. The SYM axom states that the NS does not depend on the labels,.e. f two players have the same dsagreement utlty and the same set of feasble utlty, then they wll acheve the same NS utlty. The followng theorem, frst proposed by Nash for twoplayer games [12], and later extended for more than two players [13], shows how we can fnd the unque NS for convex utlty spaces. Theorem 1: Let I = { {1,...N} u U 0,u >u 0 } be the set of players that can acheve a utlty strctly greater than the dsagreement utlty. The maxmzer of the Nash Product NP, u, s the unque NS: u = arg max u U 0 I u u 0. 1 Theorem 1 states that the convexty of the utlty space U s a suffcent condton to guarantee that the maxmzer of the NP s the unque NS. Ths condton s suffcent but not necessary. In other words, there may exst non-convex utlty spaces where the maxmzer of the NP s a unque NS. Followng s a defnton of a u 0 -comprehensve set as well as a theorem, proposed by [14], that shows that u 0 -comprehensve sets are guaranteed to satsfy some of the NS axoms. Defnton 2: u 0 -comprehensve. AsetS R N s sad to be u 0 -comprehensve f x, y R N such that u 0 y x,, then x S mples y S. Theorem 2: Let U be a non-convex, closed, bounded, and u 0 -comprehensve utlty space. The maxmzer of the NP for set U satsfes the NS axoms: INV, IR and IIA. Ths result shows that a maxmzer of the NP for a u 0 - comprehensve set satsfes most of the NS axoms. Noteworthy s the fact that the PO axom s not guaranteed to be satsfed. Ths s because a u 0 -comprehensve set may be non-convex and n game theory, all possble mxed strateges whch we may thnk of as convex combnatons of pure strateges are consdered to be avalable strateges to the players. So, t s possble for a mxed strategy to obtan an expected utlty that Pareto domnates the NP maxmzer of the u 0 -comprehensve set. In the remander of ths manuscrpt, when we refer to Pareto optmalty wth respect to only pure strateges, we call t Lmted Pareto Optmalty LPO. In the next theorem, we show that under certan condtons, the maxmzer of the NP for a non-convex set can be a unque NS. Theorem 3: Let U be a closed and bounded utlty space and U c be the smallest convex set that contans U the convex hull of U. If U has a unque maxmzer of the NP, u, whch concdes wth the NP maxmzer of U c, u c, then u s the unque NS for U. Proof: We know that U U c. Snce the unque maxmzer of the NP for both U and U c are the same, ths mples u = u c U. Thus, by the IIA axom, u s the unque NS for U. Theorem 3 tells us that f a non-convex set concdes wth ts convex hull on the maxmzer of the NP, then the NP maxmzer of the non-convex set s a NS. We wll subsequently show that the utlty space for the spectrum sharng problem s u 0 -comprehensve and not always convex, but gven the rght condtons, can be nearly convex. Thus, the NP maxmzer for the spectrum sharng utlty space satsfes most of the NS axoms and can satsfy all axoms under the condtons where t s nearly convex. III. SPECTRUM SHARING The spectrum sharng problem addresses the ssue of how to allocate the lmted avalable spectrum among multple wreless devces. The problem has two mportant, orthogonal goals: effcency and farness. The allocaton of spectrum should utlze as much of the resource as possble. However, when utlzaton s maxmzed, farness can be compromsed. Followng s a detaled dscusson of the spectrum sharng problem. We propose a game model and dscuss our assumptons. We then analyze the resultng utlty space of the game, examne ts propertes and show that when the avalable spectrum s dvded nto a large enough number of channels, effcent spectrum allocaton s acheved wth a pure strategy. 5283

3 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2007 proceedngs. A. Game Model The spectrum sharng problem can be modeled as follows. The avalable bandwdth s dvded equally nto multple channels. Each wreless devce referred to as node can transmt n any combnaton of channels at any tme and can set ts transmt power on each channel. Each transmttng node s only nterested n communcatng to a sngle recever node. Recever nodes do not transmt and thus are not consdered players n the game snce they wll act n coordnaton wth the transmtter. Let χ = {1,...,} be the set of avalable channels, be the aggregate bandwdth, wth each channel havng bandwdth, and N be the number of transmtter nodes n the network. We formulate the spectrum sharng game as follows: M = {1,...,N}, P χ = {p k k χ p k 0, k χ pk P max} and P χ = P χ 1... P χ N.Letp Pχ and u p =C p, where C p s the Shannon capacty: C p = log 2 1+ H k pk + j Hk j pk j where p k s the power transmtted by node on channel k, P max s the maxmum transmt power, Hj k s the channel gan from j to the recever of on channel k, and s the thermal nose for the entre bandwdth.. Utlty Space Ths secton dscusses the propertes of the utlty space for the spectrum sharng game. Specfcally, we explore the effect of ncreasng the number of channels on the utlty space. We show that gven enough channels, for any mxed strategy we can fnd a pure strategy that acheves a utlty at least as hgh as the mxed strategy. Ths result mples that to acheve effcent spectrum use we need not employ mxed strateges. Ths also mples that n the cases where the utlty space s not convex some mxed strateges are not ncluded n the space, ncreasng the number of channels ncreases the number of mxed strateges that are ncluded n the set the space becomes closer to beng convex. Theorem 4: For some χ and fnte subset S χ P χ, consder a mxed strategy defned by probablty dstrbuton π, such that πs s ratonal for all s S χ. For any such mxed strategy, there exsts a pure strategy t P χ, wth P χ assocated wth a set of channels χ, that yelds the same utlty as the mxed strategy. The proof s n Appendx A. Theorem 4 says that we can replcate any mxed strategy wth a pure strategy 2. Ths result mples that, gven enough channels, we do not need to employ mxed strateges to acheve effcent spectrum utlzaton. Another less evdent, yet mportant, mplcaton s that we can make the utlty space closer to convex by ncreasng the number of channels. If we can replcate mxed strateges wth pure strateges, we reduce the number of convex combnatons not ncluded n the utlty space and thus make t closer to convex. 2 We can only replcate mxed strateges wth ratonal probabltes, but snce ratonals are dense n the reals, we can approxmate any mxed strategy arbtrarly closely. 2 Fg. 1. u = 1 = 1, w/ Mxed Strateges = 4 = 4, w/ Mxed Strateges = 4, Pareto Optmal = 16 = u 1 Example utlty space for 2 player game under hgh nterference In the remander of ths secton, we nvestgate the behavor of the utlty space as the number of channels ncreases. For purposes of llustraton, we assume a game wth only two players. For readablty, when we graph a utlty space, we only graph the upper boundary of the set that s, the set ncludes all the ponts enclosed by the upper boundary, the x axs and the y axs, as the LPO ponts are all n ths boundary. Fgure 1 shows the utlty space for the case where both nodes experence nterference stronger than ther receved sgnal strength H21 k H11,H k 12 k H22, k k, for ncreasng values of. The fgure also shows the utlty space when all possble mxed strateges are ncluded convexfed space for some values of. y examnng the utlty space for =1, we can clearly see that the boundary of the mxed strateges domnates the boundary of the pure strateges. Ths shows that under the hgh nterference case for =1, usng mxed strateges can be more effcent than adoptng pure strateges. Now examne the utlty space for =4. We notce that the boundary of ths case domnates most of the boundary for the =1case wth mxed strateges. Specfcally, the LPO ponts for =4 overwhelmngly domnate the mxed strateges for = 1. These ponts are not domnated by the mxed strateges for the case =4, whch mples that these ponts are PO. From the results of Theorem 4 we would expect the boundary for =4 to be at least as effcent as some mxed strateges for the = 1 case. The overwhelmng domnance of the =4over = 1case s due also n part to an effect that s most notceable n hgh nterference envronments. When nterference s hgh, optmal mxed strateges usually nvolve only a sngle node transmttng at a tme. y ncreasng the number of channels, we allow for frequency separaton of transmssons and thus decrease nterference. The decrease n avalable bandwdth to each player s more than offset by the decrease n nterference. Corollary 1 formally presents ths effect. Corollary 1: Consder a mxed strategy as Theorem 4, such that s k > 0 for some k χ mples sk j =0for all j and k χ. Also,forevery there exsts s S χ such that s k > 0 for some k. For any such mxed strategy, there exsts a pure strategy t P χ, wth P χ assocated wth a set of channels χ, that yelds utlty greater than the mxed strategy. The proof s n Appendx. Now compare the boundares for =4wth and wthout mxed strateges. We can see that usng mxed strateges can 5284

4 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2007 proceedngs. acheve PO ponts that are not achevable wth pure strateges. Although these ponts do not domnate the pure strategy PO ponts, t may be desrable to acheve them to meet farness objectves. Fnally, examne the utlty spaces for >4. We can see that as ncreases, the pure strategy utlty space has more PO ponts. A more subtle, yet mportant effect of ncreasng channels s that the utlty space becomes closer to ts convex hull. As the utlty space becomes convex, we can acheve more PO ponts wth pure strateges. IV. DISTRIUTED ALGORITHM Our goal s to desgn a dstrbuted algorthm that acheves the NS for the spectrum sharng game. We need the algorthm to operate only wth local nformaton and no centralzed control. In ths secton we show that nodes can be aggregated nto overlappng groups, whch we can then leverage to dstrbute the computaton of the NS. Nodes wthn each group are n close proxmty, whch allows nodes to only use local nformaton. Fnally, we propose an algorthm for computng an approxmaton to the NS and prove ts convergence. We make the followng assumptons: 1 There s an underlyng method for nformaton exchange such that nodes wthn two hops can communcate wthn a tme scale shorter than the tme scale for updates to channel allocaton. 2 Nodes run the algorthm at random ntervals such that the probablty that two or more nodes wthn two hops of each other run the algorthm smultaneously s small. 3 The executon tme of the algorthm s small relatve to the nterval between executons of the algorthm. 4 The ntal agreement pont s 0. The NS soluton s based on the assumpton that all players n the game bargan as a group to reach a cooperatve soluton to the game. Ths means that n the spectrum sharng game a node cooperates wth all nodes n the network. That s, we must consder the utlty acheved by all nodes n the network n order to mplement the NS. However, we know that a node s effect s lmted to other nodes wthn close proxmty. Ths allows us to lmt the scope of a node s barganng to a subset of the network. Consder the followng concept: Defnton 3: Let R>0 and Rx be the set of recever nodes of node. Thenterference zone IZ of node j, wth nterference radus R, s defned as IZ R j = { k Rx, dstancej, k <R,j k} The nterference zone for node j s the set of transmtter nodes such that one of ther recevers s wthn dstance R of node j. IfwesetR to a large enough value, then node / IZ R j can gnore node j s actons. Thus, we can approxmate the utlty functon of node as follows: ũ p = log 2 1+ H k pk + l J H k l pk l where J = {j IZ R j}. Ths approxmaton drops the nterference terms from nodes that are far enough away from node s recevers such that they cause neglgble nterference. Then, the utlty functon of node / IZ R j s ndependent of node j s actons. 3 Algorthm 1 Dstrbuted NS Computaton 1: IZ = {j dstancerxj, <R,u j > 0} 2: oldnp = j IZ u jp 3: ˆp = MaxmzeNP, IZ, δ 4: newnp = j IZ u jˆp 5: f newnp > 1 + tol oldnp then 6: p k =ˆpk 7: end f Consder node j maxmzng the NP whle the other nodes actons reman constant. Let p P be the current strategy employed n the network, ρ j = {p p P,p k = p k, j} be the set of strateges such that only node j has an acton dfferent from p, and IZ + R j =IZ Rj {j}. Then, max ũ p = ũ l p max ũ p. 4 p ρ j p ρ j l/ IZ Rj IZ + R j Equaton 4 demonstrates that, by usng the approxmaton for the utlty functon, nodes need only consder nodes n ther IZ when maxmzng the NP. Ths result allows us to desgn an algorthm to calculate the NP for the entre network only usng local nformaton at each node. We propose Algorthm 1 for calculatng the NP, whch conssts of nodes choosng ther actons so as to maxmze the NP of ther nterference zone. The algorthm only updates the node s actons f the value of the NP s ncreased by at least tol percent. The algorthm calls the functon MaxmzeNP, IZ, δ, whch calculates the maxmum of the NP for all nodes n the IZ wth respect to node s actons, wthn a neghborhood of the current operatng pont of sze δ. Followng s a proof of convergence for the algorthm. The proof shows that the algorthm converges, but not to what value t converges to. In the smulaton secton, we show that the algorthm ndeed converges to a value close to the NS. Proof: Convergence of Algorthm 1. Snce P s compact, ũ p s upper-bounded. Snce ũ p 0, thenthenps also upper-bounded. Eq. 4 shows that node, by maxmzng the NP for nodes n ts IZ, actually maxmzes the NP for the entre network. So, after a node executes the algorthm, the value of the NP does not decrease. The value of the NP s upper-bounded and non-decreasng and thus converges. V. SIMULATION RESULTS In ths secton we present our smulaton results. All smulatons consst of the followng setup. There are N transmtterrecever pars of nodes placed randomly n an R a meter by R a meter square. A recever s no more than 100m away from ts transmtter. The total bandwdth,, s evenly dvded nto channels. The propagaton loss exponent s 4 and the root-mean-square RMS delay spread s 1µs. The antenna gan s 0.01, the maxmum transmsson power s 100mW, and the nose level s -80dm for the entre bandwdth. All capacty numbers are reported as multples of and are the actual capacty acheved, not the approxmaton used by the algorthm. Unless otherwse stated, all smulatons are averaged over 100 randomly chosen network topologes, N = 10 and tol =2%. 5285

5 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2007 proceedngs Average Capacty Average 95% Confdence Interval Farness Score Score Iteratons Round Robn Iteratons Fg. 2. Average capacty per node acheved by the NS. R a = 200m TALE I SPECTRUM ALLOCATION SCHEME COMPARISON N =5,R a = 200m NS MaxSum MaxMn WF Avg u Mn u Farness Score In our smulatons, we am to show that the NS obtans a far and effcent spectrum allocaton. To acheve ths, we compare the NS to three other spectrum allocaton schemes: MaxMn, MaxSum and Water-fllng. Water-fllng s an teratve scheme where each node maxmzes ts own utlty wthout regard to the utlty other nodes acheve. The NS and Water-fllng algorthms are smulated n a random round robn fashon. To measure the relatve farness of a spectrum allocaton, we need a metrc that captures how close the allocaton obtaned by the algorthm s to the value of the NS. We propose a metrc, whch we call the farness score, based on the NS. The metrc attans values between 0 and 1.0, where a value of 1.0 ndcates a utlty allocaton equal to the NS. As the value decreases, the allocaton becomes skewed and thus, less far. Defnton 4: Let u be the utlty acheved by the nodes at the true NS and u be some other utlty allocaton. The farness score of u s: u u 0 u u0 A. NS Effcency and Farness 1 N In ths secton we nvestgate the spectrum allocaton acheved by the NS and compare ts farness and effcency to those of the other spectrum allocaton schemes. In ths secton, all NS values presented are the true NS ponts, not the approxmatons by the proposed algorthm. Table I shows the average and mnmum of the capacty acheved for all nodes and the average farness score, wth R a = 200m and =8. We can observe that the average capacty acheved by the NS s close to that by MaxSum and much more than by MaxMn. The NS also acheves a mnmum capacty very smlar to MaxMn and sgnfcantly hgher than MaxSum. Ths shows that NS balances effcency and farness effectvely and sgnfcantly better than MaxSum or MaxMn. Water-fllng acheves average utlty smlar to a δ = P /a max Fg. 3. Effect of delta on average score. =16,R a = 200m NS, but a much lower mnmum capacty and farness score. All schemes acheve farness scores sgnfcantly lower than the optmal. Notable s the score of MaxSum, whch s near zero, as n many cases one player s assgned zero utlty. Fgure 2 shows the average capacty per node acheved by the NS, for several values of, wth R a = 200m. Snce R a s small relatve to the transmsson range, we expect sgnfcant contenton for the spectrum. As we showed n the prevous secton, we see that under a hgh nterference envronment, the average capacty per node ncreases as the number of channels ncreases. We see that after a certan pont =9, the average capacty stops ncreasng. Ths s not surprsng, as =9s close to the pont where every node can utlze a sngle channel exclusvely and thus avod sgnfcant nterference.. Algorthm Performance Fgure 3 examnes the effect of the parameter δ, whch controls the amount of the space the algorthm can search at a gven teraton. We can see that as δ the algorthm performs rather poorly. Ths s because the search space s unconstraned and the frst node to execute the algorthm wll skew the allocaton n ts favor. y lmtng the space each node can explore at any teraton, t lmts the skewng of the allocaton. The fgure shows how makng δ smaller, the algorthm converges to a hgher farness score. However, ths comes at a cost of slower convergence. The fgure shows the average number of round robn teratons requred for the algorthm to converge. It clearly shows the algorthm requres more teratons to converge, as δ decreases. For the rest of the smulatons we set ths parameter to δ = P max /5, as that s the pont where we have the best compromse between farness score and convergence speed. In Fgure 4, we show the performance of the algorthm as a functon of the nterference radus. Clearly, we would lke ths radus to be as small as possble, so as to mnmze the nformaton exchange requred. As expected, the farness score ncreases as R ncreases. However, we notce that even when R = 100, the farness score acheved by the algorthm s greater than Ths result s encouragng as t tells us that we can sgnfcantly lmt the number of nodes nvolved n the barganng process and stll acheve a reasonable outcome. 5286

6 Ths full text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for publcaton n the ICC 2007 proceedngs. Fg. 4. Farness Score = 3 = 6 = 9 = Interference Radus R Performance for varous values of Interference Radus R VI. CONCLUSIONS The spectrum sharng problem conssts of dvdng a gven amount of spectrum among many nodes n a way that s effcent and far. In ths manuscrpt we have addressed ths problem by formulatng a cooperatve game model of the spectrum sharng problem. We analyzed the utlty space of the spectrum sharng game and showed that effcency s maxmzed by ncreasng the number of channels. We also showed that the utlty space convexfes and thus maxmzng the NP gves us an allocaton that approxmately satsfes the NS axoms. Consequently, we show that the NS allocaton provdes a reasonable compromse between effcency and farness, as t acheves allocatons wth mnmums close to MnMax and effcency close to the MaxSum. Fnally, we proposed an algorthm that, wth only local nformaton, approxmates the maxmzaton of the NP and show that t converges quckly to a value close to the true NS. A. Proof of Theorem 4 APPENDIX y constructon, for any s S χ and mxed strategy π, there exsts postve numbers as, such that πs = a s s S a χ s =. We wll now fnd a pure strategy that yelds the same utlty as ths pure strategy. Let χ = {1,...,}. Let us partton χ nto dsjont sets φs,k, s S χ and k χ, wth s S χ,k χ φ s,k = χ and wth φs,k = as. Itsassumed that for all k φs,k,hj k = Hj k for all, j M. Let us construct a pure strategy, t P χ, as follows: for each k χ, fnd s,k such that k φs,k and for each M set t k = sk. Note that t s a vald strategy, as: k =1 t k s S χ k φs,k s S χ k φs,k t k s S χ πs s k P max The utlty acheved by player for strategy t, u t, s, log H k 2 1+ tk s S χ k φs,k as s S χ log 2 log l Hk l tk l H k sk + l Hk l sk l H k sk + l Hk l sk l s S χ πsu s =E π [u s]. Proof of Corollary 1 Let S χ be the set of strateges such that for some k χ, s k > 0. y constructon, S χ > 0 and M Sχ = S χ, and thus S χ s a proper subset of S χ. Now consder the pure strategy t P χ as defned n the above proof. Let = as < and let us construct another pure strategy, s S χ v P χ, such that v k = tk >t k. Note that v s a vald strategy, as: k =1 v k s S χ k φs,k v k s S χ as t k P max The utlty acheved by player for strategy v, u v, s, log H k 2 1+ vk s S χ k + l φs,k Hk l vk l log 2 1+ Hk s k s S χ k φs,k > as s S χ log 2 1+ Hk sk REFERENCES = E π [u s] [1] C. Saraydar, N. Mandayam, and D. Goodman, Effcent power control va prcng n wreless data networks, Comm., IEEE Trans., vol. 50, no. 2, pp , [2] Z. Han and. Lu, Noncooperatve power-control game and throughput game over wreless networks, Comm., IEEE Trans., vol. 53, no. 10, pp , [3] Z. Han, Z. J, and. Lu, Power mnmzaton for mult-cell OFDM networks usng dstrbuted non-cooperatve game approach, n IEEE Global Telecomm. Conf. GLOECOM 2004, vol. 6, pp , [4] R. Etkn, A. Parekh, and D. Tse, Spectrum sharng for unlcensed bands, n 1st IEEE Int l Symp. on New Fronters n Dynamc Spectrum Access Networks DySPAN 2005, pp , [5] W. Yu, W. Rhee, S. oyd, and J. Coff, Iteratve water-fllng for gaussan vector multple-access channels, Informaton Theory, IEEE Transactons on, vol. 50, no. 1, pp , [6] J. E. Hcks, A.. Macenze, J. O. Neel, and J. H. Reed, A game theory perspectve on nterference avodance, n IEEE Global Comm. Conf. GLOECOM 2004, December [7] A. Jalal, R. Padovan, and R. Pankaj, Data throughput of CDMA- HDR a hgh effcency-hgh data rate personal communcaton wreless system, n Vehcular Technology Conf., vol. 3, pp , [8] Z. Han, Z. J, and. Lu, Far multuser channel allocaton for OFDMA networks usng Nash barganng solutons and coaltons, Comm., IEEE Trans., vol. 53, no. 8, pp , [9] L. Cao and H. Zheng, Dstrbuted spectrum allocaton va local barganng, n 2nd Ann. IEEE Comm. Soc. Conf. on Sensor and Ad Hoc Comm. and Networks SECON 2005, pp , [10] F. P. elly, A. Maulloo, and D. Tan, Rate control n communcaton networks: shadow prces, proportonal farness and stablty, J. of the Operatonal Research Soc., vol. 49, pp , [11] L. Tassulas and S. Sarkar, Maxmn far schedulng n wreless networks, n INFOCOM Twenty-Frst Annual Jont Conference of the IEEE Computer and Communcatons Socetes. Proceedngs. IEEE, vol. 2, pp , [12] J. Nash, Two-person cooperatve games, Econometrca, vol. 21, no. 1, pp , [13] H. J. M. Peters, Axomatc arganng Game Theory. luwer Academc Publshers, [14] L. Zhou, The Nash barganng theory wth non-convex problems, Econometrca, vol. 65, pp , May