TAHES: Truthful Double Auction for Heterogeneous Spectrums

Transcription

1 The 31st Annual IEEE Internatonal Conference on Computer Communcatons: Mn-Conference TAHES: Truthful Double Aucton for Heterogeneous Spectrums Xaojun Feng, Yanjao Chen, Jn Zhang, Qan Zhang, and Bo L Department of Computer Scence and Engneerng, Hong Kong Unversty of Scence and Technology, Hong Kong {xfeng, chenyanjao, jnzh, qanzh, bl}@cse.ust.hk Abstract Aucton s wdely appled n wreless communcaton for spectrum allocaton. Most of pror works have assumed that spectrums are dentcal. In realty, however, spectrums provded by dfferent owners have dstnctve characterstcs n both spacal and frequency domans. Spectrum avalablty also vares n dfferent geo-locatons. Furthermore, frequency dversty may cause non-dentcal conflcts among spectrum buyers snce dfferent frequences have dstnct communcaton ranges. Under such realstc scenaro, exstng spectrum aucton schemes cannot provde truthfulness or effcency. In ths paper, we propose a Truthful double Aucton for HEterogeneous Spectrum, called TAHES. TAHES allows buyers to explctly express ther personalzed preferences for heterogeneous spectrums and also addresses the problem of nterference graph varaton. We prove that TAHES has nce economc propertes ncludng truthfulness, ndvdual ratonalty and budget balance. I. Introducton Spectrum s valuable resource for wreless communcaton. Measurement results have shown that spectrum utlzaton s hghly dynamc n dfferent geo-locatons[13]. In some places, spectrums may be under-utlzed and are usually referred to as whte spaces. More effcent utlzaton of whte spaces of dfferent frequences s consdered as a potental soluton for the next generaton of wreless networkng. A promsng way to better utlze whte spaces s to enable spectrum owners to lease ther spectrums to secondary servce provders n the geo-locatons where the prmary users wll not be nterfered. In return, the spectrum owners can get pad from secondary servce provders. Company such as SpectrumBrdge[1] has already launched an onlne platform called SpecEx for spectrum owners to sell ther unused spectrums to potental buyers. In such a new busness model, spectrums from multple sellers resde n dfferent frequency bands and have varous avalablty n dfferent locatons. Also, spectrum buyers may express dfferent preferences for dfferent spectrums. Another nterestng aspect n ths scenaro s the reusablty of spectrum. Two buyers that are far enough from each other may reuse the same spectrum concurrently. The problem of spectrum redstrbuton between multple spectrum owners and multple secondary servce provders can be modeled as a sngle round mult-tem double aucton. In our case, the spectrum owners are sellers; secondary servce provders are buyers; spectrums are goods; and the auctoneer can be a thrd party provdng the aucton platform such as SpectrumBrdge or regulators such as FCC. Although aucton has been wdely appled to spectrum allocaton n wreless communcaton, exstng double aucton schemes[6]-[9] cannot be drectly appled to our scenaro. There are three major challenges n our problem. The frst challenge s the spatal heterogenety of spectrum. Spectrums offered by dfferent spectrum owners are avalable to dfferent buyers. However, exstng works[6]-[9] consder only the scenaro where all spectrums are avalable to all buyers. The second challenge comes together wth frequency heterogenety and spectrum reusablty. The spectrums may resde n varous frequency bands. Lower-frequency bands have larger nterference ranges than hgher-frequency bands do. In ths case, dfferent buyers may have dfferent nterference relatonshps n dfferent frequences. Nevertheless, exstng works consderng spectrum reusablty[8][9] usually assume the same conflct relatonshp among all buyers throughout all frequences. The thrd challenge comes from the aucton mechansm desgn. A well desgned aucton scheme should preserve the most crtcal property: Truthfulness (or strategyproofness). A truthful aucton nctes all bdders to voluntarly reveal ther true valuaton for the tems they are bddng. The auctoned tems are fnally assgned to bdders who value t the most. Unfortunately, wth heterogenous tems, the truthfulness cannot be guaranteed when smply applyng exstng schemes. In ths paper, we propose TAHES, a Truthful double Aucton scheme for HEterogeneous Spectrum, to address the above-mentoned challenges. TAHES groups spectrum buyers consderng ther non-dentcal conflct relatonshps n heterogeneous spectrums to explore spectrum reusablty. To guarantee truthfulness, TAHES employs a matchng procedure between buyer groups and sellers. In summary, the man contrbutons of the paper are as follows: To the best of our knowledge, TAHES s the frst double aucton mechansm for heterogeneous spectrum transacton, allowng buyers to express dversfed preferences for spectrums of dfference frequences. TAHES mproves spectrum utlzaton by takng nto consderaton of the spectrum reusablty. As far as we know, TAHES s the frst aucton mechansm that can deal wth the problem of nterference graph varaton caused by spectrum frequency heterogenety. TAHES ensures that bddng truthfully s the domnant strategy for all bdders. Despte ths, TAHES also con /12/$ IEEE 3076

2 forms wth ndvdual ratonalty and budget balance. The rest of the paper s organzed as follows. Secton II presents the aucton model of heterogeneous spectrum tradng and then sets the desgn objectves for the proposed aucton mechansm. In secton III, challenges n heterogeneous aucton desgn are further explaned. We gve detaled descrpton of TAHES n secton IV. We brefly dscuss related works n SectonVandsummarzethewholeworknSectonVI. II. Model Descrpton and Desgn Targets In ths secton, we frst formulate the problem of heterogeneous spectrum exchange between spectrum owners and servce provders as a double aucton. Then, we overvew deal economc propertes of an aucton mechansm and state our aucton desgn target. A. Problem Formulaton We consder the scenaro where N secondary servce provders need to purchase spectrum resource from M spectrum owners. A sngle-round double aucton conssts of M seller and N buyers s held to serve ths purpose. Let S = {s 1, s 2,, s M } denotes the set of sellers and W = {w 1, w 2,, w N } denotes the set of buyers. A thrd-party acts as the auctoneer and decdes the wnnng bdders and the payment. Each seller contrbutes one dstnct channel. Each buyer would lke to purchase one channel, but they have dfferent valuatons for the channels. Therefore, the buyers bds are channel specfc. We assume that the aucton s sealedbd, prvate and colluson-free. In other words, n the aucton, all bdders smultaneously submt sealed bds so that no bdder knows the bd of any other partcpants. In addton, bdders do not collude wth each other to mprove the utlty of the coaltonal group. We use C to denote the bd of s. C = (C 1, C 2,, C M )s the bd matrx of all sellers and C denotes the bd matrx wth s s bd removed. We use b j to denote the bd of w for seller s j s channel. B = (b 1, b2,, bm ) s the bd vector of w. B = (B 1, B 2,, B N ) s the bd matrx of all buyers. Let B denote the bd matrx wth buyer w s bd B excluded. The true valuaton of s for ts channel s V s and the true valuaton of w for seller s j s channel s v j. Vw = (v 1, v2,, vm )sthe valuaton vector of buyer w.ifs j s channel s unavalable to w, v j s zero. The true valuaton of both buyers and sellers may or may not equal to ther bds. In the aucton, the auctoneer determnes the payment P s for seller s and the prce p w that buyer w should pay. P s and p w are not necessarly equal. Therefore, the utlty of the seller s s defned as: { P U s s = V s, f s wns (1) 0, Otherwse Smlarly, the utlty of buyer w s defned as: { u w V θ() = p w, f w wns 0, Otherwse n whch θ() s the channel that w wns. (2) B. Desgn Target Truthfulness, budget balance and system effcency cannot be acheved n any double aucton at the same tme, even wthout the consderaton of ndvdual ratonalty[5][19]. In our scenaro, we need to warrant that spectrum owners have ncentves to lease ther spectrum and also the thrd party (spectrum transacton platform, government) s wllng to partcpate as an auctoneer. Therefore, n ths paper, we set our desgn target as achevng truthfulness, ndvdual ratonalty and budget balance, whch are also selected by many exstng double auctons[9][19]: Truthfulness. Nether buyers nor sellers can get hgher utlty by msreportng ther true valuaton,.e., C j V s j or B V w. Indvdual ratonalty. A wnnng seller s pad more than ts bd and a wnnng buyer pays less than ts bd. Budget balance. The auctoneer s proft s non-negatve. Ths proft equals to the prce pad by the buyers mnus the payment to the sellers. III. Challenges of Heterogeneous Spectrum Aucton Desgn In ths secton, we brefly llustrate the challenges of desgnng a truthful aucton mechansm for heterogeneous spectrum. We wll frst ntroduce the heterogeneous nature of spectrum. Then we show that exstng mechansms are unsuccessful n meetng our desgn targets when drectly appled to heterogeneous spectrum aucton. A. Spatal Heterogenety Spatal heterogenety means that spectrum avalablty vares n dfferent locatons. For example, one TV channel s avalable only f there are no nearby TV statons or wreless mcrophones occupyng the same channel. Tradtonal aucton desgn nvolvng spectrum reusablty usually groups buyers by fndng ndependent sets on ther nterference graphs [8][9], and then set the group bd accordng to the mnmum bd n the group. However, f two buyers wth no common avalable channels are grouped together, the group bd should be 0 for all channels and ths group can never wn n the aucton. B. Frequency Heterogenety Frequency heterogenety means that dfferent frequences have dfferent transmsson ranges. Accordng to the propagaton model recommended by ITU[10], the center frequency of one spectrum band can mpact the path lose between two nodes: L = 10 log f 2 + γ log d + P f (n) 28 (3) where L s the total pass loss n decbel(db), f s the frequency of transmsson n megahertz(mhz), d s the dstance n meter(m), γ s the dstance power loss coeffcent and P f (n) s the floor loss penetraton factor. In our model, the spectrums offered by spectrum owners may consst of a wde range of frequences. For example, n the German spectrum aucton held n 2010, the hghest frequency (2.6GHz) are more than 3077

3 b 1 2 b 2 1 b 2 C1 C2 C3 C b3 b 4 Fg. 1: Buyer b 2 can manpulate ts non-wnnng bd to ncrease the utlty three tmes hgher than the lowest frequency (800MHz)[12]. Ths huge gap leads to non-dentcal nterference relatonshps among spectrum buyers n dfferent channels. However, based on our knowledge, no exstng aucton schemes address nonunform nterference relatonshps among buyers caused by frequency heterogenety. C. Market Manpulaton In double aucton for homogeneous tems, the two wellknown aucton algorthms VCG[2]-[4] and McAfee[11] can both ensure truthfulness. But truthfulness cannot be guaranteed f we drectly apply these algorthms for heterogeneous tems. In the McAfee double aucton, the auctoneer sorts the buyers bds n non-ncreasng order and the sellers bds n non-decreasng order: B 1 B 2 B N and C 1 C 2 C M. Then the auctoneer fnds the largest k such that B k C jk. The McAfee scheme dscard (B k, C jk )whchs the least proftable transacton and set the unform prce for buyers to be B k and the unform payment for sellers to be C jk. In double aucton for heterogeneous tems, although one buyer can wn at most one tem, t can manpulate ts bds to acheve hgher utlty. In Fg. 1, there are four buyers among whch buyer w 2 has two non-zero bds b 2 2 and b1 2. Both the buyers and sellers bds have been sorted accordng to the McAfee mechansm. We can see that the ndex of the least proftable transacton marked by the blue frame s k = 3. So the prce charged for the wnnng group s b 1 2 = 8. If the buyer w 2 lowers ts bd for seller s 1 to make b 1 2 < 8, say b 1 2 = 7, ths msconduct wll not change the results of the aucton, but can contort the prce from 8 to 7. As a result, the utlty of w 2 can be ncreased. Snce the VCG double aucton uses a smlar way to determne wnners and prces, t suffers the same defect. IV. Aucton Desgn: TAHES In ths secton, we propose TAHES, a Truthful double Aucton for HEterogeneous Spectrum. A. Overvew To handle both spectrum heterogenety and spectrum reusablty, we desgn three key steps n TAHES: (1) Buyer Groupng: Spectrum can be reused by non-conflct buyers n dfferent locatons. By non-conflct, we mean when two buyers w and w j are usng the same channel h, they are out of h s nterference range of each other. However, the conflct relatonshps between one par of buyers are non-dentcal n dfferent frequences. In ths step, the auctoneer uses a groupng algorthm consderng non-dentcal nterference graphs to form non-conflct buyer groups so that they can purchase the same channel. Our buyer groupng algorthm s bd-ndependent. The nput of the groupng algorthm s the channel avalablty nformaton of each buyer. Ths knd of nformaton can be calculated accordng to the path loss model gven the locatons of buyers and sellers or can be obtaned from a geo-locaton database[13]. We assume the auctoneer can get such locaton nformaton of buyers and sellers. The bd-ndependent property of the groupng algorthm s crtcal to ensure truthfulness n the aucton[9]. (2) Matchng: After the frst step, each buyer group may stll purchase channels from multple sellers f the buyers n a group have more than one common channel. Whle ndeed, each group can at most wn one channel. We have shown that only one bd n a bd vector should be kept for further wnner determnaton. Otherwse, the aucton can be vulnerable to market manpulaton. Some buyers can strategcally change some of ther bds to lower the group bds so that they can change the prce they need to pay and ncrease ther utlty. In ths matchng step, the auctoneer chooses one conventonal matchng algorthm to match each buyer group to only one buyer based on only the channel avalablty for each group. Therefore, the matchng step s also bd-ndependent. (3) Wnner Determnaton and Prcng: After buyer groupng and matchng, the remanng problem of wnner determnaton and prcng are smlar to that n the McAfee double aucton desgn. We can drectly apply the McAfee s scheme to use the k th par of buyer group and seller to determne the wnners. B. Aucton Procedure TAHES comprses the followng steps: 1) Buyer Groupng: Suppose the set of channels from sellers s H = {h 1, h 2,, h M }. h s communcaton range s R(h ). Wthout loss of generalty, we assume R(h 1 ) R(h 2 ) R(h M ). Let A = {a, j a, j {0, 1} N M }, an N by M matrx, represent the buyers channel avalablty. a, j = 1 means that channel h j s avalable for buyer w.let E = {e, j,k e, j,k {0, 1} M N N }, an M by N by N matrx, represent the conflct relatonshps between buyers n each channel. e, j,k = 1 means that buyers w j and w k are conflct n h. In ths step, the nputs are A and E, whch are both bdndependent. After groupng, we get a set of l (l n) buyer groups denoted as G = {g 1, g 2,, g l } and the canddate channel set for each group denoted as F = { f 1, f 2,, f l }. f contans the channels that g can purchase, whch s assgned by the auctoneer. G and F are the outputs of the groupng algorthm. The groupng algorthm should satsfy the followng constrants: Common Channel Exstence Constrant: There exsts at least one channel that s avalable for all buyers n the same 3078

4 Algorthm 1 Buyer-Groupng(A, E, H) 1: // L represents the set of grouped buyers 2: L =, G =, F = 3: whle L W do 4: for all h H do 5: Canddate buyers to be grouped: Q = 6: for all w j {w k A k, = 1 w k L} do 7: Q = Q w j 8: end for 9: Fnd ndependent set IS on buyer set Q based on E. 10: end for 11: Fnd IS, such that IS s maxmzed 12: g = 13: for all w j IS do 14: g = g w j, L = L w j 15: end for 16: f = 17: for all h j H do 18: f R(h j ) <= R(h ) then 19: f = f h j 20: end f 21: end for 22: G = G g, F = F f 23: end whle 24: return (G, F) group. g, h j, s.t. w k g h j f A k, j = 1 (4) Conflct Free Constrant: Any two buyers n the same group do not mutually nterferng wth each other n any channel n the canddate channel set. w j, w k g, h l f A j,l = A k,l = 1 E l, j,k = 0 (5) The groupng algorthm frst fnds an ndependent set of buyers n each channel. Then t selects one such set wth maxmum group sze and contnues to fnd the next group untl all buyers are classfed nto one group. The groupng procedure s shown n Algorthm 1. In Algorthm 1, we can use any exstng algorthms to fnd ndependent sets, for example, the algorthms descrbed n [14]. From the procedure of Algorthm 1, t s obvous that all buyers n IS have a common avalable channel h. And from lne 16-20, we only consder canddate channel wth smaller or equal communcaton range of h. Therefore, the groupng results G and F also satsfy the Conflct Free Constrant. Theorem 1. The buyer groups and canddate channel sets returned by Algorthm 1 satsfy the Common Channel Exstence Constrant and the Conflct Free Constrant. 2) Matchng: After step 1, we have formed a group set G. Suppose the number of buyers n group g s n and the group bd vector s δ = (δ 1,δ2,,δM ). We follow the dea n [9] Algorthm 2 Buyer-Group-Matchng(G, F, S ) 1: // Let Δ be an G by S matrx representng the weghted adjacent matrx between G and S 2: Δ={0} M N 3: for all g x G, s y S do 4: f δ y x > 0andh y f x then 5: Δ x,y =Δ y,x = g x 6: end f 7: end for 8: (G C, S C,σ) = MATCH(X, Y, Δ) 9: return (G C, S C,σ) and assgned the group bd to be the mnmum bd tmes the group sze as: δ j = mn{b j k w k g } n (6) Here, we can magne each buyer group as one super buyer. In δ, there may be more than one non-zero entres, say δ j 1 and δ j 2. If not well desgned, the aucton may be untruthful, as shown n Secton III-C. Such type of market manpulaton s caused by the multple non-zero group bds. To tackle ths challenge, n TAHES, we apply a channel matchng scheme to match one buyer group to an dentcal seller. We call the results after matchng the canddate wnnng group set G C and canddate wnnng sellers set S C. The matchng procedure s shownnalgorthm2. In ths algorthm, σ records the matchng result. For example, σ(g x ) = s y ndcates that buyer group g x s assgned wth seller s y. MATCH(X, Y, Δ) matches nodes set X to Y wth weghted edges n Δ. It can be any matchng algorthm for bpartte graphs specfed by the auctoneer, for example, maxmum matchng or maxmum weghted matchng. Ths matchng step here s also ndependent of bds of both buyers and sellers. 3) Wnner Determnaton and Prcng: In the wnnerdetermnaton and prcng stage, we can apply the mechansm used n McAfee and drectly use the buyer group-seller par (g k, s jk ) to determne wnners and prce. The detaled algorthm s shown n Algorthm 3. C. Economc propertes Theorem 2. TAHES s ndvdually ratonal. Proof: For each buyer w n wnnng group g k : p w = pw δσ(k) k b σ(k) For each wnnng seller s j : P s j C k P s Theorem 3. TAHES s budget-balanced. = b σ(k) Proof: Accordng to the sortng n the wnner determnaton algorthm, we have p w P s and G W = S W, therefore the budget for the auctoneer s: G W p w S W P s 0 Theorem 4. TAHES s truthful. 3079

5 Algorthm 3 Wnner-Determnaton-and-Prcng(G C, S C,σ) 1: G W =, S W = // The set of wnnng groups and sellers 2: // Sort all buyer groups n G C and sellers n S C 3: Construct X = {g 1, g 2,, g l }, such that δ σ( 1) 1 δ σ( 2) 2 δ σ( l) l 4: Construct Y = {s j1, s j2,, s jm }, such that C j1 C j2 C jm 5: Fnd the largest k, s.t.δ σ( k) 6: f k < 2 then 7: return (G W, S W, 0, 0) 8: end f 9: G W = groups δ 1 to δ k 1 k C jk 10: S W = sellers C j1 to C jk 1 11: // Determne the prce and payment 12: p w = δ σ( k) k, P s = C jk 13: for Any buyer w n g τ() G W do 14: p w = p w /n τ() 15: end for 16: for all Any seller s j S W do 17: P s j = Ps 18: end for 19: return (G W, S W, P S, p w ) Due to lmtaton of space, we do not present the proof n ths paper. V. Related Works Aucton has been extensvely studed n the scope of spectrum allocaton[6]-[9]. However, most exstng works faled to consder spectrums as non-dentcal tems. In [9], Zhou and Zheng frst address spectrum reusablty n ther aucton desgn: TRUST. In [8], the authors also consder spectrum reusablty for buyers, and they assume buyers can have multple rados. The proposed TAHES scheme also consder spectrum reusablty, moreover, TAHES can tackle the case when spectrums are heterogeneous. In [16], an aucton desgn for heterogeneous TV whte space spectrums s proposed. In that paper, the spectrum allocaton problem has been defned as an optmzaton problem where maxmum payoff of the central tradng entty (called spectrum broker) s the optmzaton goal. However, [16] s not a double aucton scheme and ts desgn goal s dfferent from TAHES. Recently, n [19], Yang et. proposed a double aucton desgn for cooperatve communcatons wth heterogeneous relay selectons. However, there s no reusablty n ther scenaro. Dfferent from our sngle-round aucton model, recently, there are also works consderng spectrum aucton n an onlne fashon[15][17]. In an onlne spectrum aucton, buyers may arrve n dfferent tme and they can request the spectrum for a partcular duraton. However, exstng onlne double aucton schemes consder only homogeneous spectrum. VI. Conclusons In ths paper, we have desgned TAHES, a truthful double aucton scheme for heterogeneous spectrums. TAHES allows multple spectrum owners wth avalable spectrums n dfferent locatons and dfferent frequency bands to partcpate n the spectrum leasng to secondary servce provders. TAHES can not only solve unque challenges caused by spectrum heterogenety but also perverse nce economc propertes: Truthfulness, Budget Balance and Indvdual Ratonalty. Acknowledgement Ths research was supported n part by Hong Kong RGC grants , , Huawe-HKUST jont lab, Natonal Natural Scence Foundaton of Chna wth grants no. as , , by a grant from NSFC/RGC under the contract N HKUST610/11, by grants from HKUST under the contract RPC11EG29, SRFI11EG17-C and SBI09/10.EG01- C, by a grant from Huawe Technologes Co. Ltd. under the contract HUAW18-15L /PN, by a grant from Guangdong Bureau of Scence and Technology under the contract GDST11EG06. References [1] Spectrum Brdge, Inc. [2] W. Vckrey, Counterspeculaton, auctons, and compettve sealed tenders, The Journal of Fnance, 16:8-37, [3] E. H. Clarke, Multpart prcng of publc goods, Publc Choce, 11:17-33, [4] T. Groves, Incentves n teams, Econometrca, 41(4):617-31, July [5] R. B. Myerson and M. A. Satterthwate, Effcent mechansms for blateral tradng, Journal of Economc Theory, 29(2): , Aprl [6] X. Zhou, S. Gandh, S. Sur, AND H. Zheng, ebay n the sky: Strategyproof wreless spectrum auctons, n ACM MobCom [7] J. Ja, Q. Zhang, Q. Zhang, and M. Lu, Revenue generaton for truthful spectrum aucton n dynamc spectrum access, n ACM Mobhoc [8] F. Wu and N. Vadya, SMALL: A Strategy-Proof Mechansm for Rado Spectrum Allocaton, n IEEE INFOCOM [9] X. Zhou and H. Zheng, Trust: A general framework for truthful double spectrum auctons, In IEEE INFOCOM [10] Propagaton Data and Predcton Methods for the Plannng of Indoor Radocomm. Systems and Rado Local Area Networks n the Frequency Range 900 MHz to 100 GHz. Recommendaton ITU-R P , [11] R. P. McAfee. A domnant strategy double aucton. Journal of Economc Theory 56, 2 (Aprl 1992), [12] German spectrum aucton ends but prces low, [13] X. Feng, J. Zhang, and Q. Zhang, Database-asssted mult-ap network on TV whte spaces: system archtecture, spectrum allocaton and AP dscovery, n IEEE DySPAN [14] S. Saka, M. Togasak, and K. Yamazak, A note on greedy algorthms for the maxmum weghted ndependent set problem, Dscrete Appled Mathematcs 126 (2003) [15] S. Wang, P. Xu, X. Xu, S. Tang, X. L, and X. Lu, TODA: Truthful Onlne Double Aucton for Spectrum Allocaton n Wreless Networks, n IEEE DySPAN [16] M. Parzy and H. Bogucka, Non-dentcal objects aucton for spectrum sharng n TV whte spaces C the perspectve of servce provders as secondary users, n IEEE DySPAN [17] L. Deek, X. Zhou, K. Almeroth, and H. Zheng, To Preempt or Not: Tacklng Bd and Tme-based Cheatng n Onlne Spectrum Auctons, n IEEE INFOCOM [18] M. Al-Ayyoub adn H. Gupta, Truthful Spectrum Auctons Wth Approxmate Revenue, n IEEE INFOCOM [19] D. Yang, X. Fang, and G. Xue, Truthful Aucton for Cooperatve Communcatons, n ACM Mobhc