SPRITE: A Novel Strategy proof Multi unit Double Auction Framework for Spectrum Allocation in Wireless Communications Abstract Keywords:

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1 SPRITE: A Novel Strategy proof Mult unt Double Aucton Framewor for Spectrum Allocaton n Wreless Communcatons He Huang*, Ka Xng +, Hongl Xu +, Lusheng Huang + *. School of Computer Scence and Technology, Soochow Unversty, Suzhou , Chna huangh@suda.edu.cn +. School of Computer Scence and Technology, Unversty of Scence and Technology of Chna, Hefe, Chna {xng, xuhongl}@ustc.edu.cn Abstract Aucton s wdely used for spectrum resource allocaton n wreless communcatons. Many exstng wors assume that the spectrum resource s sngle-unt and ndvsble, whch greatly lmts users capablty to utlze the spectrum. Furthermore, most of them fal to tae nto account of buyer/seller s dstnctve demands n aucton and consder spectrum allocaton as a sngle unt or sngle-sded aucton. In ths paper, we consder the mult-unt double aucton problem under the context that multple buyers/sellers have dfferent demands to buy/sell. Partcularly, we present a novel strategy-proof mult-unt double aucton framewor (SPRITE). SPRITE establshes a seres of bd-related buyer group constructon and wnner determnaton strateges. It mproves the spectrum reusablty and acheves sound spectrum utlzaton, farness, and essental economc propertes at the same tme. Furthermore, we theoretcally prove the correctness, effectveness and economc propertes of SPRITE and show that SPRITE s strategy-proof. In our evaluaton study, we show that SPRITE can acheve mult-unt spectrum aucton wth better aucton effcency compared wth exstng double aucton mechansms. To the best of our nowledge, SPRITE s the frst mult-unt double aucton approach for wreless spectrum allocaton. Keywords: Wreless communcaton, spectrum aucton, mult-unt, strategy-proof Ⅰ Introducton As the ncreasng popularty of wreless devces and applcatons, the ever-ncreasng demand of traffc poses a great challenge n spectrum allocaton and usage. However, large companes and organzatons occupy many spectrum resources by means of long-term and regonal leases [1] wthout consderng spectrum reuse, whle new applcants, e.g., non-contract users, new applcant, etc., are n great shortage of spectrum resources. Therefore, t s mperatve to provde an effectve soluton to redstrbute the under-utlzed spectrum resources to the ones n shortage of spectrum. Aucton, n whch the spectrum owners could gan utltes to lease ther dle spectrum n economc perspectve [2,3] whle new applcants could gan access to the spectrum, may serve as such a promsng way that could better ncrease the effcency, effectveness and economc propertes of the spectrum. However, n tradtonal one-to-many sngle-sded aucton style (smlar to FCC method) the rare resources are n centralzed control of the seller/buyer, whch s resource domnant sde that has the rghts to establsh rule of transactons. Ths aucton style may cause colluson or maret manpulaton problem. Compared to sngle-sded aucton [4,5], double aucton mechansm s more sutable for spectrum redstrbuton owng to ts farness and allocaton effcency. Both of the buyer and seller group wll lose ther relatve domnant poston n double aucton procedure, and ther relatonshp becomes supply and demand coordnaton [6]. Consequently, double aucton mechansm s more lely to acheve maxmum spectrum reuse under the premse of protectng the profts of buyers and sellers. TRUST [7] s regarded as the frst wor to tghtly ntegrate spectrum allocaton and prcng components by usng double aucton mechansm. However, t only consders sngle-unt double aucton ssue, and thus lacs the ablty to support aucton n mult-rado wreless networs, whch s taen as the enablng technology of next generaton wreless networ communcatons [8,9]. Besdes, buyers n spectrum aucton could share the same spectrum f they don t nterference wth each other n spectrum aucton, e.g., heterogeneous geo-locaton could enable spectrum reuse. However, dfferent buyer groups stand for dfferent purchasng power and dfferent payoff. It s hard to ensure that all the buyers bd truthfully by usng the exstng clearng prce rule. Ths requres double aucton mechansm consder economc effects not only n the process of transacton set constructon (just le TRUST) but also n buyer group constructon secton. To solve these ssues, we propose a novel Strategy-Proof mult unt double spectrum aucton (SPRITE) whch satsfes the economc propertes, spectrum reuse and maret clearng. The framewor of the SPRITE mechansm can be descrbed n Fg. 1. Compared wth exstng tradtonal sngle-unt double aucton approaches, the major contrbutons of SPRITE can be dentfed as follows: SPRITE jontly consders economc propertes wth spectrum allocaton problem. It could consttute a NASH Equlbrum through the whole aucton process, better mproves spectrum reuse, and further leads buyers and sellers partcpatng the aucton n an honest and far manner. SPRITE provdes a novel clearng prce mechansm to assure the strategy-proof property and other essental

2 economc propertes, whch s sgnfcantly dfferent from tradtonal spectrum aucton methods. Compared wth sngle-unt aucton, SPRITE s the frst wor that acheves mult-unt double aucton that satsfes the needs of users n mult-rado wreless networs and mproves aucton effcency at the same tme. Fg. 1. Framewor of SPRITE mechansm The rest of paper s organzed as follows: Secton II ntroduces prelmnares and surveys the most related wor. Secton Ⅲ proposes the algorthm desgn of SPRITE. In secton Ⅳ, we prove the correctness, effectveness, and economc propertes of our desgn. Secton Ⅴ evaluates the performance of our approach. In the last secton, we conclude the paper. Ⅱ Prelmnares and Bacground A. Assumptons and Termnologes Suppose that each seller contrbutes mult-unt homogeneous channels to sell and each buyer have mult-unt channels to buy. We assume the double aucton process s sealed-bd and prvate. Thus the bdders wll not collude. We also assume that all mult-unt bds are dvsble : A buyer/seller wllng to buy/sell unts at a specfed prce-per-unt would also be wllng to trade at that prce, where. Notatons: : Group of all buyers; : Group of all sellers, where ; : The bd of buyer,, can be deemed as maxmum prce t s wllng to pay for a channel; : Number of channels requrement for buyer ; : The bd of seller,, can be deemed as mnmum prce t s requred to sell a channel; : Number of channels provded by seller ; : For a buyer, ts true valuaton of the channel; : If buyer wns the aucton, the prce t needs to pay for each channel by bddng ; : The utlty of a buyer, ; : For a seller, ts true valuaton of the channel; : If seller wns the aucton, the actual payment t receved for each channel; : The utlty of a seller, ; : The success rate for buyers by bddng ; We also assume that there are partcpants, where,, represents the number of buyers and sellers, respectvely. To mae a mult-unt double aucton robust and practcal, the mechansm should be strategy-proof, ex-post ndvdual-ratonal and ex-post budget-balanced. (1) Strategy-proof. In a double aucton, f no buyer or seller can mprove ts own utlty by bddng untruthfully, we say the aucton s strategy-proof. In other words, truthfully bddng s the domnant strategy for each partcpant. Proposton 1: For buyer:, : Untruthful bd for buyers (1) For seller:, : Untruthful bd for sellers (2) (2) Ex-post ndvdually ratonal. No partcpant s expected utlty s not less than ts utlty from non-partcpaton. Proposton 2: For buyer:, (3) For seller:, (4) (3) Ex-post budget-balanced. The expected payoff of the auctoneer s non-negatve. Proposton 3: Auctoneer s Expected Payoff: (5) (4) Aucton effcency. The valuaton of the buyers s optmzed.

3 Exstng aucton mechansm Table 1. Comparng of dfferent double aucton mechansms Strategy-proof Ex-post Indvdual Spectrum budget-balanced ratonalty reuse Mult-unt goods tradng VCG McAfee BC-LP Wurman TRUST SPRITE B. Related Research Double aucton can be classfed nto two categores: contnuous double aucton (CDA) and perodc double aucton (PDA). The PDA mechansm collects bds over a specfed nterval of a tme, and then clears the maret at the expraton of the bddng nterval. In ths paper, we consder the PDA model to study the dynamc spectrum aucton. The lterature on strategy-proof mechansm desgn starts from the classcal method by Vcrey-Clare-Groves (VCG) mechansm. The mproved VCG double aucton [10] mechansm s strategy-proof, ex-post ndvdual-ratonal, and effcent. However, the VCG aucton method s not budget balanced wth multple buyers and sellers. McAfee [11] proposes a strategy-proof, budget-balanced double aucton mechansm for a smple exchange envronment, n whch all the partcpants exchange only one unt good. In [12], the author desgned an extended verson, n whch all partcpants could exchange mult-unt goods. Wurman [13] examnes a general famly of aucton mechansms that admt multple buyers and sellers, and proposes a mechansm whch transforms the buyers mult-unt goods demand to sngle-unt transacton. Babaoff [14] proposes a budget-balanced and strategy-proof double aucton mechansm for a blateral exchange scenaro wth sngle output restrcton, where each buyer desre for a bundle of goods. Chu and Shen [15] propose an asymptotcally effcent truthful double aucton mechansm called BC-LP, whch acheves bundle of commodtes transacton for buyers. TRUST [7] s the frst strategy-proof and spectrum reuse double aucton method and the most related wor to the study n ths paper, but t only refers to one-to-one channel tradng. (To be added) Ⅲ Algorthm Desgn In ths part, we propose the desgn and detals of our proposed mult-unt double spectrum aucton mechansm. Partcularly, our mechansm can be separated nto three steps. Frstly, we categorze the buyers nto dfferent groups based on each buyer s bd and ts channel request. The buyer group formaton algorthm should maxmze each group s total bd and acheve the aucton effcency smultaneously. After that, we construct the bd set on the bass of each partcpant s bd and decde the transacton set. Fnally, we determne the clearng prce for the wnnng buyers and sellers accordng to the maret clearng demand. A. Buyer Group Constructon (1) Conflctng graph constructon The buyer group s formed by a set of non-conflctng buyers. Namely buyers wthn each other s communcaton range cannot use the same spectrum due to nterference. Specally, we model the buyers nterference relatonshp n the networ as an undrected conflct graph, where each buyer n the networ s represented by a node n the graph, represents the node set n the graph, represents the edge set n the graph. There s an edge between nodes and, f two nodes and nterfere wth each other. The node weght s defned as the buyer s per-channel bd prce. Fg.2 shows an example of conflct graph. Fg. 2. Conflct graph (2) Constructon of non-conflctng buyer groups The prmary goal of the tradtonal spectrum allocaton methods s to acheve maxmum spectrum reuse, that s to say, let more user share the same channel together. The constructon of buyers maxmum ndependent set s deemed as an effectve method to acheve the goal. Nevertheless, the role of auctoneer s to devse a far and reasonable mechansm for each agent n a robust double aucton mechansm. Consder each buyer as a selfsh and ratonal agent, who only wants to wn the aucton wth lower payment, but none of them care about the spectrum reuse ssue. However, the bddng success rate for each buyer wll be ncreased along wth the total bd of buyer group that t joned. We wll prove n Theorem 6 that there s no relatonshp between charge and the total bd of group for the wnnng buyers n the aucton. Thus, all buyers are wllng to jon nto a group wth hgher bd so as to mprove ts compettveness n the aucton process. At the same tme, the group wth the hghest bd could be regarded as the max-weght ndependent set. As analyzed above, fndng maxmum non-conflctng buyer bd group problem can be consdered as choosng the

4 max-weght ndependent set problem. The max-weght ndependent set (MWIS) problem s the followng: gven a graph wth postve weghts on the nodes, fnd the heavest set of mutually nonadjacent nodes. MWIS s a well-studed combnatoral optmzaton problem that naturally arses n many applcatons. It s nown to be NP-hard, and hard to approxmate []. Due to the real-tme problem s not an essental ssue n our spectrum aucton framewor, enumeraton method s adopted to solve ths problem. Let represents an ndependent set n. We use to denote the mnmum weght of vertces n the ndependent set and the weght of the ndependent set s defned as multpled by the number of vertces n the ndependent set. Algorthm 1 shows the detaled procedure of non-conflctng buyer group constructon (NCBC). Let denotes the max-weght ndependent set ncludng buyer, and s the bd of. In the th teraton of NCBC, we frst traverse all the remanng buyers n the buyer group, and compute the bd of each ndependent set formed by the remanng buyers n buyer group. And then, the ndependent set wth maxmum wll be chosen and nomnated as the wnner n ths teraton. As shown n the conflctng graph Fg.3, there are 8 canddate buyer groups n the frst teraton, {1,4}, {2,4}, {3,5,8}, {4,2}, {5,2,4}, {6,3,8}, {7,2,5}, {8,2,5}. The buyer group {2,4} possess the hghest group bd and wll be chosen as the wnner n Iteraton 1. Let and represent buyer s number of channel demand and mnmum buyer s channel demand n respectvely. Durng the th teraton, we use to subtract for each buyer n MG, f the updated value of equal to 0, then we wll delete buyer from group I. Therefore, we wll delete buyer 4 n the end of teraton 1, and get the conflctng graph shown n Fg.3 (b) at the begnnng of the next teraton. The teraton lasts untl group I s empty. Fg. 3. Buyer groups formaton procedure Algorthm1. Non-Conflctng Buyer Group Constructon 1. Max _ bd 0; 2. for 1 to remanng buyers n group I do 3. Temp _ MG Max _ weght _ IS( ); 4. Temp _ Bd Bd ( Max _ weght _ IS( )); 5. f ( Temp _ Bd Max _ bd) 6. Max _ bd Temp _ Bd ; 7. MG Temp _ MG; 8. end f 9. end for 10. foreach buyer n MG 11. ( MG ) 12. end for 13. delete the buyer s ( Satsfed ( MG ) 0) n set MG from group I 14. ncreased by 1, goto 1 whle group I nonempty Algorthm 1 Theorem 1: Soluton of buyer group formaton game can be characterzed by Nash Equlbrum (NE). Snce a buyer group wth hgher bd has hgher wnnng probablty, buyers always want to jon nto a group wth hgher bd. Assume that there exsts a buyer n buyer group, but jonng nto s not the domnant strategy for buyer. Accordng to the buyer group constructon process, f there exsts a buyer group ncludng whch maes >, then must be generated before. There are two possble cases: 1) wll be decreased when buyer jons. Although buyer could obtan the maxmum value when

5 jonng, but the proft of others n wll be decreased. Thus wll not admt buyer s entry. 2) Some buyers n have joned nto other groups whose bd prce s hgher than. In ths case, group does not exst and buyer also cannot jon nto. Thus, jonng nto s the domnant strategy for buyer, whch contradcts the assumpton. Therefore, NCBC for buyer group formaton game can be characterzed by Nash Equlbrum. Corollary 1: NCBC s far. (3) Decson of buyer groups propertes We use,,, to denote the formed non-conflctng buyer groups. A buyer group can be vewed as a super buyer. There are two parameters wll be used to descrbe the super buyer s characterstcs. One s super buyer s bd for unt commodty,, whle the ; The other s one super buyer s channel demand, and the. After 5 rounds all the buyers n Fg. 2 are grouped nto fve dfferent super buyers, Table 2 shows the buyer group formaton results by executng Algorthm 1. Table 2. Buyer groups formaton result (Alg. 1) Round Buyer Group Channel Group Bd Demand 1 {2,4} {2,5,7} {1,8} {3,5} {6} 2 3 B. Decson of bd set and transacton set Wthout loss of generalty, we requre that all partcpants prce-per-unt bds are arranged n descendng order n the bd set. We use postve quanttes correspond to demands of buyers and negatve quanttes correspond to offers to sell for each seller. Here transacton set means the remanng super buyers and sellers at the end of a tme round. Consderng a scenaro wth M sellng offers and N buyng offers after bd set has been establshed. The (M+1) st -prce means the (M+1) st hghest offer among all (M+N) bds. We use represents the poston of a bd n the bd set. The (M+1) st -prcng rule s gven under the followng two condtons: Condton 1: The (M+1) st -prce and M th -prce belong to dfferent partcpants n the bd set. If the (M+1) st -prce comes from a seller, the transacton set constructon rule can be depcted as followng (6) If the (M+1) st -prce comes from a buyer, the transacton set constructon rule wll be (7) Condton 2: The (M+1) st -prce and M th -prce come from the same partcpant n bd set. If the (M+1) st -prce comes from a seller, the transacton set s the same as that n Condton 1. However, f the (M+1) st -prce comes from a buyer, the transacton set wll be dfferent from that n Condton 1. Partcularly, after ntal transacton set s constructed based on rule n Condton 1, we delete the resdual buyer s bd that equals to the (M+1) st -prce n the bd set, and also delete the same number of seller s bd n descendng order. For example, f there are four sellers are wllng to sell 8 channels, of whch seller1 sells only one channel at prce 13, seller2 sells 3 channels at prce 10, seller3 sells 2 channels at prce 7 and seller4 sells 1 channels at prce 6. After the (M+1) st -rule has been executed, bd set and transacton set can be depcted as followng. bd set: 13,(12,12),(10,10,10),9,(8,8),(7,7), { { { { 6 {, 4,(2, { 2, 2) transacton set: (12,12),9,(7,7), { { 6{ C. Choosng clearng prce and maret clearng Most of the well-nown double aucton mechansms are categorzed nto two classes: Dscrmnatory Aucton (DA) and Unform Prce Aucton (UPA). The man dfferences are the clearng prce decson mechansm between these two classes. The payments and charges for all the wnnng bdders n DA wll be ts actual bddng prce, and the hghest rejected prce wll be nomnated as clearng prce for wnnng bdders n UPA. The McAfee aucton mechansm adopted n TRUST and the Wurman s mult-unt double aucton mechansm n our wor all belong to unform prce aucton. However, dfferent from the tradtonal aucton methods, buyer partcpates nto the spectrum aucton n a grouped form. Thus, the strategy-proof property wll not hold n our SPRITE framewor by applyng clearng prce mechansms n DA or UPA drectly. As we have mentoned above, f bddng truthful s the domnant strategy for each agent, we can say the aucton s

6 strategy-proof. Let parameter represents the proft of buyer by bddng. If buyer lose n the aucton, then =0; If buyer wns the aucton by bddng a value greater than, the <0, and >0 when buyer wns the aucton by bddng a value less than prce mechansms.. We now prove the reason of untruthfulness by ntroducng exstng clearng Theorem 2:SPRITE s not strategy-proof f t chooses the clearng prce mechansm n UPA drectly. For spectrum aucton, we could only charge each buyer group the same payment by usng the clearng prce mechansm n UPA. Suppose buyer formed nto Group 1 when t bds, and grouped nto Group 2 when t bds untruthfully. If buyer bds f Vf, we can easy get that Bd(Group 1 ) Bd(Group 2 ),thus f Vf However, the charge for each buyer only related to the number of buyers n the group, thus the actual charge for each buyer may not be affected when t bds f Vf. The expected payoff can be regarded as E( PVf ) E( P ) and we can get: f U U ( Vf P ) ( Vf P ) f Vf f f Vf Vf ( Vf E( P )) ( Vf E( P )) f f Vf Vf ( Vf E( P )) ( Vf E( P )) f f Vf f ( )*( Vf E( P )) f Vf f ff,. All the buyers are satsfed wth, thus we cannot assure when. Therefore, bddng truthful s not the domnate strategy for buyers, and the proposed SPRITE s not strategy-proof. Smlarly, f we charge each buyer the bddng prce of ts own ( ), then the payoff for the truthful buyer always goes 0 ( ). When the buyer bds, the payoff can be depcted as U ( Vf P ) ( Vf f ) 0 f f f f Therefore, we can get and clearng prce mechansm n DA could not be used n the proposed SPRITE spectrum aucton drectly. Our SPRITE framewor proposed a novel clearng prce mechansm whch combnes the characterstcs n UPA and DA. For buyers, all the buyer groups are deemed as super buyers, and adopt clearng prce mechansm n DA. All the wnnng buyer groups wll be charged ts bddng prce: PMG Bd( MG ) (8) For the sngle wnnng buyer, ts actual charge can be descrpted as followng: Pf MnBd ( MG ) MG (9) We choose the clearng prce mechansm n UPA for the wnnng sellers n SPRITE, and the actual payment s gven as: Pg mn( MG ), j MG Transacton set (10) The clearng prce strategy n SPRITE could acheve the strategy-proof, ex-post budget-balanced and maret clearng. We wll gve the proof detals n the next paragraph. Ⅳ Proofs and correctness of our algorthm We now analyze the propertes of our proposed strategy-proof mult-unt double spectrum aucton mechansm n terms of strategy-proof, ex-post ndvdually ratonal, ex-post budget-balanced and maret clearng. Observaton 1: For each buyer, f buyer wns the aucton by bddng, then t also wns by bddng ; For each seller, f seller wns the aucton by bddng, then t also wns by bddng. Observaton 2: The observaton 1 shows the monotoncty of wnnng rules. It mples there exstng crtcal value for wnnng sellers. For each seller, f seller wns the aucton by bddng or, and these two bddngs are all less than seller group s crtcal value, the payment to seller s the same for both. A reasonable aucton framewor should follow the strategy-proof gudelne for each bdder, whch requres each bdder could get the maxmum payoff when they bd truthfully n the aucton. Based on the combnaton of agent s actons, there are

7 four possble outcomes lsts n Table 3. Table 3 Aucton results based on dfferent confguratons Case Agent les lose lose wn wn Agent s truthful lose wn lose wn Lemma 1:Buyers can t beneft from bddng n case 3. There are two outcomes may happen when the buyer rased ts bd. 1)The buyer group formaton process s not affected when the buyer bd and respectvely. In other words, buyer stll be formed nto the same super buyer group when buyer rased ts bd to. Because buyer changes the aucton results by bddng a fae prce, t demonstrates that the bd changes ths group s bddng prce. Namely, the strategy-proof bddng prce must be the lowest bd n ts group. When buyer wns the aucton, the actual prce charges buyer should satsfed nterval. Thus, the <0 when buyer bd. Because the utlty for buyer s when t bds truthfully, < =0. So buyer can t beneft from bddng n ths case. 2) Buyer formed nto when t bds, and grouped nto when t bds untruthfully. On the bass of above analyss, holds. That s to say, buyer cannot be formed nto when t bds. It shows that f buyer formed nto, wll be decreased. Namely,. Therefore, f buyer wns the aucton and be formed nto by bddng, t should pay more than for the desgnated channel. Then, and bddng truthful s also the domnant strategy for buyer. Lemma 2: When buyer wns n the aucton, the wll not be decreased wth the where. We can see from the formula (9), f, the actual charge for buyer s determned by the. At the same tme, the total bd of buyer group s determned by and number of buyers n. The lowest bd of a hgher bd buyer group wth large amount of buyers may smaller than the lowest bd of a lower bd buyer group wth fewer buyers. Thus, the relatonshp between and shows non-monotoncty. That s to say, jonng nto a buyer group wth hgher bddng prce may not decrease ts own payoff when the buyer wns n the aucton. Lemma 3: Buyers can t beneft from bddng n case 4. No matter how buyer bd, t always wns the aucton n Case 4. So the calculaton formula can be depcted as: where stands for the utlty for buyer who bds truthfully. Compared to bddng truthfully, and denote the decreasng probablty and reduced cost of when buyer bds untruthfully. And and denote the ncreasng probablty and addtonal cost respectvely. Based on Lemma 2, we can consder that, formula (11) can be rewrtten as: It s clearly shows that, thus bddng truthful s the domnant strategy. Lemma 4: Buyers can t beneft from bddng n case 2 and case 4. When,. The abbrevaton descrpton of can be wrtten as. However, the ncrease of the buyer s bd may result n the appearance of Case2. Sum up Case 2 and Case 4, when, the proft for buyer can be rewrtten as: Owng to, bddng truthful s stll the domnant strategy for buyers. Theorem 3: SPRITE mechansm s truthful for buyers. The demonstrate process s categorzed nto buyers respectvely on the bass of case 1~4. Case 1: No matter how buyer bd, t always lose n the aucton n Case 1. We can conclude that the utlty for buyer always goes zero. Case 2: On the bass of observaton 1, ths case happens only f the buyer decreases ts bd n aucton process, namely. Buyer s utlty s zero f t les n ths case, and the truthful acton maes ts utlty no less than zero. (13) (11) (12)

8 Case 3: Ths condton happens only f. Accordng to lemma 1, buyers can t beneft from bddng n case 3. Case 4: No matter how buyer bd, t always wns the aucton n Case 4. If the buyer bd, bddng truthful s the domnant strategy accordng to Lemma 3. Smlarly, bddng truthfully s stll the domnate strategy for buyers when. Theorem 4: SPRITE mechansm s truthful for sellers. Case 1 and Case 2: The proof procedure s the same wth buyer case. Case 3: Ths condton happens only f on the bass of observaton 1. We use the P g j stands for the payment to the aucton wnners. Owng to seller loses n ths case when t bds truthfully, we can get the concluson that P g j <. If the seller wns the aucton by bddng a lower prce, the payment to seller must smaller than P g j. Thus, the payment to seller also smaller than, and the utlty for seller s negatve when t bds untruthfully. Case 4: SPRITE mechansm pays each seller the mnmum buyer group s bd n the transacton set. It s the crtcal value of seller group we mentoned n observaton 2. Accordng to observaton 2, no matter the bddng prce s or, the payment for seller s all the same f t s wn n the aucton. That s to say, utlty wll not change n both condtons. Theorem 5: SPRITE mechansm s strategy-proof. As we have mentoned above, f no buyer or seller can mprove ts own utlty by bddng untruthfully no matter how other agents bd, we can say the aucton s strategy-proof. Thus, SPRITE mechansm s strategy-proof accordng to Theorem 3 and Theorem 4. Theorem 6: SPRITE mechansm s ex-post ndvdually ratonal. For each seller, the proposed SPRITE mechansm pays each seller the mnmum buyer group s bd n the transacton set. Thus, ( MG Transacton set ) no less than anyone else wnnng seller s actual bd. We have SPRITE mechansm s ex-post ndvdually ratonal for seller. For each buyer, the actual prce charges each wnnng buyer s, where. represents the lowest bd n each wnnng buyer group. Therefore, s no more than each wnnng buyer s actual bd. SPRITE mechansm s also ex-post ndvdually ratonal for buyer. Theorem 7: SPRITE mechansm s ex-post budget-balanced. In the desgned transacton set, all partcpants bds are sorted n descendng order. Based on the defnton of clearng prce decson n SPRITE, all the wnnng buyers bd are no less than the bds of wnnng sellers. From Proposton 3, t s straghtforward to show that. Therefore, SPRITE mechansm s ex-post budget-balanced. Theorem 8: SPRITE mechansm acheves the maret clearng. Let S denotes the wnnng sellers bd set and B stands for the wnnng buyers bd set. 1) The (M+1) st -prce comes from a seller: On the bass of SPRITE transacton rules, Set S can be depcted as: and Then, we have the equaton, where M represents the total quantty provded by sellers. From the rules, we can also get the descrpton of Set B:. Because, and. Thus, we can get the concluson that. It s easy to show that, combne two equatons, we have. In other words, quantty suppled by wnnng sellers equals the quantty demanded by wnnng buyers at the end of aucton. 2) The (M+1) st -prce comes from a buyer: Based on transacton rules, the (M+1) st -prce buyer cannot get nto transacton set. Thus, the wnnng buyer bd set B s the same as condton 1. At the same tme, the wnnng seller bd set s. Because the (M+1) st -prce comes from a buyer, the S can also depcted as. S s the same as condton 1. Therefore, SPRITE could acheve maret clearng. Ⅴ Smulaton Study In ths secton, we conduct smulaton study to evaluate the performance of SPRITE under the metrcs of spectrum utlzaton, number of transacted channels, per-channel utlzaton, average success rat, degradaton, group ran score, and further compare SPRITE wth the exstng mechansm. All the smulaton results are averaged over 1000 runs. In our smulatons, all the buyers are deployed n a square 100*100 area under ether random topology or clustered topology (hot spot), and any two buyers wthn 20 unt dstance wll conflct wth each other, thus the conflctng buyers cannot bd same channels. In the clustered topology, we set 50% of the whole buyers are dstrbuted n a small area. In our smulaton, all the bdders wll bd at ther true valuaton, and the bds are unformly dstrbuted n an nterval. To compare wth TRUST [7], we mplement two versons of TRUST: TRUST-1 (sngle round aucton) and TRUST-2 (mult-round aucton).

9 Fg. 4. Degradaton (a) Group ranngs n random topology (b)group ranngs n clustered topology Fg. 5. Group ranngs n dfferent topologes We frst consder the mpact of economcs factors on spectrum utlzaton. Tradtonal channel allocaton algorthms are pursung the maxmzaton of the spectrum utlzaton, whle the spectrum aucton mechansms stll have to tae bdder s purchasng power nto consderaton. Therefore, compared wth the PA (Pure Allocaton), varous aucton mechansms wll experence dfferent degradaton. We wll use ths parameter to reflect the mpact of economcs factors. In our smulaton, we choose the greedy allocaton way to represent pure allocaton method. Assumng that there are 30 buyers and 5 sellers deployed n both unform and clustered topologes. We set each buyer requres 1~3 channels n the aucton process, and each seller provdes 1~3 channels at the same tme. The bddng prces for buyers are unformly dstrbuted n nterval [10,35], and prces for sellng channels are unformly dstrbuted n nterval [35,60]. Fg. 4 plots the degradaton performance of SPRITE, TRUST-1 and TRUST-2 under random and clustered topologes respectvely. In Fg. 2, SPRITE suffers less degradaton than TRUST-1 and TRUST-2. Ths s because SPRITE constructs the buyer groups wth each buyer s bd, thus the groups wth hgher bddng prce are more lely to be generated. Therefore, SPRITE could ncrease the buyer groups opportunty to successfully bd the channels. Fg. 5(a)-(b) shows the trend of group ran score correspondng to buyer s bd. We can see that n SPRITE the wnnng probablty for a sngle buyer s correspondng to ts bd, whle n TRUST there s no such relatonshp because chooses rand dvson method n the buyer group constructon process and thus the bddng prce for a buyer does not have postve connecton wth the ran of group t belongs to. Ths ndcates that SPRITE provdes a more reasonable soluton n the realstc envronment. In the clustered topology, there are more buyer groups wll be formed compared to unform topology, thus the ran score curves are hgher than unform topology. Fg. 6 shows that SPRITE better meets the requrements of farness prncple than TRUST, whch demonstrates our theoretcal analyss. Fg. 7 shows that the average success rate for buyers ncreases from 28% up to 43% along the ncrease of maxmum bddng prce. In the clustered topology, the average success rate for sellers reaches 86% when the buyers maxmum bd equal to sellers hghest offer. In random topology, the average success rate for seller reaches 93%. Fg. 8 shows that the hgher the maxmum bddng prce of buyers, the better the buyers purchasng power, the larger the number of transacted channels, and the hgher the average success rate and spectrum utlzaton.

10 Fg. 6. Success rate of buyers Fg. 7. Average success rate Fg. 8. Spectrum utlzaton and # of transacted channels We wll concentrate on the aucton effcency between SPRITE, TRUST and McAfee n Fg. 9. The aucton effcency s defned as number of transacted channels dvded by the total channels provded by sellers. McAfee s regarded as the most classc double aucton mechansm whch does not consder spectrum reuse. In order to encourage channel tradng, we change the sellers bddng prce nterval at [20, 45] for facltate comparson. We can learn from the comparng results that the aucton effcency of SPRITE and TRUST sgnfcantly outperform than and McAfee. As we analyze before, buyer group formaton prompts buyer s purchasng power, thus the number of transacted channels for McAfee obvously less than SPRITE and TRUST. Fg. 9 demonstrates that the buyer group formaton process could effectvely mprove the purchasng power for each buyer so as to let each buyer wllng to jon nto group. At the same tme, SPRITE performs better than TRUST because of the SPRITE maxmzes buyer group s purchasng power. The aucton effcency converges to 98% when the number of buyers reaches to 30. We can get the concluson from Fg. 10 and Fg. 11 that the seller s utlty n SPRITE obvously better than TRUST and McAfee. The maxmzaton of buyer s purchasng power not only mproves the aucton effcency, but also promotes the mprovement of the actual payment to the seller. In addton, Fg. 11 shows the buyers total purchasng power and total utlty of sellers wll be ncreased wth ncreasng of number of buyers. Fg. 9. Aucton effcency Ⅵ Concluson Fg. 10. Utlty of each seller Fg. 11. Total utlty of sellers In ths paper, we propose SPRITE mechansm, a strategy-proof mult-unt double aucton framewor for spectrum allocaton n wreless networs. To our best nowledge, SPRITE s the frst wor that enables mult-unt commodtes tradng n spectrum allocaton n wreless networs. It not only assures strategy-proof but also maret clearng property. More mportantly, buyers wnnng probablty s n a proper economc way, whch s sgnfcantly dfferent from exstng double aucton approaches, e.g., TRUST. Besdes, the relatonshp among buyers could consttute a Nash Equlbrum. The correctness, effectveness and economc propertes of SPRITE are well studed n our theoretcal analyss. The smulaton study also show that SPRITE could acheve better performance under varous metrcs. The future wor ncludes the extenson of the concrete effectveness analyss of spectrum double aucton and the study of tradeoff between economc mpacts and effcency degradaton.

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