Auction Mechanisms Toward Efficient Resource Sharing for Cloudlets in Mobile Cloud Computing

Transcription

1 1 Auction Mechanim Toward Efficient Reource Sharing for Cloudlet in Mobile Cloud Computing A-Long Jin, Wei Song, Ping Wang, Duit Niyato, and Peijian Ju Abtract Mobile cloud computing offer an appealing paradigm to relieve the preure of oaring data demand and augment energy efficiency for future green network. Cloudlet can provide available reource to nearby mobile device with lower acce overhead and energy conumption. To timulate ervice proviioning by cloudlet and improve reource utilization, a feaible and efficient incentive mechanim i required to charge mobile uer and reward cloudlet. Although auction ha been conidered a a promiing form for incentive, it i challenging to deign an auction mechanim that hold certain deirable propertie for the cloudlet cenario. Truthfulne and ytem efficiency are two crucial propertie in addition to computational efficiency, individual rationality and budget balance. In thi paper, we firt propoe a feaible and truthful incentive mechanim (TIM), to coordinate the reource auction between mobile device a ervice uer (buyer) and cloudlet a ervice provider (eller). Further, TIM i extended to a more efficient deign of auction (EDA). TIM guarantee trong truthfulne for both buyer and eller, while EDA achieve a fairly high ytem efficiency but only atifie trong truthfulne for eller. We alo how the difficultie for the buyer to manipulate the reource auction in EDA and the high expected utility with truthful bidding. Index Term Mobile cloud computing, cloudlet, double auction, incentive deign, truthfulne, efficiency. 1 INTRODUCTION In recent year, mobile network operator are facing unremitting demand for high date rate and ever-emerging new application. While thi growth lead to increaing revenue to the global mobile indutry, there are riing concern on environmental impact and high capital/operating expenditure. Meanwhile, cloud computing i achieving great ucce in empowering end uer with rich experience by leveraging reource virtualization and haring. The merging of cloud computing into the mobile domain create the appealing paradigm of mobile cloud computing (MCC). MCC offer a promiing olution not only to extend the limited capabilitie of mobile device, but alo to reduce energy conumption if deigned in a green manner [1]. A illutrated in Fig.1, the traditional centralized cloud [2] hot hared reource in remote data center and act a an agent between the original content provider and mobile device. To acce reource/ervice at the data center, mobile device often need to go through the backbone network. The long latency and high energy conumption can hamper the capability of the cloud to upport interactive application demanded by uer. In contrat, the lightweight cloudlet [3] can balance the cale of hared reource and the acce overhead. A cloudlet i a truted, reource-rich, Internetconnected computer or a cluter of computer, which can be utilized by mobile device via a high-peed wirele local area A. Jin, W. Song (correponding author), and P. Ju are with the Faculty of Computer Science, Univerity of New Brunwick, Fredericton, Canada ( along.jin@unb.ca, wong@unb.ca, i1689@unb.ca). Thi reearch wa upported in part by Natural Science and Engineering Reearch Council (NSERC) of Canada. P. Wang and D. Niyato are with School of Computer Engineering, Nanyang Technological Univerity, Singapore ( wangping@ntu.edu.g, dniyato@ntu.edu.g). BS AP Centralized cloud ISP backbone Mobile device Cloudlet Fig. 1: Typical MCC architecture: Centralized cloud and cloudlet. network (WLAN). In thi architecture, mobile device function a the client and cloudlet a the ervice provider. Seamle interaction between them can be more eaily achieved in the cloudlet phyical proximity with the low one-hop communication latency. Due to the patial ditribution of cloudlet and their ditinct capabilitie or hoted reource, mobile device have different preference over the cloudlet. On the other hand, the cloudlet need to be incentivized to hare their reource, e.g., through gaining monetary value paid by the mobile device for uing the ervice. A een, there exit a trade between the mobile device requeting the ervice and the cloudlet providing uch ervice. Auction i a popular trading form that can efficiently ditribute reource of eller to buyer in a market at competitive price. Auction theory [4] i a well-reearched field in economic and ha been applied to other domain, e.g., radio reource management for wirele ytem [5]. If the

2 2 reource trading ytem with buyer and eller i viewed a an ecoytem, the auction mechanim need to appropriately addre the conflicting interet of buyer (e.g., minimizing charge for highet valuation) and eller (e.g., maximizing reward for leat cot), and internal competition among buyer/eller. The mechanim hould fairly allocate the trading reource, determine the pricing, and optimize the payoff that each buyer/eller can achieve with the preence of competition. A uch, the buyer and eller can be incentivized to participate in the auction and the ecoytem can reach an equilibrium. Such an auction mechanim i expected to hold certain deirable propertie, uch a individual rationality, budget balance, truthfulne or incentive compatibility, ytem efficiency, and computational efficiency. Individual rationality enure that a buyer i never charged more than it bid, while a eller i paid not le than it ak. Budget balance require that the auctioneer, which act a an intermediate agent between buyer and eller, hot and run the auction without a deficit. Truthfulne i eential to reit market manipulation and enure auction fairne. An auction mechanim i truthful (alo known a incentive compatible) if revealing the private valuation truthfully i alway the weakly dominant trategy for each participant to receive an optimal utility, irrepective of what trategie other participant are taking. There are variou definition of efficiency in economic from different perpective, among which allocative efficiency i a main type that aim to maximize ocial welfare, i.e., the um of valuation of the buyer who receive their deired commoditie. In thi paper, we are particularly intereted in double auction, in which buyer and eller ubmit to an auctioneer their bid and ak, repectively. For uch bilateral trade, a mechanim i efficient if whenever a buyer bid i greater than the eller ak, the correponding commodity i allocated to the buyer [4]. In addition to the above economic propertie, computational efficiency enure that the auction outcome i tractable with a polynomial time complexity, which i important to enable feaible implementation. Many exiting auction mechanim cannot be directly applied to the cloudlet cenario without jeopardizing certain deired propertie. For example, the multi-round auction tudied in [6] [8] are not uitable due to the high communication and computation overhead. For double auction, no mechanim can be efficient, truthful, individually rational, and at the ame time balance the budget [4]. Conidering only homogeneou commoditie, McAfee double auction [9] can achieve three deirable propertie, i.e., individual rationality, budget balance, and truthfulne. The Truthful Auction Scheme for Cooperative communication (TASC) propoed in [1] extend McAfee double auction by taking into account heterogeneou trading commoditie, i.e., ervice of relay node. Although TASC addree a imilar cenario, it i found that the efficiency of TASC can be further improved. In [11], the Vickreybaed double auction exploit the participant uncertainty of bid/ak of other participant to reduce manipulation and boot allocative efficiency, while enforcing budget balance and individual rationality. Thi mechanim can be fairly efficient and fairly truthful, but not computationally efficient. A een, it i impoible to imultaneouly atify all the aforementioned deirable propertie in an auction mechanim. In many cae, certain propertie (e.g., computational efficiency or truthfulne) have to be relaxed to trade for other propertie (e.g., ytem efficiency). A a reult, thi leave room for improving the relaxed propertie. In thi paper, we focu on deigning feaible and efficient double auction mechanim to timulate cloudlet to erve nearby mobile device. Here, a feaible auction mechanim need to atify computational efficiency, individual rationality, budget balance, and truthfulne. Conequently, we have to relax the requirement for ytem efficiency, which i ub-optimal but ufficiently cloe to the optimum. Firt, we propoe a truthful incentive mechanim (TIM), which i alo computationally efficient, individually rational and budget-balanced. To further improve the ytem efficiency, we extend TIM to a more efficient deign of auction (EDA), by involving randomne and bidding uncertainty. EDA maintain all deired propertie of TIM except for lightly relaxed truthfulne. EDA guarantee trutfulne of eller but it i not trongly truthful for buyer. We provide rigorou analyi regarding thee propertie of both mechanim. Numerical reult confirm the analyi and demontrate their deirable propertie, epecially the high ytem efficiency achieved by EDA. It i alo hown that EDA i truthful in expectation for buyer. The remainder of thi paper i tructured a follow. Firt, we briefly review related auction mechanim in Section 2. Then, Section 3 provide the problem formulation a a double auction and an example demontrating the deign rationale. After that, we give the deign detail of the propoed TIM and EDA mechanim in Section 4 and Section 5, repectively. Numerical reult are preented in Section 6, followed by concluion in Section 7. 2 RELATED WORK A a promiing paradigm, mobile cloud computing ha attracted coniderable reearch attention and effort. There have been a number of tudie addreing variou apect of MCC, uch a virtual machine migration [12], ervice enhancement with MCC [13], and emerging application with MCC [14,15]. However, the reearch on incentive deign for MCC i limited, although there have been many incentive mechanim propoed for wirele cooperative communication [1], radio reource allocation [5], device-to-device communication [16], martphone ening [17], and mart grid [18]. In [1], Yang et al. propoe a truthful double auction mechanim, TASC, to timulate relay node to forward packet for other wirele node. TASC ha two tage, namely, Aignment and Winner-Determination & Pricing. In the aignment tage, the auctioneer applie an aignment algorithm to determine the winning buyer candidate (ource node), the winning eller candidate (relay node), and the mapping between thee buyer and eller. Depending on the deign objective, the auctioneer can chooe a different aignment algorithm. For example, the optimal relay aignment algorithm [19] can maximize the minimum valuation among all buyer; the maximum weighted matching algorithm [2] can maximize the total valuation; and the maximum matching algorithm can

3 3 maximize the number of ucceful trade (final matching). In the winner-determination & pricing tage, TASC tightly integrate the winner determination and the pricing operation. Baed on the return of the aignment tage, the auctioneer applie McAfee double auction [9] to determine the winning buyer, the winning eller, and the correponding clearing price and payment. TASC overcome the limitation of the original McAfee double auction [9] that conider only homogeneou commoditie, i.e., buyer have no preference over auction item. When TASC i ued in the cloudlet cenario, it can atify individual rationality, budget balance, and truthfulne for the eller. However, we ue an example in Section 3.4 to illutrate that the ytem efficiency of TASC can be further improved. Another cloely related work in [21] addree pecifically reource haring with MCC. In [21], cloud reource are categorized into everal group (e.g., proceing, torage, and communication). The reource allocation problem i formulated a a combinatorial auction with ubtitutable and complementary commoditie. Thi combinatorial auction mechanim i not applicable for the cloudlet architecture ince it key problem i the allocation of M reource of G group in one MCC ervice provider to N uer. In contrat, our ytem model with cloudlet focue on ditinct valuation of cloudlet to mobile uer. Different from [21], we further conider computational efficiency and budget balance, which are alo critical to an auction mechanim. It i noted that many exiting incentive mechanim emphaize the property of truthfulne, which prevent market manipulation and eliminate the trategic overhead of the participant. However, there are few work on double auction deign to improve ytem efficiency. In [11], Parke et al. propoe a Vickrey-baed double auction, which can achieve individual rationality and budget balance. The aignment between buyer and eller i determined to maximize ocial welfare (ytem efficiency), while the participant utility equal the incremental contribution to the overall ytem, i.e., the difference between the ocial welfare with and without the participation. However, the Vickrey-baed double auction in [11] i only fairly efficient and fairly truthful. Moreover, thi mechanim cannot atify computational efficiency. 3 PROBLEM FORMULATION 3.1 Sytem Model A depicted in Fig. 1, the cloudlet offer reource pool cloer to the network edge. Thu, the cloe proximity of cloudlet can be exploited to reduce the acce overhead of mobile uer in term of energy conumption and communication latency. The reource at the cloudlet may be valued differently by the mobile uer depending on variou factor [22], uch a computation capability, communication cot, and wirele link performance (e.g., throughput, latency, and link variation). Such valuation of a mobile uer toward a cloudlet i alo aociated with the application demand. For intance, when a mobile uer offload a computation-intenive tak, it value high a cloudlet with rich computing reource of memory and CPU capacity. In contrat, a mobile uer with a real-time tak prefer a cloudlet with a hort communication latency, which require large network bandwidth and cloe patial location. A een, the valuation of a mobile uer toward a cloudlet actually take into account the ervice quality that the cloudlet can provide to the uer. On the other hand, the cloudlet can be paid for haring reource a compenation for it computation and communication cot. The cot cover the cloudlet expenditure on acquiring computation reource (e.g., computing equipment, energy, and torage), and leaing communication facility from ISP, mobile carrier, or network operator. Note that the cot implicitly incorporate the reource contraint of the cloudlet. Clearly, the trading between the cloudlet and the mobile uer hould meet certain requirement to benefit both the cloudlet and mobile device. The cloudlet need to be incentivized to provide the reource, and the demand of the mobile uer hould be atified. In particular, a cloudlet cannot be paid le than it cot, while the allocated reource of the cloudlet mut fulfill a mobile uer ervice requet. To maximize the reource utilization, the incentive mechanim hould properly aign the matching between the cloudlet reource and the mobile uer demand. 3.2 Auction Model To ait the matching between mobile uer and cloudlet, a truted third party i neceary to adminiter the trading between them, e.g., in the form of auction. In particular, a double auction fit well the bilateral nature of thi cenario. The truted third party in a double auction i the auctioneer between mobile uer (buyer) and cloudlet (eller). The auctioneer need to determine the matching of winning buyer and winning eller, the price it charge the buyer and the price it reward the eller. Apparently, the auctioneer hould at leat run the auction without a deficit and preferably benefit from the proce. Taking the dominant video treaming ervice a an example, we can identify ome network entitie that would upport the auctioneer functionality. In practice, content provider often pay content delivery network (CDN) operator to deliver their content, while a CDN operator in turn pay ISP, mobile carrier, or network operator for hoting it erver in their data center [23]. Beide, cloudlet can offer ervice cloer to network edge, thu complementing traditional CDN with lower delivery cot and better performance. Hence, the end uer need to pay the content provider for content conumption, while the content provider in turn pay CDN operator or cloudlet for content delivery. Becaue cloudlet when available can deliver content with lower cot and higher quality, the content provider itting between the end uer and cloudlet can have a trong motivation and uitable poition to run an auctioneer erver to reduce their expenditure and utain competitivene. Conidering m cloudlet (eller) that provide available reource for n mobile device (buyer), we formulate the underlying reource allocation problem a a ingle-round multiitem double auction imilar to [1]. Each buyer (rep. eller) can ubmit it bid (rep. ak) privately to the auctioneer o that everyone ha no knowledge of other.

4 4 Symbol b i j n m B S B w S w σ( ) D j i D i D j D A j A A j V j i V i C j P b i P j P b ij P ij U b i U j U b ij TABLE 1: Important notation. Definition Buyer (mobile device) Seller (cloudlet) Total number of buyer Total number of eller Set of buyer (mobile device) Set of eller (cloudlet) Set of winning buyer Set of winning eller Mapping function from the indice of S w to B w Bid of buyer b i on eller j Bid vector of buyer b i Bid vector of all buyer for eller j Bid matrix of all buyer Ak of eller j Ak vector of all eller Ak vector of all eller except j Valuation of buyer b i on ervice from eller j Valuation vector of buyer b i Cot of eller j for providing ervice Price charged to buyer b i Payment rewarded to eller j Price charged to buyer b i for ervice of eller j Payment rewarded to eller j with aigned buyer b i Utility of buyer b i Utility of eller j Utility of buyer b i with aigned eller j buyer et B w B, the winning eller et S w S, the mapping between B w and S w, i.e., σ : {j : j S w } {i : b i B w }, the price Pi b that the winning buyer b i B w i charged, and the payment Pj that the winning eller j S w i rewarded 1. To highlight the utilitie for the particular matching between b i and j, we alo ue Pij b and P ij in certain cae to denote the price and payment, repectively. In addition to the price and payment, the utilitie of the buyer and eller further depend on the valuation of the buyer toward the acquired ervice and the cot for providing uch ervice by the eller. Let V j i be the valuation to buyer b i for having the ervice from eller j, and C j be the cot to eller j for providing the ervice. The valuation vector of buyer b i i denoted by V i = (Vi 1,V i 2,...,V i m ). Given a buyer-eller mapping, i = σ(j), the utility of buyer b i and that of eller j are repectively defined a follow: { Ui b V j = i Pi b, if b i B w, otherwie { P Uj = j C j, if j S w, otherwie. Note that we alo ue Uij b and U ij when neceary to capture that the utilitie are with repect to the matching between buyer b i and eller j. A een, a utility Ui b > mean that the mobile uer b i a a buyer i allocated the reource of a cloudlet with a valuation greater than the charged price. Thu, Ui b indicate the level of atifaction of the mobile uer on the allocated reource. On the other hand, a utility Uj of the cloudlet a a eller repreent the urplu of the received payment over it cot. In other word, Uj characterize the profit of a cloudlet for haring it reource. Some important notation are ummarized in Table 1. U ij Utility of eller j with aigned buyer b i For each buyer b i B, B = {b 1,b 2,...,b n }, it bid vector i denoted by D i = (D 1 i,d2 i,...,dm i ), where Dj i i the bid for eller j S, S = { 1, 2,..., m }. The bid matrix coniting of the bid vector of all buyer i defined a D = (D 1 ;D 2 ;...;D n ). For all eller in S, the ak vector i denoted by A = (A 1,A 2,...,A m ), where A j i the ak of eller j S. A een, the ak of eller do not differentiate among buyer ince the eller only aim at collecting payment for uing their reource. On the other hand, the bid of buyer differ with repect to eller a mobile device have preference over the cloudlet. Although the bid of buyer are private information to the eller, the cloudlet do need to releae certain information, uch a their hoted reource, computation capabilitie, network bandwidth and communication latency, to nearby mobile uer, o that the uer can determine their valuation toward the cloudlet. The cloudlet, however, keep their ervice cot confidential. The auctioneer that hold the private information thu need to apply ecurity mechanim [24] to guarantee protection of privacy. Given B,S,D and A, the auctioneer decide the winning 3.3 Deirable Propertie and Deign Objective The auction model introduced in Section 3.2 i repreented by Ψ = (B,S,D,A). Accordingly, the auctioneer hould follow an auction mechanim to determine the et of winning buyer B w, the et of winning eller S w, the mapping σ between B w and S w, the et of clearing price Pw b charged to the winning buyer, and the et of clearing payment Pw rewarded to the winning eller. A feaible auction mechanim hould firt atify the following three deirable propertie. Computational Efficiency: The auction outcome, which include the winning et of buyer and eller, their mapping, and the clearing price and payment, i tractable with a polynomial time complexity. Individual Rationality: No winning buyer i charged more than it bid and no winning eller i rewarded le than it ak. With repect to the auction model Ψ, thi mean that for every winning matching between b i B w and j S w, we have P b i D j i and P j A j. 1. To ditinguih the price charged to buyer and the payment rewarded to eller, we ue b and in the normal form a the upercript, repectively. The ame naming convention i alo applied to the utilitie of buyer and eller.

5 5 Budget Balance: The total price that the auctioneer charge all winning buyer i not le than the total payment that the auctioneer reward all winning eller, o that there i no d- eficit for the auctioneer. That i, b i B w Pi b j S w Pj. Enforcing the hard contraint on the preceding three propertie, we further conider two other crucial propertie which can be trictly or fairly atified. Sytem Efficiency: Referring to allocative efficiency in e- conomic, we evaluate ytem efficiency by the number of ucceful trade (the number of final matching between winning buyer and winning eller) and the ocial welfare (the total valuation of winning buyer). There are many other definition of efficiency, uch a maximizing the revenue, i.e., the total payment to winning eller, minimizing the total charge to winning buyer, and even maximizing the profit of the auctioneer, i.e., the urplu between total charge to buyer and total reward to eller. In our double auction model, the number of final matching fit the bilateral trade nature, ince a ucceful trade mean that both the requirement of the eller (cloudlet) and the buyer (mobile uer) are atified. Maximizing the number of ucceful trade can involve a many cloudlet a poible in the trading, o that the reource utilization of cloudlet can be booted. Thi metric ha been conidered in many exiting work [1] to evaluate ytem efficiency. Compared with maximizing eller revenue or minimizing buyer charge, maximizing the number of ucceful trade i more realitic and deirable to maintain a table ytem from which both buyer and eller can benefit. If the ytem i deigned toward the interet of only one ide, e.g., maximum revenue to eller or minimum charge to buyer, the bia may eventually lead to opting-out of underprivileged participant which are not offered ufficient incentive. Truthfulne: An auction mechanim i truthful if playing (bidding or aking) truthfully i a weakly dominant trategy for each player (buyer or eller) who i only concerned with it own utility. In other word, no buyer can improve it utility by ubmitting a bid different from it true valuation, and no eller can improve it utility by ubmitting an ak different from it true cot. Specifically, it implie the following for our auction model: b i B, Ui b i maximized when the bid D i = V i ; and j S, Uj i maximized when the ak A j = C j. Thi truthfulne i very retrictive for a randomized mechanim due to the uncertainty of the actual outcome. A weaker truthful notion i truthfulne in expectation, which guarantee that a player expected utility for truthful bidding i at leat it expected utility for bidding any other value. With the property of truthfulne, an auction mechanim can be free from market manipulation, and the trategie of the participant can be implified accordingly. Since telling the truth produce the highet utility for each player, no rational buyer or eller would play untruthfully, even though the global utility might be improved. What i more appealing i that no player would deviate from the truth-telling trategy and the ytem thu reache an equilibrium. Each player only need to apply the imple truth-telling trategy and doe not have to TABLE 2: An illutrative example. (a) Bid matrix of 5 buyer b b b b b (b) Ak vector of 7 eller. Seller Ak b 1 b 4 b 5 b 2 b Fig. 2: Aignment reult with truthful bid and ak: A bipartite graph of matched winning candidate. adapt to other behavior. Though truthfulne preent all the aforementioned merit, unfortunately, there i no double auction mechanim that i truthful, efficient 2, individually rational, and at the ame time balance the budget [4]. In thi paper, we deign two feaible auction mechanim that are computationally efficient, individually rational, and budget-balanced. The firt mechanim TIM further atifie truthfulne in addition to thee three propertie. Extending TIM by involving randomne, the econd mechanim EDA can achieve a higher ytem efficiency than that of TIM but truthfulne i guaranteed in a weak ene, i.e., trongly truthful for eller but truthful in expectation for buyer a demontrated in the experiment reult. Nonethele, the difficultie in computing an effective lie, combined with the backfire rik of untruthful bid will convince the buyer not to bother to lie. 3.4 Deign Rationale A mentioned in Section 2, there i till room to improve the ytem efficiency of many exiting truthful auction mechanim. To illutrate thi, we take the TASC double auction in [1] a an example, and conider a bid matrix of 5 buyer with true valuation in Table 2(a) and the ak vector of 7 eller with true cot in Table 2(b). 2. Thi paper addree both ytem efficiency and computational efficiency. To avoid confuion, we ue the term efficient and efficiency to indicate pecifically the ytem efficiency, while we ue the term computationally efficient and computational efficiency to ditinguih from ytem efficiency.

6 6 Suppoe that the auctioneer ue maximum matching in the aignment tage to maximize the number of matching. According to the aignment algorithm of TASC, the winning buyer candidate, the winning eller candidate and the mapping between them are hown in Fig. 2. Then, following the TASC trategy for winner-determination & pricing, we have the et of winning buyer B w = {b 1,b 4 }, the et of winning eller S w = { 6, 2 }, the clearing price Pw b = {8}, and the clearing payment Pw = {6}. A een, the number of ucceful trade i only 2. On the other hand, if the true valuation of buyer and true cot of eller are known a priori, an optimal trategy can be applied to maximize the number of ucceful trade. The auctioneer can aign a eller j to a buyer b i a long a V j i C j. Then, the auctioneer can elect a value between C j and V j i a the price charged to buyer b i and the payment rewarded to eller j, i.e., V j i Pij b = P ij C j. In the given example, it can be eaily hown that thi optimal trategy can maximize the number of ucceful trade to 5. Thu, TASC only achieve 4% of the ytem efficiency of the optimal trategy. Inpired by thi obervation, we propoe two feaible auction mechanim. TIM in Section 4 give a truthful verion, while EDA in Section 5 can achieve a higher ytem efficiency. 4 TRUTHFUL INCENTIVE MECHANISM (TIM) In thi ection, we preent a truthful auction mechanim TIM for reource haring of cloudlet, which improve the ytem efficiency over TASC. We firt introduce the algorithm of TIM and then give an illutrative example. The propertie of TIM are alo analyzed in term of computational efficiency, individual rationality, budget balance and truthfulne. 4.1 Detail of TIM TIM in Algorithm 1 contain two ub-procedure pecified in Algorithm 2 and Algorithm 3, which correpond to two tage, Candidate-Determination & Pricing and Candidate- Elimination, repectively. Algorithm 1 TIM(B, S, D, A). Input: B, S, D, A Output: B w,s w,σ,p b w,p w 1: (B c,s c,ˆσ,p b c,p c) TIM-CD&P(B,S,D,A); 2: (B w,s w,σ,p b w,p w) TIM-CE(B c,s c,ˆσ,p b c,p c,d); 3: return (B w,s w,σ,p b w,p w); In the candidate-determination & pricing tage, the candidate determination and pricing operation are coupled together. Firt, the auctioneer determine the buyer candidate for each eller j with ak A j. Then, the price charged to the buyer candidate and the payment paid to the eller will be determined accordingly. Let A j denote the ak vector excluding the ak of j, and A o j be the median of A j. The median of a vector i the value of the middle element in a non-decreaing order of the element. If there i an even number of value, the mean of the two middle value i defined a the median. According to Algorithm 2, there are two cae in thi tage to determine the winning buyer candidate for eller j. Algorithm 2 TIM-CD&P(B, S, D, A). Input: B, S, D, A Output: B c,s c,ˆσ,p c,p b c 1: B c, S c, Pc b, Pc ; 2: for j S do 3: Find the median ak A o j of the ak vector A j; 4: B j = {b i : D j i Aj, bi B}; 5: if B j = 1 then 6: if D j i Ao j and A j A o j then 7: ˆσ(j) = i,b c B c {b i},s c S c { j}; 8: Pij b = Pj = A o j; 9: 1: Pc b Pc b {Pij},P b c Pc {Pj}; end if 11: ele if B j > 1 then 12: Sort B j to B j uch that D j i (1) D j i (2) ; 13: if D j i (1) A o j then 14: if the firt t (t 2) bid of B j are the ame then 15: Randomly elect b i from firt t buyer of B j ; 16: ele 17: Select firt b i of B j with the highet bid; 18: end if 19: ˆσ(j) = i,b c B c {b i},s c S c { j}; 2: Pij b = Pj = max{a o j,d j i (2) }; 21: P b c P b c {P b ij},p c P c {P j}; 22: end if 23: end if 24: end for 25: return (B c,s c,ˆσ,p b c,p c); Only one buyer b i with a bid not le than A j : If D j i Ao j and A j A o j, b i i added into the buyer candidate et B c with price A o j, and j i added into the eller candidate et S c with payment A o j ; otherwie, b i cannot win j. Two or more buyer with bid not le than A j : If the highet bid i le than A o j, no buyer win the ervice of j ; otherwie, the buyer with the highet bid (or a randomly elected buyer when there i a tie) i added into the buyer candidate et B c, and j i added into the eller candidate et S c. The price to the elected buyer and the payment to the correponding eller i the ame a the maximum of A o j and the econd highet bid. In the candidate-elimination tage, ince a buyer candidate in B c may win two or more eller in the eller candidate et S c, Algorithm 3 i run to chooe only one bet eller for uch a buyer. Specifically, the auctioneer elect the eller o that the correponding buyer achieve the highet utility. Likewie, when there i a tie in term of the achievable utilitie, one eller i randomly elected. At the end of the candidate elimination tage, every buyer b σ(j) B w ha a one-to-one mapping with only one winning eller j S w. 4.2 A Walk-Through Example To illutrate the procedure of TIM, we ue an example with the bid matrix in Table 2(a) and the ak vector in Table 2(b). 1) Candidate-Determination & Pricing: B c =,S c =,P b c =,P c = ; 1 : We have the ak vector A 1 = {2,5,6,4,1,7}, and the median ak A o 1 = 4.5. A D1 1 > A 1 and D 1 2 > A 1, we have B 1 = {b 1,b 2 }. Since B 1 > 1 and D 1 1 > A o 1 > D 1 2, b 1 i elected a the winning buyer candidate for 1 with

7 7 Algorithm 3 TIM-CE(B c,s c,ˆσ,p b c,p c,d). Input: B c,s c,ˆσ,p b c,p c,d Output: B w,s w,σ,p b w,p w 1: B w B c, S w S c, P b w P b c, P w P c, σ ˆσ; 2: for any two eller α, β S w,α β do 3: if σ(α) = σ(β) then 4: U b σ(j)j = D j σ(j) Pb σ(j)j,j = {α,β}; 5: if U b σ(α)α = U b σ(β)β then 6: j randomly elected from {α,β}; 7: ele 8: j argmin j {α,β} {U b σ(j)j }; 9: end if 1: S w S w \{ j }; 11: P b w P b w \{P b σ(j )j },P w P w \{P j },σ(j ) = ; 12: end if 13: end for 14: return (B w,s w,σ,p b w,p w); P11 b = P 1 = Ao 1. Therefore, we have ˆσ(1) = 1,B c B c {b 1 },S c S c { 1 },P11 b = P1 = A o 1, Pc b Pc b {Pb 11 },P c P c {P 1 }. A a imilar procedure i applied to examine other eller in S, we kip the detail in the following to avoid repetition; 2 : B 2 = {b 4,b 5 }, A o 2 = 4.5. Since D2 5 < Ao 2 < D2 4, we have ˆσ(2) = 4,B c B c {b 4 },S c S c { 2 },P42 b = P2 = A o 2, Pc b Pc b {P42},P b c Pc {P2}; 3 : B 3 = {b 3,b 5 }, A o 3 = 3.5. Since Ao 3 < D3 3 < D3 5, we have ˆσ(3) = 5,B c B c {b 5 },S c S c { 3 },P53 b = P3 = D3 3, Pb c Pb c {Pb 53 },P c P c {P 3 }; 4 : B 4 = {b 5 }, A o 4 = 3.5. Since Ao 4 < A 4, no buyer win the ervice from 4 ; 5 : B 5 = {b 1 }, A o 5 = 4. Since A 5 = A o 5 < D5 1, we have ˆσ(5) = 1,B c B c {b 1 },S c S c { 5 },P15 b = P 5 = A o 5,Pc b Pc b {P15},P b c Pc {P5}; 6 : B 6 = {b 1,b 3 }, A o 6 = 4.5. Since Ao 6 < D6 3 < D6 1, we have ˆσ(6) = 1,B c B c {b 1 },S c S c { 6 },P16 b = P6 = D6 3, Pb c Pb c {Pb 16 },P c P c {P 6 }; 7 : B 7 = {b 2,b 4 }, A o 7 = 3.5. Since Ao 7 < D7 4 < D7 2, we have ˆσ(7) = 2,B c B c {b 2 },S c S c { 7 },P27 b = P7 = D7 4, Pb c Pb c {Pb 27 },P c P c {P 7 }; In ummary, Algorithm 2 give the following output: B c = {b 1,b 2,b 4,b 5 }; S c = { 1, 2, 3, 5, 6, 7 }; Pc b = {Pb 11 = 4.5,Pb 42 = 4.5,Pb 53 = 6,Pb 15 = 4,Pb 16 = 9,P27 b = 7}; Pc = {P1 = 4.5,P2 = 4.5,P3 = 6,P5 = 4,P6 = 9,P7 = 7}; ˆσ = {ˆσ(1) = 1,ˆσ(2) = 4,ˆσ(3) = 5,ˆσ(5) = 1,ˆσ(6) = 1,ˆσ(7) = 2}. 2) Candidate-Elimination: The output of Algorithm 2 include ˆσ(1) = ˆσ(5) = ˆσ(6) = 1, which mean buyer candidate b 1 win three eller candidate, i.e., 1, 5 and 6. Algorithm 3 i run ubequently to retain one bet eller for buyerb 1. The utilitie ofb 1 with repect to the three eller candidate are computed and given by: U11 b = D1 1 Pb 11 = 6 4.5= 1.5,Ub 15 = D5 1 Pb 15 = 5 4= 1, U16 b = D6 1 Pb 16 = 1 9 = 1. Clearly, 1 reult in the highet utility for b 1. Thu, 5 and 6 are eliminated from the winning eller et. In concluion, Algorithm 3 produce the following auction outcome: B w = {b 1,b 2,b 4,b 5 }; S w = { 1, 2, 3, 5, 6, 7 }\{ 5, 6 } = { 1, 2, 3, 7 }; P b w = {P b 11,P b 42,P b 53,P b 15,P b 16,P b 27} \ {P b 15,P b 16} = {P b 11,Pb 42,Pb 53,Pb 27 } = {Pb 1 = 4.5,P b 4 = 4.5,P b 3 = 6,P b 2 = 7}; P w = {P 1,P 2,P 3,P 5,P 6,P 7 } \ {P 5,P 6 } = {P 1 = 4.5,P 2 = 4.5,P 3 = 6,P 7 = 7}; σ = {σ(1) = 1,σ(2) = 4,σ(3) = 5,σ(7) = 2}. 4.3 Proof of Deirable Propertie In the following, we prove that TIM hold the propertie of computational efficiency, individual rationality, budget balance and truthfulne. Theorem 1. TIM i computationally efficient. Proof. To implement Algorithm 2 for the candidatedetermination & pricing tage, we can firt ort the eller in S with a time complexity of O(mlogm). Then, within the for-loop in Algorithm 2, the median of the ak vector A j can be obtained in O(1) time. In Line 4, finding B j ha a time complexity of O(n), while orting B j to B j in Line 12 ha a time complexity of O(nlogn). Since there are at mot n buyer in B j, the buyer candidate for each eller can be determined in O(n) time. Thu, each round of the forloop take O(nlogn) time. In total, Algorithm 2 ha a time complexity of O(mlogm+mnlogn). In the candidate-elimination tage, the input to Algorithm 3 apparently atifie S c S m. Hence, the for-loop in Algorithm 3 pendo( Sc ( Sc 1) 2 ) = O(m 2 ) time. Therefore, the overall time complexity of TIM i polynomial in the order of O(m 2 +mnlogn). Theorem 2. TIM i individually rational. Proof. In Algorithm 2, there are two cae for buyer b i to be elected a a buyer candidate (b i B c ) and for eller j to become a eller candidate ( j S c ). B j = 1: Thi cae reult in a buyer candidate b i and a eller candidate j only if D j i Ao j A j. The clearing price charged to buyer b i and the payment to eller j are both A o j. B j > 1: In thi cae, buyer b i mut have a bid D j i not le than A o j and Dj i i the highet bid for eller j, which implie that A o j Dj i and Dj i (2) D j i. The clearing price and payment are the maximum of D j i (2) and A o j. Beide, every qualified buyer in B j mut bid not le than A j. A een, each buyer candidate determined in Algorithm 2 i never charged a price greater than it bid, while each eller candidate i rewarded a payment not le than it ak, which enure individual rationality for both buyer and eller. It i poible for Algorithm 2 to aign multiple eller to one buyer candidate b i B c. Algorithm 3 eliminate redundant eller and keep only one bet eller providing buyer b i the highet utility. It i evident that thi procedure doe not change the charging price Pij b to the winning buyer. Thu, the buyer in B w after the candidate-elimination tage

8 8 till atify individual rationality. On the other hand, a eller candidate j i either a winning eller with the ame payment Pj determined in Algorithm 2, or eliminated with zero payment at zero cot. Therefore, individual rationality till hold for the winning eller in S w reulted from Algorithm 3. In concluion, TIM i individually rational. Theorem 3. TIM i budget-balanced. Proof. After the candidate-elimination tage, every winning buyer b i B w ha only one winning eller j S w. Conidering thi one-to-one mapping between B w and S w, we have B w = S w. The clearing price and payment atify Pi b = Pj for each winning buyer b i and it aigned matching eller j, i.e., σ(j) = i. Therefore, it can be eaily hown that i P ( j = P b σ(j) P ) j = j S w j S w b i B w P b which complete the proof. Theorem 4. TIM i truthful. Theorem 4 can be proved by the following Lemma 1 and Lemma 2, which how that TIM i truthful for eller and buyer, repectively. Lemma 1. TIM i truthful for eller. Proof. To compare the auction outcome with truthful or any arbitrary ak, we add tilde in the notation for the general (untruthful) cae. To prove Lemma 1, we divide the eller et S into three ubet: S w, S c \S w and S \S c. 1) For eller j S w : There are two cae when j ak untruthfully. Seller j / S w : Ũ j = U j. Seller j S w : There are four ubcae. B j = 1 and B j = 1: P j = P j = Ao j. B j = 1 and B j > 1: à j < C j, P j = P j = Ao j. B j > 1 and B j = 1: à j > C j, P j = P j = Ao j. B j > 1 and B j > 1: P j = P j = Ao j or Dj i (2). Thu, we have Ũ j = P j C j = Pj C j = Uj. 2) For eller j S c \S w : j cannot be in S w no matter what value Ãj i. Thu, we have Ũ j = = U j. 3) For eller j S \S c : There are two cae when j ak untruthfully. Seller j / S w : Ũ j = = U j. Seller j S w : Let b i be the buyer that win j at price P ij b = P j when j ak untruthfully. Here, we conider two ubcae to compare the utilitie. B j = : We know that C j i greater than any bid of the buyer. According to Theorem 2, the buyer atify individual rationality o that D j i P ij b. Thu, we have C j > D j i P ij b = P j, and Ũ j = P j C j < = Uj. B j = 1, and A o j < A j = C j D j i : The reaon that j S w when j ak untruthfully i that à j A o j and/or B j 2. Additionally, buyer b i need to pay P ij b = Ao j or D j i (2), where D j i (2) i the econd highet bid of the buyer in B j. However, we know that A o j < A j = C j and D j i (2) < A j = C j. Thu, we have P j = P b ij < C j, and Ũ j = P j C j < = U j. A een, telling truth provide the maximum utility for each eller in TIM. Therefore, truthful aking i a weakly dominant trategy for eller in TIM, which prove Lemma 1. Lemma 2. TIM i truthful for buyer. Proof. Similarly, we add tilde in the notation for the outcome with general (untruthful) bid. Here, we divide the buyer et B into two ubet: B w and B\B w. 1) For buyer b i B w : Auming buyer b i win eller j with truthful bidding, we conider three cae when b i bid untruthfully. Buyer b i loe: Ũb i = Ui b. Buyer b i till win j : According to Algorithm 2, ince the clearing price i independent of the bid of buyer b i, we have P ij b = Pb ij, and thu Ũb i = Ũb ij = V j i P ij b = Ui b. Buyer b i win eller j : There are two ubcae. Seller j S c when buyer b i bid truthfully: ˆσ(j ) = i with truthful bid D i : According to Algorithm 2, we have P ij b = Pb ij. The reaon that buyer b i win eller j when it bid truthfully i Uij b Ub ij, whereub ij = V j i Pij b. A a reult, we have Ũb i = Ũb ij = V j i P ij b = Ub ij Ub ij = Ub i. ˆσ(j ) i with truthful bid D i : According to Algorithm 2, there exit a buyer candidate i with D j i Dj i = V j i. When bidding untruthfully, buyer b i win eller j with price P ij b Dj i V j i. Thu, Ũb i = Ũb ij = V j i P b ij Ub i. Seller j / S c when buyer b i bid truthfully: Truthful bid D j i i le than ak A j : When buyer b i bid untruthfully, it pay P ij b to win eller j. According to Theorem 2, the eller atify individual rationality o that P ij b A j > Dj i = V j i. A a reult, Ũb i = Ũb ij = V j i P ij b < Ub i. Truthful bid D j i i not le than ak A j, and A o j > Dj i = V j i : When buyer b i bid untruthfully, it pay P ij b = Ao j to win eller j. A a reult, we till have Ũb i = Ũb ij = V j i P ij b < Ub i. 2) For buyer b i B\B w : Obviouly, Ui b =. There are two cae when b i bid untruthfully. Buyer b i till loe: Ũb i = = U b i. Buyer b i win eller j : There are three ubcae. Truthful bid D j i i le than ak A j: When buyer b i with untruthful bid win eller j at price P ij, the eller hould atify individual rationality according to Theorem 2, o that Pb ij A j > D j i = V j i. A a reult, we have Ũb i = Ũb ij = V j i P ij b < = Ub i. Truthful bid D j i i not le than ak A j, and A o j > D j i = V j i : When buyer b i bid untruthfully, it win eller j at price P ij b. According to Algorithm 2, we

9 9 know that Pb ij A o j > V j i, and thu Ũb i = Ũb ij = V j i P ij b < = Ub i. Truthful bid D j i i not le than ak A j, D j i > Ao j, and there exit buyer b i with bid D j i Dj i = V j i : When buyer b i bid untruthfully, it pay P b ij Dj i to win eller j. Thu, we have Ũi b = Ũij b = V j i P ij b = U b i. Therefore, telling truth maximize the utility of each buyer in TIM and Lemma 2 i proved. 5 EFFICIENT DESIGN OF AUCTION (EDA) In Section 4, we introduce the truthful incentive mechanim TIM, which i proved to hold the deirable propertie of computational efficiency, individual rationality, budget balance and truthfulne. Unfortunately, a double auction mechanim cannot further achieve ytem efficiency (e.g., maximizing ocial welfare) at the ame time [4]. In thi ection, we preent another more efficient deign of auction EDA, which lightly relaxe the truthfulne contraint and enure truthfulne in a weak ene. We follow the ame equence a Section 4, giving the detailed algorithm of EDA, then an illutrative example, and the analyi of it propertie at lat. 5.1 Detail of EDA A illutrated in Section 4.2, the buyer candidate b 1 win three eller candidate ( 1, 5 and 6 ), and only the eller candidate 1 become a winning eller in S w. The other eller candidate 5 and 6 are not uccefully matched to potential buyer, which degrade the ytem efficiency and thu jeopardize the utilization of MCC reource. EDA improve the ytem efficiency by involving randomne and more uncertainty in the auction mechanim. The detail of EDA are given in Algorithm 4. According to EDA, the auctioneer firt contruct a randomly ordered lit S of the eller et S, then determine the winning buyer for each eller j S following the order of S. Recall that A j denote the ak vector without the ak of j, and A o j i the median ak of A j. Let B j be the ublit of buyer in B\B w with bid not le than the ak of eller j. Then, Algorithm 4 handle two different cae a follow. B j = 1: Aume that b i i the buyer in B j with bid D j i. Among all the bid of buyer for eller j, denoted by D j, let D j i be the highet bid in Dj that i le than D j i, and P b be the maximum of A o j and Dj i. If Dj i Pb A j, then b i i added into B w and aigned a clearing price P b, while j i added into S w with P b a the clearing payment. If D j i < Pb or A j > P b, b i cannot win the ervice of j. B j > 1: If the highet bid of the buyer in B j i le than A o j, no buyer obtain the ervice of j; otherwie, the buyer b i B j with the highet bid (or a randomly elected buyer when there i a tie) win j. Then, buyer b i and eller j are added into B w and S w, repectively. Let D j i (2) be the econd highet bid of the buyer in B j, and D j i be the highet bid in D j that i le than D j i. Note that Dj i the bid vector of all buyer for eller j, while B j only include Algorithm 4 EDA(B,S,D,A). Input: B, S, D, A Output: B w,s w,σ,pw,p b w 1: B w, S w, Pw b, Pw ; 2: Randomly order S to obtain S = { r(1), r(2),..., r(m) }; 3: for j S in the order of S do 4: Find the median ak A o j of the ak vector A j; 5: B j = {b i : D j i Aj, bi B \Bw}; 6: if B j = 1 then 7: 8: Find highet bid D j i Dj that i not greater than D j i ; P b = max{a o j,d j i }; 9: 1: if D j i Pb and A j P b then σ(j) = i,b w B w {b i},s w S w { j}; 11: Pij b = Pj = P b ; 12: 13: Pw b Pw b {Pij},P b w Pw {Pj}; end if 14: ele if B j > 1 then 15: Sort B j to get B j uch that D j i (1) D j i (2) ; 16: Find highet bid D j i Dj that i not greater than D j i (1) ; 17: if D j i (1) A o j then 18: if the firt t (t 2) bid of B j are the ame then 19: Randomly elect b i from firt t buyer of B j ; 2: ele 21: Select firt buyer b i of B j with the highet bid; 22: end if 23: σ(j) = i,b w B w {b i},s w S w { j}; 24: Pij b = Pj = max{a o j,d j i }; 25: 26: Pw b Pw b {Pij},P b w Pw {Pj}; end if 27: end if 28: end for 29: return (B w,s w,σ,pw,p b w); the remaining buyer that have bid not le than A j and are till competing. The clearing price charged to b i and the payment rewarded to j are both the maximum of A o j and D j i. A een in Line 5, the winning buyer in B w are eliminated in the buyer ublit B j. Thu, a winning buyer b i B w will not compete with other buyer for the remaining eller. Hence, it i not neceary to include candidate elimination a in TIM and the ytem efficiency i improved a a reult. 5.2 A Walk-Through Example For eay comparion, the following illutrative example for EDA i alo baed on the bid matrix in Table 2(a) and the ak vector in Table 2(b). Applying Algorithm 4, we have B w =,S w =,Pw b =,P w = ; Suppoe that the eller et S i randomly ordered to S = { 3, 5, 7, 1, 4, 6, 2 }; 3 : Similar to the walk-through example for TIM, we only provide the detail for thi cae to ave pace. Firt, we have the ak vector A 3 = {3,2,6,4,1,7}, and it median A o 3 = 3.5. Since b 3 B\B w with D3 3 > A 3, and b 5 B\ B w with D5 3 > A 3, we obtain B 3 = {b 3,b 5 }. A B 3 > 1, D5 3 = 6 and D3 5 > D3 5 = D3 3 > Ao 3, b 5 i elected a the winning buyer for 3 with P53 b = P3 = D5 3. Therefore, we have σ(3) = 5,B w B w {b 5 },S w S w { 3 },P53 b = P3 = D3 5, Pb w Pw b {P53 b },P w Pw {P3 }; 5 : B 5 = {b 1 }. Since D 5 1 =,Ao 5 = 4,P b = A o 5 and A 5 = P b < D 5 1, we have σ(5) = 1,B w B w

10 1 {b 1 },S w S w { 5 },P b 15 = P 5 = Pb,P b w Pb w {P b 15 },P w P w {P 5 }; 7 : B 7 = {b 2,b 4 }. Since A o 7 = 3.5 < D 7 4 = D 7 2 < D 7 2, we have σ(7) = 2,B w B w {b 2 },S w S w { 7 },P b 27 = P 7 = D7 2, Pb w P b w {P b 27 },P w P w {P 7}; 1 : B 1 =. No buyer win the ervice of 1 ; 4 : B 4 =. No buyer win the ervice of 4 ; 6 : B 6 = {b 3 }. Since D 6 3 =,Ao 6 = 4.5,Pb = A o 6 and A 6 < P b < D 6 3, we have σ(6) = 3,B w B w {b 3 },S w S w { 6 },P b 36 = P 6 = P b, P b w P b w {P b 36 },P w P w {P 6 }; 2 : B 2 = {b 4 }, ince D 2 4 = 2,Ao 2 = 4.5,Pb = A o 2 and A 2 < P b < D 2 4, we have σ(2) = 4,B w B w {b 4 },S w S w { 2 },P b 42 = P 2 = Pb, P b w Pb w {P b 42},P w P w {P 2}; In ummary, Algorithm 4 give the auction outcome: B w = {b 1,b 2,b 3,b 4,b 5 }; S w = { 2, 3, 5, 6, 7 }; P b w = {P b 42 = 4.5,Pb 53 = 6,Pb 15 = 4,Pb 36 = 4.5,Pb 27 = 7}; P w = {P 2 = 4.5,P 3 = 6,P 5 = 4,P 6 = 4.5,P 7 = 7}; σ = {σ(2) = 4,σ(3) = 5,σ(5) = 1,σ(6) = 3,σ(7) = 2}; A een, the ytem efficiency of EDA i improved to 5 ucceful matching a oppoed to 4 in TIM. It i worth noting that the performance of Algorithm 4 may depend on the ordered lit S. The auctioneer can generate multiple random lit to obtain an outcome of the highet ytem efficiency. 5.3 Proof of Deirable Propertie Similar to TIM, EDA alo atifie computational efficiency, individual rationality and budget balance. Nonethele, a mentioned earlier, EDA achieve a higher ytem efficiency but at the cot of weaker truthfulne. Theorem 5. EDA i computationally efficient. Proof. According to the ak vector of eller, we can firt ort the eller in S in a non-decreaing order of their ak, which ha a time complexity of O(mlogm). In each round of the for-loop in Algorithm 4, we can pend O(1) time to determine the median of the ak vectora j, ando(n) time to obtain B j. In addition, finding D j i run in O(n) time. When B j > 1, orting B j to B j further take O(nlogn) time, while the winning buyer can be elected in O(n) time ince there are at mot n buyer in B j. Thu, each round of the for-loop pend O(n log n) time. The overall time complexity of Algorithm 4 i then O(mlogm+mnlogn). Theorem 6. EDA i individually rational. Proof. Conider a pair of matched buyer and eller, b i and j, in the winning et B w and S w. There are two cae for Algorithm 4 to aign thi matching. B j = 1: In thi cae, b i win the ervice of j only when D j i Pb A j. The clearing price and payment are both P b, which i not greater than the bid of b i and not le than the ak of j. B j > 1: In thi cae, b i mut have a bid D j i not le than A o j and Dj i i the highet bid in Bj. Thu, we have A o j D j i and D j i Dj i. Since the clearing price and payment are the maximum of A o j and Dj i, the preceding condition imply that the price charged to buyerb i i not greater than it bid D j i. On the other hand, we have Dj i A j. Thi can be eaily inferred from the definition that D j i i the highet bid not greater thand j i indj, and the fact thatd j i i the highet bid in B j. Since B j > 1 and every buyer in B j ha a bid not le thana j, we have D j i = Dj i (1) D j i Dj i (2) A j. Here, D j i (1) and D j i (2) are the firt and econd highet bid in B j, repectively. Thu, the clearing payment to j hould be not le than A j. A een in both cae, each winning buyer determined in Algorithm 4 never pay more than it bid, while each winning eller i paid not le than it ak, which guarantee individual rationality for buyer and eller. Theorem 7. EDA i budget-balanced. Proof. In the auction outcome, EDA keep one-to-one mapping between B w and S w a TIM, while the clearing price charged to a winning buyer i equal to the clearing payment to the matched winning eller. Thu, we can eaily how that EDA i budget-balanced by applying the proof of budget balance of TIM Truthfulne of EDA for Seller Theorem 8. EDA i truthful for eller. Proof. Similar to the proof of truthfulne of TIM for eller, we conider two ubet of the eller et S: S w and S \S w. 1) For eller j S w : There are two cae when eller j ak untruthfully. Seller j / S w : Ũ j = U j. Seller j S w : There are four ubcae with repect to B j = 1 or B j > 1, and B j = 1 or B j > 1. In all thee ubcae, we can infer from Algorithm 4 that P j = Pj = max{ao j,dj i }. Therefore, we have Ũj = P j C j = Pj C j = Uj. 2) For eller j S\S w : There are two cae when eller j ak untruthfully. Seller j / S w : Ũ j = = U j. Seller j S w : Let b i be the buyer that win eller j at price P ij b = P j. There are two ubcae to examine the utilitie of different ak. B j = : That i D j i < A j = C j. According to Theorem 6, the buyer atify individual rationality, o that D j i P ij b. A a reult, we have C j > D j i P ij b = P j, and Ũ j = P j C j < = Uj. B j = 1, and max{a o j,dj i } < A j = C j D j i : When eller j ak untruthfully, j S w only if à j max{a o j,dj i } and/or B j 2. The price charged to buyer b i i then P ij b = max{ao j,dj i }. Since max{a o j,dj i } < A j = C j, we have P j = P ij b < C j, and Ũ j = P j C j < = Uj.

11 11 A een, telling truth i a weakly dominant trategy for eller in EDA, which prove Theorem Truthfulne of EDA for Buyer Theorem 8 how that EDA i trongly truthful for eller. However, EDA cannot maintain trong truthfulne for buyer. Fortunately, the following analyi alo how that the uncertainty and randomne in EDA introduce high rik and difficultie for buyer to bid untruthfully to improve their utilitie. A truthful bid can alway lead to a non-negative utility, wherea bidding untruthfully may reult in a nonpoitive utility. Therefore, EDA can enure truthfulne for buyer in a weak ene. A further demontrated in the experiment reult, EDA achieve truthfulne in expectation. When bidding truthfully, a buyer in B can fall into either ubet B w or ubet B\B w. Firt, we prove that telling truth i a weakly dominate trategy for buyer b i B \ B w a it maximize the buyer utility. There are two cae when b i bid untruthfully. Buyer b i till loe: Ũb i = = Ui b. Buyer b i win eller j : There are three ubcae. Truthful bid D j i i le than ak A j : According to Theorem 6, the eller atify individual rationality, o we have P ij b A j > D j i = V j i. A a reult, we have Ũi b = Ũb ij = V j i P ij b < = Ub i. Truthful bid D j i i not le than ak A j, and A o j > Dj i = V j i : When buyer b i bid untruthfully, it win eller j at price P ij b b. According to Algorithm 4, we know that P ij A o j > V j i, and thu Ũb i = Ũb ij = V j i P ij b < = Ub i. Truthful bid D j i i not le than ak A j, D j i > Ao j, and there exit a buyerb i B\B w with bidd j i Dj i = V j i : When buyer b i bid untruthfully, it pay P ij b Dj i to win eller j. A a reult, we till have Ũb i = Ũb ij = V j i P ij b = Ub i. Second, we how two ituation that buyer b i B w can improve it utility by bidding untruthfully. 1) When buyer b i win eller j with truthful bid D j i = V j i and price Pij b = Dj i, buyer b i can ubmit an untruthful bid D j i which i lightly maller than D j i, o that buyer b i till win eller j but lead to a new price P ij b = Ao j. 2) When bidding truthfully, buyer b i ha the highet bid for eller α and β among all buyer in B \B w, and eller α appear before eller β in S. In thi ituation, buyer b i can bid untruthfully with D i α = to loe eller α, o that it win eller β for a higher utility. Although the buyer in B w can improve utility in principle by bidding untruthfully, there are ome difficultie in computing an effective lie. Without the knowledge of the other bid, a buyer ha no way to determine the auction outcome, uch a whether the buyer could win with truthful bid, the matched eller if the buyer win, and the clearing price. Thu, it i very hard for a buyer to improve it utility by bidding untruthfully. A hown in the preceding dicuion, when buyer b i B\B w with truthful bid, bidding untruthfully cannot reult in a poitive utility to exceed the zero utility with truthful bid. When buyer b i B w with truthful bid, we take n = 1 m = 1 n = 1 m = 1 TABLE 3: Computational efficiency. (a) Computation time of TIM. m Time (m) n Time (m) (b) Computation time of EDA. m Time (m) n Time (m) the econd ituation above a an example. If buyer b i want to loe eller α to improve it utility, buyer b i mut ubmit a zero bid, i.e., Dα i =, ince buyer b i doe not know the bid of other buyer. There i alo a rik that buyer b i may achieve zero utility Ũb i =, if it cannot win a eller other than α. Furthermore, uppoing complete information of the bid matrix D and the ak vector A i publicly known, due to the randomne introduced in EDA, an untruthful bid may even backfire to a buyer who hould win with truthful bid but loe the auction with a lie. Take the firt ituation above a an example. Even with the knowledge of D and A, buyer b i cannot determine B j, ince B j depend on the randomly ordered lit S. Thu, if buyer b i want to improve it utility with a bid D j i < Dj i, it may experience Ũb i =, ince b i i till in B \B w. 6 NUMERICAL RESULTS Thi ection preent the numerical reult to evaluate the performance of TIM and EDA. A een in the proof in Section 4.3 and Section 5.3, TIM and EDA atify deirable propertie including computational efficiency, individual rationality, budget balance, and truthfulne. The proof doe not et any preumption on the bid of buyer or the ak of eller. Thu, the concluion are valid for any poible data et of the bid and ak. Thu, without lo of generality, we randomly generate the bid of buyer and the ak of eller according to a uniform ditribution within (, 1] unle otherwie pecified. For the imulation regarding each deirable property, we alo vary the number of buyer or eller, the bid of buyer, or the ak of eller, which are detailed in each ubection. 6.1 Computational Efficiency To confirm our analyi on time complexity of TIM in Theorem 1 and that of EDA in Theorem 5, we give the computation time of TIM and EDA with different etting in Table 3. For each etting, we randomly generate 1 intance and average the reult. All the tet run on a Window PC with 3.16 GHz Intel R Core TM 2 Duo proceor and 4 GB memory. A een, both TIM and EDA are ubject to polynomial computation time with repect to n and m, which are the number of buyer and eller, repectively.

12 Bid Pricing Ak Utility Utility Bid Bid.4 (a) Buyer b i B w. (b) Buyer b i / B w Winning buyer eller pair (a) TIM. Utility Utility Bid Pricing Ak Ak (c) Seller j S w Ak (d) Seller j / S w. Fig. 4: Truthfulne of buyer and eller with TIM Winning buyer eller pair (b) EDA. Fig. 3: Individual rationality of TIM and EDA. 6.2 Individual Rationality and Budget Balance To validate Theorem 2 and Theorem 6 regarding individual rationality of TIM and EDA, we how the bid, pricing, and ak in Fig. 3. Since the price charged to each winning buyer i equal to the payment rewarded to each winning eller, the pricing here preent both. A een, for both TIM and EDA, each winning buyer i charged a price not higher than it bid, while each winning eller receive a payment not le than it ak from the auctioneer. Therefore, both TIM and EDA are individually rational. The reult demontrate that the cloudlet receive ufficient compenation to be incentivized to hare their reource. On the other hand, the mobile uer are allocated the demanded reource and pay no more than their valuation of thee reource. Thu, the mobile uer are alo timulated to requet ervice from the cloudlet. Pij b Pj In addition, ince Pij b = P j for all the winning pair in TIM and EDA, budget balance i alo achieved in both mechanim, which confirm Theorem 3 and Theorem 7 for budget balance of TIM and EDA. Hence, the auctioneer can ait the reource allocation without a deficit. 6.3 Truthfulne of TIM To verify the truthfulne of TIM, we randomly chooe everal buyer/eller to examine how their utilitie change when they bid or ak different value. The reult are depicted in Fig. 4. Fig. 4(a) how the cae that buyer b i win eller j and gain utility U b i =.2885 when it bid truthfully with D j i = Utility Ak (a) Seller j S w. Utility Ak (b) Seller j / S w. Fig. 5: Truthfulne of eller with EDA. V j i = It can be een that buyer b i cannot improve it utility no matter what other bid it take. Fig. 4(b) how a different cenario that buyer b i doe not win eller j when it bid truthfully with D j i = V j i =.437. Thu, b i achieve zero utility without having the ervice. Fig. 4(b) how that the utility cannot be greater than zero even when buyer b i bid untruthfully. Fig. 4(c) how an example that eller j win when aking truthfully with A j = C j =.3841 and achieve utility Uj = A een, the utility with a truthful ak i the highet among all poible ak. Fig. 4(d) how that eller j loe when aking truthfully with A j = C j =.551 and thu obtain zero utility. For all other ak, the achievable utility i either zero or negative, but cannot be more than zero. In ummary, TIM guarantee truthfulne for both buyer and eller ince the utility cannot be improved by bidding or aking untruthfully. In addition, a een in Fig. 4(a) and Fig. 4(c), the winning mobile uer and the cloudlet that are matched uccefully gain poitive utilitie, which mean that both benefit from uing or providing the demanded reource. 6.4 Truthfulne of EDA Fig. 5 and Fig. 6 evaluate the truthfulne of EDA for eller and buyer, repectively. Fig. 5(a) how an example with a randomly choen winning eller j that ak truthfully with A j = C j =.926 and achieve utility U j =.464. A een, the utility with a truthful ak i the highet among all poible ak. Fig. 5(b)

13 13 Expected utility improvement Bid Fig. 6: Truthfulne in expectation of buyer with EDA. how the cae that eller j loe when aking truthfully with A j = C j =.698 and thu obtain zero utility. For all other ak, the achievable utility cannot be more than zero. A analyzed in Section 5.3.2, EDA cannot enure trong truthfulne for buyer. Conidering everal randomly choen buyer, Fig. 6 how their utility improvement with different bidding value over truthful bid. Since EDA involve a randomly ordered eller lit in the auction, we here calculate the expected utility improvement with 1 random intance. A een, no buyer can achieve a poitive improvement in it expected utility by bidding untruthfully. The obervation in Fig. 5 and Fig. 6 confirm our concluion on the truthfulne of EDA for eller and it weak truthfulne for buyer. 6.5 Sytem Efficiency A dicued in Section 3.3, thi work aim to improve ytem efficiency in term of the number of ucceful trade or the total valuation of winning buyer. Intuitively, the ytem efficiency depend on the range of buyer bid and that of eller ak. In practice, the bid and ak may be widely dipered in extreme cae. For example, if the bid of buyer are far below the ak of eller, there can be few ucceful trade that lead to low total valuation. To eliminate the data dependency in comparing the ytem efficiency of different mechanim, we normalize the ytem efficiency with repect to that of the optimal trategy with complete auction information. Thi normalized ytem efficiency clearly demontrate the gap to the optimal achievable performance. Fig. 7 compare four incentive mechanim, including TIM, EDA, and TASC [1] with maximum weighted matching (TASC-MWM) or maximum matching (TASC-MM). Fig. 7(a) and Fig. 7(b) how the normalized ytem efficiency of thee mechanim in term of the number of ucceful trade and the total valuation of winning buyer, repectively. Similar trend are oberved in the two figure, ince the total valuation of winning buyer i proportional to the number of ucceful trade. A een, TASC-MM achieve the lowet ytem efficiency, while EDA achieve the highet ytem efficiency of around 8%. The high ytem efficiency demontrate that EDA improve the reource utilization of the cloudlet and the atifaction of the mobile uer ervice demand. One main reaon for the low efficiency of TASC-MM i that the maximum matching algorithm ued in the aignment b 7 b 49 b 38 b 15 b 25 tage may aign a eller to a buyer of a low bid. However, uch a matching i eventually invalidated in the winner determination & pricing tage, which heavily depend on the ordered tatitic of the bid and ak. Beide, TASC ue a univeral threhold to determine the winner, which may eliminate poible matching unnecearily. On the other hand, TIM and EDA improve the efficiency by applying multiple threhold (i.e., a differenta o j for each eller j) to determine the winner. Due to different threhold ued in TIM and EDA, more complex pricing i thu required to enure truthfulne. Another intereting obervation of Fig. 7 i that the curve are not monotonic but fluctuating, which eem counterintuitive ince the number of ucceful trade and the total valuation of winning buyer hould increae with the number of eller. Indeed, more eller can better atify the divere demand of buyer and thu lead to more ucceful trade. Here, however, we normalize the ytem efficiency with repect to that of the optimal trategy in order to meaure the gap to the highet achievable performance. A a reult, the curve are not monotonic becaue the performance of the optimal trategy alo varie with the number of the eller. 7 CONCLUDING REMARKS Thi paper ha conidered an appealing MCC application paradigm for mobile device to acquire the reource of nearby cloudlet. The cloe proximity of cloudlet can reduce the energy conumption and acce latency of mobile device. We propoed two double auction mechanim, TIM and EDA, which coordinate the reource trading between mobile device a ervice uer (buyer) and cloudlet a ervice provider (eller). Both mechanim are proved to be feaible, which enure computational efficiency, individual rationality and budget balance. Furthermore, TIM guarantee truthfulne for both buyer and eller. EDA achieve fairly high ytem efficiency but atifie truthfulne in a weaker ene. EDA till maintain truthfulne for eller, while preventing untruthful bidding of buyer with increaed difficulty of computing an effective lie. The numerical reult validated our theoretical analyi and demontrated improvement in ytem efficiency. REFERENCES [1] M. Rahman, J. Gao, and W.-T. Tai, Energy aving in mobile cloud computing, in Proc. IEEE International Conference on Cloud Engineering (IC2E), Mar. 213, pp [2] Z. Liu, Y. Feng, and B. Li, Socialize pontaneouly with mobile application, in Proc. IEEE INFOCOM, Mar. 212, pp [3] M. Satyanarayanan, P. Bahl, R. Cacere, and N. Davie, The cae for VM-baed cloudlet in mobile computing, IEEE Pervaive Computing, vol. 8, no. 4, pp , 29. [4] V. Krihna, Auction Theory, 2nd ed. Academic Pre, Aug. 29. [5] Y. Zhang, C. Lee, D. Niyato, and P. Wang, Auction approache for reource allocation in wirele ytem: A urvey, IEEE Communication Survey & Tutorial, vol. 15, no. 3, pp , 213. [6] L. M. Auubel, An efficient dynamic auction for heterogeneou commoditie, American Economic Review, vol. 96, no. 3, pp , 26. [7] G. Demange, D. Gale, and M. Sotomayor, Multi-item auction, Journal of Political Economy, vol. 94, pp , [8] D. Mihra and R. Garg, Decending price multi-item auction, Journal of Mathematical Economic, vol. 42, pp , 26. [9] R. P. McAfee, A dominant trategy double auction, Journal of Economic Theory, vol. 56, pp , Apr

14 Normalized ytem efficiency EDA TIM TASC MWM TASC MM Normalized ytem efficiency EDA TIM TASC MWM TASC MM Number of eller (a) With repect to the number of ucceful trade Number of eller (b) With repect to total valuation of winning buyer. Fig. 7: Normalized ytem efficiency. [1] D. Yang, X. Fang, and G. Xue, Truthful auction for cooperative communication, 211, djyang/doc/ MOBIHOC211-TASC.pdf. [11] D. Parke, J. Kalagnanam, and M. Eo, Achieving budget-balance with Vickrey-baed payment cheme in exchange, in Proc. International Joint Conference on Artificial Intelligence, 21, pp [12] L. Gkatziki and I. Koutopoulo, Migrate or not? Exploiting dynamic tak migration in mobile cloud computing ytem, IEEE Wirele Communication, vol. 2, no. 3, pp , Jun [13] S. Abolfazli, Z. Sanaei, E. Ahmed, A. Gani, and R. Buyya, Cloudbaed augmentation for mobile device: Motivation, taxonomie, and open challenge, IEEE Communication Survey & Tutorial, vol. 16, no. 1, pp , 214. [14] Y. Xu and S. Mao, A urvey of mobile cloud computing for rich media application, IEEE Wirele Communication, vol. 2, no. 3, pp , Jun [15] L. Zhou, Z. Yang, J. Rodrigue, and M. Guizani, Exploring blind online cheduling for mobile cloud multimeida ervice, IEEE Wirele Communication, vol. 2, no. 3, pp , Jun [16] L. Song, D. Niyato, Z. Han, and E. Hoain, Game-theoretic reource allocation method for device-to-device (D2D) communication, IEEE Wirele Communication, vol. 21, no. 3, pp , Jun [17] B. Di, T. Wang, L. Song, and Z. Han, Incentive mechanim for collaborative martphone ening uing overlapping coalition formation game, in Proc. IEEE GLOBECOM, Dec [18] J. Ma, J. Deng, L. Song, and Z. Han, Incentive mechanim for demand ide management in mart grid uing auction, IEEE Tranaction on Smart Grid, vol. 5, no. 3, pp , May 214. [19] Y. Shi, S. Sharma, Y. T. Hou, and S. Kompella, Optimal relay aignment for cooperative communication, in Proc. ACM MobiHoc, 28. [2] D. Wet, Introduction to Graph Theory, 2nd ed. Prentice Hall, [21] Y. Zhang, D. Niyato, and P. Wang, An auction mechanim for reource allocation in mobile cloud computing ytem, in Proc. Intl. Conf. on Wirele Algorithm, Sytem, and Application, 213, pp [22] C. Lee, P. Wang, and D. Niyato, A real-time group auction ytem for efficient allocation of cloud Internet application, IEEE Tranaction on Service Computing, 214, in pre. [23] D. D. Clark, W. Lehr, and S. Bauer, Interconnection in the Internet: The policy challenge, Telecommunication Policy Reearch Conference, 211. [24] J. Trevathan and W. Read, Privacy and ecurity concern for online hare trading, in Proc. of IASK International Conference: E-Activity and Leading Technologie, Dec. 217, pp A-Long Jin received hi B.Eng. degree in communication engineering from Nanjing Univerity of Pot and Telecommunication, Nanjing, China, in 212. He received hi M.Sc. degree in computer cience from Univerity of New Brunwick, Fredericton, NB, Canada, in 215. Hi reearch interet include cooperative wirele network, game theory for wirele network, machine learning, and optimization. Wei Song (M 9-SM 14) received the Ph.D. degree in electrical and computer engineering from the Univerity of Waterloo, Waterloo, ON, Canada, in 27. From 28 to 29, he wa a Potdoctoral Reearch Fellow with the Department of Electrical Engineering and Computer Science, Univerity of California, Berkeley, CA, USA. In 29, he joined the Faculty of Computer Science, Univerity of New Brunwick, Fredericton, NB, Canada, where he i now an Aociate Profeor. Her current reearch interet include cooperative wirele networking, energy-efficient wirele network, and device-to-device communication. She wa a co-recipient of a Bet Student Paper Award from IEEE CCNC 213, a Top 1% Award from IEEE MMSP 29, and a Bet Paper Award from IEEE WCNC 27. She i the Communication/Computer Chapter Chair of IEEE New Brunwick Section. Ping Wang (M 8-SM 15) received the Ph.D. degree in electrical and computing engineering from the Univerity of Waterloo, Waterloo, ON, Canada, in 28. She joined the School of Computer Engineering, Nanyang Technological Univerity, Singapore in June 28, where he i now an Aociate Profeor. Her current reearch interet include reource allocation in multimedia wirele network, cloud computing, and mart grid. She wa a co-recipient of the Bet Paper Award from IEEE Wirele Communication and Networking Conference (WCNC) 212 and IEEE International Conference on Communication (ICC) 27. Duit Niyato (M 9-SM 15) i currently an Aociate Profeor in the School of Computer Engineering, at the Nanyang Technological Univerity, Singapore. He received the Ph.D. degree in electrical and computer engineering from the U- niverity of Manitoba, Winnipeg, MB, Canada in 28. Hi reearch interet are in the area of the optimization of wirele communication and mobile cloud computing, mart grid ytem, and green radio communication. Peijian Ju received M.Eng. and B.Eng. degree from Huazhong Univerity of Science and Technology, Wuhan, China, in 211 and 29, repectively. He i working toward the Ph.D. degree at the Univerity of New Brunwick, Fredericton, NB, Canada. Hi reearch interet include cro-layer deign and game theory for cooperative wirele network.