WIRELESS service providers are facing critical spectrum

Transcription

1 3012 TAMES: A Truthful Double Auction for Multi-Demand Heterogeneous Spectrums Yanjiao Chen, Jin Zhang, Kaishun Wu, and Qian Zhang Abstract To accommodate the soaring mobile broadband traffic, the Federal Communications Commission (FCC) in the U.S. sets out to retrieve under-utilized spectrum (e.g., TV Whitespace) and lay the groundwork for spectrum redistribution. Auction is an efficient way to allocate resources to those who value them the most. The large pool of spectrums to be released, especially the ones in TV Whitespace, consist of wide-range frequencies. Apart from spatial reuse, spectrum heterogeneity imposes new challenges for spectrum auction design: 1) Wireless service providers with different targeted cell coverages have different spectrum frequency preferences; 2) interference relationship is frequency-dependent due to frequency-selective signal fading. Unfortunately, existing spectrum auction mechanisms either assume spectrum valuation is homogeneous or use homogeneous interference graph to group buyers who can reuse the same spectrum. In this paper, we propose TAMES, an auction framework for heterogeneous spectrum transaction. We consider a multi-seller-multi-buyer double auction, in which every buyer submits a bid, consisting of the spectrum demand and a bidding profile of prices for spectrums contributed by all sellers. A novel buyer grouping approach is proposed to tackle the problem of heterogeneous interference graph. TAMES is proved to be truthful as well as individually rational. The simulation results show that TAMES significantly improves spectrum utilization, sellers revenue and buyers utility by making smart use of spectrum heterogeneity, while keeping low running time comparable with existing auction mechanisms. Moreover, via simulation, we show how to help buyers obtain continuous spectrums which further improves buyers satisfaction. Index Terms Heterogeneous spectrum auction, multiple item double auction, economic robustness, spectrum continuity Ç 1 INTRODUCTION WIRELESS service providers are facing critical spectrum crunch. In 2010, users worldwide downloaded 5 billion mobile applications, 15 times more than the figure (300 million) in In the U.S., the number of subscribers to mobile services increased by 20 million in 2011 alone, amounting to 294 million [1]. Realizing that more spectrum resources are needed to meet the ever-increasing mobile broadband demand, FCC plans to retrieve spectrum from current spectrum licensees and auction these spectrum for emerging wireless broadband services [1]. Existing works on spectrum auction mostly treat spectrum as identical objects with homogeneous attributes, which is acceptable when a small amount of spectrum is considered. But with the forthcoming freeing-up of large pool of spectrums, especially the ones used in TV service that have wide range frequencies, the homogeneous spectrum auction will suffer from severe efficiency loss and interference problems. Spectrum is intrinsically heterogeneous because signal propagation is frequency-selective. In other words, spectrum. Y. Chen, J. Zhang, and Q. Zhang are with the Department of Computer Science and Engineering, Hong Kong University of Science and Technology. {chenyanjiao, jinzh, qianzh}@ust.hk.. K. Wu is with the Guangzhou HKUST Fok Ying Tung Research Institute, and also with the National Engineering Research Center of Digital Life, State-Province Joint Laboratory of Digital Home Interactive Applications, Sun Yat-sen University. kwinson@ust.hk. Manuscript received 24 Aug. 2013; revised 29 Oct. 2013; accepted 9 Nov Date of publication 24 Nov. 2013; date of current version 15 Oct Recommended for acceptance by S. Guo. For information on obtaining reprints of this article, please send to: reprints@ieee.org, and reference the Digital Object Identifier below. Digital Object Identifier no /TPDS of different frequencies have different path losses, thus different transmission ranges. Heterogeneous spectrum auction is challenging due to the following reasons.. Interference relationship among users is variant when different spectrums are used for transmission. Users who are interference-free on high-frequency spectrum may generate harmful interference to each other on low-frequency spectrum.. The co-existence of macro/micro/pico/femto cellular networks indicates that network operators have different targeted cell coverages, therefore different preferences for different frequencies. Macrocell operators may favor low frequency spectrum with long transmission range while femtocell operators are satisfied with high-frequency spectrum whose transmission range is enough to cover the indoor area and generates less cross-tier interference. A naive way to deal with spectrum heterogeneity is to conduct a separate auction for each heterogeneous spectrum. However, this is not only inefficient (e.g., 10 auctions for 10 heterogeneous spectrum), but also problematic: for a buyer who wants to buy one spectrum, he has to submit different bids to each auction; if he bids the highest prices in multiple auctions, he will win more than he wishes for. Therefore, we need an integrated auction where the buyers can bid for all heterogeneous spectrums simultaneously. This motivates us to propose a new auction framework tailored for multi-demand heterogeneous spectrum double auction. However, it is naturally quite challenging, because a well-designed auction mechanism should not only be economic-robust, but Ó 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See for more information.

2 CHEN ET AL.: TAMES: A TRUTHFUL DOUBLE AUCTION FOR MULTI-DEMAND HETEROGENEOUS SPECTRUMS 3013 also be efficient in terms of spectrum utilization, buyers utility and sellers revenue.. Economic-robustness. Truthfulnessisoneofthemost essential economic properties. It guarantees that no seller or buyer has incentive to cheat in the auction and the spectrum is allocated to those who value it the most. 1 Individual rationality ensures that all auction participants achieve non-negative utility so that they have incentive to participate in the auction.. Efficiency. Spectrum heterogeneity makes it difficult to design an economic-robust auction mechanism that achieves high spectrum utilization. If bidders are forced to bid a common price for all spectrums, they may overpay for a less-valued spectrum when bidding high, which violates individual rationality; or lose a high-valued spectrum when bidding low. If bidders are granted the right to bid different prices for different spectrums, they can manipulate the price for one spectrum to influence the winning results of another spectrum, which violates truthfulness. In addition, interference graph heterogeneity makes it hard to exploit spectrum reusability while ensuring interference-free spectrum allocation. Traditional auction mechanisms, such as second-price or VCG [1], [2], [3] mechanisms, have been proved to be untruthful or individually irrational when applied to spectrum auction. The major reason is that, unlike other commodities, spectrum can be reused and assigned to multiple buyers who are interference-free. Existing spectrum auction mechanisms, though having resolved the problem of spectrum reusability, failed to consider spectrum heterogeneity. Most of existing work treated spectrum as identical objects [4], [5], [6], [7], [8], [9] and assumed that buyers submit uniform price for all spectrums. In this way, buyers either fail to bid higher and have higher chance to win a higher-valued spectrum or become unsatisfactory for obtaining a lower-valued spectrum. Due to a lack of flexible bidding, the sellers revenue is also affected. When determining whether two buyers can reuse the same spectrum, previous work either assumed that the interference graph is a complete graph [4] or used the same interference graph [6], [7], [8], [9], [10] for all spectrums. In the former case, no user can reuse the same spectrum and spectrum utilization is extremely low. In the latter case, if the interference graph is constructed based on the long-transmission-range spectrum, interference constraint is over-strict, thus spectrum utilization is low; if the interference graph is constructed based on the short-transmission-range spectrum, the auction results cannot guarantee interferencefree spectrum allocation. In order to take advantage of spectrum heterogeneity as well as address the above-mentioned challenges, we propose TAMES, a Truthful double Auction Mechanism for heterogeneous Spectrum allocation. We consider the double auction with multiple sellers, multiple buyers and a 1. In this paper, we assume that demand truthfulness is obeyed by all buyers, i.e., no buyer lies about his spectrum demand. third-party auctioneer. One seller owns multiple spectrums, and assigns each spectrum to multiple noninterfering buyers. TAMES allows buyers to submit a bidding profile to express their diversified valuations for different spectrums. Unlike existing grouping algorithms that cluster all buyers based on the same interference graph, TAMES groups buyers based on the frequencyspecific interference graphs. Group bids are determined by the group size and the minimum bid in the group so that no buyer will pay more than his bids (individual rationality) or intend to misreport his true valuation (truthfulness). Finally, the auctioneer determines the winning buyers, the winning sellers, and corresponding payments. The key contributions of this work are as follows:. To the best of our knowledge, TAMES is the first auction mechanism that jointly considers the influence of spectrum frequency heterogeneity on spectrum reusability and diversified spectrum preference.. TAMES provides an economic-robust auction mechanism which is proved to be truthful and individually rational.. TAMES significantly improves spectrum utilization while ensuring interference-free spectrum allocation by exploring reuse opportunities on frequencyspecific interference graphs.. The simulation results show that TAMES can further improve buyers satisfaction by increasing the spectrum continuity for buyers who win multiple spectrums. An earlier version of this paper is published in the IEEE INFOCOM 2013 mini-conference [11]. Different from the conference paper, we 1. consider the double auction instead of simple auction; 2. consider that each buyer may require multiple spectrums rather than restrict their demand to one spectrum; 3. analyze the running time of the proposed auction mechanism; 4. add simulation to compare the auction results of two typical topologies: the uniform topology and the hotspot topology. The rest of the paper is organized as follows. We define and formulate the problem of heterogeneous spectrum auction in Section 2. The challenges of designing auction mechanism for heterogeneous spectrum is discussed in Section 3. We develop the TAMES auction mechanism in Section 4, explaining the main algorithms and analyzing its properties. In Section 5, we conduct comprehensive simulations to evaluate TAMES performance. Related works are reviewed in Section 6 and we finally summarize our work in Section 7. 2 PROBLEM DEFINITION AND FORMULATION To begin with, we outline the framework of heterogeneous spectrum auction and introduce the criterion used to evaluate the auction mechanism.

3 Auction Participants The auction participants include three parties: the spectrum owners who are the sellers, the wireless service providers who are the buyers and a third-party auctioneer who hosts the auction The Sellers in the Auction We assume there are M spectrum owners who are sellers. Spectrum owner i contributes a set of spectrums, denoted by ðs i;1 ;s i;2 ;...;s i;ni Þ. Let K ¼ P M i¼1 n i denote the total number of spectrums contributed by all sellers. Seller i submits his ask to the auctioneer, denoted by ðr i;1 ;r i;2 ;...;r i;ni Þ. Seller i s ask is based on his true valuation for the spectrum, denoted by ðv s i;1 ;vs i;2 ;...;vs i;n i Þ. We use ðw s i;1 ;ws i;2 ;...;ws i;n i Þ to represent seller i s winning profile and ðp s i;1 ;ps i;2 ;...;ps i;n i Þ to represent seller i s payment profile. w s i;j ¼ 1 means seller i successfully sells spectrum s i;j ;j2½1;n i Š; otherwise w s i;j ¼ 0. After the auction, the utility of seller i is the payment he receives minus his true valuation for all the sold spectrum u s i ¼ Xn i j¼1 p s i;j Xn i j¼ The Buyers in the Auction w s i;j vs i;j : (1) We assume there are N wireless service providers who are buyers. Each buyer submits a bid to the auctioneer. Buyer i s bid consists of two parts: d i is the number of spectrums wanted and ðb i;1 ;b i;2 ;...;b i;k Þ is a bidding profile in which b i;j is buyer i s bidding price 2 for s j ;j2½1;kš. d i 1 since one buyer may want more than onespectrums.weassumethatthebuyerscannotsubsell their spectrums. In this case, if a buyer lies about his spectrum demand as di 0 9 d i, he may end up paying for more spectrums than he actually needs; if a buyer lies about his spectrum demand as di 0 G d i, he never has the chance to win enough spectrum as he wants to. So the buyers are discouraged to be untruthful about their spectrum demands. Therefore, we only focus on the truthfulness regarding the sellers asks and the buyers bids. The bid of a buyer is based on his true valuation for the spectrum, denoted by ðv b i;1 ;vb i;2 ;...;vb i;kþ. Since buyers scatter at different areas and not all spectrums are available in each area, we assume that v b i;j ¼ 0 if spectrum s j ;j2½1;kš is not available to buyer i. 3 We use ðw b i;1 ;wb i;2 ;...;wb i;kþ to represent buyer i s winning profile and ðp b i;1 ;pb i;2 ;...;pb i;kþ to represent buyer i s payment profile. w b i;j ¼ 1 means buyer i wins spectrum s j ;j2½1;kš; otherwise w b i;j ¼ 0. After the auction, the utility u b i of the buyer i is the aggregation of his true 2. The buyers have different preferences for spectrums, not for the spectrum owner who sells it. Hence the bids are spectrum-oriented, not spectrum owner-oriented. 3. In this paper, we assume that the spectrum availability to a buyer is known by the auctioneer, hence he cannot lie about it. This is reasonable since the location of the buyer and the spectrum availability in that location can be easily detected by the auctioneer. valuation for the procured spectrum minus his total payment [8], [9] 4 u b i ¼ XK j¼1 w b i;j vb i;j XK j¼1 p b i;j : (2) In this paper, we consider a single-round, sealed-bid, collusion-free auction. The auction mechanism is evaluated from the following criterion:. Truthfulness. Assume that seller i gets u s i and us0 i for bidding truthfully and untruthfully respectively; buyer i gets u b i and u b0 i for bidding truthfully and untruthfully respectively. In a truthful auction, u s0 i u s i and ub0 i u b i always holds.. Individual rationality. A seller is paid more than his ask; a buyer pays less than his bid, i.e., p s i;j r i;j and p b i;j b i;j.. Buyers utility.. Sellers revenue.. Running time of the auction algorithm. In this paper, the goal of the proposed heterogeneous spectrum auction mechanism is defined as follows: Given seller s asks and buyers bids, how to find a truthful, individually rational auction mechanism to allocate heterogeneous spectrum to noninterfering buyers while trying to improve spectrum utilization, buyers utility and sellers revenue with acceptable running time? 5 3 CHALLENGES OF HETEROGENEOUS SPECTRUM DOUBLE AUCTION While spectrum reusability has been addressed by many of the previous works, frequency diversity brings new challenges. 3.1 Heterogeneous Preference According to the propagation model recommended by ITU [12], the path loss is highly dependent on the central frequency. Therefore, spectrums with different frequencies have different transmission range, different coverage, and are suitable for different cellular networks. In homogeneous auction, sellers are tend to overbid because they are afraid that the payment is less than the higher-valued spectrum; on the contrary, buyers are tend to underbid because they are afraid that they pay more for the 4. In this paper, we assume that the utility of the buyer does not depend on whether his demand is fully satisfied or not. The proposed auction mechanism guarantees that a buyer will not win more than he demands. However, he may win less than he demands. We assume that his utility from the winning spectrum does not change if his total spectrum demand is not fully satisfied. In the future, we will consider the utility difference between the situations when the buyers demand is partly or fully satisfied. 5. In this paper, we assume that no buyers or sellers collude with each other. A collusion-resistant auction mechanism is our future direction.

4 CHEN ET AL.: TAMES: A TRUTHFUL DOUBLE AUCTION FOR MULTI-DEMAND HETEROGENEOUS SPECTRUMS 3015 preference and heterogeneous interference graph while taking into consideration the spectrum utilization, buyers utility and sellers revenue. Fig. 1. Heterogeneous interference graph. The left and right figures are based on high-frequency spectrum s h and low-frequency spectrum s l respectively. lower-valued spectrum. In either case, the spectrum market cannot be efficient. 3.2 Heterogeneous Interference Graph The distinctive property of spectrum is spatial reusability. The auctioneer can pick multiple non-interfering winners for one spectrum based on the interference graph. However, the interference graphs for spectrum with different frequencies are no longer the same. If the highest frequency is three times the lowest frequency, the path loss difference can be as large as 10 db. This will change theinterferencegraphstructureandmakethespectrum allocation more challenging. Fig. 1 shows interference graphs of three buyers in terms of high-frequency spectrum s h and low-frequency one s l. 6 If we use the interference graph of s h and assign s l simultaneously to buyers 1 and 3, they end up interfering with each other. If we use the interference graph of s l and assign s h to only one user, the spectrum utilization becomes low. With numerous frequencies and buyers, the problem is even more complicated. 3.3 Bid Manipulation A buyer may want more than one spectrums. One solution is to replace the original buyer with multiple dummies and each dummy requests one spectrum. After dummy creation, grouping can be performed in the same way as a single-demand auction. However, this simple conversion leaves the chance for buyers to manipulate their bid to gain higher utility. A buyer with multiple dummies can manipulate bidding price for one spectrum to influence the winning result of another spectrum. Existing works on singledemand, collusion-free auction do not have this loophole because different bidders would not conspire with each other to cheat in the auction. Existing works on multidemand homogenous auction also circumvent this problem because bidders bid a common price for all spectrums so that bid manipulation does not exist. 4 DETAILED TAMES AUCTION MECHANISM The analysis above motivates us to design a truthful, individually rational multi-demand double auction mechanism that can address the problem of heterogeneous 6. Fig. 1 is an illustrative example based on the propagation model in [12]: low-frequency spectrum has longer transmission range; highfrequency spectrum has shorter transmission range. 4.1 Main Algorithms The proposed auction mechanism consists of three steps: grouping, allocation and pricing Grouping We use G j ¼ðV j ;E j Þ to denote the interference graph of s j ;j2½1;kš, in which the vertices-set V j stands for all buyers except those for whom s j is unavailable; and the edge-set E j stands for all interference relations on s j.there exists an edge between two buyers if they interfere with each other when transmitting via s j. To form buyer groups is equivalent to finding independent sets on the interference graph. However, as the transmission range varies for different frequencies, it is problematictouseacommon interference graph. Note that if the transmission range is long, more buyers will conflict with each other and there are more edges in theinterferencegraph.thegroupsizeshrinksbecausethe size of independent set decreases. And the group bid decreases as well no matter how the group bid is decided under the constraint of individual rationality. Therefore, it becomes harder for the group to win a spectrum. This motivates us to use sequential grouping to mitigate the problem. The algorithm for sequential grouping is shown in Algorithm Create dummies for buyers with multiple demand. Every dummy inherits the interference relationship and the bidding profile of the original buyer. There is an edge between any two dummies of the same buyer so that they will not be grouped together. 2. Sequentially group buyers for each spectrum in nondecreasing frequency (non-increasing transmission range) manner. We assume that the spectrums are sorted in the frequency ascending order as s 1 ;s 2 ;...;s K. For each spectrum s j, we construct the interference graph G j and find an independent set on the graph. 7 In fact, it is not necessary to construct interference graph from scratch every time. We only have to 1) add buyers who cannot access s j but can access s jþ1 ;2)eliminateverticesof already grouped buyers and corresponding edges; 3) remove edges that no longer exist since the transmission range decreases for a spectrum with higher frequency. 3. Calculate the group bid as the lowest bid in the group for s j multiplies the group size minus one. 4. Repeat 2) 3) until there are no buyers to be grouped or there are no sellers spectrums left. 7. In this paper, we leverage existing works on finding independent set. The auctioneer can choose different independent set selection policies for different purposes, e.g., maximal independent set may increase spectrum reusability, random picking independent set may ensure fairness. In the future, we will consider the influence of independent set selection policies on the auction results.

5 3016 Algorithm 2 Allocation & Pricing. 1: for all g j ;j¼ 1; 2;...do 2: if F j 9 r i;j (i is the seller who owns s j ) then 3: Group members in g j except the the lowest-bid one are assigned spectrum s j ; 4: Group g j is charged the group bid F j. For buyer k 2 g j,thepaymentis: p b k;j ¼ F j jg j j 1 ¼ min k2g j b k;j ; (4) 5: Pay seller i the group bid: p s i;j ¼ F j; (5) Fig. 2. A simple example. Algorithm 1 Sequential Grouping. 1: Unassigned buyer set V ¼ N; 2: Create d i dummies for buyer i if d i 9 1. Delete the original buyer and insert the dummies in V ; 3: for all s j ;j¼ 1; 2;...;K do 4: if V is non-empty then 5: V j is the set of unassigned buyers for whom s j is available. 6: Construct interference graph based on s j for buyers in V j ; 7: Find one independent set g j and match g j to s j ; 8: The group bid of g j is F j ¼ min b i;j jg j j 1 ; (3) i2g j in which jg j j is the number of buyers in the group; 9: Eliminate members in g j from V. 10: end if 11: end for Allocation & Pricing Allocation and pricing procedure is shown in Algorithm 2. After the group formation and spectrum matching, we compare each group bid with the seller s ask for the matched spectrum. If the group bid is higher than the ask, the auctioneer assigns the corresponding spectrum to every member in the group except the one with the lowest bid. The auctioneer then charges exactly the group bid, which will be split equally among all winners, i.e., each winner pays the same amount as the lowest bid of the group. The group member with the lowest bid for is assigned nothing and pays nothing either. If there are more than one group members who have the lowest bid, randomly pick one to be the loser. The auctioneer will pay the seller of the spectrum the amount of the group bid We assume that the auctioneer is not profit-oriented, for example, governmental bodies. 6: else 7: All buyers in g j lose and pay nothing. 8: The seller i loses and is paid nothing. 9: end if 10: end for A Simple Example Now, we give a simple example to show how TAMES works. In Fig. 2, there are 6 buyers. Buyer d requests two spectrums. There are 2 spectrum owners: one spectrum owner has 2 spectrums with asks r 1;1 ¼ 6;r 1;2 ¼ 3; the other spectrum owner has 1 spectrum with ask r 2;1 ¼ 2. We assume that the frequencies of the spectrums are sorted in the increasing order as we mentioned above, so we renumber the 3 spectrums as 1, 2, 3. We also assume that each spectrum is available to all buyers. The bidding profile of each buyer is shown in the figure. Interference graph for s 1 isshowninthetopmostgraph. Buyer a; e form g 1 with group bid 5, the bid of e for s 1.Inthe second iteration, a; e are eliminated from the graph, along with the edge between ðb; d 2 Þ and ðc; d 1 Þ because these two pairs no longer interfere with each other on spectrum s 2, which has a higher frequency and shorter transmission range. g 2 ¼fb; d 2 g with group bid 4. Verticesb; d 2 and edge ðd 1 ;fþ are further eliminated in the third iteration, leaving c; d 1 ;f forming g 3 with group bid 4. After comparing group bids with the asks of the sellers, we know that d 2 is assigned s 2 and pays 4, seller 1 who owns s 2 gets payment of 4; c; d 1 are assigned s 3 and pay 2 respectively, seller 2 who owns s 3 gets payment of 4. d wins two spectrums via its two dummies. Others lose in the auction. 4.2 Economic Properties In this section, we give proofs for the truthfulness and individual rationality of TAMES. Proposition 1. The proposed auction mechanism is individually rational. Proof. On the buyers side, it can be easily proved that each winning buyer pays no more than its bid because they are

6 CHEN ET AL.: TAMES: A TRUTHFUL DOUBLE AUCTION FOR MULTI-DEMAND HETEROGENEOUS SPECTRUMS 3017 charged the price of the lowest bid in their group. On the sellers side, it can also be easily proved that each winning seller gets paid more than its ask because the group bid is greater than the ask in order to win. g As for truthfulness, we first prove the truthfulness on the buyers side from two aspects: 1) The bidding price for one spectrum does not affect the winning result of another spectrum, i.e., for buyer i, b i;j and w b i;l ;l6¼ j are irrelevant; 2) A buyer cannot misreport the bidding price for one spectrum to achieve higher utility from that spectrum. Lemma 1. A buyer cannot manipulate the bidding price for one spectrum to affect the winning result of another spectrum. Proof. The buyers have no control over the grouping process. After group formation, the group members are matched to a specific spectrum. It is not sure whether the matched spectrum will be assigned to them or not but it is definitely sure that other spectrums will not be assigned to them. The group bid only relies on the bid of each group member for the matched spectrum, independent of their bids for other spectrums, i.e., for buyer i; b i;j and w b i;l ;l6¼ j are independent. g Lemma 2. A buyer cannot misreport the bidding price for one spectrum to increase its utility gained from that spectrum. Proof. Take buyer i s bid b i;j as an instance. When i bids truthfully, the winning result is w b i;j and i gains u b i;j ; otherwise, the winning result is w b0 i;j and i gains ub0 i;j.we will prove that u b i;j ub0 i;j always stands. If b i;j 6¼ v b i;j,therearefourcases: 1. w b i;j ¼ 0;wb0 i;j ¼ 0. Inthiscase,ub i;j ¼ ub0 i;j ¼ w b i;j ¼ 1;wb0 i;j ¼ 1.Inthiscase,ub i;j ¼ ub0 i;j ¼ vb i;j pb i;j,in which p b i;j is the lowest bid in the group for s j.since buyer i does not bid the lowest in the group (otherwise it loses the spectrum), p b i;j is independent of b i;j. 3. w b i;j ¼ 1;wb0 i;j ¼ 0. In this case, ub0 i;j ¼ 0, ub i;j ¼ vb i;j p b i;j 0 due to individual rationality. 4. w b i;j ¼ 0;wb0 i;j ¼ 1.Inthiscase,ub i;j ¼ 0. There are three possibilities why a buyer loses: 1) The buyer doesn t belong to any group after the grouping process; 2) The buyer belongs to a group whose group bid is lower than the reserve price of the matched spectrum; 3) The buyer has the lowest bid in the group. Bidding untruthfully cannot help a buyer win thespectruminsituation1).insituation2)and3),ifa buyer wants to win the spectrum, it must bid higher than the second lowest bid in the group. Therefore p b0 i;j 9 vb i;j and ub0 i;j ¼ vb i;j pb0 i;j G 0 ¼ ub i;j. In summary, buyer i does not have incentive to be untruthful since he cannot gain higher utility by misreporting his true valuation. g Nowweprovethetruthfulnessonthesellers side. Lemma 3. TAMES is truthful on the sellers side, i.e., to report true valuation is the (weakly) dominant strategy for the sellers. Proof. It can be easily proved that a seller cannot misreport his true valuation for one spectrum to affect the winning result of another spectrum because his ask for one spectrum does not affect the winning result of another spectrum. Now we prove that the seller cannot misreport the ask for one spectrum to increase his utility obtained from that spectrum. Take seller i s ask r i;j ;j2½1;n i Š as an instance. When i asks truthfully, he ; otherwise, he gains us0 i;j gains u s i;j u s i;j us0 i;j always stands.. We will prove that If r i;j 6¼ v s i;j, there are four cases: 1. w s i;j ¼ 1;ws0 i;j ¼ 1. In this case, the seller is paid the group bid, independent of its ask. u s i;j ¼ us0 i;j ¼ g j v s i;j,inwhichg j is the group bid. 2. w b i;j ¼ 0;wb0 i;j ¼ 0. Inthiscase,us i;j ¼ us0 i;j ¼ w b i;j ¼ 1;wb0 i;j ¼ 0.Inthiscase,us0 i;j ¼ 0, us i;j ¼ g j v s i;j 0 due to individual rationality. 4. w b i;j ¼ 0;wb0 i;j ¼ 1. It can only happen when r0 i;j G g j G r i;j ¼ v s i;j.inthiscase,us0 i;j ¼ g j v s i;j G 0 ¼ us i;j. In summary, the seller cannot improve its utility gained from a certain spectrum by misreporting the ask for that spectrum. Therefore, to report true valuation is the (weakly) dominant strategy for the sellers. g Proposition 2. TAMES is truthful. Proof. By combining lemmas 1, 2, and 3, we can establish that TAMES is truthful on both buyers and sellers side. g 5 PERFORMANCE EVALUATION In this section, we use simulation to evaluate the performance of TAMES. First, we compare the auction results of TAMES with those of the following mechanisms: 1) Single-demand auction; 2) homogenous interference graph; 3) homogenous bids. Then, we analyze how the bids of buyers affect the auction results. We study two situations: 1) Buyers have similar preference, which means that all buyers prefer long-transmissionrange spectrums to short-transmission-range spectrums; 2) buyers have complementary preference, which means some buyers are more fond of long-transmission-range spectrums while others are the opposite. We also study how the buyers topology affects the auction results. We consider two situations: 1) Uniform topology, where the buyers are located uniformly in the concerned area. 2) Hotspot topology, where there are some locations that are compacted by several buyers who are close to each other. We assume that the sellers and the buyers are within a 1 1 area. The number of contributed spectrums of each seller follows uniform distribution on the range [1, 8]; the demand of each user follows uniform distribution on the range[1,10].oneseller sspectrumsareavailablefora buyer if they are within 0.5 unit distance. The sellers asks and the buyers bids follow uniform distribution on the range[0,1].wedefinethefollowingmetricstoevaluatethe auction results:

7 3018 Fig. 3. TAMES vs 1) Single-demand; 2) homogenous interference graph; and 3) homogenous valuation auctions with different sellers. (a) Buyers satisfaction. (b) Ratio of winning buyers. (c) Revenue. (d) Spectrum reusability degree. (e) Ratio of sold spectrums. (f) Average group size.. Buyers Satisfaction (BS), the ratio of winning bids to the total demands P N P K i¼1 j¼1 BS ¼ wb i;j P N i¼1 d : (6) i. Ratio of Winning buyers, the ratio of the number of winning buyers (who win at least one spectrum) to the total number of buyers.. Seller s Revenue, total winning sellers revenue.. Spectrum Reusability degree (SR), the ratio of winning profiles to the number of sold spectrums. P N P K i¼1 j¼1 CR ¼ wb i;j n o s j 2 Sj9i; w b i;j ¼ 1 (7) in which jjdenotes the size of the set.. Ratio of sold spectrums, the ratio of the number of sold spectrums to the total number of on-sale spectrums.. Average group size of the sequential grouping.. Running time. The running time of the main algorithm, i.e., the grouping, allocation and pricing processes. 5.1 Comparison of Different Auction Mechanisms We consider three benchmark auction mechanisms: 1. Single-unit Auction in [13], in which each buyer is only allowed to demand one spectrum. 2. Homogenous Interference Graph Auction in [14]. We assume that the adopted interference graph is the one constructed on the lowest-frequency

8 CHEN ET AL.: TAMES: A TRUTHFUL DOUBLE AUCTION FOR MULTI-DEMAND HETEROGENEOUS SPECTRUMS 3019 Fig. 4. TAMES vs 1) single-demand; 2) homogenous interference graph; and 3) homogenous valuation auctions with different buyers. (a) Buyers satisfaction. (b) Ratio of winning buyers. (c) Revenue. (longest-transmission-range) spectrum to ensure interference-free auction results. 3. Homogenous Valuation Auction in [6] with modifications for multi-demand and heterogeneous interference graph. We assume that the auctioneer compresses the bidding profile of each bidder to the lowest value to ensure individual rationality. In Fig. 3, with the increase of sellers, the spectrums on sale increases, TAMES has a sharp surge in buyers satisfaction (as shown in Fig. 3a) and the ratio of winning buyers (as shown in Fig. 3b). Sellers revenue goes up (as shown in Fig. 3c) since more spectrums are sold. But the ratio of sold spectrum decreases (as shown in Fig. 3d) because the total number of spectrum on sale increases. The spectrum reusability decreases (as shown in Fig. 3e) as the average group size decreases (as shown in Fig. 3f) because the abundant spectrum resource makes spectrum reusability less important. In Figs. 4a and 4b, as the number of buyers increases, the buyers satisfaction and the ratio of winners first increase and then decrease. This is because, at first, the spectrum resource is relatively abundant; when the number of buyers rises, it is easier to form large-size groups, the group bids are higher and more buyers become winners. As the number of buyers increases further, the number of winning groups is bounded by the number of spectrums. The sellers revenue keeps increasing as shown in Fig. 4 since more spectrums are sold as shown in Table 1. Also, the average group size increases, leading to higher spectrum reusability degree as shown in Table TAMES vs Single-Unit Auction In comparison, the single-demand auction cannot efficiently improve buyers satisfaction even though the winning buyer ratio is relatively high. Because it only grants one spectrum to buyers who actually have multiple demand. The group size is very small because there are no extra TABLE 1 TAMES vs Other Mechanisms

9 3020 TABLE 2 Common vs Complementary Preference dummies. So the group bid will be low and fewer spectrums are sold, generating lower income for the seller TAMES vs Homogenous Interference Graph Auction The most significant drawback of homogenous interference graph is that its spectrum reusability degree is the lowest since redundant edges exist in theinterferencegraphwhen forming groups for high-frequency spectrums. Therefore, theaveragegroupsizeisgenerallythesmallest.thesmall group-size weakens the buying power of groups and leads to low ratio of sold spectrums and low revenue for the seller. The buyers satisfaction and ratio of winning buyers is quite low for homogenous interference graph, further underlying the importance of considering heterogeneous interference graph TAMES vs Homogenous Valuation Auction The homogenous valuation does not affect the grouping process of the auction mechanism.therefore,theaverage group size is almost the same as that of TAMES. Nevertheless, the group bid is much lower since only the lowest bid of each buyer is considered, resulting in low ratio of sold spectrum and sellers revenue. Buyers satisfaction and ratio of winning buyers are both low as they are not able to freely express their bidding preference. 5.2 Influence of Buyers Bids According to Table 2, the average group size are independentfrombuyers bidsbecausethebidsdoesnotaffectthe grouping process. However, the ratio of sold spectrums of common preference situation are higher than those of complementary preference, partly because the group bid relies on the minimum bid in the group. In the common preference situation, the lowest bids of individual buyers are put in the same group for the least-valued spectrum, affecting the group bid of that specific spectrum. In the complementary preference situation, the lowest bids of individual buyers scatter in various groups, dragging the group bids of many groups. With fewer sold spectrums (as shown in Table 2) and winning buyers (as shown in Fig. 5b), the seller s revenue and buyers satisfaction for complementary preference situation are lower as shown in Figs. 5a and 5c. Nevertheless, the spectrum reusability degree of complementary preference situation is higher than that of common preference situation (as shown in Table 2), which indicates that the spectrum utilization of the former situation is higher. This is because in the complementary preference situation, as the minimum bid of the group tends to be lower, the group size has to be larger to win the spectrum. The spectrum reusability degree is primarily Fig. 5. Auction results of common preference vs complementary preference situations. (a) Buyers satisfaction. (b) Ratio of winning buyers. (c) Revenue.

10 CHEN ET AL.: TAMES: A TRUTHFUL DOUBLE AUCTION FOR MULTI-DEMAND HETEROGENEOUS SPECTRUMS 3021 Fig. 6. Auction results of uniform vs hotspot topologies. (a) Buyers satisfaction. (b) Ratio of winning buyers. (c) Revenue. decided by the average winning group size since all members in a group reuse the same spectrum. 5.3 Influence of Buyers Topology The previous simulations are all based on the uniform topology. In this section, we examine the influence of different buyers topologies on the auction results. The uniform and hotspot topologies are generated in the following way: Uniform topology. We divide the area into N ¼ 64 grid cells of the same size and place one buyer randomly in each cell. Hotspot topology. We create 4 hotspots. For each hotspot, there are N=4 buyers who are within areaofeach other. Fig. 6 and Table 3 show the simulation results. In the hotspot topology, buyers are close to each other, so they generate interference to each other severely. Therefore, according to Table 3, the group size in the hotspot topology is lower than that in the uniform topology where there are many reuse opportunities for buyers. As a result, the spectrum reusability degree in the hotspot topology is also lower (as shown in Table 3). Hence, the group bid will be low in the hotspot topology, leading to smaller ratio of sold spectrums and smaller ratio of winning buyers (as shown in Fig. 6b). Therefore, the buyers satisfaction is lower (as shown in Fig. 6a) and the sellers revenue is lower (as shown in Fig. 6c) in the hotspot topology. 5.4 Spectrum Continuity In this section, we tweak the grouping algorithm to improve the spectrum continuity, and verify by simulation that the modified grouping algorithm increases the chance that buyers who win multiple spectrums get continuous spectrums. Naturally, buyers with multiple spectrum demand prefer to have continuous spectrums because 1) some devices can only work with continuous spectrums; 2) even if multiple devices are involved, it is still more desirable to have continuous spectrums due to easy management. However, during the grouping process, dummies of the same buyer are assigned randomly to different groups for different spectrums. So the auction results do not guarantee the spectrum continuity of the buyers who win multiple spectrums. A naive solution is to let buyers exchange spectrums after the auction. This is feasible if buyers have homogenous valuation for all spectrums. However, since buyers TABLE 3 Uniform vs Hotspot Topologies

11 3022 TABLE 4 Non-CT Grouping vs CT-Grouping Fig. 7. Non-CT spectrum allocation. have heterogeneous valuations, they are unwilling to exchange high-preferred spectrums for low-preferred spectrums. Also, the payment for different spectrums are different. It is impossible to carry out such spectrum exchange to accommodate buyers need for continuous spectrums. In order to mitigate the problem of spectrum discontinuity, we make changes to the algorithm in [15], which is used to find the maximal independent set in the 7th step of Algorithm 1. The major idea is: When seeking for the group g j for spectrum s j,ifanodevhas a brother dummy (we refer to dummies of the same buyer as the brother dummies) in g j 1, we increase the probability that v will be added in g j. But such bonus will be deprived when v is considered for the next group g jþ1.figs.7and8compare the spectrum continuity when using the original algorithm in [15] (referred to as Non-CT algorithm) and the modified algorithm (referred to as CT algorithm). In both figures, square dots marks the spectrums that a buyer has won; there is a line between two dots if they are continuous spectrums. In Fig. 7, most of the spectrums of a buyer are non-continuous. In comparison, in Fig. 8 with the modified algorithm, most buyers get continuous spectrums. To quantify the degree of spectrum continuity, we define a Continuity Index CI. CI is defined as the sum of intervals between two consecutive spectrums won by a buyer divided by the number of spectrums obtained. Let s m1 ;s m2 ;...;s mw denote the spectrums won by buyer m P w 1 i¼1 CI m ¼ ðm iþ1 m i 1Þ : (8) w 1 The higher the continuity index CI is, the less continuous all the spectrums are. Table 4 shows that CT algorithm significantly reduce the continuity index, with little sacrifice of buyers satisfaction, revenue and spectrum reusability. In fact, buyers satisfaction will be improved by winning continuous spectrums. 5.5 Running Time The most time-consuming step in TAMES is to find independent set on the interference graph. And the computational complexity increases as the number of sellers or the number of buyers increases because the number of iterations in the grouping processes increases. Figs. 9 and 10 show the running time of TAMES and adaptation of the existing auction mechanisms. We can see that the running time of TAMES is comparable with that of [14] and [6]. The running time of [13] is the smallest because there is no dummy creation, reducing the number of vertices in the interference graph. Fig. 8. CT spectrum allocation. Fig. 9. Running time when the number of sellers increases.

12 CHEN ET AL.: TAMES: A TRUTHFUL DOUBLE AUCTION FOR MULTI-DEMAND HETEROGENEOUS SPECTRUMS 3023 bid and single-unit demand are allowed or homogenous interference graph is used for grouping. We modified the grouping process to improve buyers spectrum continuity at little cost of spectrum utilization. The running time of TAMES is comparable with existing auction mechanisms. Fig. 10. Running time when the number of buyers increases. 6 RELATED WORK Many works have studied one-seller-multi-buyer forward auction. In [16], [17], the authors designed auction for spectrum allocation, but spectrum reusability is not considered. In [5], [8], [9], [18], spectrum reusability is exploited by assigning the same spectrum to non-interfering neighbors, but they don t consider that the set of interfering neighbors is frequency dependent. Some works deal with multi-seller-multi-buyer double auction. A truthful double spectrum auction is proposed in [6], but it is restricted that each seller or buyer trades only one spectrum. In [4], the interference graph is assumed to be complete, therefore no spectrum reusability. In [19], the authors exploit spectrum locality but haven t considered how different transmission range would impact the allocation results. The paper [20] considers heterogeneous reserve price and bidding price but not the influence of spectrum frequency heterogeneity on the interference graph structure. The paper [13] proposes an auction mechanism for heterogeneous spectrum. However, it is restricted that each buyer can only demand one spectrum. In fact, the auction in [13] will become untruthful when extended to multi-demand heterogeneous spectrum auction by simply creating dummies. Instead of truthfulness, some works targets at revenue maximization [7], [10], [21]. Some works used interference temperature instead of interference graph [22], [23]. Major drawback is that spectrum information between any transmitter-receiver pair is needed, which is difficult to obtain. 7 CONCLUSION In this paper, we proposed TAMES, an double auction mechanism tailored for heterogeneous spectrum transaction. TAMES takes sellers multi-unit supply and heterogeneous asks, buyers multi-unit demands and heterogeneousbidsasinputandusessequentialgrouping to tackle the problem of variant interference graphs. We proved that TAMES preserves the nice economic properties of truthfulness and individual rationality. The simulation results show that TAMES generates higher revenue, buyer satisfaction, and spectrum utilization compared with the situation where only homogeneous REFERENCES [1] W. Vickrey, Counterspeculation, Auctions, Competitive Sealed Tenders, J. Finance, vol. 16, no. 1, pp. 8-37, [2] E. Clarke, Multipart Pricing of Public Goods, Public Choice, vol. 11, no. 1, pp , [3] T. Groves, Incentives in Teams, Econometrica: J. Economet. 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13 3024 Yanjiao Chen received the BE degree in electronic engineering from Tsinghua University, China, in 2010 and is currently pursuing the PhD degree from Hong Kong University of Science and Technology, Hong Kong. Her research interests include spectrum management for Femtocell networks and network economics. Jin Zhang received the BE and ME degrees in electronic engineering from Tsinghua University, Beijing, in 2004 and 2006, respectively, and the PhD degree in computer science from Hong Kong University of Science and Technology, Hong Kong in She is currently a Research Assistant Professor with Fok Ying Tung Graduate School in Hong Kong University of Science and Technology. Her research interests include cooperative communication and networking, cognitive radio networks, network economics, body area network and mobile healthcare. Qian Zhang joined HKUST in Sept where she is a full Professor in the Department of Computer Science and Engineering. Before that, she was in Microsoft Research Asia, Beijing, from July 1999, where she was the Research Manager of the Wireless and Networking Group. He has published about 300 refereed papers in international leading journals and key conferences in the areas of wireless/internet multimedia networking, wireless communications and networking, wireless sensor networks, and overlay networking. Dr. Zhang has received MIT TR100 (MIT Technology Review) worlds top young innovator award. She also received the Best Asia Pacific (AP) Young Researcher Award elected by IEEE Communication Society in year Her current research is on cognitive and cooperative networks, dynamic spectrum access and management, as well as wireless sensor networks.. For more information on this or any other computing topic, please visit our Digital Library at center networks. Kaishun Wu received the BS, MS, and PhD degrees from Wuhan University, China, in 1994, 1996, and 1999, respectively, all in computer science. He received the PhD degree in computer science and engineering from Hong Kong University of Science and Technology, Hong Kong, in He is currently a research assistant professor in Fok Ying Tung Graduate School with the HKUST. His research interests include wireless communication, mobile computing, wireless sensor networks and data