A Truthful Double Auction for Device-to-device Communications in ...

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A Truthful Double Auction for Device-to-device Communications in Cellular Networks Peng Li, Member, IEEE, Song Guo, Senior member, IEEE, and Ivan Stojmenovic Fellow, IEEE,

Abstract—As the emergence of new wireless applications, data traffic in cellular networks has dramatically increased in recent years, which imposes an immediate requirement for large network capacity. Although many efforts have been made to enhance the wireless channel capacity, but far from being solved due to physical limit and cost. Device-to-device (D2D) communication is recently proposed as a novel and promising solution for increasing network capacity. However, most existing work on D2D communications focuses on optimizing throughput or energy efficiency, without considering economic issues. In this paper, we propose a truthful double auction for D2D communications (TAD) in multi-cell cellular networks for trading resources in frequency-time domain, where cellular users with D2D communication capability act as sellers, and other users waiting to access the network act as buyers. Both intra-cell and inter-cell D2D sellers are accommodated in TAD while the competitive space in each cell is extensively exploited to achieve a high auction efficiency. With a sophisticated seller-buyer matching, winner determination and pricing, TAD guarantees individual rationality, budget balance, and truthfulness. Furthermore, we extend our TAD design to handle a more general case that each seller and buyer ask/bid multiple resource units. Extensive simulation results show that TAD can achieve truthfulness and good performance in terms of seller/buyer sanctification ratio and auctioneer profit.

I. I NTRODUCTION S the booming growth of wireless applications, data traffic in cellular networks has dramatically increased in recent years. For example, AT&T reports that traffic grew 30fold from 2009 to 2010 in United States, and NTT DoCoMo’s mobile data traffic grew 60% from year to year in Japan [1]. Such an explosion of data traffic has imposed an immediate need of large network capacity for modern cellular networks such that more users and new applications can be accommodated with satisfied performance. To increase network capacity, many efforts focus on improving wireless channel capacity by exploring new coding schemes or advantages of multiple antennas. For example, turbo coding/processing [2], [3] is proposed to approach the Shannon limit on channel capacity, while space-time coding [4], [5] increases the possible channel capacity by exploiting the rich multipath nature of fading wireless environments. Although these technologies are successful in increasing wireless channel capacity, they are far from solving the network capacity enhancement problem because: (1) wireless channel capacity has physical limit and cannot be increased infinitely,

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P. Li and S. Guo are with the School of Computer Science and Engineering, The University of Aizu, Japan, e-mail: {pengli, sguo}@u-aizu.ac.jp I. Stojmenovic is with School of Electrical Engineering and Computer Science, University of Ottawa, Canada, e-mail: [email protected],ca.

Fig. 1. An illustration of D2D communications in multi-cell networks.

(2) equipping multiple antennas incurs additional hardware cost, which is not adopted by most of mobile devices, and (3) traffic growth is faster than the progress of communication technologies. As an alternative, Device-to-Device (D2D) communication [6] provides a new communication model to the network capacity enhancement problem in cellular networks by letting two devices in proximity of each other establish a direct local link for data transmission. After offloading some data traffic to D2D links, multiple users can transmit simultaneously in a single-collision domain. An example of D2D communication is shown in Fig. 1, where solid arrows represent the transmissions under traditional cellular mode, and dashed arrows represent D2D links. Under cellular mode, nodes 1 and 2 communicate over an uplink and a downlink while all the other nodes (e.g., nodes 3 and 4) in the same cell should keep silence. When D2D communication is enabled, node 1 can send data to its destination node 2 over a direct link if the link quality is good enough. Meanwhile, node 3 or 4 can access the network if they cause no interference to that D2D pair. The D2D communications can happen not only between a pair of nodes within a single cell, which is referred to as an intra-cell D2D pair, but also between those located in difference cells, which is called an inter-cell D2D pair, such as nodes 5 and 6 in Fig. 1. Usually, D2D communication is managed by base stations, i.e., base stations determine whether a pair of users communicate under cellular mode or D2D mode.

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Although there are many existing work on D2D communications, all of them are based on optimization frameworks, i.e., base stations, as central controllers, determine the transmission modes of all users in cellular networks with the objective of optimizing throughput or energy efficiency. In practice, however, user transmissions are affected by economic considerations. Traditionally, each cellular user occupies some network resources, in form of spectrum and time slots, exclusively by making a payment to the network operator. It is reasonable to reward a D2D user because of its vacated cellular links for space multiplexing. In other words, existing cellular users can sell their network resources by changing their transmissions into the D2D mode, e.g., nodes 1 and 2 in Fig. 1. On the other hand, the waiting users can access the cellular network by buying network resources from D2D pairs, e.g., node 3 or 4 in Fig. 1. When we study D2D communications from an economic perspective, a market-based approach is desired to enhance the utilization of network resources such that cellular users are willing to use D2D communications with financial incentives, and others can gain access to cellular networks with an acceptable payment. Auction is a popular market-based allocation mechanism that allows fair and efficient resource allocation. In this paper, we propose a truthful double auction scheme for D2D communications (TAD) in multi-cell wireless networks, in which existing cellular users that can potentially use D2D communications act as sellers and other users that would like to access the network act as buyers. An auctioneer accepts asks from sellers and bids from buyers, based on which it then allocates network resources and determines the trading price for both sellers and buyers. Although several general double auction schemes have been proposed in [7]–[9], they cannot be directly applied for D2D communications considered in this paper because they fail to guarantee the interferencefree between sellers and buyers. Moreover, multi-cell cellular network environment imposes new challenges for auction design since both intra-cell and inter-cell sellers should be taken into consideration in an efficient way. Our double auction design guarantees the following economic properties that are necessary for a practical auction. (1) Individual rationality: all participants in the auction obtain a non-negative utility. (2) Budget balance: the auctioneer has a non-negative profit after the auction. (3) Truthfulness: all participants in the auction should ask/bid according to their true valuation for the network resources. The main contributions of this paper are summarized as follows. • To the best of our knowledge, TAD is the first truthful double auction scheme for D2D communications in multi-cell networks. It works as an incentive mechanism to improve network resource utilization by encouraging existing cellular users to switch to the D2D mode, and unsatisfied ones to exploit the cellular transmission opportunities provided by D2D pairs. • We propose a novel design that accommodates both intra-cell and inter-cell D2D sellers while extensively exploiting the competitive space in each cell. With a sophisticated seller-buyer matching, winner determination and pricing design, TAD guarantees individual rationality,

budget balance, and truthfulness. We extend TAD to handle a more general case that each seller would contribute multiple resource units and each seller would purchase multiple ones as well. All three economic properties are still achieved in our extensions. The rest of this paper is organized as follows. We review related work in Section II. Section III introduces the system model considered in this paper. The basic auction design is presented in Section IV. We extend our work in Section VI. Simulation results are presented in Section VII, followed by conclusions in Section VIII. •

II. R ELATED WORK Device-to-device (D2D) communication becomes a hot research topic recently due to its benefits of offloading data traffic at base station in cellular networks. In their early work, Janis et al. [10] propose to facilitate local peer-to-peer communication by a D2D radio that operates as an underlay network to an IMT-Advanced cellular network. Later, they have studied D2D communication in three modes, i.e., reuse mode (D2D links share common channels with cellular links), dedicated mode (D2D links use dedicated channels), and cellular mode (all communication is relayed by base station), and designed mode selection algorithm for a three-user (one D2D pair and a cellular user) cellular network [11]. Based on a similar network model, Yu et al. [12] have investigated the throughput optimization problem over shared resources while fulfilling prioritized cellular service constraints. For more general models, D2D communications have been extensively investigated from aspects such as interference management, power control, spectrum sharing, and so on. For example, Janis et al. [13] have proposed a practical and efficient scheme for generating local awareness of the interference between cellular and D2D users at the base station, which then exploits the multiuser diversity inherent in the cellular network to minimize the interference. Kaufman et al. [14] have developed a distributed dynamic spectrum protocol, in which ad-hoc D2D users opportunistically access the spectrum actively in use by cellular users. A new interference management scheme is proposed to improve the reliability of D2D communication in [15]. They derive outage probability in close form and design a mode selection algorithm to minimize outage probability. Auction has been extensively studied in economics literature [7]–[9]. Recently, it becomes to show its power in solving networking problems. For example, a novel scheduling algorithm is proposed in [16] based on a two-phase combinatorial reverse auction with the objective of maximizing the number of satisfied users . Zhou et al. [17] have proposed a truthful and computation-efficient spectrum auction to support an eBay-like dynamic spectrum market. An auction scheme for the cooperative communications is designed in [18], where wireless node can trade relay services. Chen et al. [19] have proposed a Vickery-Clarke-Grove (VCG) based reverse aucntion for access permission transaction to promote hybrid access in femtocell networks. The most related work is [20], in which a sequential second price auction is proposed for D2D underlay cellular networks. Later, they have proposed a reverse iterative

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Fig. 2. Illustration of network resource unit.

combinatorial auction based on a similar model [21]. However, they are fundamentally different from our work on system model, studied problem, and auction design. III. S YSTEM

MODEL

In this section, we first present our system and auction models. We then introduce three economic properties that should be guaranteed in our auction design. A. Network Model In this paper, we consider a multi-cell multi-channel wireless network consisting of a number of cellular users as shown in Fig. 1. Each channel can be shared among several users by time division multiplexing. The network resources in both frequency and time domains can be represented by a matrix as shown in Fig. 2, where each time frame under any channel is referred to as a network resource unit. When all users work under traditional cellular mode, we suppose a feasible resource allocation scheme such that each user is assigned some network resource units according to its payment to the network operator. Meanwhile, a set of users W = {w1 , w2 , ..., wn } are waiting to access the network. Some may be the users rejected by the network operator due to limited network capacity, and others may be existing cellular users but still seeking more resource units to improve their performance. In the network, some existing cellular users form a set of source-destination (S-D) pairs that can potentially use D2D communications, which is denoted by S = {s1 , s2 , ..., sm }. When a local D2D link is enabled between a pair of nodes, other waiting users can access the network on the vacated cellular channel as long as they do not cause interference. To characterize the interference in the network, various interference models, such as protocol and physical interference models [22], [23], can be accommodated in our design. All nodes in the network have a single antenna and work under half-duplex mode that they cannot transmit and receive at the same time. B. Auction Model The TAD is designed as a single-round double auction, in which the S-D pairs in S with D2D communication capability are sellers, and users in W waiting to access the cellular network are buyers. If the S-D pair of seller si ∈ S is within a

single cell, si is referred to as an intra-cell seller. Accordingly, it is called an inter-cell seller if the S-D pair across two cells. Each user in W would like to use a cellular link. Each seller si ∈ S contributes one resource unit in each cell by submitting an ask ai based on its true valuation vis . All asks from sellers form a vector a = {a1 , a2 , ..., am }, and the vector without ask ai is denoted by a−i , i.e., a−i = {a1 , ..., ai−1 , ai+1 , ..., aM }. By letting pi be the payment of a winning intra-cell seller si from the auctioneer, its utility is defined by:  pi − vis , if si wins, usi = 0, otherwise. Accordingly, each buyer wj ∈ W submits a bid bj to purchase one resource unit. All bids from buyers form a vector b = {b1 , b2 , ..., bN }. Similarly, b−j denotes the bid vector without bj , i.e., b−j = {b1 , ..., bj−1 , bj+1 , ..., bN }, vjb denotes wj ’s valuation of a resource unit, and qj denotes the price needed to be paid to the auctioneer if wj wins. Finally, the utility of wj can be defined by:  b vj − qj , if wj wins, ubj = 0, otherwise. After collecting all ask/bid information from sellers and buyers, the auctioneer determines resource allocation, payment to winning sellers, and charges to winning buyers. The auction is sealed-bid, private and collusion-free, i.e., all participants simultaneously submit sealed asks/bids such that no one knows others’ asks/bids, and they do not collude with each other to improve the utility of the coalitional group. Moreover, all asks and bids from participants are static and will not change during the auction. C. Economic Properties To be a practical and useful design, TAD should satisfy the following three economic properties: (1) Individual rationality. A double auction is individual rational if no winning seller will be paid less than its ask and no winning buyer will be charged more than its bid, i.e., pi ≥ ai , ∀si ∈ S, qj ≤ bj , ∀wj ∈ W.

(1) (2)

(2) Budget balance. An auction is budget-balanced if the total income from buyers is no less than the payment to sellers, i.e., X X Φ= qj − pi ≥ 0, (3) wj ∈W

si ∈S

where Φ is referred to as profit of the auctioneer. (3) Truthfulness. A double auction is truthful if any participant (a seller or a buyer) cannot improve its utility by submitting a bid different from its true valuation, no matter how other participants ask/bid, i.e., usi (ai , a−i ) ≥ usi (a′i , a−i ), ∀a′i 6= ai = vis , ubj (bj , b−j ) ≥ ubj (b′j , b−j ), ∀b′j 6= bj = vjb .

(4) (5)

Note that an ideal auction design should also optimize the network resource utilization. Unfortunately, it has been proved

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that no double auctions can simultaneously achieve above three economic properties while maximizing resource efficiency [24]. Considering above three properties are mandatory for a meaningful auction, we focus on a design that satisfies all of them, while making best efforts to improve system efficiency. IV. AUCTION D ESIGN In this section, we first present several design choices and the basic idea of TAD. Then, we present the auction algorithm of TAD in details. Finally, an example is given to show the working process of TAD. A. Basic idea An intuitive design for an auction in a multi-cell wireless network is to conduct auction in each cell independently. In such a design, the base station in each cell acts as the auctioneer and only intra-cell D2D pairs contribute resource units as sellers. This simple design is referred to as independent local auction (ILA) in this paper. Although ILA is easy to be implemented with low time-complexity because the auctions of multiple cells can be conducted simultaneously, its auction efficiency is low since all resources provided by potential intercell sellers are eliminated. For example, nodes 5 and 6 in Fig. 1 cannot join the auction in either side in ILA, and their D2D opportunity is wasted. A natural improvement is to treat multiple cells as a single market, and let all participants compete in the auction. We call this design global auction (GA) in this paper. Unfortunately, since each buyer is only interested in resources in its own cell, mandatorily including them all in the same competitive space would lead to unnecessary losses, and thus low efficiency. We will show this phenomenon using an illustrative example later. As a more sophisticated auction strategy, TAD conquers the weaknesses in above design by letting two cells associated with each inter-cell seller conduct local auctions sequentially. The ask of each inter-cell seller in the successor cell is determined by the auction results in its predecessor cell. An inter-cell seller can successfully sell its resources if and only if it wins in both cells. In such a way, inter-cell sellers can join the auction, while the influence among buyers and intra-cell sellers from different cells is eliminated. In the following, we present our auction algorithm in details. B. Auction algorithm The simplest way to implement sequential local auctions for inter-cell sellers is to conduct auction in each cell one by one, unfortunately, which would lead to high time-complexity. We observe that sequential local auctions are imposed only for neighbouring cells involving inter-cell sellers, which motivates us to divide the cells into several independent sets as shown in Algorithm 1, such that the cells in each set are not neighbouring with each other, and can conduct local auctions simultaneously. After all local auctions have been finished, we check the auction results of each inter-cell seller si in its associated cells c and c′ , which are returned by function cell(). If it losses in any cell, we remove it from the winner set of

the other cell as shown in lines 5 to 10. Otherwise, it can successfully sell its resources to buyers and get payment from the two cells. Algorithm 1 Auction Algorithm 1: We divide the cells into several independent sets denoted by {I1 , I2, , I3 , ...IK }, where the cells in each set are not neighboring with each other. 2: for k = 1 to K do 3: the cells in Ik execute the LCA algorithm simultaneously. 4: end for 5: for each inter-cell seller si do 6: (c, c′ ) = cell(si ) 7: if si wins in c, but losses in c′ then 8: remove si and its associated buyer wσc (i) from the corresponding winner set in c; 9: end if 10: end for We then present the details of function LCA shown in Algorithm 2, which consists of three main steps including sellerbuyer matching, ask-determination, winner determination and pricing. 1) Step 1: seller-buyer matching. Recall that a buyer can take the cellular channel yielded by a seller only if they cause no interference. We construct a conflict graph for cell c by including all sellers and buyers in cell such that their conflict relationship can be described by a matrix E = {exy } with binary elements, where x ∈ S c , y ∈ W c , and  1, if user x conflicts with user y, exy = 0, otherwise. All sellers and buyers in a cell forms a bipartite graph, where there is a link between a seller si and a buyer wj if esi wj = 0, i.e., wj can purchase the resource unit provided by si . Finally, the matching results are stored in σc . We also notice that the matching process is bid-independent such that it can avoid bid manipulation, i.e., sellers/buyers cannot strategically submit their asks/bids to change the formed groups, and hence their utilities. 2) Step 2: ask-determination. Each intra-cell seller will use their original ask in the auction. For each inter-cell seller si , it is easy to see that its associated cells c and c′ are included in two different independent sets. Without loss of generality, we suppose that the auction in c is conducted before that in cell c′ , i.e., c ∈ Ij , c′ ∈ Ik , and j < k. We let si first join the auction in cell c with ask aci = 0, and obtain payment Psc that may be less than original ask ai . To compensate the unsatisfied ′ portion, si asks with aci = max{ai − Pc′ , 0} in the auction of cell c′ . The ask-determination process is represented by lines 4 to 12 in Algorithm 2. 3) Step 3: winner-determination and pricing. Winning canc c didate sets Swin and Wwin are initialized to be empty first. We c sort all sellers in S according to their asks in non-decreasing order and all buyers in W c according to their bids in a nonincreasing order as shown in lines 13 and 14. The sorted lists are denoted as S c and W c , respectively. Then, we find the

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Algorithm 2 Local Cell Auction (LCA) 1: construct a bipartite graph based on S c and W c such that there is a link between si and wj if esi wj = 0; 2: find a matching in this bipartite graph and store the results in σ c ; 3: for each si ∈ S c do 4: if si is an inter-cell seller then 5: (c, c′ ) = cell(si ); 6: if c′ ∈ Ij and j < k then aci = max{ai − Pc′ , 0}; 7: else aci = 0; end if 8: else 9: aci = ai ; 10: end if 11: end for c c 12: Swin = ∅, Wwin = ∅; 13: Sort all sellers in S c to get a list S c = {si1 , si2 , ...si|S c | } such that aci1 ≤ aci2 ≤ ... ≤ aci|Sc | ; 14: Sort all buyers in W c to get a list W c = {wj1 , wj2 , ...wj|W c | } such that bj1 ≥ bj2 ≥ ... ≥ bj|W c | ; 15: Find the largest k such that bjk ≥ acik ; 16: if k < 2 then 17: all participants loss in cell c, and return; 18: end if 19: Qc = wjk , Pc = aik ; c c = 20: Swin = {si1 , si2 , ..., sik−1 }, and Wwin {wj1 , wj2 , ..., wjk−1 }; 21: for each si ∈ Swin do c c c 22: if wσc (i) ∈ / Wwin then Swin = Swin − {si }; 23: end for c 24: for each wj ∈ Wwin do c c c 25: if sσc−1 (j) ∈ − {wj }; = Wwin / Swin then Wwin 26: end for

maximum k that satisfies bjk ≥ aik . The first k − 1 sellers and buyer in the sorted lists become winning candidates, while other participants are sacrificed to guarantee truthfulness. We let the ask of the k-th seller be the payment to all winning sellers, and the bid of the k-th buyer be the price charged to all winning groups as shown in line 19. After putting winner candidates into sets Swin and Gwin , we continue to check each one in them to determine the final winners. Note that sellers and buyers are associated by function σc , and we use σc−1 to denote the reverse function. For each candidate si ∈ Swin , if its corresponding buyer wσc (i) is not included in Wwin , it should be removed as a loser. Otherwise, it becomes the final winner and obtain the payment Ps . Similar operations are conducted for each one in Wwin from line 24 to 26. C. An illustrative example For a better understanding, we use an example to show the working process of TAD, ILA, and GA, respectively, and compare their results. We consider a two-cell cellular network including 4 sellers and 5 buyers, among which s2 is an intercell seller. The corresponding conflict graph is shown in Fig. 3, where the number at each node is its valuation of a resource unit. All participants ask/bid truthfully in this example.

Fig. 3. An example of conflict graph.

Fig. 4. Sorting results of two cells in TAD.

1) TAD: Obviously, the example network can be divided into two independent sets: I1 = {c}, and I2 = {c′ }. We first consider the auction in cell c1 . The seller-buyer matching results are denoted by solid lines in Fig. 4. By sorting sellers in a non-decreasing order and buyer in a non-increasing order in cell 1, we find the maximum k = 3, which results in two sets c c of winner candidates Swin = {s2 , s1 } and Wwin = {w2 , w3 }. Seller s3 and buyer w1 lose in the auction, and their ask and bid become the payment and charging price for winning sellers and buyers, respectively. Since associated buyer s1 is c c not included in Wwin , s1 is removed from Swin . For a similar reason, w3 also loses. We then consider the auction in cell c′ by updating s2 ’s ′ ask as ac2 = a2 − Pc = 8 − 5 = 3. As shown in Fig. 4(b), we find the maximum k = 2, such that s4 and w4 loss, and their ask/bid becomes the payment Pc′ = 4 for winner s2 and charging price Qc′ = 5.5 for winner w5 . Finally, inter-cell seller s2 wins in both cells and obtain the payment Pc + Pc′ = 5 + 4 = 9. The winning buyers are w3 and w5 that are charged with price Qc = 7 and Qc′ = 5.5, respectively. The profit of auctioneer is (7 + 5.5) − (5 + 4) = 3.5. 2) ILA: The inter-cell seller s2 is eliminated from auctions in both cells. After a similar matching process with TAD, we sort all sellers and buyers in two cells and show the results in Fig. 5. In cell c, although s1 and w2 become winning candidates because k = 2, they finally lose because their associated buyer/seller are not included in winning sets. All participants in cell c′ loss because k = 1. Thus, the profit of auctioneer and utility of all participants is zero in ILA. 3) GA: We conduct seller-buyer matching, and continue to sort sellers and buyers in Fig. 6. Unfortunately, all participants lose according to the winner determination rule.

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Fig. 5. Sorting results of two cells in ILA. Fig. 7. An illustration for bid-monotonic.

Fig. 6. Sorting results in GA.

V. E CONOMIC A NALYSIS In this section, we provide the proof of individually rationality, budget-balance and truthfulness for TAD. Since the local auction in each cell is independent, these three economic properties are achieved if they are hold in each cell. Theorem 1: TAD is individually rational. Proof: We first consider winning intra-cell sellers and winning buyers in each cell c. According to Algorithm 2, it is c easy to see that for each intra-cell seller si ∈ Swin and each c buyer wj ∈ Wwin , we have: psi = Pc ≥ ai , pbj = Qc ≤ bj .

(6) (7)

Then, we prove the individually rationality for each winning inter-cell seller si . Without loss of generality, we suppose that si first participate the auctions in cell c, and then in cell c′ . ′ Based on the fact that aci = ai − Pc ≤ Pc′ , we have: psi = Pc + Pc′ ≥ Pc + ai − Pc ≥ ai .

(8)

Thus, individual rationality is achieved in TAD. Theorem 2: TAD is budget-balanced. Proof: According to Algorithm 2, we have Qc = wjk ≥ aik = Pc in the local auction of any cell c. Also, the number of winning sellers is the same with that of winning buyers, i.e., c c c c |Swin | = |Wwin |, which leads to |Wwin |Qc − |Swin |Pc ≥ 0. Thus, the aggregated profit of all cells is no less than zero. Lemma 1: Given b−j and a, if buyer wj wins by bidding bj , it also wins by bidding ebj > bj . Proof: Since the seller-buyer matching is independent of bids and asks, we suppose that buyer wj is matched with the same seller si with bj and ebj . As shown in Fig. 7, if buyer wj wins with bj , it must locate before the k-th position in the sorted list W. When wj bids ebj > bj , its position in the

Fig. 8. An illustration for ask-monotonic.

f is before than that in W. Since the resulting sorted list W order of other participants does not change, wj also wins in f W. Lemma 2: Given a−i and b, if seller si wins in a cell by asking ai , it also wins by asking e ai < ai . Proof: When si is an intra-cell seller, we suppose asks ai e respectively. Since and e ai result in two sorted lists S and S, sellers in both lists are sorted in non-decreasing order and e ai < ai , the position of ai in S is before that in S ′ as shown in Fig. 8. Thus, if seller si wins with ask ai , it also wins with ask e ai . We then consider that case that si is an inter-cell seller that wins first in cell c and then in cell c′ with ai . By submitting e ai , si still wins in cell c because aci = e aci = 0. Then, si c′ c c′ ′ asks e ai = e ai − Pc < ai in cell c . According to the similar ′ analysis for intra-cell sellers, si will also win with e aci in cell c′ . Lemma 3: Given b−j and a, if buyer wj wins by bidding bj and ebj , respectively, it is charged with the same price pbj . Proof: Without loss of generality, we assume bj < ebj , and buyer wj is located in cell c. As shown in Lemma 1, seller-buyer matching is bid-independent and buyer wj cannot change the auction result with ebj if it wins with bj . Therefore, the buyer in the k-th position is the same under both bids. According to the pricing strategy, buyer wj will be charged the same price pbj = Qc . Lemma 4: Given a−i and b, if seller si wins by asking ai and e ai , respectively, it is paid the same payment psi in this cell.

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Fig. 9. An illustration of truthfulness for buyers. Fig. 10. An illustration of truthfulness for sellers.

Proof: We first consider the case that seller si is an intracell seller. According to Lemma 2, the k-th seller in the sorted seller list will not change under asks ai and e ai if they both win in the auction. Therefore, each intra-cell seller si will be paid the same price by asking ai and e ai , respectively. We then consider the case that seller si is an inter-cell seller that wins first in cell c and then in cell c′ . Since si wins in cell c under both ai and e ai because aci = e aci = 0, it obtains ′ the same payment Pc . In cell c , si wins by asking aci and e aci , respectively. According to the similar analysis for intra′ cell sellers, si is paid the same payment Psc . Therefore, each inter-cell seller will be paid the same price by asking ai and e ai , respectively. Theorem 3: TAD is truthful for buyers. Proof: Without loss of generality, we suppose that buyer wj located in cell c bids bj = vjb and ebj 6= vjb , and the resulting utility is denoted by ubj and u ebj , respectively. Case 1: buyer wj loses under both bj and ebj . It leads to the same utility of zero, i.e., ubj = u ebj = 0. Case 2: buyer wj wins under bj and losses under ebj . Because of Theorem 1, wj obtains non-negative utility ubj ≥ 0, while ebj leading to zero utility u ebj = 0. Case 3: buyer wj loses under bj and wins under b′j . As e b ≥ Qc > bj . Therefore, shown in Fig. 9, there must be ebj > Q ′ b e c < 0. Since bid the utility when wj bids bj is u ej = vjb − Q b b b bj loses with utility uj , we have uj > u ej . Case 4: buyer wj wins under both bj and ebj . Because of Lemma 3, wj is charged with the same price pbj under both bids, which lead to the same utility, i.e., ubj = u ebj = vj − pbj .

Theorem 4: TAD is truthful for intra-cell sellers. Proof: Suppose seller si located in cell c asks aj = vis and e ai 6= vis , and the resulting utility is denoted by usi and u esi , respectively. Case 1: seller si loses under both ai and e ai . Obviously, we have usi = u esi = 0. Case 2: seller si wins under ai and loses under e ai . Since utility is always non-negative as guaranteed by Theorem 1, we have usi ≥ 0 and u esi = 0, thus, usi ≥ u esi . Case 3: seller si loses under ai and wins under e ai . Since si loses with ai , the payment P s in auction result is larger ai , we must have than ai , i.e., Pc > ai = vis . If si wins with e e ai > Pec ≥ Pc > ai as shown in Fig. 10. The resulting utility

is u esi = Pc − vis ≤ 0. Thus, we have usi ≥ u esi . Case 4: seller si wins under both ai and e ai . According to Lemma 4, they are charged with the same price psi , which leads to the same utility, i.e., usi = u esi . If seller si is an inter-cell seller across cells c and c′ , its utility in these two cells can be analyzed in a similar way. Therefore, TAD is truthful for sellers. Theorem 5: TAD is truthful for inter-cell sellers. Proof: We need to show that any inter-cell seller si cannot obtain higher utility by asking e ai 6= ai = vis . The resulting utility under e ai and aj is denoted by u esi and usi , respectively. Without loss of generality, we suppose that si first joins the auction in cell c, and then in cell c′ . We analyze the cases shown in Table. I, where ’L’ means losing, and ’W’ means winning. Case 1: seller si loses in two cells under both ai and e ai . Note that although si ask aci = 0 in cell c, it may loss when there exists multiple inter-cell sellers asking with zero in this cell. We have usi = u esi = 0. Case 2: seller si wins with ai and losses with e ai . It is easy to see that ui ≥ 0 and u ei = 0. Case 3: seller si losses with ai , and wins with e ai . We first consider the subcase that si loss in cell c′ . According aci = 0, and to Lemma 4, si wins in cell c by asking aci = e ′ obtains the same payment Pc = Pec . In cell c , si losses with ′ ′ aci , but wins with e aci . According to the similar analysis for ′ ′ aci . Based on the intra-cell sellers, we have aci > Pc′ ≥ Pec > e ′ fact that aci = ai − Pc , the utility u ei can be calculated by: u ei

= Pec + Pec′ − vis = Pc + Pec′ − ai ′ = Pc + Pec′ − (aci + Pc ) ′ = Pec′ − aci < 0

Under the subcase that si losses in cell c by asking aci = 0, si cannot win with another ask e aci ≥ 0, which makes this subcase impossible. Therefore, we have ui ≥ u ei . Case 4: We have shown in Lemma 4, when si wins with esi ; aj and e ai , it obtains the same payment. Thus, usi ≥ u VI. E XTENSION

Until now, all our discussions are based on a basic model where each seller contributes one time slot and each buyer

8

TABLE I D ATA CHARACTERISTICS OF DIFFERENT JOBS

ai

eai

Case 1 WL L* WL L*

Case 2 WW WL L*

Case 3 WL L*

Case 4

WW

WW

WW

would like to purchase one time slot in a single-cell cellular network. In this section, we extend TAD to handle a more general case that each participant asks/bids multiple resource units. Specifically, we consider that each seller si ∈ S contributes mi time slots by transmitting under D2D mode. Seller si has the same valuation vis for all mi time slots, which leads to a ask ai for each of them. Let psik denote the payment of its k-th time slot from the auctioneer, the utility function of si is redefined as follows. usi =

 Pmi

s k=1 (pik

0,

Fig. 11. An illustrative conflict graph with dummies.

− vis ), if si wins, otherwise.

Accordingly, each buyer wj ∈ W would like to purchase nj time slots with uniform valuation vjb and bid bj . We use pbjk denote the charging price of k-th time slot and wj ’s utility function is:  Pni b b k=1 (vj − pjk ), if wj wins, ubj = 0, otherwise. Under such a model, TAD can be designed as a multi-unit double auction based on previously developed techniques with minor modifications. Specifically, we create dummies for the multiple supplies and demands from sellers and buyers, respectively. The dummies of the same participant have identical interference relationship, and any two of them are connected by an edge in conflict graph. We then follow the seller-buyer matching in Algorithm ?? as well as winner determination and pricing in Algorithm ?? by treating each dummy as an independent participant. We use an example to illustrate how this extension works. For simplicity, we consider the auction in a single-cell network whose conflict graph is the same with the one of Cell 1 in Fig. 3, and suppose seller s2 would contribute 2 resource units and buyer w1 would purchase 2 resource units. The conflict graph with dummies s′2 and w1′ is shown in Fig. 11. Following Algorithm ??, we create a bipartite graph and find a maximum matching between sellers and buyers as shown in Fig. 12. Finally, we sort sellers and buyers according to the rules in Algorithm 2 and show the sorting results in Fig. 13. The maximum value of k is 4 and two seller-buyer pairs wins.

VII. P ERFORMANCE E VALUATION In this section, we conduct extensive simulations to evaluate the performance of TAD. For comparison, we also show the results of ILA and GA. The simulation settings are first introduced, and then we present the simulation results.

Fig. 12. Matching results with dummies.

A. Simulation Settings In our simulation settings, sellers and buyers are randomly distributed in 4-cell wireless networks. When we study the influence of number of participants, sellers’ asks and buyers’s bids are in uniform distribution between [0, 20] and [10,30], respectively. All results are averaged over 100 network instances. We consider the following metrics in our simulations: (1) seller satisfaction ratio in terms of percentage of winning sellers, (2) buyer satisfaction ratio in terms of percentage of winning buyers, and (3) auctioneer profit. B. Simulation results We verify the truthfulness of both sellers and buyers in TAD by considering a random network instance with 50 sellers and 50 buyers in each cell. We first randomly choose a seller with true valuation 5.52 and change its ask from 1 to 20. As shown in Fig. 14, its utility is 22.5785 under any ask between 1 and 16. The utility decreases to 11.89 when it asks 17 and further increasing its ask leads to utility of zero. We also change the asks of other sellers and obtain the similar observation. It indicates that a seller cannot increase its utility by asking untruthfully. We then randomly choose a buyer with true valuation 24.04. As shown in Fig. 15, its utility is 0 when the bid grows from 10 to 18. When the bid is set to the value greater than 18, we always have a utility 5.18. We also test other buyers using a similar method and all results show that a buyer cannot achieve a higher utility by biding untruthfully. After that, we study the influence of number of buyers on seller satisfaction ratio. The number of sellers in each cell is

9

6

5

Utility

4

3

2

Fig. 13. Sorting results with dummies. 1

0

25

−1 10

15

20

25

30

Fig. 15. Buyer utility under different bids.

15

Utility

20

Buyer bid

10

0.65

5

0 2

4

6

8

10

12

14

16

18

20

Seller ask Fig. 14. Seller utility under different asks.

fixed to 100. As shown in Fig. 16, seller satisfaction ratio grows as the number of buyers in each cell increases from 80 to 180 in all schemes. TAD outperforms ILA and GA under all cases. For example, seller satisfaction ratio of TAD is 0.45 in networks with 80 buyers in each cell, while the corresponding results of ILA and GA are 0.32 and 0.26, respectively. When the number of sellers in each cell grows to 180, TAD has a seller satisfaction ratio of 0.58 that is higher than 0.41 in ILA and 0.28 in GA. We also observe that the number of buyers has little influence on GA since the participants of different cells affect each other seriously. We show the buyer satisfaction ratio under different number of sellers in Fig. 17. The number of buyers in each cell is fixed to 100. The buyer satisfaction ratio shows as an increasing function of number of sellers for all auction schemes. In TAD, the ratio value grows from 0.5 to 0.72 as the seller number in each cell increases from 80 to 180, which is always higher than the corresponding results of ILA and GA. Moreover, the performance increasing rate becomes smaller when the number of buyer in each cell is larger than 140. That is because most sellers have been satisfied and further increasing the number of buyers has little contribution to performance. We study the affect of auction size on auctioneer profit by changing the number of both sellers and buyers in each cell from 80 to 180. As shown in Fig. 18, the profit of all auction schemes shows as an increasing function of auction sizes. That is because there are more winning sellers and buyers

Seller Satisfaction Ratio

0.6

TAD ILA GA

0.55 0.5 0.45 0.4 0.35 0.3 0.25 80

100

120

140

160

180

Number of buyers Fig. 16. Seller satisfaction under different number of buyers.

in auctions with larger size. In TAD, the profit grows from 166.62 to 292.97 as the auction size increases from 80 to 180. The profit of ILA and GA is always lower than that in TAD. They perform closely when auction size is less than 160. Although we have observed that ILA always outperforms GA in both seller and buyer satisfaction ratios in Fig. 16 and Fig. 17, respectively, GA has higher profit than that of ILA when auction size is greater than 140. Next, we study the influence of bid range on the performance metrics. We first consider the seller satisfaction and show the corresponding results in Fig. 19. In all schemes, the influence of bid range is not significant. For example, seller satisfaction ratio in TAD is 0.45 when bid range is [4,24]. Even after increasing the bid range to [12,32], the ratio is only 0.48. We attribute this phenomenon to the fact that auction results are also affected by the seller-buyer matching that is ask/bidindependent. Although increasing buyers’ bid can potentially increase the value of k in winner determination, many candidates are removed from winner sets due to the matching relationship. We have similar results for buyer satisfaction ratio

10

0.8

TAD ILA GA

0.55

Seller Satisfaction Ratio

Buyer Satisfaction Ratio

0.9

0.7

0.6

0.5

0.4

0.3

0.2 80

TAD ILA GA

0.5 0.45 0.4 0.35 0.3 0.25

100

120

140

160

0.2 [4,24]

180

[6,26]

Number of sellers

[8,28]

[10,30]

[12,32]

Bid range

Fig. 17. Buyer satisfaction under different number of sellers.

Fig. 19. Seller satisfaction under different bid ranges.

300 TAD ILA GA

0.8

Buyer Satisfaction Ratio

280 260

Profit

240 220 200 180 160 140 120 80

0.7

TAD ILA GA

0.6 0.5 0.4 0.3 0.2 0.1

100

120

140

160

180

Auction size

0 [4,24]

[6,26]

[8,28]

[10,30]

[12,32]

Bid range

Fig. 18. Auctioneer profit under different auction sizes.

Fig. 20. Buyer satisfaction under different bid ranges.

as shown in Fig. 20. Finally, we show the auctioneer profit under different bid ranges in Fig. 21. In contrast to Fig. 19 and Fig. 20, buyers’ bid has significant influence on auctioneer profit. For example, the profit of TAD increases from 67 to 230 as the bid range grows from [4,24] to [12,32]. That is because although the number of winning sellers and buyers has no big change, the larger bid may lead to a lower payment and a greater charging price. The performance of ILA and GA is close except under [12,32], where the profit of ILA is higher than that of GA.

extensively exploited to achieve a high auction efficiency. With a sophisticated seller-buyer matching, winner determination and pricing, TAD guarantees individual rationality, budget balance, and truthfulness. Furthermore, we extend our TAD design to handle a more general case that each seller and buyer ask/bid multiple resource units. Extensive simulation results show that TAD can achieve truthfulness and good performance in terms of seller/buyer sanctification ratio and auctioneer profit.

VIII. C ONCLUSION

R EFERENCES

In this paper, we study the D2D communication from an economic perspective. We propose a truthful double auction for D2D communications (TAD) in multi-cell multi-channel networks for trading resources in frequency-time domain, where cellular users with D2D communication capability act as sellers, and other users waiting to access the network act as buyers. Both intra-cell and inter-cell D2D sellers are accommodated in TAD while the competitive space in each cell is

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240 220

TAD ILA GA

200 180

Profit

160 140 120 100 80 60 40 [4,24]

[6,26]

[8,28]

[10,30]

[12,32]

Bid range Fig. 21. Auctioneer profit under different bid ranges.

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