DIARY: A Differentially Private and Approximately Revenue Maximizing Auction Mechanism for Secondary Spectrum Markets

Transcription

1 DIARY: A Differentially Private and Aroximately Revenue Maximizing Auction Mechanism for Secondary Sectrum Markets Chunchun Wu, Zuying Wei, Fan Wu, and Guihai Chen Shanghai Key Laboratory of Scalable Comuting and Systems Deartment of Comuter Science and Engineering Shanghai Jiao Tong University, China {bubble chun, zu ying hi}@sjtu.edu.cn, {fwu, gchen}@cs.sjtu.edu.cn Abstract It is urgent to solve the contradiction between limited sectrum resources and the increasing demand from the ever-growing wireless networks. Sectrum redistribution is a owerful way to mitigate the situation of sectrum scarcity. In contrast to existing truthful mechanisms for sectrum redistribution which aim to maximize the sectrum utilization and social welfare, we roose DIARY in this aer, which not only achieves aroximate revenue maximization, but also guarantees bid rivacy via differential rivacy. Extensive simulations show that DIARY has substantial cometitive advantages over existing mechanisms. I. INTRODUCTION Recent years have witnessed the fast develoment of wireless technology, which has brought about a great increase in demand of sectrum resources. Sectrum is a scarce commodity controlled by governmental agencies(e.g., Federal Communications Commission (FCC) in US). Traditionally, static sectrum allocation scheme is adoted to determine resources distribution. However, studies show that such a static sectrum allocation scheme is inefficient due to the dramatic changes of sectrum utilization in both satial and temoral dimensions [12]. The current sectrum allocation roblem is that, large chunks of allocated sectrum are left idle most of the time at lots of laces, meanwhile unlicensed secondary users are badly in need of sectrum to carry out their work. To solve this roblem and imrove sectrum utilization, a number of auction-based dynamic sectrum allocation schemes are roosed [7], [17], [21], [23]. In fact, there are already some comanies like Sectrum Bridge [1] conducting sectrum auctions, which makes such dynamic schemes no longer merely theoretical. This work was suorted in art by the State Key Develoment Program for Basic Research of China (973 roject 14CB333 and 12CB3161), in art by China NSF grant , , and , in art by Shanghai Science and Technology fund 12PJ149 and 12ZR14149, and in art by Program for Changjiang Scholars and Innovative Research Team in University (IRT1158, PCSIRT) China. The oinions, findings, conclusions, and recommendations exressed in this aer are those of the authors and do not necessarily reflect the views of the funding agencies or the government. F. Wu is the corresonding author. Auctions are well-studied rotocols in economic theory for allocating scarce resources, and attractive because of their well-defined notions for various objective. A major difference between economic auctions and sectrum auctions is the satial reusability of sectrum. A single chunk of sectrum can be leased to multile secondary users as long as they do not interfere with each other. A natural goal of such auctions is to maximize the revenue for rimary users, since rimary users may not be willing to share their own sectrum resources unless there are sufficient incentives. Another major difference between sectrum auctions and traditional auctions is that sectrum auctions are held reeatedly due to the dramatic changes of sectrum utilization. Mostly revious studies on this issue neglect the reeatability of sectrum auctions. Clues about others rivate information may be concluded form historical records or revious rounds, so that siteful secondary users can use such information to cheat or collusion. These malicious maniulations not only abate victims enthusiasm for articiation in auctions, which reduces the long-term revenue of rimary users, but also cause some vindictive actions from secondary users who never have a chance to win [24]. The roblem of analyzing sensitive data with an eye towards maintaining its rivacy has existed for some time. However, most of the existing mechanisms cannot guarantee the bid rivacy. For instance, lots of revenue maximization mechanisms [11], [19], which make articiators to bid truthfully in order to guarantee strategy-roofness, obviously violate the bid rivacy. The recent notion of differential rivacy [2], [3], in addition with its own intrinsic virtue, can assure that articiants have limited effect on the outcome of the mechanisms. As a consequence, articiants will have limited incentive to lie and little worry about rivacy violations. For the reasons mentioned above, designing a channel auction mechanism, which could maximize the revenue of the rimary users and guarantee the bid rivacy, is recisely the goal of our work. Obviously, designing such an auction mechanism has its own challenges: Comuting Comlexity: We all know that to find otimal solutions to the general roblem of the channel allocation

2 is NP-comlete [], which means that the the otimization roblem of revenue maximization is NP-hard. It s imossible to find a deterministic solution to the revenue maximization roblem in olynomial time. Bid Privacy: Due to the satial reusability of the sectrum, differential rivate mechanisms belonging to the family of exonential mechanisms [1], [15] cannot be adoted directly. In this aer, we roose DIARY, which is a DIfferentially rivate and Aroximately Revenue maximizing auction mechanism for secondary sectrum markets. DIARY not only achieves aroximate revenue maximization, but also guarantees bid rivacy. To our best knowledge, we are the first to investigate the auction mechanisms which achieve both bid rivacy and aroximate revenue maximization. Our key contributions are listed as follows. First, we model the channel redistribution roblem as an auction, and roose a novel non-deterministic mechanism, DIARY, to achieve aroximate revenue maximization and bid rivacy via differential rivacy. Second, we rove that DIARY satisfies all requirements of our goal. Third, we conduct extensive exeriments to comare the erformance of DIARY with existing mechanisms RGTS [11], TSA [19] and PFR [8]. Results show that DIARY has substantial cometitive advantages over existing mechanisms, esecially when the cometition for channels is fierce. The rest of this aer is organized as follows. In Section II, we introduce our auction model and review some solution concets. In Section III, we show the design of our auction mechanism DIARY. Extensive exeriments are resented in Section IV to comare the erformance of the roosed mechanism with others. In Section V, we discuss the related works. In the end, we draw our conclusion and introduce our future work in Section VI. II. PRELIMINARIES In this section, we show our roblem model and introduce some solution concets about differential rivacy. A. Model We model the roblem of sectrum allocation as a sealedbid auction. Usually, in a sectrum auction, we refer to secondary users as bidders or buyers, and the rimary user as the seller. In this auction model, there are a number of buyers and a single seller. The seller, who has m idle channels, wants to lease the channels out to get rofit. We denote the m channels by C = {c 1,c 2,...c m }. A channel can be leased to multile buyers, if these buyers can communicate simultaneously and send/receive signals with an adequate Signal to Interference and Noise Ratio (SINR). We assume that there are n buyers, such as access oints(ap), who want to lease/buy channels to carry out their works, denoted by N = {1,2,...,n}. The buyers bids are reresented as b = (b 1,b 2,...,b n ). Each buyer has a erchannel valuation, which is rivate to himself, reresented by v = (v 1,v 2,...,v n ). We use r = (r 1,r 2,...,r n ) to indicate the demands rofile of the buyers. We use a matrix T to indicate the allocation of the channels, where T ik = 1 indicates that we allocate channel c k to buyer i. What s more, we use ik to indicate the rice of buyer i for using channel c k. Utilities of the buyers are defined as the difference between his valuation and the rice for using the channel. For examle, rice of buyer i is m k=1 ik. His utility after articiating in this auction will be: m (v i ik )T ik. k=1 Following the tradition, we assume buyers are selfish and rational, which means they select strategies to maximize their utilities. The seller wisher for a channel allocation without interference, and charges ingeniously to maximize his own revenue. The revenue of the seller is the sum of the charges to the buyers, denoted by f( b,): f( n m b,) = ik T ik. B. Solution Concets i=1 k=1 We review some solution concets used in this aer about differential rivacy here. Definition 1 (Differential Privacy [13]): A randomized function M gives ǫ-differential rivacy if for any inut vector b = (bi, b i ) and b = (b i, b i ) differing on a single item b i s bid, where b i indicates the bid vector of other buyers excet i, and all P P = Range(M), Pr[M( b) P ] e ǫ Pr[M( b ) P ]. Definition 2 ( f Sensitivity [14]): For user i, when b i b i = v i, let f be the difference of f((b i, b i ),) and f((b i, b i ),) over all P. If for all (b i, b i ) over all P, the following inequality holds, we say that the objective function f( b,) is f Sensitivity. f((b i, b i ),) f((b i, b i ),) f. (1) The solution concet of differential rivacy is roosed by Dwork [2]. McSherry and Talwar [13] combined differential rivacy and mechanism design for the first time. f Sensitivity guarantees that a single bidder s misreorting has limited effects on the outut. If the objective function is f Sensitivity, f will be a deterministic value, and it cannot be maniulated by the buyer. III. DESIGN OF DIARY In this section, we show the design details of DIARY. DIARY uses a novel method to achieve bid rivacy and aroximate revenue maximization in channel allocation. For clear illustration, we discuss the scenario in which each buyer is equied with a single radio, and can just request one channel. Multi-radio scenario can be easily extended by using elementary buyers to indicate each radio equied.

3 A. Design Details To achieve bid rivacy, we use a robabilistic mechanism M ǫ q to determine the rice for the winners. The mechanism can be divided into three hases: grouing, rice determination, and winner selection. Phase 1: Grouing Fig. 1. A simle conflict grah, in which there are 7 buyers A, B, C, D, E, F and G. The link between two buyers indicates a confliction. As the channel has satial reusability, we can model the confliction constraints by a conflict grah. In the conflict grah, each node indicates a buyer and each link between two buyers in the conflict grah indicates a confliction. In other words, the two buyers linked by an edge cannot work on the same channel simultaneously. With the method roosed in [22], such a conflict grah can be figured according to an adequate Signal to Interference and Noise Ratio(SINR). We divide the buyers into λ grous in a bid indeendent way, using existing grah coloring algorithms (e.g., [16]). The λ grous are denoted by G = {g 2,g 2,...,g λ }. Fig. 1 shows a toy examle, in which there are seven buyers who want to lease the channels, denoted by A, B, C, D, E, F and G. We divides all the buyers into grous. For examle, in Fig. 1, the 7 elementary buyers can be divided into 3 grous: g 1 = {A,D,F}, g 2 = {B,E} and g 3 = {C,G} or g 1 = {A,E}, g 2 = {B,C,F} and g 3 = {D,G}. Phase 2: Price Determination After dividing the buyers into non-conflicting grous, we determine the rice for the winners in each grou. A too low rice will cause a loss to final revenue, but increasing the offered rice casually has the otential to send all buyers home emty-handed. So the key oint is to find an adequate rice for each grou to maximize the revenue of the seller. Here, we use three stes to determine the rice for each buyer grou κ. Ste 1: We declare rices set P κ. The rices set, which is exactly the bids set in each grou κ, is enumerated as follows. P κ = {b i i g κ }. Ste 2: We calculate all otential revenues of grou κ according to its all otential rices. For each P κ, we use ϕ i () to indicate whether buyer i in grou κ wins (can afford the rice) or not. { if b i <, ϕ i () = 1 otherwise. Algorithm 1 Price Determination Inut: A set of elementary buyers N, a vector of bids b. Outut: A vector of rice. 1: (G,λ) = Grouing(N). 2: for all g κ G do 3: for all j P κ do 4: R κ ( j ) = j i g κ ϕ i ( j ). 5: ER κ ( j ) = ex(ǫr κ ( j )). 6: end for 7: Prκ() =. 8: for all j g κ do ER κ( θ ) Pκ ERκ(). 9: Prκ(j) = θ j 1: end for 11: tm = rand(, 1). 12: for all j P κ do 13: if Prκ(j) < tm Prκ(j +1) then 14: κ = j. 15: break. 16: end if 17: end for 18: end for 19: return. We define the function q(g κ, b,) to calculate the revenue of grou κ when the rice is : q(g κ, b,) = i g κ ϕ i (). (2) Ste 3: We determine the final rice of each grou κ. A robabilistic mechanism is adoted to determine the final rice of each grou, which is denoted by M ǫ q : M ǫ q := Pr κ () e ǫq(gκ, b,). Intuitively, the robability of each rice being chosen increases exonentially with its corresonding revenue. But a single articiant s bid change just have limited multilicative influence on the robability of the relevatn rice being chosen. We choose κ as the final rice for the winners in grou κ, the revenue of this grou will be q(g κ, b, κ) = κ ϕ i ( κ). (3) i g κ For all the buyer grous, we use the above three stes to determine the rice for the winners. A vector is used to indicate final rices determined for the λ grous: = ( 1, 2,..., λ). Algorithm 1 shows the seudo code for the rice determination rocess. R κ indicates the set of all the otential revenues, where the rices is in the set P κ : R κ = {R κ () P κ }. Line 3-6 calculates all the otential grou revenues for grou g κ. Line 7-1 calculates the robability of being chosen.

4 Algorithm 2 Winner Selection Inut: A set of buyers N, a vector of bids b, a set of the idle channels C, the number of idle channels m, a vector of rice. Outut: A set of winners W, an allocated matrix of channels T. 1: T = n,m. 2: cnt = min{m,λ}. 3: G = Sort(G) based on grou revenue. 4: for all g κ G do 5: if cnt then 6: cnt = cnt 1. 7: for all i g κ do 8: if ϕ i ( κ) == 1 then 9: T iκ = 1. 1: end if 11: end for 12: WS = g κ {i i g κ,ϕ i ( κ) = }. 13: W = W WS. 14: end if 15: end for 16: return W and T. Then, Line determines the final rice κ deending on a random variable. According to the mechanism, we can see that the ossibility of the rice being chosen will enjoy a exonential growth with the increase of corresonding revenue. Intuitively, the mechanism achieves aroximate revenue maximization, which will be roved in Section III-B. Phase 3: Winner Selection We now determine the winning grous and winners in each winning grou. There are m idle channels to be leased out and λ grous waiting to lease channels. If λ m, then all the grous are winning grous. If λ > m, we sort the grous in non-increasing order according to the grou revenue, denoted by G. In case of a tie, we break it randomly. G : g 1 g 2... g κ... g λ. B. Analysis Now we are going to rove that DIARY achieves differential rivacy and aroximate revenue maximization. Differential Privacy We assume that for any grou g κ, a single buyer s bid change in b can change q(g κ, b,) by q, which means that the objective function q(g κ, b,) is q sensitivity. Lemma 1: When there is at most one siteful bidder, M ǫ q gives (2ǫ q)-differential rivacy. Proof: When there is one siteful bidder, he just can belong to one grou. Without loss of generality, we assume the siteful bidder belongs to grou g κ. Here, P κ = {b 1,b 2,...,b i,...,b n }. Assume a buyer i misreorts his bid b i as b i, for any P κ, the density of M ǫ q at is equal to e ǫq(gκ,(bi, b i),) P κ e ǫq(gκ,(bi, b i),) e ǫ q e ǫq(gκ,(b i, b i), ) e ǫ q P e κ ǫq(gκ,(b i, b i), ) =e 2ǫ q e ǫq(gκ,(b i, b i), ) P e. κ ǫq(gκ,(b i, b i), ) This gives a factor of at moste ǫ q in the numerator and at least e ǫ q in the denominator, giving e 2ǫ q in total. Intuitively, M ǫ q gives (2ǫ q)-differential rivacy. Lemma 2: When there are at most τ siteful bidders, M ǫ q gives (2ǫτ q)-differential rivacy. Proof: The bid change of a single buyer can change q by q. When there are at most τ siteful bidders, we assume that k buyers will be divided into one grou. A single buyer s bid change in a grou can change the revenue by q, then k(k 1) bidders bid changes in a grou can change q by k q. Then, no matter how the τ buyers will be divided, τ buyers bid change can change the revenue by at most τ q. Here, b = (b 1,b 2,...,b i,...,b i+τ 1,...,b n) and the rice set P κ is: P κ = {b 1,b 2,...,b i,...,b i+τ 1,...,b n }. (5) We choose the first m grous with higher revenue as the winning grous in grou set G. In each winning grou g κ, buyers whose bid is higher than the final rice are winners. Algorithm 2 shows the seudo code for the winner selection rocess. At the end of the auction, the seller collects all the ayments as his revenue. To illustrate clearly, we define Q(G, b) as the final revenue: Q(G, b) = min{m,λ} κ=1 q(g κ, b, κ). (4) Using Theorem 1, for any P κ, we can get that: e ǫq(gκ, b,) P κ e ǫq(gκ, b,) e τǫ q e ǫq(gκ, b,) e τǫ q P e ǫq(gκ, b, ) κ = e 2τǫ q e ǫq(gκ, b,). P κ eǫq(gκ, b, ) So (2ǫτ q)-differential rivacy have been guaranteed. (6)

5 We can get the following theorem according to the Lemma 1 and Lemma 2: Theorem 1: DIARY achieves ǫ -differential rivacy. Aroximate Revenue Maximization Lemma 3: Let q(g κ, b,) be a q sensitivity objective function and P κ. Then for any b and < ǫ < 1, E M ǫ q [q(g κ, b,)] (1 ǫ)max q(g κ, b,) δ, where δ = 1 ǫ ln(1 ǫ g κ ). Proof: For a fixed vector of bids b, we denote by P ˆ κ = {ˆ P κ : q(g κ, b, ˆ) < max q(g κ, b,) δ}. Then, for any ˆ P ˆ κ, the following holds: M ǫ q(ˆ) = e ǫq(gκ, b,ˆ) P κ e ǫq(gκ, b,) eǫ(max q(gκ, b,) δ) e ǫmax q(gκ, b,) = e ǫδ. Then we can get thatmq( ǫ P ˆ κ ) = ˆ P ˆ κ Mq(ˆ) ǫ P ˆ κ e ǫδ g κ e ǫδ. What s more,mq(p ǫ κ \ P ˆ κ ) 1 g κ e ǫδ. The above calculation results imly: E [q(g κ, b,)] Mq ǫ (max q(g κ, b,) δ)mq(p ǫ κ \ P ˆ κ ) (max q(g κ, b,) δ)(1 g κ e ǫδ ). We substitute for δ, then we get: E [q(g κ, b,)] Mq ǫ (max q(g κ, b,) δ)(1 ǫ) (1 ǫ)maxq(g κ, b,) δ. Then we draw the conclusion that E M ǫ q [q(g κ, b,)] (1 ǫ)maxq(g κ, b,) δ. (7) We use Equation 7 to all winning grous, which gives min(m,λ) E Mq ǫ κ=1 min(m,λ) κ=1 [q(g κ, b,)] (1 ǫ)max q(g κ, b,) δ) min(m,λ) =(1 ǫ)maxq(g, b) δ. κ=1 We can draw the following theorem according to Lemma 3: Theorem 2: DIARY achieves revenue aroximate maximization. IV. NUMERICAL RESULTS In this section, we do extensive exeriments to comare the erformance of DIARY with the existing mechanisms. A. Evaluation Setu In our evaluation setu, we assume that bidders are deloyed in a large geograhic area randomly, and then aly a distancebased interference model to roduce the corresonding conflict grah. In our exeriment, we choose meters as default terrain. Any two bidders within 425 meters are suosed to conflict with each other. We assume that all bidders true valuation is uniformly distributed over (,1]. The results are averaged over runs to obtain exected results. Revenue of Seller Revenue of Seller Revenue of Seller 3 1 DIARY-6C RGTS-6C TSA-6C PFR-6C Buyers (a) There are 6 channels available. DIARY-12C 1 RGTS-12C TSA-12C PFR-12C Buyers 8 6 (b) There are 12 channels available. DIARY-24C RGTS-24C TSA-24C PFR-24C Buyers Fig. 2. B. Evaluation Results (c) There are 24 channels available. Revenue of DIARY, RGTS, TSA and PFA In our first set of exerimental results, we comare the revenue of DIARY with existing mechanism RGTS [11], TSA [19] and PFR [8]. In RGTS [11], the valuation of bidder is relaced by virtual valuation. We use the normal distribution here. In TSA [19], bidders are groued into several cells, adjacent cells conflict with each other. In PFR [8], an iterative algorithm is used to determine the winner and the rice. As shown in Fig.2, DIARY outerforms other mechanisms in most cases. Fig.2 Also shows that when the number

6 of buyers is very small or channels are sufficient, DIARY achieves similar revenue with RGTS and PFR. We can see that the more fierce the cometition is, the better erformance DIARY gets. Revenue Fig DIARY RGTS TSA PFR 5x5 1x1 Terrain 15x15 x Revenue of DIARY, RGTS, TSA, PFR in different terrains Then we comare the erformance of DIARY with existing auction mechanisms in different terrains. The terrains range from 5 5 meters to meters while remaining the density of bidder. We assume that there are, 8, 18 and 3 buyers in 5 5, 1 1, and meters resectively. The result is shown in Fig.3. DIARY roduces better erformance than other mechanisms in different terrain areas. As the terrain area increases or the cometition gets more fierce, DIARY s advantages over RGTS, TSA and PFR become more significant. V. RELATED WORKS In this section, we review related works on channel allocation schemes mechanism design via differential rivacy. Recently, dynamic mechanisms of sectrum auctions has been widely concerned, esecially in asect of strategy-roof [6], [17], [21], [23] and revenue maximization [5], [11], [19]. However, the existing revenue maximization mechanisms all lose sight of the imortance of bid rivacy, which may cause a series of negative influence. Only a handful of work [9] guarantee the rivacy of bidders, but comromising the revenue unfortunately. Since the revenue maximization is well-studied, the novel idea of our work is that we achieve bid rivacy via differential rivacy. So here we focus on the related works about differential rivacy. Dwork [2] roosed the solution concet of differential rivacy for the first time, then McSherry and Talwar [13] combined the differential rivacy and mechanism design, and gave a general method to deal with revenue maximization in unlimited suly auction, attribute auctions and constrained ricing roblems. Dwork [4] introduced the definition of differential rivacy and two basic techniques for achieving differential rivacy, meanwhile showed some interesting alications of the techniques. Recently, Xiao [18] argued that the study of rivacy must be couled with the study of incentives, and introduced a model combining differential rivacy with truthfulness and efficiency. Nissim et al. [15] designed an general aroximately otimal mechanism via differential rivacy. Nissim et al. [14] also modeled the rivacy-aware agents and designed a rivacy-aware mechanism via differential rivacy. VI. CONCLUSION In this aer, we have designed DIARY, which is a differentially rivate and aroximately revenue maximizing auction mechanism for secondary sectrum markets. We have roven that DIARY not only achieves differential rivacy but also achieves aroximate revenue maximization. The exerimental results have shown that DIARY outerforms the existing mechanisms. REFERENCES [1] Sectrum bridge, htt:// [2] C. 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