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Ascending-Price Progressive Spectrum Auction forCognitive Radio Networks with Power-Constrained

Multi-Radio Secondary UsersChangyan Yi, Student Member, IEEE, and Jun Cai, Senior Member, IEEE

Abstract—In this paper, we investigate spectrum sharing withpower-constrained multi-radio secondary users (SUs) in cognitiveradio networks. The scenario under consideration consists ofa primary spectrum owner who runs auctions for leasing heridle channels, and multiple SU bidding for winning the usageof spectrum channels. Different from existing works in theliterature with an assumption of single-minded SUs, in thispaper, SUs can benefit from flexible quantity of channels. Inaddition, since each SU is ordinarily equipped with a fixednumber of radios, she cannot utilize the amount of channelsthat exceeds her radio capacity. Moreover, each SU has a certainpower limitation so that the quality of service (QoS) of hertransmission may also be constrained even though the numberof allocated channels is increased. To jointly address all thesechallenges, a novel ascending-price progressive auction algorithmis proposed, where the spectrum allocation decisions are made bygradually increasing the unit channel price. Theoretical analysesprove that the proposed algorithm meets the properties ofQoS satisfaction, individual rationality, incentive compatibility, andachieves Pareto optimality. Simulation results further demonstratethat the proposed auction algorithm can improve both the auctionrevenue and the social welfare, and increase the number ofwinning SUs compared to the counterparts.

Index Terms—Spectrum auction, cognitive radio networks,power limitation, quality of service, Pareto optimality.

I. INTRODUCTION

RECENT studies [1] revealed that radio spectrum scarcityhas become a critical limitation for future development

of wireless communications. Meanwhile, the existing rigidspectrum regulatory policy based on exclusive spectrum usageleads to significant spectrum under-utilization [2], [3]. Thus,to alleviate the burden of spectrum shortage through promot-ing utilization efficiency, cognitive radio (CR) [4]–[6] wasintroduced as a promising technology, which could providesecondary users (SUs) dynamic spectrum access to licensedchannels owned by primary users (PUs). The implementationof CR requires participation of not only SUs but also PUs,and thus motivates market-driven spectrum sharing [7]–[9],where PUs can obtain economic incentives for temporarilyleasing their unused spectrum while SUs are willing to payfor fulfilling their spectrum usage requests.

Auction mechanism is a natural and effective way in con-structing economic models for market-driven spectrum sharing

Copyright (c) 2015 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

C. Yi and J. Cai are with the Department of Electrical and ComputerEngineering, University of Manitoba, Winnipeg, MB, Canada R3T 5V6. E-mail: changyan.yi@umanitoba.ca, jun.cai@umanitoba.ca.

[10]. Generally, spectrum auction design needs to satisfy somefundamental properties, such as incentive compatibility andindividual rationality. Or in other words, all participants inspectrum auctions have to behave truthfully (i.e., their bestresponses are to report their truthful private information), eachindividual has to be guaranteed with non-negative utility afterthe service.

Although spectrum auction has been extensively studiedin the literature [11]–[14], some critical issues, especiallythose related to practical implementations, have not been welladdressed. One is the availability of multiple radios on a singlesecondary wireless device, which allows each SU to accessmultiple channels at the same time. This feature motivates theemployment of multi-unit combinatorial auction [15], whereSUs can request a bundle of radio channels. However, almostall existing multi-unit spectrum auction mechanisms [16]–[18]exploited a common assumption of single-minded SUs (i.e.,SUs will get exactly the amount of channels they want ornothing), which significantly limits the achievable spectrumefficiency. In fact, there are practical secondary applications,such as secondary file exchange (HTTP/FTP), that can alwaysbenefit from transmissions on any number of channels. Thus,different from single-minded demands, SUs need to evaluatethe valuations of their transmission services based on theactual number of channels allocated to them.

Another important issue, which has been omitted by allexisting spectrum auction designs, is the ordinary constrainton the maximum transmission power at each SU. Such powerconstraint will definitely affect the quality of service (QoS)(which is defined as the actual achievable channel capacityor transmission rate in this paper) that each SU can achieve.Intuitively, SUs with higher payments will expect better QoS.For example, consider that a user requires to download a fileby using wireless connections. In case that the channel accessis granted to this user (i.e., her downloading request can besuccessfully fulfilled), her valuation can be seen as the valueof the file. Naturally, this user would expect to have a higherdownloading rate (or a shorter delay) if she pays a higher price.Here, we define a new term, called “QoS expectation”, whichis SU’s expected QoS depending on her payment. However,due to the power limitation, obtaining more channels by payingmore may not necessarily result in a better QoS.

Thus, it is imperative for us to design a new spectrumauction mechanism with joint consideration of both flexiblespectrum demands and the satisfaction of SUs’ QoS expec-tations under power constraints. However, the introduction of

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these new features makes the spectrum auction design muchmore challenging due to the following aspects:

• Due to the multi-radio settings, each SU is enabled todemand flexible amounts of channels, and such flexibilityinduces a large solution space of spectrum allocationso that the complexity in finding an efficient auctionoutcome is greatly increased.

• SUs show different attitudes to the spectrum auctionbecause of not only their different valuations but alsotheir heterogeneous limitations on transmission powers.

• Because of the implicit relationship between the achievedQoS (which is constrained by the power limit) and theQoS expectation of SUs (which depends on the actualpayment), the formulated utility functions of SUs maybecome non-quasilinear so that Vickrey-Clarke-Groves(VCG) or any VCG-like mechanisms [19], which havebeen widely used in the spectrum auction literature, areno longer applicable.

Therefore, to address all these concerns, in this paper, wereinvestigate multi-unit spectrum auction in CR networks withpower-constrained multi-radio SUs. In the considered systemmodel, a primary spectrum owner (PO) owns multiple homo-geneous channels and seeks to auction off the idle ones (i.e.,channels that are not occupied by any PU) among multipleheterogeneous SUs. Each SU has a private valuation for eachsingle channel. But her total valuation won’t continuouslyincrease with the number of allocated channels and is boundedby her radio capacity, i.e., the number of radios equipped.Furthermore, by taking into account the power constraints,SUs may not be willing to demand as many channels aspossible within her radio capacity, because doing so mayresult in a compromised QoS worse than her expectationwith respect to the payment. Considering that the hardwareparameters (such as radio capacities and power constraints)are public information for all SUs and the PO, SUs will bid bystrategically reporting their unit valuations, and then the POwill make spectrum allocation and payment decisions basedon the received bids. To tackle the challenges as identified inthis trading procedure and the non-quasilinear utility functionsresulted from the consideration of power constraints, wepropose a new mechanism, called ascending-price progressiveauction algorithm, where the PO makes spectrum allocationdecisions at each auction round by gradually increasing theunit channel price, and the payment of each winning SU iscalculated as the total price over all channels she obtains at dif-ferent unit prices. Theoretical analyses prove that the proposedauction algorithm can meet the properties of QoS satisfaction,individual rationality, incentive compatibility, and achievesPareto optimality. Simulation results further demonstrate thatthe auction revenue and social welfare can be improved, andthe number of winning SUs can be increased by the proposedauction mechanism.

The main contributions of this paper are as follows:

• A new framework of spectrum auction is introduced forCR networks with power-constrained multi-radio SUs.

• The flexibility of channel demands from SUs is consid-ered in the design of spectrum auction mechanism to relax

the common assumption on single-minded demands.• The impacts of both limited radio capacities and power

constraints are explored mathematically in the formula-tion of SUs’ utility functions.

• To avoid the complexity in achieving social welfare orauction revenue maximization, Pareto optimality is intro-duced as the measurement for evaluating the efficiencyof spectrum auction design.

• A novel spectrum auction mechanism, called ascending-price progressive auction algorithm, is proposed.

• Both theoretical analyses and numerical results are pre-sented to prove that the proposed mechanism is econom-ically robust and efficient.

To the best of our knowledge, this work is the first thatjointly considers flexible spectrum demands and wirelessusers’ power constraints in designing an efficient spectrumauction mechanism.

The rest of this paper is organized as follows: SectionII summarizes recent related works. Section III presents thesystem model and problem description. Section IV showsthe spectrum auction design for power-constrained multi-radioSUs in detail. Simulation results are illustrated in Section V.Finally, we give a brief conclusion in Section VI.

II. RELATED WORK

Market-driven spectrum sharing for self-interested wirelessusers in CR networks has been widely studied in the pastdecade. For instance, Niyato et al. [20] considered the problemof spectrum trading among multiple competitive PUs and SUs,where the dynamic behaviors of SUs were analyzed by usingevolutionary game model. In [21], Yang et al. proposed apricing-based spectrum access control, where the competitionswere modeled by a three-stage Stackelberg game. Gao etal. [22] designed a spectrum sharing framework with qualitydiscrimination, where the PU could offer a set of quality-price contracts for leasing channels to certain SUs’ types.The authors in [23] developed a contract-based approach forcooperative spectrum sharing between single PU and SU.

Different from these traditional game and contract theoret-ical approaches, auction mechanism determines the allocationoutcome based on the interaction among multiple strategicindividuals with privately possessed information. This featurewell fits the characteristics of CR networks, where SUs arealways intelligent, competitive and selfish, and information iscommonly asymmetric. Zhou et al. [11] introduced a generalframework of double spectrum auction to support dynamicmulti-agent spectrum trading. The authors in [12] incorporatedmarket locality in the design of spectrum auction and provedits truthfulness. Both works focused on the auction modelswith single channel demand from SUs and ignored the poten-tial multi-channel demands. Chen et al. [16] investigated anauction framework for heterogeneous spectrum transactions,where each buyer can bid for any specific amount of channelsshe desired. In [17], Li et al. presented a truthful auctionscheme in which the spectrum bandwidth was allocated ina time-frequency division manner. Yi et al. [18] proposed anovel multi-item spectrum auction algorithm by considering

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that the PU could recall the auctioned channels from winningsecondary buyers for protecting its own primary spectrumusages. However, all these works assumed that SUs had single-minded utility functions so that they are inapplicable if partialsatisfaction of SUs’ channel demands is allowed.

Spectrum auction design for scenarios with multi-radiospectrum buyers has been recently studied [24]–[27], wherethe authors relaxed the restriction of single-minded requestby allowing spectrum buyers to demand flexible numbers ofchannels. Specifically, Feng et al. [24] considered a spectrumauction among secondary wireless service providers, whereeach of them could flexibly determine the channel demand andthe corresponding bidding value according to the requirementof its end users. Zhan et al. [25] developed a short-intervalspectrum auction framework, where each SU was allowedto win part of her spectrum request. Both of these worksassumed that the utility function of each spectrum buyer wasan increasing function with respect to the number of allocatedchannels. However, this assumption may not be true in reality.By considering the hardware restrictions, each SU may have alimit on her demand due to her radio capacity. The authorsin [26] and [27] formulated the problem of heterogeneousspectrum allocation as multi-minded combinatorial auctions,where each secondary spectrum buyer was enabled to bidfor multiple channel bundles within her radio capacity. How-ever, all aforementioned works ignored a fact that each SU’sachievable QoS may be further constrained by her availabletransmission power. In other words, the utility functions ofSUs may need to be reformulated by taking into account theimpact from their power constraints.

In summary, this paper differs from [20]–[23] by consid-ering spectrum trading among multiple strategic SUs withprivately possessed valuation information. Unlike [11]–[13]where each SU can only demand a single channel in spectrumauctions, we allow SUs to have heterogeneous multi-channeldemands. In addition, this work does not require SUs to besingle-minded, which is different from the models in [16]–[18]. Besides, the problem addressed in this paper can betteraddress practical applications in wireless networks than [24]–[27] by jointly considering both radio capacity and powerconstraint on each SU. Last but not least, although ourdesigned auction approach is inspired by the framework ofclinching auction [28], [29], it is more delicate and applicablefor spectrum sharing with the consideration of hard demandlimit (i.e., radio capacity) and non-linear relationship betweenpower constraint and payment.

III. SYSTEM MODEL AND PROBLEM DESCRIPTION

In this section, we describe the system model under con-sideration and present the corresponding spectrum auctionframework along with the design requirements.

A. System Model

Consider a CR network with N SUs and one primaryspectrum owner (PO) who owns multiple homogeneous andindivisible additive white gaussian noise (AWGN) channels toserve subscribed PUs [24], [30]. Because of the complicated

problem formulation resulted from flexible spectrum adoptionand power-constrained QoS expectation considered in thispaper, we limit our discussions to the scenario with a single POand leave the multi-PO scenarios in our future works (somemore detailed discussions on this extension can be found inend of this section). If at some periods there are idle channelsavailable, the PO will lease out these temporarily unusedchannels to SUs who require opportunistic spectrum accessby running spectrum auctions. Assume that such spectrumredistributions (i.e., auctions) are conducted periodically, andthe primary spectrum usage is static within each period. Suchassumption has been widely used for short-interval secondaryspectrum market [16], [17], [24]–[26], [30], [31]. In this sce-nario, we can limit our discussion on the spectrum allocationin one period and consider that there are M idle channelsavailable for leasing at the beginning of the considered period.

Due to hardware limitations, we assume that each SU i isequipped with qi ∈ Z+ radios (where qi is finite) so thatshe can support the transmissions on at most qi channelsin any auction period. Similar to existing spectrum auctiondesigns [16], [32], [33], we assume that each SU i has auniform user-specific valuation vi ∈ R+ (which may dependon the specific wireless applications, transmission capacitiesand power costs) for each channel. However, different fromtraditional designs, due to the limitation on the number ofradios, the total valuation of SU i won’t keep increasing whenthe number of channels allocated to SU i is larger than qi. Bytaking this into account, the valuation function of any SU ican be defined as

Vi(ci) =

civi, if 0 < ci ≤ qi,ciqi, if ci > qi,

0, otherwise,(1)

where ci ∈ Z represents the number of channels allocated toSU i. Notice that the defined valuation function Vi(ci) is non-linear with ci. In addition, unlike most of existing works withrestriction of single-minded valuation functions, the definitionin (1) implies that SUs can benefit from the transmissions onany number of channels.

The transmission power of each SU i is bounded by Pi ∈R+. Similar to [18] and [34], we define the achievable QoSof SU i as the maximum capacity (or transmission rate) byobtaining ci channels, i.e.,

Ri(ci) =

ci log2

(1 + Pi

cin0

), if 0 < ci ≤ qi,

qi log2

(1 + Pi

qin0

), if ci > qi,

0, otherwise,

(2)

where n0 represents the noise power spectral density, and thespectrum bandwidth of each channel has been normalized to1. Each SU has a desired QoS expectation which dependson her payment for transmission service. To characterize therelationship between the QoS expectation and the payment,we introduce a pre-defined increasing function1 f(·) to match

1For explanation purpose, the defined function is assumed to be homoge-neous for all SUs. However, our designed auction mechanism can also beapplied to the scenarios with heterogeneous fi() for each SU i.

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Fig. 1. An illustration on the QoS expectation of SU i.

the intuition that the more money the SU pays, the better QoSshe expects. Let Θi and θi be the total payment and averageunit payment of SU i for accessing ci channels, respectively.Then, the following constraint has to be satisfied regardless ofthe channel demand limit qi,

Ri(ci) ≥ f(Θi) = f(θici). (3)

Equation (3) indicates that the achieved QoS of SU i shouldnot be worse than her expectation from the payment Θi. Eventhough both Ri(·) and f(·) are increasing functions of ci, welimit our discussions on the situation that there always existsan allocation threshold c∗i ≤ qi such that Ri(ci) ≥ f(θici)for any ci ≤ c∗i , and Ri(ci) < f(θici) otherwise. Note that,Ri(ci) is a concave function of ci due to the power limitation[35]. Moreover, if such c∗i does not exist, we have only twotrivial cases: i) Ri(ci) ≥ f(θici) for all ci, which implies thatPi is considerably large (i.e., the impact of power constraint isnegligible) so that the QoS expectation can always be satisfied;or ii) Ri(ci) < f(θici) for all ci, which means that the QoSexpectation can never be met. An example on the relationshipbetween Ri(ci) and f(θici) is shown in Fig. 1.

For explanation purpose, we assume that all SUs are locatedwithin the strong interference range of each other so that onechannel cannot be accessed by multiple SUs simultaneously.This assumption has been commonly adopted for spectrumauctions within small regions [18], [24], [25], [32], [36], [37].For large-scale networks where SUs are weakly interfered witheach other, the users’ clustering methods [11], [33], [38] can beused to group multiple SUs with negligible mutual interferenceas super spectrum buyers in the spectrum auction to achievepotential spectrum spatial reuse. In this case, the interferencedue to frequency reuse can be modeled as a noise term incalculating SUs’ data rates. Since this is not the main focusof this paper, we omit the associated discussions, and referinterested readers to [11], [33], [38] for more details.

Furthermore, it is expected that our proposed auction mech-anism can be extended to multi-PO scenarios by decouplingthe multi-PO network into multiple single-PO subsystemsthrough an additional association process [39], [40]. Or, wecan consider to redesign the auction mechanism by incor-porating any one of the following approaches: i) Similar to[12], [13], [16], we can consider that there is a centralizedspectrum regulator who is able to collect information from allPOs and SUs, and then runs a double-sided spectrum auction;

TABLE IIMPORTANT NOTATIONS IN THIS PAPER

Symbol MeaningN number of SUsM number of auctioned channelsqi radio capacity of SU ivi SU i’s private valuation for each channelci number of channels allocated to SU iVi(ci) SU i’s valuation with regard to ciPi power constraint of SU iRi(ci) achievable QoS of SU i with regard to ciΘi total payment of SU if(Θi) expected QoS of SU i by paying Θi

Ui utility of SU iθ updated unit price of each channelm remaining number of unallocated channelsDi(θ) demand of SU i at price θ

and ii) Following the existing analytical framework in [32],a distributed multi-auctioneer progressive spectrum auctioncan be designed, in which each PO (acted as an auctioneer)systematically raises the trading price of her own channels,and each spectrum buyer subsequently chooses one PO forreporting her spectrum request.

For convenience, Table I lists all important notations usedin this paper.

B. Auction Model and Design Requirements

At the beginning of an auction period, the PO, who actsas the seller, runs a spectrum auction for leasing M idlechannels, and N SUs bid for obtaining appropriate amountof channels. Since the radio capacities and power constraintsof SUs are system factors and cannot be changed arbitrarily,it is reasonable to treat both qi and Pi, 1 ≤ i ≤ N , as publicinformation for the PO and all SUs. However, the user-specificvaluation vi is a private information and belongs to SU i only.In fact, vi implies the SU i’s willingness to pay. Hence, thebidding process only includes the report of SUs’ valuations.

Without loss of generality, we assume that all SUs are selfishand risk-neutral. Then, the utility function of each SU i canbe formulated as

Ui =

{Vi(ci)−Θi, if Ri(ci) ≥ f(Θi);

infeasible, otherwise.(4)

The interpretation of this utility function is as follows: whenthe QoS expectation can be met for SU i, Ui equals thedifference between SU i’s total valuation, i.e., Vi(ci), and hertotal payment Θi; otherwise, the resulting spectrum allocationoutcome is infeasible. Notice that even though it is intuitivethat each SU i will be unwilling to demand/adopt more chan-nels than her radio capacity, the formulated utility function in(4) does not rule out the possibility of having ci > qi for amore comprehensive consideration.

As the seller and the auctioneer, the PO determines thechannel allocation and the payment for each SU after receiv-ing all bids. The outcome of the spectrum auction can bedenoted as (c,Θ), where vectors c = (c1, c2, . . . , cN ) andΘ = (Θ1,Θ2, . . . ,ΘN ). In order to eliminate some trivialresults, we assume that the total radio capacity of SUs is no

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less than the total number of auctioned channels so that thecompetition among SUs always exists, i.e.,∑

1≤i≤N

qi ≥M. (5)

Inequality (5) also matches the fact in practical CR networksthat the total channel demand from SUs is always larger thanthe number of idle primary channels. Then, we can defineour design goals for the spectrum auction among power-constrained multi-radio SUs in the following.

Definition 1: An auction outcome (c,Θ) is generally fea-sible if the following properties can be satisfied.

• Allocation feasibility: The total amount of channels al-located to SUs is no more than the total number ofauctioned channels provided by the PO, i.e.,∑

1≤i≤N

ci ≤M. (6)

• QoS satisfaction: The achievable QoS of each SU cannotbe worse than her desired expectation, i.e.,

Ri(ci) ≥ f(Θi), for 1 ≤ i ≤ N. (7)

• Individual rationality: No SU will be charged more thanher total valuation, i.e.,

Vi(ci)−Θi ≥ 0, for 1 ≤ i ≤ N. (8)

With these properties, the PO can allocate her idle channelsfor economic profit without harming her own PUs, and allSUs can obtain corresponding QoS satisfaction with regardto their payments and non-negative utilities. In summary, allindividuals are incentivized to participate in the auction.

However, as we have mentioned earlier, the outcome ofthe auction (c,Θ) is determined based on the received bids(i.e., the reported valuations from all SUs). Since SUs areselfish for maximizing their own utilities, they may not revealtheir private valuation information truthfully (i.e., potentiallymisreport their valuation for each channel) in the biddingprocess if and only if they can benefit from such behaviors. Tomaintain a robust auction equilibrium which can resist marketmanipulations, we define the property of incentive compatibil-ity (or truthfulness) for our auction design as follows.

Definition 2: An auction mechanism (c,Θ) is truthful orincentive-compatible if for all SUs with private valuationinformation (v1, v2, . . . , vN ) and every potential misreport ofbidding information (v1, v2, . . . , vN ), we always have

Ui = Vi(ci)−Θi ≥ Vi(ci)− Θi = Ui, for 1 ≤ i ≤ N, (9)

where (ci,Θi) and (ci, Θi) are decisions of channel allocationand payment for SU i with the bid vi and vi, respectively.Inequality (9) implies that no SU i can obtain extra utilitygain by misreporting vi 6= vi.

The objective of our considered spectrum auction frame-work, i.e., Pareto optimality, is defined as follows.

Definition 3: An auction outcome (c,Θ) is Pareto-optimalif for decision (ci,Θi), 1 ≤ i ≤ N , there is no other decision(c′i,Θ

′i), 1 ≤ i ≤ N , such that utilities of all SUs and the

profit of the PO can be further improved, i.e., (c,Θ) is Pareto-

optimal if and only if

Ui = Vi(ci)−Θi ≥ Vi(c′i)−Θ′i = U ′i , for 1 ≤ i ≤ N, (10)

and ∑1≤i≤N

Θi ≥∑

1≤i≤N

Θ′i, (11)

with at least one strict inequality.Simulations in Section V will show that by achieving Pareto

optimality in the considered auction framework, our proposedalgorithm can also improve both the auction revenue and socialwelfare, and increase the number of winning SUs comparedto other existing works.

IV. ASCENDING-PRICE PROGRESSIVE SPECTRUMAUCTION ALGORITHM

From the expression of (4), we can easily observe thatthe utility functions of SUs are non-quasilinear. Thus, anyVCG-like mechanisms cannot be applied since they cannotguarantee the economic robustness of auctions under non-quasilinear settings [19], [41]. To address this issue, in thissection, we propose an ascending-price progressive spectrumauction algorithm and then prove that it can satisfy all desiredrequirements as defined in Section III.

A. Design of the Auction Algorithm

In the proposed ascending-price auction algorithm, the POdetermines the allocation of her channels to SUs by graduallyincreasing the unit price of each channel. Thus, channels thatare allocated to SUs may have different prices, and the totalpayment of a SU equals the sum of prices for all channelsshe obtains. Notice that the ascending-price procedure onlyinvolves in the determination of channel allocations at the PO.In other words, i) there is no price negotiation among the POand SUs; ii) no decision information will be revealed beforethe end of the whole process; and iii) the PO who acts as thecentral controller will solely determine the auction outcomes.

To formally present the algorithm, some notations are firstintroduced in the following. Let ci denote the number ofchannels that have been already allocated to SU i, Θi representher accumulated payment, and Qi = qi − ci be her remainingdemand limit. Let θ and m denote the current unit price ofa channel and the number of remaining unallocated channels,respectively. Initially, we set θ = 0 and m = M . We furtherdefine the individual demand of SU i at price θ as

Di(θ) =

{min{bdi(θ)c, Qi}, for θ < vi,

0, otherwise,(12)

where di(θ) is computed by solving

Ri(ci + di(θ)) = f(Θi + θdi(θ)),

which is equivalent to (i.e., by substituting (2))

(ci+di(θ)) log2

(1+

Pi(ci + di(θ))n0

)=f(Θi+θdi(θ)). (13)

Therefore, the obtained di(θ), which can satisfy this condition,represents the maximum number of channels that can be

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further allocated to SU i at the unit price θ without breakingthe condition of her QoS expectation (i.e., Ri(ci + di(θ)) ≥f(Θi + θdi(θ))). Clearly, di(θ) can be explicitly expressedif a closed-form expression of f(·) is applied. However,without loss of generality, we do not limit f(·) to be anyspecific function throughout the analyses. According to therequirement of QoS satisfaction in (7), bdi(θ)c represents thelargest number of channels that SU i can further request withinher QoS expectation. In other words, ci + bdi(θ)c is exactlythe allocation threshold (i.e., a upper bound) such that theQoS expectation of SU i can still be met. In addition, di(θ)is obviously a decreasing function of θ due to the definitionof f(·) in (3). Therefore, the demand function Di(θ) of eachSU i is non-increasing with the unit price θ. Note that, thedemand of each SU i at different price θ, i.e., Di(θ), can bedirectly calculated by the PO after she receives the bids fromall SUs, and thus no further information exchange betweenSUs and the PO is required after the bidding process.

The general procedure of the proposed spectrum auctionalgorithm can be described as follows. When auction begins,all SUs bid by reporting the PO their private valuations foreach channel. After collecting all bids, the PO graduallyascends the unit price θ, and θ keeps increasing as long as thetotal channel demand from all SUs with regard to the currentunit price is strictly larger than the number of remainingunallocated channels m. If for any SU i and a certain valueof θ, the total demand of all the other SUs becomes strictlybelow m, i.e., δ = m −

∑j 6=iDj(θ) > 0, then the PO

determines to allocate δ channels to SU i with a charge of δ ·θ.After that, all relevant variables are updated as ci ← ci + δ,θi ← θi+δ ·θ, θi ← θi−δ and m← m−δ, where the symbol“←” refers to “be updated as”. When the value of θ satisfiesthe condition that m =

∑iDi(θ), all SUs will be assigned by

their demands, and the auction ends.Since the demand function defined in (12) is apparently

non-continuous with θ, the total demand from SUs will dropintegrally with the increase of θ and may even drop by severalchannels at once. This leads to a possible situation that m <∑iDi(θ) but m >

∑iDi(θ

+)2. This causes the situation thatthe available channels cannot be completely shared among SUsand the requirement of Pareto optimality is broken. Similar tothe discussions in [29], two kinds of potentially problematicpoints can be identified: i) when the unit price θ reaches vi forsome SU i, an abrupt change of demand will happen accordingto (12); and ii) when the price θ reaches a threshold θ∗i forsome SU i, where θ∗i is calculated from (ci + 1) log2

(1 +

Pi(ci+1)n0

)= f(Θi + θ∗i ). This implies that Di(θ) = 0 for all

θ > θ∗i . In order to avoid problems caused by these specialpoints, we define

D−i (θ) = limx→θ−

Di(x), (14)

D+i (θ) = lim

x→θ+Di(x), (15)

so that when θ = vi ≤ θ∗i , Qi > 0, we have Di(θ) = 0,but D−i (θ) > 0, and when θ = θ∗i ≤ vi, Qi > 0, we have

2In this paper, “+” and “−” are used to indicate right- and left-handedlimit, respectively.

Algorithm 1: Ascending-Price Spectrum AuctionInitialization: θ ← 0,m←M ;

ci ← 0,Θi ← 0, Qi ← qi for all SUs.1 Compute Di(θ), D

−i (θ), D+

i (θ) for each SU i;2 while

∑Ni D+

i (θ) > m do3 if there exists a SU i such that D+

−i(θ) =∑

j 6=iD+j (θ) < m

then4 ci = ci + (m−D+

−i(θ)),5 Θi = Θi + θ · (m−D+

−i(θ)),6 Qi = Qi − (m−D+

−i(θ)),m = D+−i(θ);

7 Recompute D+i (θ) and then go back to step 3.

8 else9 Increase θ by a predetermined increment ∆θ and then go

back to step 1.

10 otherwise11 if there exists a SU i such that D+

i (θ) > 0 then12 ci = ci +D+

i (θ),Θi = Θi + θ ·D+i (θ),

13 m = m−D+i (θ);

14 else if there exists a SU i such that Di(θ) > 0 and m > 0 then15 ci = ci + min{Di(θ),m},16 Θi = Θi + θ ·min{Di(θ),m},17 m = m−min{Di(θ),m};18 else if there exists a SU i such that D−i (θ) > 0 and m > 0 then19 ci = ci + min{D−i (θ),m},20 Θi = Θi + θ ·min{D−i (θ),m},21 m = m−min{D−i (θ),m};22 else23 Stop and return ci,Θi for all SUs.

Di(θ) > 0, but D+i (θ) = 0.

With the use of (14) and (15), the detailed procedure of theproposed auction algorithm can be summarized in Algorithm1. Note that, since the total demand from SUs and theremaining unallocated channels will reduce identically duringthe allocation process, any SU who is able to be allocatedby a certain number of channels at a given unit price willbe definitely granted with these channels before the priceincreases. Or in other words, there is no ordering requirementin the allocation process for a given unit price θ. Moreover,the condition in step 2 of the algorithm is examined based onD+i (θ) rather than Di(θ). This can ensure that the auction will

not terminate at a price that is just a bit higher than θ∗i of anySU i.

Complexity of Algorithm 1: Obviously, the key condition(in step 2), i.e.,

∑Ni D

+i (θ) > m, will be broken only when

θ becomes sufficiently large. Initially (with θ = 0), the differ-ence between

∑Ni D

+i (θ) and m is

∑Ni=1 qi−M . By increas-

ing θ with a predetermined increment of ∆θ,∑Ni=1 qi−M de-

creases by at least N ·min{(2∆θ−1)Pi/n0} (which can be eas-ily derived from (13)). Thus, the condition of

∑Ni D

+i (θ) > m

will lead to at most |∑Ni=1 qi−M

N ·min{(2∆θ−1)Pi/n0}| iterations before

violation. Therefore, the overall complexity of Algorithm 1 canbe eventually expressed asO(N2|

∑Ni=1 qi−M

N ·min{(2∆θ−1)Pi/n0}|+N3),

which is polynomial with respect to the network scale.From this algorithm, we can immediately observe that

both allocation feasibility and QoS satisfaction have beenguaranteed since the maximum number of auctioned channelsfrom the PO is set as M and the number of channels allocatedto each SU will not exceed her maximum individual demand

7

Fig. 2. An illustration of the time-line (T � τ ).

calculated by (12). In next subsections, we will further provethat the proposed auction algorithm can meet the properties ofindividual rationality, incentive compatibility, and can achievePareto optimality.

For convenience, the time-line of implementing the pro-posed auction algorithm is shown in Fig. 2, and its detailedinterpretation is summarized as follows:• At the beginning of an auction period T , the PO hosts a

spectrum auction for temporarily leasing M idle channelsand asks all SUs located within the radio coverage tosubmit their bids (i.e., vi for each SU i).

• The PO receives all bids from SUs and runs an ascending-price spectrum allocation algorithm (i.e., Algorithm 1)to determine the auction outcomes including channelallocations and charges.

• The PO announces the final auction outcome (c,Θ) andcollects the payments from all winning SUs.

• All winning SUs will then access the allocated channelsduring the rest of T .

B. Proofs of Individual Rationality and Incentive Compatibil-ity

Since the proposed auction algorithm runs by graduallyincreasing θ, we define a progress index t and denote θt asthe unit price at instant t. Accordingly, we use cti and Θt

i todenote the number of allocated channels and the total paymentof SU i at t, respectively. If SU i is granted some channels att, we define µti as her marginal utility, i.e., her utility gain fora single channel obtained at t. µti can be calculated as

µti =

{vi − θt, if cti ≤ qi and θt ≤ θ∗i , (16a)< 0, otherwise. (16b)

Notice that when cti > qi or θt > θ∗i , the unit value vi is non-positive and thus µti must be negative. Next, we first prove animportant property of the defined marginal utility.

Lemma 1: For any SU i, her marginal utility µti is non-negative if and only if θt ≤ vi. Or in other words, if θt ≤ vi,equation (16a) must hold.

Proof: Let δt denote the number of channels that arenewly allocated to SU i at instant t. According to Algorithm1, δt 6= 0 if and only if θt ≤ vi. In addition, the channelallocation can only happen in steps 3, 11, 14 or 18. Weconsider these possibilities in two cases.• Case I: δt channels are allocated to SU i in steps 3, 11

or 14. Specifically,

i) if this happens in step 3, we must have∑iD

+i (θt) >

m, and thus δt = m −∑j 6=iD

+j (θt) = m −∑

j D+j (θt) +D+

i (θt) < D+i (θt) ≤ Di(θ

t);ii) if this happens in step 11, we have δt = D+

i (θt) ≤Di(θ

t);iii) if this happens in step 14, we must have δt =

min{Di(θt),m} ≤ Di(θ

t).In summary, δt ≤ Di(θ

t) always holds in this case.Assume that SU i is granted with δt channels just aftert− ε, where ε is arbitrarily small. Then, we have

cti = ct−εi + δt ≤ ct−εi +Di(θt)

= ct−εi + min{bdi(θt)c, qi − ct−εi }≤ ct−εi + qi − ct−εi = qi.

(17)

• Case II: δt channels are allocated to SU i in step 18.In this case, we must have m > 0, D−i (θt) > 0 andDi(θ

t) = 0. With the definition (14), such case happensonly when θt = vi. Since δt ≤ D−i (θt) and D−i (θt) =D−i (vi) = limx→v−i

D(x) = min{bdi(vi)c, Qi} =

min{bdi(θt)c, qi − ct−εi }, where v−i indicates the left-handed limit of vi, we have

cti ≤ ct−εi + min{bdi(θt)c, qi − ct−εi } ≤ qi. (18)

Both Case I and Case II indicate that cti ≤ qi. Besides,δt 6= 0 also implies that f(θt−εi + θ∗i ) = (ct−εi + 1) log2(1 +

Pi(ct−εi +1)n0

) ≥ f(θt−εi + θt). Recall that f(·) is an increasingfunction, and thus θ∗i ≥ θt. Hence, θt ≤ vi can automaticallylead to cti ≤ qi and θt ≤ θ∗i , which are conditions for (16a).Then, we have µti = vi − θt and µti ≥ 0, when θt ≤ vi.

With Lemma 1, we can now prove the properties of indi-vidual rationality and incentive compatibility in the followingtheorems.

Theorem 1: The proposed ascending-price spectrum auctionalgorithm is individual-rational.

Proof: We can observe from Algorithm 1 that any truthfulSU i will be granted with channels only when the unit priceis lower or equal to her unit value, i.e., θ ≤ vi. Since themarginal utility of SU i is always non-negative under suchcircumstance (based on Lemma 1), we can conclude that theutility of each SU is non-negative throughout the auction andthe individual rationality can be guaranteed.

Theorem 2: The proposed ascending-price spectrum auctionalgorithm is incentive-compatible.

Proof: According to the procedure of Algorithm 1, thevalue declared by each SU i determines the unit price at which(or when) she will be precluded from the decision process ofAlgorithm 1. According to Lemma 1, the marginal utility ofchannels granted to SU i is non-negative when θt ≤ vi andstrictly negative when θt > vi. Hence, the utility of SU ithroughout the auction will increase with vi for θt ≤ vi, anddecrease with vi for θt > vi. In conclusion, the utility of eachSU is maximized by declaring her truthful value, which provesthe incentive compatibility.

8

C. Proofs of Pareto Optimality

Obviously, it is difficult to directly prove the achievement ofPareto optimality by checking all possible auction outcomesaccording to Definition 3. Thus, we first transform the generaldefinition of Pareto optimality to a more explicit form (whichis easier to work with) by carefully considering the featuresof the proposed spectrum auction framework and the requiredauction properties explained in Definitions 1 and 2.

Proposition 1: The outcome (c,Θ) is Pareto-optimal forthe considered spectrum auction if and only ifa)∑

1≤k≤N ck = M ;b) ck ≤ qk, for 1 ≤ k ≤ N ;c) for any 1 ≤ i, j ≤ N such that vi > vj and cj > 0, we

must have either ci = qi or f(Θi+vj) > (ci+1) log2

(1+

Pi(ci+1)n0

).

Proof: Please see Appendix.The physical meaning of Proposition 1 can be interpreted

as follows: an auction outcome is Pareto-optimal if and only ifi) all idle channels of the PO are completely allocated to SUs;ii) no SU is granted more channels than her radio capacity;and iii) no SU can obtain a non-zero number of channelsunless the other SUs with higher unit valuations have alreadyreached their bounds (i.e., either their radio capacities havebeen exhausted or their QoS satisfactions can no longer bemet for adopting more channels).

Before applying Proposition 1 for proving the Pareto opti-mality of the proposed auction algorithm, some prerequisitesneed to be examined. For notation simplicity, we denote∑iDi(θ),

∑iD

+i (θ) and

∑iD−i (θ) as D(θ), D+(θ) and

D−(θ), respectively. Since the individual demand is a non-increasing function of the unit price θ, D(θ), D+(θ) andD−(θ) will also be non-increasing with θ. Moreover, withdefinitions in (14) and (15), for any continuity point of D(θ),we have D(θ) = D+(θ) = D−(θ). Whereas, for discontinuitypoints of D(θ), we have the following lemma.

Lemma 2: If θd is a discontinuity point of D(θ) such thatD+(θ) > m for all θ < θd, we have D−(θd) > m.

Proof: If D(θ) is discontinuous at θ = θd, it is expectedthat D(θ) is continuous in the interval [θd − εθ, θd) for anysmall value of εθ. Since D(θ) =

∑iDi(θ), Di(θ) is also

continuous in [θd − εθ, θd) for any SU i. With previous

definitions, we have D−i (θd) = limx→θd− Di(x) = Di(θd −

εθ) = D+i (θd − εθ), and thus D−(θd) = D+(θd − εθ).

Furthermore, D+(θ) > m for θ < θd directly indicates thatD+(θd − εθ) > m. Hence, we have D−(θd) > m under suchcondition.

Next, we show that no channel remains unallocated afterAlgorithm 1.

Lemma 3: In the proposed ascending-price progressivespectrum auction, all available channels of the PO will becompletely allocated to SUs.

Proof: Suppose that the auction enters step 10 of Al-gorithm 1 at a unit price θ. It is required to prove thatD−(θ) ≥ m by the end of the auction. First, we prove thatD−(θ) ≥ m at the beginning of step 10. Since θ is the pricethat makes the auction algorithm enter step 10, we must haveD+(θ) > m at the beginning of step 2 for any θ < θ.

Considering the fact that D+(θ) and m will both decreaseby the same amount δ, if δ channels are allocated in step 3,D+(θ) > m can be maintained before step 7. However, whenθ increases to θ, we may still have D+(θ) ≥ m if θ is acontinuity point or D+(θ) < m if θ is a discontinuity point.For the former case, step 2 will be repeated (if D+(θ) > m) orall channels will be allocated in step 11 (if D+(θ) = m). Forthe latter case, we must have D−(θ) > m because of Lemma2. After step 10, channels may be allocated in steps 11, 14 or18. Similar to step 3, steps 11 and 14 will keep the relationshipof D−(θ) > m. In step 18, min{D−i (θ),m} channels will beallocated to SUs with D−i (θ) > 0. Since D−(θ) > m beforestep 18, m channels must be completely allocated.

With all above lemmas, we can finally prove the Paretooptimality of the proposed auction algorithm.

Theorem 3: The proposed ascending-price spectrum auctionalgorithm can produce an outcome that is Pareto-optimal.

Proof: We can prove this theorem by checking the con-ditions stated in Proposition 1. Condition a) has already beenproved by Lemma 3. Besides, we have proved in Theorem 1that the marginal utility of each SU i is always non-negative,which implies ci ≤ qi, and thus condition b) is satisfied. Forproving condition c), we have to show that if SU j withvj < vi receives at least one channel, then Qi = 0 orf(Θi + vj) > (ci + 1) log2

(1 + Pi

(ci+1)n0

)for SU i at the

end of the auction. Assume that θ is the highest unit price ofchannels that have been allocated to SU j. We need to considerthe following two cases.

• Case I: θ is not the price that ends the decision processof the auction. This implies that SU j is granted withher last channel in step 2 of Algorithm 1 with θ < vj .After the channel allocation for SU j, we have m =D+−j(θ) so that any SU i 6= j will be then allocated

D+i (θ) channels at some unit price higher than or equal to

θ. Consider D+i (θ) > 0 for any SU i, we have D+

i (θ) =min{bdi(θ)c, Qi}. Apparently, Qi will be zero eventuallyif bdi(θ)c ≥ Qi. Otherwise, we have

(ci + 1) log2

(1 +

Pi(ci + 1)n0

)= f(Θi + θ), (19)

This condition is similar to (13) except that di(θ) is set as1 (i.e., the smallest positive integer). Thus, θ∗ representsa maximum threshold for θ, such that when θ increasesto any value larger than θ∗, no more channels can beallocated to SU i. Since θ < vj and f(·) is an increasingfunction, we have

(ci + 1) log2

(1 +

Pi(ci + 1)n0

)< f(Θi + vj). (20)

On the other hand, for any SU i with D+i (θ) = 0, we

have either Qi = 0 or θ ≥ θ∗i . Obviously, Qi = 0 can bemaintained till the end of Algorithm 1 and

(ci+1) log2

(1 +

Pi(ci + 1)n0

)= f(Θi + θ∗i ) ≤ f(Θi + θ) < f(Θi + vj).

(21)

Thus, condition c) can always be satisfied in this case.

9

• Case II: θ is the price that ends the auction. Under thiscase, SU j will be granted with her last channel after step10, and θ ≤ vj . Now, consider any SU i with vi > vj . IfD+i (θ) > 0, SU i can obtain all her demand in step 11.

By the previous argument, this leads to

Qi = 0, (22)

or

(ci+1) log2

(1+

Pi(ci + 1)n0

)= f(Θi+θ) < f(Θi+vj).

(23)If D+

i (θ) = 0 and Di(θ) > 0, we have θ = θ∗i andQi > 0. Then, if SU i receives her demand in step 14,condition c) must hold. Otherwise, we have m = 0 at thebeginning of step 18 and SU j gets her last channel insteps 11 or 14. Thus, vj > θ = θ∗i and

(ci+1) log2

(1+

Pi(ci + 1)n0

)=f(Θi+θ

∗i ) < f(Θi+vj).

(24)If Di(θ) = 0 and D−i (θ) > 0, this can happen onlywhen θ = vj , which clearly contradicts the assumptionthat vi > vj . Finally, if D−i (θ) = 0, we have θ > vi orθ∗i < θ or Qi = 0. Similarly, θ > vi cannot happen dueto the assumption of vi > vj . Thus, it is only possiblethat θ∗i < θ or Qi = 0, which meets condition c).

In summary, all conditions in Proposition 1 are satisfied,and thus the proposed algorithm is Pareto-optimal.

V. NUMERICAL RESULTS

In this section, simulations are conducted to show theperformance and the efficiency of our proposed spectrumauction algorithm for CR networks with power-constrainedmulti-radio SUs.

A. Simulation Settings

Consider a CR network with one PO who runs a spectrumauction for leasing M = 40 idle channels among N = 20 SUs.For each SU i, her radio capacity qi and power limit Pi (inKW) are randomly chosen as integers within intervals of [2, 10]and [1, 6], respectively. Furthermore, the SUs’ valuations oneach channel is uniformly distributed over (0, 5], and the noisepower spectral density n0 is set as 2 × 10−3 W/channel.Similar settings have also been employed in [18], [24], [25].In addition, for explanation purpose, we assume f(·) as anincreasing linear function, i.e., f(Θi) = α · Θi for any SUi, where the default value of α is 0.7. However, similarobservations can be obtained for other forms of increasingfunctions. Notice that some parameters (including M , N andα) may vary according to different scenarios, and all resultsare obtained by taking averages over 20 runs.

To demonstrate the superiority of our proposed spectrumauction algorithm on dealing with power-constrained multi-radio SUs, we also compare its performance with two existingmulti-unit auction mechanisms, i.e., single-minded VCG [18]and FlexAuc [24]. In the single-minded VCG, each SU de-clares the channel demand equal to her radio capacity and does

TABLE IICONVERGENCE PERFORMANCE (AVG.)

∆θ = 0.1 ∆θ = 0.05 ∆θ = 0.01Convergence time 15.49 sec. 17.01 sec. 62.35 sec.Allocation efficiency 81.73% 93.26% 100%

Unit price θ0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Nu

mb

er

of

un

allo

ca

ted

ch

an

ne

ls

0

5

10

15

20

25

30

35

40 N=20N=30N=40

Fig. 3. The amount of remaining unallocated channels with increase of θ.

not accept any partial satisfaction of her requested amount.On the other hand, FlexAuc enables flexible channel demandsfrom SUs within their radio capacities. However, both of thesemechanisms ignored the power constraints on SUs and didnot consider their QoS expectations with respect to the actualpayments.

B. Performance Evaluation

Table II shows the average convergence performance ofthe proposed spectrum auction algorithm with respect todifferent values of the predetermined price increment ∆θ.The convergence time is defined as the running time of theproposed algorithm on an iMac with 3.6 GHz Intel Core i5CPU and 4GB RAM, and the allocation efficiency is definedas the ratio between the number of allocated channels and theamount of total available channels. We can see from this tablethat both the convergence time and the allocation efficiencyincreases with the decrease of ∆θ. Since when ∆θ = 0.01,the allocation efficiency becomes 100%, and the convergencetime is 62.35 sec. (which is relatively short than an entireauction period that commonly lasts for 10 minutes [17]), wetake ∆θ = 0.01 for all the following simulations.

Fig. 3 illustrates the variation of the remaining unallocatedchannels throughout the ascending-price decision procedure. Itis shown that the amount of unallocated channels decreases atsome certain values of the unit price θ, and eventually becomeszero. This matches the design goal of Algorithm 1 that thePO’s channels may be traded at different unit prices and willbe completely allocated at the end. From this figure, we canalso observe that the auction ends at a higher unit price whenN increases, which matches the intuition that the more intensecompetition leads to a higher trading price.

Fig. 4 verifies the property of incentive compatibility fora randomly selected SU i. Due to the selfishness, each SUmay strategically report her unit value so as to maximize

10

Reported unit value0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Utilit

y o

f S

U i

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5v

i=0.7

vi=1.2

vi=1.7

Truthfulness

Fig. 4. The optimal bidding decision of SU i.

Power limit (KW)1 2 3 4 5 6

Au

ctio

n r

eve

nu

e

0

5

10

15

20

25

30

35

M=30

M=40

M=50

Fig. 5. The impact of SUs’ power constraintson the auction outcome.

her individual utility. When the reported value is relativelysmall, the utility of SU i maintains zero since she cannotwin the auction. With the further increase of the reportedvalue, the SU’s utility increases since more channels can beobtained. However, after a certain point (i.e., the truthful vi),the payment becomes dominant so that the utility decreases.Such trends in Fig. 4 clearly indicate that each SU will alwaysbid truthfully by reporting her actual unit value in order to gainthe highest utility.

In Fig. 5, the impact of SUs’ power constraints is investi-gated by checking the auction revenues under different powerlimits. For evaluation purpose, we consider that all SUs havea same power limit in this scenario. It can be seen from thecurves in Fig. 5 that the auction revenue is considerably lowwhen the power limit is below a certain level, and increasesdramatically when the power constraint is relaxed. This isbecause the SUs’ QoS satisfactions can hardly be met undersevere power insufficiency, while with the increase of thepower limit, the total demand from SUs increases so that ahigher auction revenue can be produced. However, such impactof power limit on auction revenue becomes weak when thepower limit is large enough. It is because the SU’s channeldemand will be no longer dominated by her power constraintsince her QoS expectation can be easily satisfied. Besides,from the figure, it is intuitive that the more channels the PO

Number of channels20 25 30 35 40 45 50

Be

ne

fit

5

10

15

20

25

30

35

40

45

50Revenue, single-minded VCG

Revenue, FlexAuc

Revenue, proposed auction algorithm

Social welfare, single-minded VCG

Social welfare, FlexAuc

Social welfare, proposed auction algorithm

Fig. 6. Comparison on both the auction revenue and social welfare withdifferent number of auctioned channels.

α

0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Be

ne

fit

5

10

15

20

25

30

35

40

45Revenue, single-minded VCG

Revenue, FlexAuc

Revenue, proposed auction algorithm

Social welfare, single-minded VCG

Social welfare, FlexAuc

Social welfare, proposed auction algorithm

Fig. 7. Comparison on both the auction revenue and social welfare withdifferent QoS expectation parameter of SUs.

has, the higher auction revenue she obtains.The comparison on the auction revenue and social welfare

of different algorithms are presented in Figs. 6 and 7. Itis intuitive that both the auction revenue and social welfareincrease when more channels can be auctioned, and decreasewhen the QoS expectations of SUs becomes higher (i.e., αbecomes larger). Both Figs. 6 and 7 reveal that FlexAucoutperforms single-minded VCG. This is because the spectrumutilization can be improved by employing the flexible channelallocation. Moreover, it can be seen that the proposed auctionalgorithm achieves a much better performance than FlexAucand single-minded VCG on both the auction revenue and socialwelfare. The reason is that the proposed algorithm not onlyallows flexible channel adoptions, but also takes into accountthe power constraints of SUs in the allocation so that the QoSexpectations of all winners can be satisfied.

Figs. 8 and 9 show the comparison on the numbers of win-ning SUs by using different auction algorithms. Apparently,we can see that the proposed auction algorithm leads to muchmore winners than FlexAuc in both Figs. 8 and 9, and thissuperiority becomes more obvious for a larger value of α(i.e., higher QoS expectation). This is because FlexAuc losessome winners due to the dissatisfactions of their QoS, and suchproblem becomes more severe when SUs request higher QoS.

11

Number of auctioned channels M20 30 40 50

Nu

mb

er

of

win

nin

g S

Us

0

2

4

6

8

10

12

14

16

18

Proposed auction algorithm

FlexAuc

Fig. 8. Comparison on the number of winning SUs with different number ofauctioned channels.

α

0.6 0.7 0.8 0.9

Nu

mb

er

of

win

nin

g S

Us

0

2

4

6

8

10

12

14

16

18

Proposed auction algorithm

FlexAuc

Fig. 9. Comparison on the number of winning SUs with different QoSexpectation parameter of SUs.

However, the proposed auction algorithm can well balance thechannel demand and the QoS requirement of each SU so thatmore winners can be granted.

VI. CONCLUSION

In this paper, we studied a multi-unit spectrum auction forCR networks with power-constrained multi-radio SUs. Twodistinctive features of practical wireless applications have beenconsidered. The first one was the flexible spectrum adoption,which allowed SUs to fulfill transmission services on anynumber of channels within their radio capacities. The secondone was the impact of power constraints on the QoS achievedat each SU. To jointly resolve these challenges in spectrumauction design, an ascending-price progressive auction al-gorithm was proposed. Theoretical analyses and simulationresults showed that the proposed algorithm could satisfy alldesired properties, and could improve both the auction revenueand social welfare, and increase the number of winning SUscompared to the counterparts.

APPENDIXPROOF OF PROPOSITION 1

Proof: We first prove that a), b) and c) are necessaryconditions for Pareto optimality. Recall that the total radio

capacity of all SUs is assumed to be large enough, i.e.,∑1≤k≤N qk ≥M .If a) does not hold, we must have

∑1≤k≤N ck < M ≤∑

1≤k≤N qk. In this case, there will be at least one SU ` withc` < q`. Consider a modified auction decision (c′,Θ′) where

c′` = c` + 1 ≤ q`, c′k = ck, for 1 ≤ k ≤ N and k 6= `;

Θ′k = Θk, for 1 ≤ k ≤ N.

Since Ri(·) is a monotone increasing function as proved in[42], (c,Θ) is a feasible solution that satisfies (7) and

(c` + 1) log2

(1 +

Pi(c` + 1)n0

)≥ c` log2

(1 +

Pic`n0

). (25)

Then, we have

Ri(c′`) = Ri(c` + 1) ≥ f(Θ`) = f(Θ′`), (26)

which indicates that (c′,Θ′) is also feasible. Note that i)∑1≤k≤N Θ′k =

∑1≤k≤N Θk, ii) U ′k = Uk for 1 ≤ k ≤ N and

k 6= `, and iii) U ′` = c′`v` −Θ′` = (c` + 1)v` −Θ` > U` sincec′` ≤ q`. Apparently, (c′,Θ′) is a possible Pareto improvementfor (c,Θ), or (c,Θ) is not Pareto-optimal when condition a)does not hold.

If b) does not hold, there is at least one SU i with ci > qi.Since

∑1≤k≤N ck = M ≤

∑1≤k≤N qk, we must have at

least one SU j 6= i with cj < qj . Then, consider a modifiedauction decision (c′,Θ′) where

c′i = ci − 1 ≥ qi, c′j = cj + 1 ≤ qj , c′k = ck, for k 6= i, j;

Θ′k = Θk, for 1 ≤ k ≤ N.

Similarly, (c′,Θ′) is feasible because the original (c,Θ) is afeasible solution and R(c′i) = R(ci), R(c′j) ≥ R(cj). Notethat i)

∑1≤k≤N Θ′k =

∑1≤k≤N Θk, ii) U ′k = Uk for 1 ≤ k ≤

N and k 6= i, j, iii) U ′i = Vi(c′i) − Θ′i = Vi(ci − 1) − Θi ≥

Vi(ci)−Θi = Ui since Vi(·) is non-increasing for ci > qi, andiv) U ′j = c′jvj − Θ′j = (cj + 1)vj − Θj > Uj since c′j ≤ qj .Hence, (c′,Θ′) can be a Pareto improvement for (c,Θ), andthus (c,Θ) is not Pareto-optimal.

If c) does not hold, there are SUs i and j such that vi > vjand cj > 0, but ci < qi and

f(Θi + vj) ≤ (ci + 1) log2

(1 +

Pi(ci + 1)n0

). (27)

In this case, SU i is able to buy one more channel fromSU j and pays an additional price of vj while guaranteeingthat her QoS expectation can still be satisfied (this is adirect observation from (27)). Specifically, consider a modifiedauction decision (c′,Θ′) where

c′i = ci + 1, c′j = cj − 1, Θ′i = Θi + vj , Θ′j = Θj − vj ;

c′k = ck, Θ′k = Θk for 1 ≤ k ≤ N and k 6= i, j.

Note that i)∑

1≤k≤N Θ′k =∑

1≤k≤N Θk, ii) U ′k = Uk for1 ≤ k ≤ N and k 6= i, j, iii) U ′j = c′jvj − Θ′j = (cj −1)vj − (Θj − vj) = cjvj − Θj = Uj , and iv) U ′i = c′ivi −Θ′i = (ci + 1)vi − (Θi + vj) = (civi − Θi) + (vi − vj) >Ui since vi > vj . Moreover, it can be easily verified that(c′,Θ′) is also a feasible solution. Therefore, (c′,Θ′) is a

12

Pareto improvement for (c,Θ), so that (c,Θ) is not Pareto-optimal without condition c).

Next, we prove that a), b) and c) are also sufficientconditions for Pareto optimality. Assume by the way ofcontradiction that, when a), b) and c) hold for (c,Θ), thereexists another auction decision (c′,Θ′) that is the Paretoimprovement for (c,Θ), i.e.,∑

1≤i≤N

Θ′i ≥∑

1≤i≤N

Θi, (28)

andU ′i ≥ Ui, for 1 ≤ i ≤ N, (29)

with at least one strict inequality.If the Pareto optimality is achieved by (c′,Θ′) instead of

(c,Θ), (c′,Θ′) must satisfy the necessary conditions a), b)and c) as well, and thus we have c′i ≤ qi for 1 ≤ i ≤ N and∑

1≤i≤N c′i = M . Furthermore, substituting (1) and (4) into

the assumption (29) results in

Θ′i −Θi ≤ vi(c′i − ci), for 1 ≤ i ≤ N. (30)

Now, we re-index SUs according to the non-increasing orderof their unit valuations, i.e., v1 ≥ v2 ≥ . . . ,≥ vN . Conditionc) implies the existence of an index ` such that ci > 0 for1 ≤ i ≤ `, and ci = 0 for ` < i ≤ N . In other words, ` is thelargest index with c` > 0. Consider the following equation∑c′i>ci

(Θ′i −Θi) =∑

c′i>ci,i≥`

(Θ′i −Θi) +∑

c′i>ci,i<`

(Θ′i −Θi). (31)

For the first term of (31), i.e.,∑c′i>ci,i≥`

(Θ′i −Θi), we have∑c′i>ci,i≥`

(Θ′i −Θi) ≤∑

c′i>ci,i≥`

vi(c′i − ci) ≤ v`

∑c′i>ci,i≥`

(c′i − ci),

(32)where the first inequality comes from (30), and the secondinequality holds because v` ≥ vi for i ≤ ` and c′i > ci.

Let’s consider the second term of (31), i.e.,∑c′i>ci,i<`

(Θ′i−Θi). For SU i < `, according to condition c), we must haveeither ci = qi or

f(Θi + v`) > (ci + 1) log2

(1 +

Pi(ci + 1)n0

), (33)

since v` > 0 and vi > v`. Additionally, if SU i < ` andc′i > ci, we must have ci < qi. Otherwise c′i will exceed theradio limit qi. Thus, for any SU i < ` and c′i > ci, inequality(33) must hold. Naturally, (c′,Θ′) is a feasible outcome sothat

f(Θ′i) ≤ c′i log2

(1 +

Pic′in0

). (34)

Then, with ci < c′i ≤ qi and the monotone increasing trend ofRi(·), we can let c′i = ci + 1 and obtain

f(Θ′i) ≤ (ci + 1) log2

(1 +

Pi(ci + 1)n0

). (35)

By further combining inequalities (33) and (35), we have

f(Θi + v`) > f(Θ′i). (36)

Recall that f(·) is defined as an increasing function so that

(36) yields Θi + v` > Θ′i. Thus

Θ′i −Θi < v` ≤ v`(c′i − ci), (37)

where the second inequality holds since c′i > ci. Accordingly,we can draw a conclusion that∑

c′i>ci,i<`

(Θ′i −Θi) < v`∑

c′i>ci,i<`

(c′i − ci). (38)

Substituting (32) and (38) into (31), we have∑c′i>ci

(Θ′i −Θi) < v`∑

c′i>ci,i≥`

(c′i − ci) + v`∑

c′i>ci,i<`

(c′i − ci)

= v`∑c′i>ci

(c′i − ci) = v`∑c′i≤ci

(ci − c′i)

≤∑c′i≤ci

vi(ci − c′i) ≤∑c′i≤ci

(Θi −Θ′i),

(39)

where∑c′i>ci

(c′i − ci) =∑c′i≤ci

(ci − c′i) since∑c′i =∑

ci = M . From inequality (39), we can derive that∑1≤i≤N

Θ′i <∑

1≤i≤N

Θi, (40)

∑1≤i≤N

(U ′i − Ui) =∑c′i>ci

vi(c′i − ci)−

∑c′i≤ci

vi(ci − c′i)

+∑c′i>ci

(Θ′i −Θi)−∑c′i≤ci

(Θi −Θ′i) < 0,

(41)

where∑c′i>ci

vi(c′i − ci) −

∑c′i≤ci

vi(ci − c′i) ≤ 0 because∑c′i>ci

vi(c′i − ci) ≤ v`

∑c′i>ci

(c′i − ci) = v`∑c′i≤ci

(ci −c′i) ≤

∑c′i≤ci

vi(ci − c′i) (which is obtained from (32) and(39)). Obviously, (40) and (41) contradict the assumptions in(28) and (29).

In summary, a), b) and c) are necessary and sufficientconditions for guaranteeing a Pareto-optimal outcome for theconsidered spectrum auction framework.

ACKNOWLEDGMENT

The authors would like to thank the Natural Sciences andEngineering Research Council of Canada (NSERC) DiscoveryGrant.

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Changyan Yi (S’16) received the B.Sc. degreefrom Guilin University of Electronic Technology,China, in 2012, and M.Sc. degree from Universityof Manitoba, Winnipeg, MB, Canada, in 2014. Heis currently working toward the Ph.D. degree inElectrical and Computer Engineering, University ofManitoba. He was awarded Edward R. ToporeckGraduate Fellowship in Engineering in 2014, 2015,2016 (three times), University of Manitoba Gradu-ate Fellowship (UMGF) for 2015-2018, and IEEEComSoc Student Travel Grant for IEEE Globecom

2016. His research interests include algorithmic game theory, queueing theoryand their applications in radio resource management, wireless transmissionscheduling and network economics.

Jun Cai (M’04-SM’14) received the B.Sc. andM.Sc. degrees from Xi’an Jiaotong University,Xi’an, China, in 1996 and 1999, respectively, and thePh.D. degree from the University of Waterloo, ON,Canada, in 2004, all in electrical engineering. FromJune 2004 to April 2006, he was with McMasterUniversity, Hamilton, ON, as a Natural Sciences andEngineering Research Council of Canada Postdoc-toral Fellow. Since July 2006, he has been with theDepartment of Electrical and Computer Engineering,University of Manitoba, Winnipeg, MB, Canada,

where he is currently an Associate Professor. His current research inter-ests include energy-efficient and green communications, dynamic spectrummanagement and cognitive radio, radio resource management in wirelesscommunications networks, and performance analysis. Dr. Cai served as theTPC Co-Chair for IEEE VTC-Fall 2012 Wireless Applications and ServicesTrack, IEEE Globecom 2010 Wireless Communications Symposium, andIWCMC 2008 General Symposium; the Publicity Co-Chair for IWCMC in2010, 2011, 2013, and 2014; and the Registration Chair for QShine in 2005.He also served on the editorial board of the Journal of Computer Systems,Networks, and Communications and as a Guest Editor of the special issue ofthe Association for Computing Machinery Mobile Networks and Applications.He received the Best Paper Award from Chinacom in 2013, the Rh Awardfor outstanding contributions to research in applied sciences in 2012 fromthe University of Manitoba, and the Outstanding Service Award from IEEEGlobecom in 2010.