Designing Auction Mechanisms for Dynamic Spectrum Access

Mobile Netw ApplDOI 10.1007/s11036-008-0081-1

Designing Auction Mechanisms for DynamicSpectrum Access

Shamik Sengupta · Mainak Chatterjee

© Springer Science + Business Media, LLC 2008

Abstract With the increasing demands for radiospectrum, techniques are being explored that wouldallow dynamic access of spectrum bands that are under-utilized. In this regard, a new paradigm called dynamicspectrum access is being investigated where wirelessservice providers (WSPs) would dynamically seek morespectrum from the under-utilized licensed bands whenand where they need without interfering with the pri-mary users. Currently, there is little understanding onhow such a dynamic allocation will operate so as tomake the system feasible under economic terms. Inthis paper, we consider the dynamic spectrum alloca-tion process where multiple WSPs (bidders) competeto acquire necessary spectrum band from a commonpool of spectrum. We use auction theory to analyzethe allocation process when the demand from WSPsexceeds the available spectrum. We investigate variousauction mechanisms under different spectrum alloca-tion constraints to find WSPs’ bidding strategies andrevenue generated by spectrum owner. We show thatsequential bidding of bands provides better result thanthe concurrent bidding when WSPs are constrained toat most single unit allocation. On the other hand, whenthe bidders request for multiple units, (i.e., they are

S. Sengupta · M. Chatterjee (B)School of Electrical Engineering and Computer Science,University of Central Florida, Orlando, FL 32816, USAe-mail: [email protected]

S. Senguptae-mail: [email protected]

not restricted by allocation constraints) synchronousauction mechanism proves to be beneficial than asyn-chronous auctions.

Keywords auctions · dynamic spectrum access(DSA) · cognitive radio · winner determination ·knapsack

1 Introduction

Privatization of the telecommunications industry cou-pled with technological advancements and economicliberalization has stimulated competition among wire-less service providers (WSPs) and driven down theprices. In addition, the transformation from secondgeneration (2G) mobile telephony to third generation(3G) technologies has also boosted this competitionto a great extent resulting in numerous WSPs in onegeographic region.

In most countries, the competitive behavior amongWSPs was initiated by spectrum auctions held in 2000and 2001 [4]. These auctions were conducted either bythe government or under the supervision of a regula-tory body. These regulatory bodies set the rules andregulation which govern the access and use of spec-trum. Though the auctions were very successful in somecountries (e.g., United Kingdom, Germany), they wereopen to criticism in others (e.g., Austria, Switzerland,Netherlands) [4]. Through the Federal Communica-tions Commission (FCC), spectrum was auctioned inthe United States – the results of which are hotlydebated. For example, 824–849 MHz, 1.85–1.91 GHz,

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1.930–1.99 GHz are reserved for licensed cellular andPCS services and require a valid FCC license, whereas902–928 MHz, 2.40–2.50 GHz, 5.725–5.825 GHz arefree-for-all unlicensed bands (http://www.ntia.doc.gov/osmhome/osmhome.html). These spectrum allocationsare usually long–term and any changes are made underthe strict guidance of FCC.

Recent studies have shown that the spectrum usageis both space and time dependent, and therefore staticallocation of spectrum often leads to low spectrum uti-lization (http://www.sharedspectrum.com/inc/content/measurements/nsf/NYC_report.pdf). In static spectrumallocation, a large part of the radio bands are allocatedto the military, government and public safety systems.However, the utilization of these bands are significantlylow. One may argue that spectrum allocated to cellularand PCS network operators are highly utilized. But inreality, spectrum utilization even in these networks varyover time and space and undergo under–utilization.Often times, the usage of spectrum in certain networksis lower than anticipated, while there might be a crisis inothers. Static allocation of spectrum fails to address theissue of spectrum access even if the service providers(with statically allocated spectrum) are willing to payfor extra amount of spectrum for a short period of timeif there is a demand from the users they support.

In order to break away from the inflexibility andinefficiencies of static allocation, a new paradigm calledDynamic Spectrum Access (DSA) is being investigated[2]. In DSA, spectrum bands would be allocated andde-allocated dynamically from coordinated access band(CAB) [3]. Examples of such bands include the publicsafety bands (764–776 MHz, 794–806 MHz) and un-used broadcast UHF TV channels (450–470 MHz, 470–512 MHz, 512–698 MHz, 698–806 MHz). Note that,these bands will be dynamically leased to WSPs on ashort–term basis in addition to the statically allocatedspectrum that all WSPs already have.

Allocating spectrum dynamically among competingWSPs raises one important question ‘how would theoptimal allocation be achieved’. With the exact valueof spectrum unknown to both the seller and the buyers(WSPs), the use of auctions is a rational choice since thespectrum bands in the CAB is less than the usual ag-gregate demand from the WSPs. For every geographicregion, auction can be conducted in a periodic mannertaking the service providers in that region into account.The service providers are the bidders and the spectrumowner is the seller in this auction. From here onwards,we use the terms service providers and bidders in-terchangeably. Spectrum would be allocated and de–allocated every auction period, which is also known asthe lease duration. At the end of the lease period, all

WSPs would release their bands and fresh auction willbe again initiated. The service providers buy spectrumfrom auctioneer and sells the spectrum in form of ser-vices to the end users to make additional profits. Thedemands for spectrum and the revenue generated fromthe end users become the driving factors for the WSPsto participate in the auction. As WSPs compete for apart of the available spectrum and are willing to paya price for that part only, the kind of auction modelneeded must be more efficient than the traditionalauctions. Moreover, as the bidding behavior is differentfor different auction mechanisms, it is obvious that theoutcome of these auctions will be dependent on specificauction type and need to be studied separately.

In this paper, we investigate spectrum auction mech-anisms when multiple units of spectrum are availableand the demand from WSPs exceeds the availablespectrum. We study various auction mechanisms underdifferent allocation constraints to find WSPs’ biddingstrategies and revenue generated by the auctioneer.First, we investigate the special case where WSPs (bid-ders) are granted at most one spectrum chunk fromthe pool of spectrum chunks in each allocation period.We study both sequential and concurrent auctions, i.e.,when bands are auctioned one after another and whenall the bids for all the bands are submitted simultane-ously. Substitutable and non–substitutable – both typesof bands are considered and analyzed in this regard.The novelty of this research is that we focus on calculat-ing the optimal bid (bidder’s reservation price) and therevenue generated by auctioneer under the auction set-ting of single unit grant from multiple unit auction pool.We show that sequential auction provides better resultthan concurrent auction. As the more general case, wealso consider the spectrum allocation where biddersare not constrained to single unit of spectrum. We de-vise a “Dynamic spectrum allocator knapsack auction”mechanism with the help of sealed bid, second priceauction strategies that is used to determine the winningset of WSPs and dynamically allocate and de-allocatespectrum to the winning set of WSPs. The synchronousand asynchronous allocation policies are investigatedand compared in terms of average spectrum allocated,average revenue generated, and probability of winning.

The rest of the paper is organized as follows. InSection 2, we discuss the relevant works that deal withspectrum auctions. Basics of auctions are presented inSection 3. In Section 4, we propose the auction modelfor single unit grant and analyze sequential and concur-rent auctions. In Section 5, auction design for multipleunit grant is proposed and analyzed. Simulation modeland results are presented in Section 6. Conclusions aredrawn in the last section.

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2 Related work

Auction theory has been used to determine the valueof a commodity that has an undetermined or variableprice. A large number of Internet auction sites havebeen set up to process both consumer–oriented andbusiness–oriented transactions. Currently, most auc-tion sites (e.g., eBay, http://www.ebay.com/) support abasic bidding strategy through a proxy service for asingle-unit auction where ascending bidding continuestill a winner emerges. In this type of auction, there isonly one item for auction and all the bidders bid for thatonly item. In such single unit auction, Vickrey provedthat “English” and “Dutch” type auctions yield thesame expected revenue under the assumptions of riskneutral participants and privately known value drawnfrom a common distribution [16].

With the emergence of spectrum markets [1, 17],single unit auction models are no longer valid. Multi–unit auctions have been used to investigate pricingpolicies of network resources (e.g., transmission rate,bandwidth or link capacity) in [5–8, 12, 15] and ref-erences therein. The key issue addressed in [5] con-cerns how the available bandwidth within the networkshould be shared between competing streams of elas-tic traffic; the stability and fairness of a class of ratecontrol algorithms are also investigated. The implica-tions of flat pricing and congestion pricing for capac-ity expansion are studied in [6]. A bandwidth pricingmechanism based on second-price auctions that solvescongestion problems in communication networks hasbeen proposed in [7, 8]. A decentralized auction-basedapproach to price edge-allocated bandwidth in a dif-ferentiated services Internet is presented in [12]. Mostworks done so far on auctions are extensions of Vickreyauction [16] with somewhat strong assumptions. First,the auctions are designed in such a way such that thebidders with higher bids are always favored, e.g., in anyclassical auction. But favoring higher bidders does notalways necessarily maximize the revenue. Moreover,FCC’s intention is not only to maximize but also tobe fair to the market, where bidders have varying de-mands. Bidders in these auctions may either look forsingle unit or bundle of units from the available pool ofmultiple units of resources. Second, a major part of theliterature assumes the objects in multi-object auctionsto have a common value, which may not be true forspectrum auctions. This is because revenue generatedfrom the same spectrum band through the services forend–users may be different for different WSPs due tomany factors such as their locations, interference fromothers etc. Considering such constraints, a spectrumarchitecture called DIMSUMnet was proposed in [2].

In [9, 10], the authors introduced a DSA scheme inwhich a spectrum manager periodically auctions short-term spectrum licenses. The spectrum is sold at a unitprice, and the assumptions underneath is that a largenumber of spectrum buyers are present and none hasenough power to influence the market clearing price.Spectrum auctioning mechanisms under heterogeneouswireless access networks have been investigated in [11].

3 Auction design and classifications

Good auction design is important for any type of suc-cessful auction and often depends on the item beingsold. For example, the auctions held in Ebay (http://www.ebay.com/) are typically used to sell an art ob-ject or a valuable item. In contrast to Ebay auctions,spectrum auction is similar to the multi–unit auction,where multiple units are up for auction. Multiple bid-ders present their bids for a part of the spectrum band,where sum of all these requests exceed the total spec-trum band capacity thus causing the auction to takeplace. Moreover, unlike classic single unit auction, mul-tiple winners evolve in this auction model constitutinga winner set. The determination of winner set oftendepends on the auction design strategy taken by thespectrum owner.

There are three important issues behind any auctiondesign. They are (i) attracting bidders (enticing biddersby increasing their probability of winning), (ii) pre-venting collusion thus preventing bidders to control theauction and (iii) maximizing auctioneer’s revenue [13].It is not at all intended that only big companies withhigh spectrum demand should acquire entire spectrum.The goal is to increase competition and bring freshnew ideas and services. As a result it is necessary tomake the small companies, who also have a demand ofspectrum, interested in taking part in the auction.

For spectrum auctions, we assume that there aremultiple service providers who are willing to buy morespectrum for short lease periods to serve more end–users and to make more profit. The WSPs determinetheir spectrum requirements and the price (bids) theyare willing to pay. Spectrum is then allocated dynam-ically by the spectrum owner depending on these bidsand the requested amounts of spectrum based on somewinner determination strategy.

To determine which WSPs must get the requestedspectrum, the auctioneer must answer couple of ques-tions. First, what is the objective of the spectrum al-locator (FCC for reference)? Apart from maximizingthe revenue, the spectrum owner must also be fair inleasing out the unused spectrum bands for the purpose

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of self-coexistence. The second question that followsis what would be the pricing and market mechanisms?In such auction models, multiple spectrum chunks areavailable simultaneously and service providers are in-terested in one or multiple units of these chunks. InFig. 1, we present the broad classification of auctionsbased on the spectrum allocation constraints and thepart shown within the circle is the focus of this research.Thus unlike classic single unit auction, multiple winnersevolve in this auction model constituting a winner set.The determination of winner set often depends on theauction design strategy taken by the spectrum owner.The spectrum band could be substitutable or non-substitutable. By substitutable band we mean a bidderwill not care about which band(s) he gets as long asall the bandwidths are equal. (We ignore the physicalcharacteristics of signals when they operate at differentfrequencies.) Non-substitutable bands are the ones withdifferent bandwidths and thus different valuations.

Within multiple units available for spectrum auctioncategory (refer to Fig. 1), we consider both the spec-trum allocation mechanisms, (i) bidders are grantedat most one spectrum unit from the available pool,(ii) bidders are not constrained and thus are grantedmultiple units. The use of constrained auction mech-anism, where bidders are granted at most one singleunit is justified by the newly proposed IEEE 802.22wireless network spectrum sharing model where IEEE802.22 devices share the spectrum in the sub-900 MHz.The available spectrum is limited and thus spectrumowner needs to ensure availability of some free spec-trum chunks for the incoming requests. For the moregeneralized scenario, where bidders are granted multi-ple spectrum bands without any allocation constraint,we only analyze the non-substitutable bands. This isbecause substitutable bands under multiple units grant

single unitBidders granted

available Single unit

Concurrentbidding

Auction

S NS S NS

Sequentialbidding

NSS

Bidders grantedmultiple units

available

Multiple units

Figure 1 Auction classifications (S substitutable, NS non-substitutable)

is isomorphous to non-substitutable bands under thesingle unit grant from multiple units for the bidders.

4 Auction design for single unit grant

Let S = {s1, s2, · · · , sm} be a vector of m substitutablespectrum bands and N = {N1,N2, · · · ,Nn} be the nbidders engaged in the auction. For proper auctionsetting, we assume n > m. Without loss of generality,we assume the WSPs to be greedy, i.e., they alwaystry to maximize profit. Let B = {b1, b 2, · · · , b n} denotethe n-bid vector from the bidders submitted to theauctioneer where bi is the bid from the ith bidder. Afterthe auction is completed, winners obtain the lease of thebands for a certain period. The service providers thenuse the total allocated band (the static band alreadyallocated plus dynamic band won) to provide service tothe end users. We follow the sealed-bid auction policyto prevent collusion. We assume all the bidders in theauction to be rational such that losing bidders in anyauction round will increase their bids by certain amountin the next round if their bids were less than the truevaluation of the bands. Similarly, winning bidder(s) willdecrease their bids by certain amount in the next roundto increase their payoff(s) till a steady state is reached.At the end of each auction round, the auctioneer onlybroadcasts the information of minimum bid submittedin that round. Note that, the justification behind notbroadcasting any other information (e.g., maximumbid) and only broadcasting minimum submitted bidinformation in the proposed model is that bidders areonly allowed to know the lower bound of the bids.Knowing the lower bound will encourage only the po-tential bidders (bidders with reservation price higherthan or equal to the broadcasted bid information) toparticipate in the next auction rounds.

The WSPs use the acquired spectrum to provide ser-vices to the end users. The revenue generated from theend users gives an indication of the true valuation priceof the band. Providers use this valuation price profileto govern their future bidding strategy for forthcomingauction periods. To complicate matters in real-worldscenario, the revenue generated even from one particu-lar spectrum band can be different for different serviceproviders depending on company policy and pricing forthe end-users. Note that, this assumption does not con-tradict the definition of substitutable band. (With thesubstitutable band assumption, one single provider seesno difference between any two bands but two providerscan have different revenues from same band.) As aresult, the common valuation price for a spectrum bandwill also be different for different service providers. We

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present this true valuation price of a substitutable bandas a vector, V = {V1, V2, · · · , Vn} for n bidders. Later,in Section 4.4, we will reform the valuation price vectorfor the non-substitutable bands. With the valuationprice and bids from a bidder formally defined, payoffof a bidder is given by,{

Vi − bi if ith bidder wins0 if ith bidder loses

(1)

We analyze and compare the sequential and concurrentbid mechanisms under the above mentioned auctionsetting. Auctions among the WSPs occur periodicallywith the periods being the DSA period. In the sequen-tial mechanism, spectrum bands are auctioned one afteranother in one DSA period and each winning biddergets at most one spectrum band, i.e., winning bidderis not allowed to participate for the remaining auctionrounds in that DSA period. Thus each DSA periodconsists of m auction rounds with decreasing number ofbidders. In contrast to the sequential bid, in concurrentbidding, each DSA period consists of only one auctionround. All the bidders submit their bids concurrentlyat the beginning of each DSA period. When one DSAperiod expires, all the bands are returned to the auc-tioneer and the process repeats for both sequential andconcurrent bids.

4.1 Sequential auction for substitutable bands

In sequential auction, m spectrum bands are auctionedone after another. First, n bidders submit their sealedbids for band s1 and the winner is determined. Winnerof s1 does not participate for the rest of the auction inthat DSA period. Remaining (n − 1) bidders then bidfor spectrum band s2 and so on till all the spectrumbands are auctioned. Let us analyze the properties ofsequential auction.

4.1.1 Probability of winning

We assume a time instance when the auction for kspectrum bands are over and k winners have emerged.As a result, there are (n − k) bidders participating for(m − k) spectrum bands. We assume that bids from allthe bidders are uniformly distributed. The probabilitydensity function of bid submissions in sequential auc-tion mechanism can be given by,

f (b) = 1

Vmax − bmin(2)

where, Vmax is the maximum valuation possible of aspectrum band and bmin is the minimum bid of all thebids submitted by the existing bidders.

Now, let us assume that bidder i submits a bid bi

at the beginning of (k + 1)th band auction. All theother (n − k − 1) bidders also submit their correspond-ing bids for the (k + 1)th band. Bidder i will win the(k + 1)th band if and only if all the (n − k − 1) bidders’bids are less than bi. Let us first find the probabilitythat any other bid bj, ( j ∈ (n − k − 1) bidders) is lessthan bi. The probability that any bid bj < bi, such that,j �= i; j, i ∈ (n − k) bidders, can be given by

P(bj <bi | j �= i; j ∈ (n − k − 1) bidders) =∫ bi

bmin

f (b)db

(3)

Substituting f (b) and integrating, we obtain,

P(bj < bi | j �= i; j ∈ (n−k−1) bidders) = bi − bmin

Vmax − bmin

(4)

If bidder i is to win the (k + 1)th band, we need tocalculate the probability that all the (n − k − 1) bidders’bids are lower than the bid bi. Thus probability ofbidder i winning the (k + 1)th band can be given by,

P(∀ bj < bi | j �= i; ∀ j ∈ (n − k − 1) bidders)

=n−k−1∏

P(bj < bi | j �= i; j ∈ (n − k − 1) bidders)

(5)

Using Eq. 4 in Eq. 5, we obtain the probability of abidder winning the (k + 1)th auction round as

Pseq(ith bidder winning) =(

bi − bmin

Vmax − bmin

)(n−k−1)

(6)

4.1.2 Optimal bid analysis

We define optimal bid of ith bidder as the bid thatwins a band and maximizes the payoff for ith bidder.In other words, optimal bid denotes the reservation bidof a bidder, exceeding which, the bidder is in the riskof obtaining low payoff. If on the other hand, the bidsubmitted is less than the optimal bid, probability ofwinning also decreases.

The ith bidder’s expected payoff is given by,

Ei = (Vi − bi) × P(ith bidder winning) (7)

Substituting Pseq(ith bidder winning) from Eq. 6 intoEq. 7, we obtain,

Ei = (Vi − bi)

(bi − bmin

Vmax − bmin

)(n−k−1)

(8)

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Let us evaluate bid b∗i that will maximize Ei. To maxi-

mize Ei, we equate the first derivative of Ei to 0, i.e.,

∂ Ei

∂bi= (Vi − bi)(n − k − 1)(bi − bmin)

(n−k−2)

(Vmax − bmin)(n−k−1)

− (bi − bmin)(n−k−1)

(Vmax − bmin)(n−k−1)= 0 (9)

We obtain the optimal bid for ith bidder in (k + 1)thauction round as

b∗iseq

= (n − k − 1)Vi + bmin

(n − k)(10)

In our auction formulation, as all the bidders are ratio-nal, the natural inclination of the losing bidders wouldbe to increase their bids (if the bids are less than the bid-ders’ true valuation prices). As the auction progresses,bmin will be non-decreasing. Thus in the steady state,with increase in auction rounds, bmin → Vmin, whereVmin is the minimum true valuation price of the bands.

4.2 Concurrent auction for substitutable bands

In concurrent auction, m spectrum bands are auctionedconcurrently where all the n bidders submit their bidstogether at the beginning of a DSA period. As allthe bands are substitutable, each bidder submits justone bid. Each of the highest m bidders win a spec-trum band. Let us analyze the properties of concurrentauction here.

4.2.1 Probability of winning

In concurrent auction setting, a bidder’s choice wouldbe to be among the highest m bidders and to maximizethe payoff profit. The probability of winning would thenboil down to the probability of generating a bid suchthat all the bids from (n − m) losing bidders are belowthis bid.

The probability of bidder i winning a band in concur-rent auction can be given by,

Pcon(ith bidder winning)

=n−m∏

P(bj < bi | j �= i; j ∈ (n − m) bidders) (11)

As a greedy bidder, the aim of the bidder is not only towin but also to maximize the profit. In other words, theaim is to win with the lowest possible bid.

Simplifying and expanding Eq. 11, we obtain theprobability of a bidder winning in concurrent auctionwith maximized profit as

Pcon(ith bidder winning) =(

bi − bmin

Vmax − bmin

)(n−m)

(12)

4.2.2 Optimal bid analysis

The expected payoff is given by

Ei = (Vi − bi) × Pcon(ith bidder winning) (13)

Substituting Pcon(ith bidder winning) from Eq. 12 intoEq. 13, we obtain,

Ei = (Vi − bi)

(bi − bmin

Vmax − bmin

)(n−m)

(14)

To maximize Ei, we take the first derivative of Ei andequate to 0,

∂ Ei

∂bi= (Vi − bi)(n − m)(bi − bmin)

(n−m−1)

(Vmax − bmin)(n−m)

− (bi − bmin)(n−m)

(Vmax − bmin)(n−m)= 0 (15)

Solving Eq. 15, we obtain the optimal bid for ith bidderin concurrent auction as

b∗icon

= (n − m)Vi + bmin

(n − m + 1)(16)

This bid is optimal in the sense that this is theminimum bid to maximize the probability of winning aspectrum band and thus maximizes the expected payoff.Next, we present a comparison between optimal bidsfor both sequential and concurrent auction to study thedominant strategies for bidders.

4.3 Dominant strategy—sequential and concurrentauction

The optimal bids for sequential and concurrent auc-tions are given in Eqs. 10 and 16 respectively. Let usconsider their difference as

bdif f = b∗iseq

− b∗icon

(17)

We consider two cases. First, under the transient stateand second, when steady state has been reached. Wedefine steady state as the state when all the bidderseventually settle down to their corresponding fixed bidsand after that bidders will have no extra payoff inunilaterally changing their bids. Transient state is thelearning phase where bidders have not reached thesteady state and are willing to experiment with theirbids. Under the transient state, we again consider twopossibilities. One at the beginning of the allocationperiod (even before the first band auction in sequentialsetting: all m bands remaining) and the other after kspectrum bands auctions are over.

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Transient state – No bands auctioned so far: The dif-ference in optimal bids b∗

iseqand b∗

iconis

bdif f = (n − 1)Vi + bmin

n− (n − m)Vi + bmin

(n − m + 1)(18)

Simplifying we obtain,

bdif f = (m − 1)(Vi − bmin)

n(n − m + 1)(19)

We know that for a bidder to win a spectrum band, thefollowing conditions must be true.

Vi ≥ b∗iseq

> bmin and Vi ≥ b∗icon

> bmin (20)

From conditions presented in Eq. 20 and for m > 1, wecan conclude that bdif f in Eq. 19 is a positive quantity(bdif f > 0). This establishes the fact that optimal bid(reservation price of the bidder) to win in sequentialauction setting is more than that in concurrent auction.It is also clear from Eq. 19 that with increase in thenumber of available bands, m, while keeping n fixed,bdif f increases, i.e., the difference between reservationprices in sequential auction and concurrent auctionincreases. Thus increasing available spectrum bands forauction, which should have been an incentive for theauctioneer, does not benefit auctioneer in real worldscenario in concurrent auction setting.

Transient state – k bands auctioned so far: All bid-ders participating in (k + 1)th auction round have thechance to iterate their bids thus increasing the mini-mum bid. Note that, compared to concurrent auction,in sequential auction, bidders get the opportunity to re-visit their bids (m − 1) times more in each DSA period.Then in concurrent auction, as the bidders have lessnumber of chances to resubmit their bidding strategies,it is clear that minimum bid submitted in concurrentauction would be less than the minimum bid submittedin sequential auction.

After k spectrum bands auctions are over let theminimum bids in sequential and concurrent auctionsbe bmin1 and bmin2 respectively; such that bmin2 ≤ bmin1 .Substituting values of b∗

iseqand b∗

iconin Eq. 17, we get

the difference in optimal bids between sequential andconcurrent auction as,

bdif f = (n − k − 1)Vi + bmin1

(n − k)− (n − m)Vi + bmin2

(n − m + 1)

(21)

Simplifying Eq. 21, we obtain,

bdif f = (m − k − 1)(Vi − bmin1)

(n − k)(n − m + 1)

+ (n − k)(bmin1 − bmin2)

(n − k)(n − m + 1)(22)

As all the terms in Eq. 22 are positive, it can be con-cluded that optimal bids in sequential auction settingis more than that in concurrent auction setting. Thus,from the auctioneer’s perspective, it is more beneficialto follow the sequential bidding mechanism for substi-tutable bands.

Steady state reached: In this case, we assume that theauction has been run for sufficient large number oftimes to reach the steady state both for sequential andconcurrent mechanisms. As we mentioned previously,auctioneer broadcasts the minimum bid submitted sothe history of minimum bids are known to all the bid-ders. Thus as we assume the auction model to achievethe steady state, minimum bid submitted both for se-quential and concurrent mechanism would be the same.

Then the difference in optimal bids between sequen-tial and concurrent auction is given as,

bdif f = (m − k − 1)(Vi − bmin)

(n − k)(n − m + 1)(23)

As all the terms in Eq. 23 are positive, it can be con-cluded again that optimal bids in sequential auctionsetting is more than that in concurrent auction setting.

4.4 Concurrent and sequential auctionsfor non-substitutable bands

In this section, we present the concurrent and sequen-tial auction models for m non-substitutable bands. Forevery bidder, the value of each of these m bands isdifferent. We assume that bidders have complete infor-mation about the valuation and rankings of the bands.Under the complete information scenario, n bidderssubmit bids concurrently at the beginning of the allo-cation period.

Let the true valuation price be in the form of a vectorof vectors,

V = {{V1}, {V2}, · · · , {Vn}} (24)

where {Vi} is the valuation price vector of ith bidder forall m spectrum bands, i.e.,

Vi = {Vi1, Vi2, · · · , Vim} (25)

Let the reservation price of ith bidder for all m spec-trum bands be

Ri = {ri1, ri2, · · · , rim} (26)

With all the values for bands known, it is obvious thata bidder i will choose to submit bid for that spectrumband which will maximize his payoff profit,

Ui = Vij − rij; j ∈ m (27)

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The dominant strategy of bidder i in concurrent auctionwould be to choose the band which will provide himthe maximum payoff profit Ui. Thus it may happenthat jth band provides the maximum payoff profit forl bidders which will result in l bidders competing for jthband excluding all other bands from the spectrum bandlist. Moreover, in concurrent auction the losing biddersdo not have chance to revisit their bid strategy even ifthere might be less valuation bands unoccupied by anybidder. This problem does not happen if the auctionis sequential as bidders get chances to revisit theirbid strategies. We compare concurrent and sequen-tial auction revenue generation from the auctioneer’sperspective.

4.4.1 Concurrent auction

Before we calculate the aggregate revenue for the auc-tioneer, let us first analyze only one band j. If l > 1bidders aim for this band j, then the revenue Revl j

generated from this band would be the maximum bidsubmitted from all these l bidders. If only 1 bidder aimsfor the band j, the revenue generated will be the bidsubmitted by the sole bidder. If no bidder aims at theband j, the revenue generated will be zero from band j.

Then the total revenue generated from all the nbidders and m bands in the concurrent auction settingcan be expressed in the following recursive way

Revcon[n, m] = Revl1 + Revcon[n − l, m − 1] (28)

where Revcon[n, m] is the total revenue generated fromn bidders and m bands and l can take values from 0to n. The disadvantage in such a concurrent setting isthat (n − l) may be 0 even if some of the bands are stillleft unoccupied. Thus all the bands are not sold out inauction even if n > m and thus auctioneer does not getfull benefit of all the bands.

4.4.2 Sequential auction

Similarly, we formulate the revenue generated from thesequential auction. The total revenue generated can bepresented as a recursive expression

Revseq[n, m] = Revl1 + Revseq[n − 1, m − 1] (29)

where l can take values from 0 to n. We find that asthe bands are sequentially auctioned, all the bands aresold out thus providing better revenue possibility thanconcurrent auction.

5 Auction design for multiple unit grant

So far, we analyzed the scenario where only a singleband would be assigned to a service provider fromthe multiple bands available. In this section, we relaxthis constraint and investigate the more generalizedcase where service providers can win multiple spectrumbands available from the common spectrum pool. Wepropose our auction model and formulate the conflictamong the service providers and spectrum broker un-der such multiple units grant.

To ensure successful auction design, we considerthree important issues on which the success of theauction depends. They are (i) maximizing auctioneer’srevenue, (ii) attracting bidders by increasing their prob-ability of winning, and (iii) preventing collusion sothat bidders can not control the auction. It is not atall intended that only big companies with high spec-trum demand should acquire these additional spectrumbands. The goal is to increase competition and bringfresh new ideas and services. As a result it is neces-sary to make the small companies, who also have ademand of spectrum, interested to take part in theauction. This way, revenue can be maximized and max-imum use of the available spectrum from the CAB canbe made.

The situation described above maps directly to the0-1 knapsack problem, where the aim is to fill the sackas much as possible maximizing the valuations of thesack. Here, we compare the spectrum bands presentin CAB as the total capacity of the sack and the bidspresented by service providers as the valuations for thespectrum amount they request. We propose this auctionprocedure as “Dynamic Spectrum Allocator KnapsackAuction”.

We formulate the above mentioned knapsack auc-tion as follows. Let us consider that there are n bidderslooking for the additional amount of spectrum fromthe CAB. All the bidders submit their demand throughsealed bids. We follow sealed bid auction strategy be-cause sealed bid auction has shown to perform wellin all-at-a-time auction bidding and has a tendency toprevent collusion. Note that, each service provider hasknowledge about its own bidding quantity and biddingprice but do not have any idea about any other serviceproviders’ bidding quantity and price. We assume thatthe spectrum band available in CAB is W. Now, if thespectrum requests submitted by some or all of the ser-vice providers exceed the spectrum available in CABthen the auction is held to solve the conflict amongthese providers.

Let, i = 1, 2, · · · , n denote the bidders (serviceproviders). We denote the strategy taken by service

Mobile Netw Appl

provider i as qi, where qi captures the demand tuple ofthis ith service provider and is given by

qi = {wi, xi} (30)

where, wi and xi denote the amount of spectrum andbidding price for that spectrum respectively.

Auction is best suited when the total demand is morethan the supply, i.e.,

n∑i=1

wi > W (31)

Our goal is to solve the dynamic spectrum alloca-tion problem in such a way so that earned revenue ismaximized from the spectrum owner’s point of view,by choosing a bundle of bidders, subject to conditionsuch that total amount of spectrum allocated does notexceed W. Thus the allocation policy of the spectrumowner would be,

maximizei

∑i

xi (32)

subject to the condition,∑i

wi ≤ W (33)

5.1 Synchronous and asynchronous auctions

Spectrum allocation with the help of proposed sealedbid knapsack auction can be done either synchronouslyor asynchronously [14]. In synchronous auction, bidsfrom all the bidders are taken simultaneously andallocation/de-allocation of spectrum from and to theCAB are done only at fixed intervals. On the otherhand, in asynchronous auction, bids are submittedby bidders asynchronously and allocation/de-allocationof spectrum from and to the CAB are not done atfixed intervals.

Asynchronous auction: As the name suggests, thisauction procedure of spectrum is asynchronous amongthe service providers as shown in Fig. 2. Whenever aservice provider comes up with a request for spectrumfrom the CAB, the spectrum owner checks to see if thatrequest can be serviced from the available pool of CAB.If the requested amount of spectrum is available, spec-trum owner assigns this chunk to the service providerfor the requested time and declines if the spectrumrequested is not available. Similarly, if more than oneservice provider come up with requests for spectrumfrom the CAB, the spectrum owner checks to see if allthe requests can be serviced from the available pool ofCAB. If they can be serviced, the spectrum is assignedbut if all the requests can not be granted, then auction

Bidder 5

Bidder 1

Bidder 2

Bidder 3

Bidder 4

t t t t ttt1 2 3 4 5 6 7

(2) (1)

(4)(1)

(2)

(2) (2)

(3)

Figure 2 Asynchronous allocation in different intervals of time

is initiated. We denote the strategy taken by serviceprovider i as qa

i . qai captures the demand tuple of this ith

service provider in asynchronous allocation mode andis given by

qai = {wi, xi, Ti} (34)

where, wi and xi denote the amount of spectrum andbidding price for that spectrum respectively. Ti is theduration for which the spectrum amount is requested.The numbers inside the parenthesis in the Fig. 2 denotethe duration Ti of the spectrum lease allocated to thecorresponding bidders. As the decision about whetherto allocate or not to allocate spectrum to a serviceprovider is taken instantly in this allocation procedureby looking at the available pool only this allocation pro-cedure is not very effective and may not maximize theearned revenue. It may happen that a service providerB is willing to pay a higher price than a service providerA who paid a lower price for the same demand, butunfortunately B’s request came up after A’s request. Inthis allocation procedure, as the spectrum owner doesnot have any idea about the future, A’s request willbe processed and B’s will be declined (assuming thatthe available pool does not change at the time of B’sarrival). Thus revenue could not be maximized throughthis allocation procedure.

Synchronous auction: In synchronous auction, spec-trum bands are allocated and de-allocated at fixed in-tervals as shown in Fig. 3. All the service providerswith a demand present their requests to the spectrumowner and the price they are willing to pay. Spectrumowner takes all the requests, processes them usingsome strategy and then allocates the spectrum bandsto the providers at the same time for the same leaseperiod. When the lease period expires, all the allocatedspectrum chunks are returned to the common poolfor future use. For example, lease periods for all thebidders are indicated as 1 in the Fig. 3.

Mobile Netw Appl

Bidder 5

Bidder 1

Bidder 2

Bidder 3

Bidder 4

t t t t ttt1 2 3 4 5 6 7

(1) (1) (1)

(1)(1) (1)

(1) (1) (1) (1)

(1)

(1) (1) (1) (1)

(1)

Figure 3 Synchronous allocation of spectrum in fixed intervals

5.2 Performance Comparison

We analyze and compare the synchronous and asyn-chronous strategies with the help of knapsack auc-tion. Below, we present two lemmas to show theperformance comparison between synchronous alloca-tion coupled with knapsack auction and asynchronousallocation of spectrum.

Lemma 1 Average Revenue generated in asynchronousallocation through knapsack auction procedure can notbe better than average revenue generated in synchronousallocation for a given set of biddings.

Proof We assume that there are n bidders competingfor W amount of spectrum. In asynchronous allocationmode, the bid strategies taken by ith service provideris given by tuple qa

i , while in synchronous mode, thetuples are represented by, qi.

We prove the above proposition with the help ofcounter-example. We arbitrarily decide two time inter-vals, t j and t j+1 for the asynchronous mode allocation.We assume that first deallocation(s) of spectrum andnew allocation(s) are happening at time t j+1 after timet j. Moreover, we assume that the asynchronous alloca-tion at time t j is maximal and provide the maximumrevenue. Let, m be the number of bidders who weregranted spectrum at time t j. Then, the maximum rev-enue generated at time t j can be given by,

m∑i

xi (35)

Now, we assume l of m bidders de-allocate at timet j+1 and rest (m − l) bidders continue to use their

spectrum. Then the revenue generated by these (m − l)bidders is given by,

m−l∑i

xi (36)

Moreover, the (n − m) bidders, who were not grantedspectrum at time t j, will also compete for the rest of thespectrum,

W −m−l∑

i

wi (37)

Now, we need to find, whether the revenue generatedin this asynchronous mode at time t j+1 can exceedthe synchronous mode revenue at the same time by thesame set of bidders. For simplicity, we assume that thebidders do not change their bidding requests in timeintervals t j and t j+1.

By the property of 0-1 knapsack auction, we knowthat the revenue generated by a subset (we denote thissubset by Q) of n − l bidders will be a local maxima, ifonly the revenue obtained from all the (n − l) biddersare considered simultaneously, i.e., synchronous allo-cation of spectrum to (n − l) interested bidders (notethat l is the set of bidders de-allocating their spectrumat time t j+1 and are not taking part in auction at timet j+1). But on the other hand, in the asynchronous mode,(m − l) bidders are already present and thus knapsackauction is conducted among (n − m) bidders for thespectrum W − ∑m−l

i wi. Then, it can be easily said fromthe property of 0-1 knapsack auction that, this asyn-chronous mode will generate the same local maximaas the synchronous mode, if and only if all (m − l)bidders (who are already present from the previoustime interval) fall under the optimal subset Q. If anyof the bidders out of (m − l) bidders do not fall underthe optimal subset Q, then it is certain that asynchro-nous mode allocation will not be able to maximize therevenue for that given set of biddings.

Let us provide a simple example to clarify the proof.An illustrative example: Let us consider that 5 bidderswho compete for a total capacity of 14 and the bid tu-ples generated by them at time interval t j are (6, 10, 2),(5, 9, 3), (7, 14, 1), (2, 8, 2) and (3, 9, 3) respectively.The first number of the tuple denotes spectrum amountrequested, while the second and third numbers denotethe price willing to pay for that spectrum request andtime duration for which the spectrum request is donerespectively. As we can see from the above tuples that

Mobile Netw Appl

bidder 3’s request has duration 1, that means, bidder 3will de-allocate first at time t j+1.

We execute both asynchronous and synchronousknapsack auction. In asynchronous mode, the revenuegenerated at time t j is 31 with the optimal subset ofbidders given by bidders 2, 3, and 4. Now at time t j+1,bidder 3 exits, while bidders 2 and 4 continue. Theremaining spectrum left in the CAB is 7 for which thebidders 1 and 5 compete. Then the revenue generatedat time t j+1 is 27 and the bidders granted are 1, 2, and 4.

On the other hand, in synchronous allocation, eachof the providers are allocated and de-allocated at fixedtime intervals. Then with the same set of bid requestsof spectrum amount and price, it is seen that maximumpossible revenue generated at time t j+1 out of the bid-ders 1, 2, 4 and 5 (as bidder 3 is not interested to takepart in auction at time t j+1) is 28, while the optimalsubset of bidders is given by Q = {1, 2, 5}. This clearlyshows that asynchronous auction may not provide themaxima all the time depending on the bidders de-allocating and requesting.

Lemma 2 Asynchronous allocation through knapsackauction procedure is sub-optimal while synchronous al-location is optimal.

Proof We define a process as optimal that always pro-vides a local maxima for a given set of values, while asub-optimal process may or may not achieve that localmaxima with the same set of values. With the help ofthis definition and the proof provided in Lemma 1, wecan similarly prove Lemma 2.

5.3 Bidders’ strategies

In knapsack auction, we investigate bidders’ strategiesfor both first and second price bidding. In first priceauction, bidder(s) with the winning bid(s) pays theirwinning bid(s). In contrast, in second price auction, bid-der(s) with the winning bid(s) do not pay their winningbid but pay some other lower winning bid according tothe strategy fixed by the auctioneer.

For investigating the bidders’ strategy, we considera particular bidder j. Let each bidder i submit the de-mand tuple qi. Then the optimal allocation of spectrumto the bidders is done by the auctioneer taking all thedemand tuples into consideration. We denote this op-timal spectrum allocation as M, where M incorporatesall the demand tuples qi and is subject to conditions pre-sented in Eqs. 32 and 33. Moreover, we assume that thejth bidder’s request falls among the optimal allocation

M, i.e., jth bidder has been granted the spectrum. Thenthe revenue generated by auctioneer is given by,∑i∈M

xi (38)

where, all the bids of bidders present in the optimalallocation M, are summed.

In contrast, let us assume a case where jth bidderdoes not exist at all and the auction is held amongthe rest of the bidders. Let the optimal allocation bedenoted by M∗ and is again subject to conditions pre-sented in Eqs. 32 and 33. Then the revenue generatedby auctioneer in this case is given by,∑i �= j,i∈M∗

xi (39)

Then the minimum winning price charged to jthbidder can be given by,

a j =∑

i �= j,i∈M∗xi −

∑i �= j,i∈M

xi (40)

It is clear from the above equation that bidder j’srequest is granted if

xj > aj, (41)

bidder j’s request is not granted if

xj < aj (42)

and bidder j is indifferent between winning andlosing if

xj = aj (43)

With these insights, we try to find the bidders’ strate-gies in first price and second price bidding under theknapsack auction model.

Lemma 3 In second price bidding, the dominant strat-egy of the bidder is to bid their reservation price.

Proof Before proving this lemma, let us explain thereservation price or true evaluation price of the bidder.When a service provider (bidder) buys spectrum fromthe spectrum broker, the service provider needs to sellthat spectrum in form of some service to the end userswho are willing to pay for that service. The revenuegenerated from the end users for that amount of spec-trum can be the true evaluation price or reservationprice for that service provider (bidder).

Let us assume jth service provider (bidder) has thedemand tuple qj = {wj, xj} and its reservation price forthat amount of spectrum requested be r j. As per Eq. 40,jth bidder’s request will be granted and hence be in theoptimal allocation M, only if the bid generated by jth

Mobile Netw Appl

bidder is more than aj. Then according to the secondprice bidding policy, jth bidder will pay the secondprice which is aj. The expected payoff obtained by jthbidder is given by,

Ej = rj − aj (44)

We proceed to show that jth bidder’s true bid is itsreservation price rj as claimed in the lemma usingcounter proof approach.

We assume that jth bidder does not bid its trueevaluation of the spectrum requested, i.e., x j �= rj. Twocases might arise depending on the relative values of xj

and rj.

Case 1 Bid is less than the reservation price, i.e.,xj < rj.

• rj > xj > aj: bidder j falls inside the optimal allo-cation M and its request is granted. The expectedpayoff obtained by jth bidder is still given by(rj − aj).

• rj > aj > xj: bidder j loses and its request is notgranted. Accordingly, the expected payoff is 0.

• aj > rj > xj: bidder j still loses and the expectedpayoff is again given by zero.

Case 2 Bid is more than the reservation price, i.e.,xj > rj.

• xj > rj > aj: bidder j falls inside the optimal allo-cation M and its request is granted. The expectedpayoff obtained by jth bidder is still given by(rj − aj).

• xj > aj > rj: though bidder j wins but the expectedpayoff becomes negative in this case. The expectedpayoff obtained by jth bidder is now given by (rj −aj) < 0. Bidder j definitely will not be interested inthis scenario.

• aj > xj > rj: bidder j loses and the expected payoffis again 0.

Thus it is clear that if bidder j wins, then the maxi-mum expected payoff this bidder can obtain is given byEj = rj − aj and bidding any other price above or belowits reservation price rj will not increase the payoff. Thusthe dominant strategy of the bidders in second pricebidding is to bid their reservation prices.

Lemma 4 In first price bidding, the bid is upperbounded by the reservation price.

Proof In contrast to the Lemma 3, in first price bidding,the expected payoff obtained by jth bidder can begiven by,

Ej = rj − xj (45)

as the price paid by the bidder is the same as the bid.Then, to increase the expected payoff, i.e., to keepEj > 0, xj must be less than rj.

At the same time, for winning, bid x j must be greaterthan a j, as specified in Eq. 40. Thus dominant strategyfor the bidders in first price auction is rj > xj > aj.

6 Simulation model and results

To prove the effectiveness of the proposed auctionmodels and the bidding strategies, we conducted sim-ulation experiments. We divided the experiments intobroad categories. Auctions with the allocation con-straint of at most one spectrum band grant are dis-cussed in Subsection 6.1. In Subsection 6.2, we presentthe results of the auction model where bidders aregranted multiple spectrum bands.

6.1 Results for single unit grant

For single unit grant, we present a comparison be-tween sequential and concurrent bidding for both sub-stitutable and non-substitutable bands. We assume thenumber of bands to be less than the number of biddersfor the auction to take place.

6.1.1 Substitutable spectrum bands

The parameters for this auction setting are as follows.We assume all the spectrum bands are of equal valueto all the bidders. Note that throughout this simulationmodel, we use the notation unit instead of any par-ticular currency. The reservation price for each bidderis assumed to follow a uniform distribution with mini-mum and maximum as 250 and 300 units respectively.Moreover, the bids presented by the bidders are alsoassumed to follow a uniform distribution between 100and 300 units.

In Fig. 4a and b, we compare the auctioneer’s rev-enue for both sequential and concurrent bidding withvarying number of bidders and spectrum bands. As dis-cussed earlier, the revenue generated in the sequentialauction setting is more than that in the concurrent one.In fact, with increase in number of bands and bidders,

Mobile Netw Appl

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Figure 4 a Revenue in sequential auction, and b revenue in concurrent auction; with number of bidders and substitutable spectrumbands

revenue generated in sequential setting is almost 200%more than the revenue in concurrent setting, thus prov-ing that sequential auction to be more beneficial fromthe auctioneer’s perspective.

In Fig. 5a, b and c, we present the revenue gener-ated by auctioneer in both sequential and concurrentbiddings with increase in DSA periods. We assume thatthe bidders use auction histories of previous roundsto submit their bids in future rounds. Thus a winningbidder in one DSA period will try to submit a lowerbid in next DSA period to increase his surplus profitwhereas a losing bidder will increase his bid providedthe previous bid was less than his reservation price.For all three results, we fixed the number of biddersas n = 100 but varied the number of bands as m = 10,

m = 50 and m = 90. We find that the difference in therevenue generated between sequential setting and con-current setting increases with number of bands (notethe y-axis scale value in Fig. 5a, b and c). Thus sequen-tial auction provides more revenue than the concurrentauction. Moreover, we find that with increasing numberof bands, sequential auction reaches steady state muchfaster than the concurrent auction. This happens due tothe fact that as more and more number of bands areavailable in the common pool for the auction (m → n),greedy bidders will get more incentive bidding less thantheir true valuation prices as was proved earlier. Thisof course will not happen in the sequential auction.Thus sequential auction is clearly a better choice forauctioneer to generate higher revenue.

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Sequential Auction setting: 100 bidders, 90 bandsConcurrent Auction setting: 100 bidders, 90 bands

(c)

x 104

Figure 5 Auctioneer’s revenue with substitutable bands: a 100 bidders and 10 spectrum bands; b 100 bidders and 50 spectrum bands;c 100 bidders and 90 spectrum bands

Mobile Netw Appl

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Figure 6 Substitutable bands: optimal bid for a specific bidder for a sequential auction, and b concurrent auction

Next, we present the optimal bid for a specific bidderto win a spectrum band for both sequential and con-current bidding in Fig. 6a and b. It can be observedthat the optimal bid for the concurrent auction is lessthan the optimal bid for the sequential auction andeven decreasing with m → n. Thus in concurrent auc-tion setting, auctioneer will not receive any incentiveincreasing the number of bands in the common poolthus reducing the whole purpose of dynamic spectrumallocation.

6.1.2 Non-substitutable spectrum bands

For non-substitutable bands, the bands are not ofequal value. We assume the band’s true value follow

a uniform distribution with minimum and maximumbeing 450 and 500 units respectively. We follow thesame distribution of bids as mentioned in the previoussubsection.

In Fig. 7a and b, we present the revenue with varyingnumber of bidders and bands. It is clear that sequentialauction provides better revenue for the auctioneer thanthe concurrent setting for non-substitutable bands.

In Fig. 8a, b and c, we present the revenue gener-ated by auctioneer in both sequential and concurrentbiddings with increase in auction rounds. Similar to theprevious case, we assume that the bidders use auctionhistories of previous rounds to submit their bids in fu-ture rounds. We find that the difference in the revenuegenerated between sequential setting and concurrent

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Figure 7 a Revenue in sequential auction, and b Revenue in concurrent auction; with number of bidders and non-substitutablespectrum bands

Mobile Netw Appl

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Figure 8 Auctioneer’s revenue with non-substitutable spectrum bands: a 100 bidders and 10 spectrum bands; b 100 bidders and 50spectrum bands; c 100 bidders and 90 spectrum bands

setting under non-substitutable bands is even morethan that of the substitutable bands of previous case(note the y-axis scale changes in Fig. 8a, b and c). Thussequential auction setting is clearly a better choice forauctioneer to generate better revenue for both types ofbands.

6.2 Results for multiple unit grant

We simulate the dynamic spectrum allocator knapsackauction model and show how the synchronous allo-cation outperforms the asynchronous allocation whenbidders are granted multiple non-substitutable spec-trum bands.

6.2.1 Spectrum auctioning methodologyand parameters

The main factors that we consider for comparing theperformance of the proposed synchronous knapsacksealed-bid auction with the asynchronous auction arethe revenue generated by spectrum owner, total spec-

Table 1 Simulation parameters

Parameter Value

Total amount of spectrum 125Minimum amount of spectrum that can be requested 11Maximum amount of spectrum that can be requested 50Minimum bid for per unit of spectrum 25Minimum time requested for spectrum leasing 1

in asynchronous allocationMaximum time requested for spectrum leasing 5

in asynchronous allocationFixed time for spectrum leasing 1

in synchronous allocation

trum usage, and probability of winning for bidders. Weconsider the following for the simulation model.

• Bid tuple: The bid tuple qi generated by bidderi in synchronous auction consists of amount ofspectrum requested, wi and the price the bidderis willing to pay, xi. In asynchronous auction, theduration is also advertised in addition to the abovetwo. Each bidder has a reservation or evaluationprice for the amount of spectrum requested and thebid is governed by this reservation price.

• Bidders’ strategies: We follow second price sealed-bid mechanism. We could have chosen the firstprice bidding policy; the only reason for choosingsecond price policy is that it has more propertiesthan first price in terms of uncertainty [16]. Aftereach round of auction, the only information bid-ders know is whether their request is granted or

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Figure 9 Revenue generated with auction rounds

Mobile Netw Appl

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Figure 10 Spectrum usage with auction rounds

not. We assume that all the bidders are presentfor all the auction rounds; bidders take feedbackfrom previous rounds and generate the bid tuple fornext round.

• Auctioneer’s strategies: Spectrum owner tries tomaximize the revenue generated from the bidders.

For better insight into the results, we compare theproposed synchronous sealed bid knapsack auctionwith the asynchronous sealed bid knapsack auctionunder the second price bidding policy, i.e., bidder(s)with the winning bid(s) do not pay their winning bidbut pay the second winning bid. Simulation parametersare shown in Table 1.

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Figure 11 Average revenue generated with number of serviceproviders

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Figure 12 Average spectrum usage with number of serviceproviders

6.3 Simulation results

Figures 9 and 10 compare revenue and spectrum usagefor both the strategies (synchronous and asynchronous)with increase in auction rounds. The number of biddersconsidered in this simulation is 15. Note that, bothrevenue and usage are low at the beginning and sub-sequently increases with rounds. When auction starts,bidders always act skeptical, thus initial bids are alwaysmuch lower than their true potential bids. With the in-crease in auction rounds, bidders get an idea of the bidsof other bidders and thus try to increase or decreasetheir bids accordingly.

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Figure 13 Average revenue generated with increase in CAB

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Figure 14 Average spectrum usage with increase in CAB

Figures 11 and 12 show the average revenue andspectrum usage with varying number of bidders forboth the auction strategies. We observe that theproposed synchronous knapsack auction generates ap-proximately 10% more revenue compared to the asyn-chronous knapsack auction and also reaches steadystate faster. The average spectrum usage is also morewith the synchronous allocation policy.

Figures 13 and 14 show the average revenue andspectrum usage with increase in capacity in CAB forboth the auction strategies. It is clear that with increasein CAB, synchronous strategy provides more revenueand makes optimal use of CAB than the asynchronous

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Number of bidders

Synchronous Knapsack AuctionAsynchronous Knapsack Auction

Figure 15 Average probability of winning spectrum with numberof bidders

strategy and thus provides more incentive for the spec-trum owner.

In Fig. 15, we look at the auction model from thebidders’ perspective. Higher revenue requires morenumber of bidders. We compare the two strategies interms of the probabilities to win a bid. We observethat the proposed synchronous auction strategy has asignificantly higher probability of winning compared toasynchronous auction strategy. This implies that bid-ders will be encouraged to take part in the synchro-nous knapsack auction; thus increasing the competitionamong the providers and increasing the chance to gen-erate more revenue.

7 Conclusions

In this research, we investigate possible auction mecha-nisms for dynamic spectrum allocation. We first focuson the scenario where there are multiple spectrumbands in the common pool of auction but each bidder isallocated at most one spectrum band. Through analysisand simulation we show that the popular conceptionof concurrent auction does not prove beneficial in thiscase. In this regard, we considered two metrics: revenuegenerated by auctioneer and optimal bid of the biddersfor comparison of sequential and concurrent auctions.We have shown that sequential auction proves to be thebetter choice for DSA auctions with spectrum alloca-tion constraint.

On the other hand, we also studied the scenario with-out any allocation constraint. We proposed an auctionmechanism for DSA that is based on the well knownknapsack problem. Both synchronous and asynchro-nous auction strategies are studied and compared inthis context. Through simulations it was found that itis in the best interest of both bidders and spectrumowner to adopt the synchronous auction. We showedhow the optimal usage of spectrum band is achievedand the revenue is maximized for the spectrum owner.The proposed mechanism yields higher probabilityof winning for the service providers and thus en-courages the providers to participate in the biddingprocess.

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Shamik Sengupta is a Post-Doctorate researcher in the De-partment of Electrical and Computer Engineering at StevensInstitute of Technology, New Jersey. Prior to that, he received hisPh.D. from the School of Electrical Engineering and ComputerScience at the University of Central Florida. He received his B.E.degree with First Class (Hons.) in Computer Science and Engi-neering from Jadavpur University, Calcutta, in 2002. His researchinterests include resource management in wireless networks, auc-tion and game theories, pricing, and WMAN technologies. He isa student member of IEEE.

Mainak Chatterjee received his Ph.D. from the department ofComputer Science and Engineering at The University of Texasat Arlington in 2002. Prior to that, he completed his B.Sc. withPhysics (Hons) from the University of Calcutta in 1994 and M.E.in Electrical Communication Engineering from the Indian Insti-tute of Science, Bangalore, in 1998. He is currently an AssociateProfessor in the school of Electrical Engineering and ComputerScience at the University of Central Florida. His research inter-ests include economic issues in wireless networks, applied gametheory, resource management and quality-of-service provision-ing, ad hoc and sensor networks, CDMA data networking, andlink layer protocols. He serves on the executive and technicalprogram committee of several international conferences.