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Signal Processing

Signal Processing 106 (2015) 15–29

http://d0165-16

n CorrE-m

visa.koi

journal homepage: www.elsevier.com/locate/sigpro

Cooperative game-theoretic approach to spectrumsharing in cognitive radios

Jayaprakash Rajasekharan n, Visa KoivunenSMARAD CoE, Department of Signal Processing and Acoustics, Aalto University, P.O. Box 13000, FI-00076 Espoo, Finland

a r t i c l e i n f o

Article history:Received 15 July 2013Received in revised form4 June 2014Accepted 16 June 2014Available online 23 June 2014

Keywords:Cooperative game theoryCognitive radioSpectrum sharingVCG auctions

x.doi.org/10.1016/j.sigpro.2014.06.01384/& 2014 Elsevier B.V. All rights reserved.

esponding author.ail addresses: jayaprakash.rajasekharan@aaltvunen@aalto.fi (V. Koivunen).

a b s t r a c t

In this paper, a framework for modeling multi-user, multi-band, spectrum sensing andsharing problem in cognitive radios as a cooperative game (CG) in a characteristic form isproposed. Secondary users (SUs) jointly sense the spectrum and cooperatively detectprimary user (PU) activity for identifying unoccupied spectrum bands. A CG is formulatedto quantify and share the benefits of cooperation by accessing identified idle channels in afair manner. The characteristic function describing the CG is based on the worth of SUs,which is calculated according to amount of work done for coalition by increasingawareness about state of spectrum that may also be seen as reduction in uncertaintyabout PU activity. Such CGs are balanced and super-additive, making resource allocationpossible and providing SUs with an incentive to cooperate and form the grand coalition.Based on their worth, SUs get payoffs that are computed using singleton solutions. SUs usepayoffs earned from sensing to bid for idle channels through a scheduling mechanism, inparticular, the socially optimal Vickrey–Clarke–Groves auction. Simulation results showthat, in comparison with other resource allocation models, the proposed CG modelprovides the best balance among fairness, cooperation and performance in terms of datarates obtained by SUs.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Current wireless networks are characterized by staticspectrum allocation policy, where spectrum is assigned tolicense holders on a long term basis. Due to continuousincrease in spectrum demand, certain bands face severescarcity and yet, a large portion of spectrum is oftenunder-utilized across time and space [1]. Apparent scarcityin spectrum arises from rigid frequency allocations ratherthan actual physical shortage of spectrum. Techniquesfacilitating flexible spectrum usage have been developedin order to solve these inefficiency problems. The keyenabler of dynamic spectrum access is cognitive radio (CR)

o.fi (J. Rajasekharan),

technology [2,3], which provides the capability for unli-censed secondary users (SUs) to opportunistically accessunused licensed bands (spectrum overlay approach) with-out causing harmful interference to primary users (PUs).

In a CR network, SUs may collaboratively sense thespectrum based on commonly used sensing methods suchas energy detection [4,5], cyclostationary-based detection[6], and matched filter to identify idle sub-bands ofspectrum referred to as spectrum holes. By combininginformation about the state of spectrum occupancy interms of local log likelihood ratios (LLRs) and side infor-mation such as observed interference levels and signal tonoise ratios (SNRs), SUs are able to improve detectionperformance and network coverage [7]. After sensing, SUsshare the available spectrum amongst themselves andcoordinate access to idle channels based on a access policy.A Fusion Center (FC) manages the coalition's sensing and

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–2916

access policy. Since spectrum access is entirely dependenton sensing results, both policies are closely related andhence, a joint spectrum sensing and access policy must bemodeled to optimize the utilization of spectral resources.Thus, there is a need to normatively model the spectrumsensing and sharing problem jointly in CRs to design,regulate, optimize and evaluate system performance.

Sharing the benefits of cooperative sensing is a non-trivial problem of great interest. In order to arrive at anaccess policy that is acceptable to all SUs, competition overspectral resources between the SUs must be resolved in afair manner. This calls for a game-theoretic approach tomodeling the problem at hand. A cooperative game (CG) isideally suited to model this scenario where all SUs benefitfrom cooperation and share the benefits amongst them-selves fairly. For example, if identification of unoccupiedspectrum bands and improving the state of awareness ofspectrum by the SUs can be construed as benefit in aquantitative and/or qualitative sense, the problem ofaccessing unoccupied spectrum bands reduces to allocat-ing this benefit fairly among SUs by means of a CG. Thus,participating in the CG guarantees benefits for all SUs andprovides themwith an incentive to cooperate in the future.The main assumption in CGs is that the grand coalition ofall SUs within a certain local area (since spectrum oppor-tunities and propagation characteristics are local in nat-ure) will form and hence, the aim of the CG is to allocateoverall benefit created by cooperating SUs in a fair andstable manner. A cooperative game in coalitional formfocuses on what SUs in a coalition can jointly achievewhile maximizing each SU's payoff. In a non-cooperativegame, each SU may have a large number of availableactions/strategies to choose from and with a large numberof SUs, the analysis may become computationally complexand intractable with a lot of communication overhead andinformation exchange. Thus, modeling spectrum sensingand sharing in CRs as a CG is both intuitively and logicallyappealing. A brief introduction to CG theory has beenprovided in [8] that is relevant in this context. For a moregeneral setting, see [9,10].

It is possible that the number of idle sub-bands is quitelimited in order to be allocated to all SUs or SUs might notwant to access idle channels immediately after sensing butat a later time. This can occur when SU–SU channelsquality is bad and/or if a SU wants to conserve power orif she does not want to interfere with PU receiver. Hence, itwould be useful to have a procedure where SUs cantranslate payoffs to some form of common currency anduse it to bid at an appropriate time depending upon theirdata rate requirements for transmission, power con-straints, estimated channel quality, etc., or simply savepayoffs for later use. A suitable scheduling mechanism isneeded to facilitate the process of coordinating bidsand allocating idle channels. Vickrey–Clarke–Groves(VCG) auction [11] is used as an example to demonstratethat a socially optimal and feasible scheduling mechan-ism exists that can allocate idle channels to SUs. How-ever, devising an optimal bidding strategy for SUs is acomplex combinatorial problem that entails a separatestudy in mechanism design and is beyond the scope ofthis paper.

Non-cooperative game theory has been used to study,design, and evaluate performance of CRs in [12]. A detailedtutorial on coalitional game theory for communicationnetworks can be found in [13]. CGs have been applied inhigher layers such as network and transportation layers tostudy routing protocols in [14], packet forwarding in [15],dynamic spectrum access in [16] and resource allocation in[17]. CGs have been used to model cooperation in wirelessnetworks [18] and coalition formation games have beenstudied with respect to cooperative spectrum sharing ininterference channels [19]. Auctioning has been studied indetail with respect to spectrum sharing in [20], whereascooperative and competitive spectrum bidding and pricinghave been studied in [21]. A joint spectrum sensing andsharing model based on a coalitional game in partitionform is proposed in [22], where the payoffs are dependenton externalities, i.e, the way network is partitioned. How-ever, in a typical CR setting, worth of a coalition does notdepend on SUs who are not a part of the coalition (exceptwhen SUs have malicious intent), thereby rendering CGs ina characteristic form more suited to modeling scenarioswhere each SU is looking to solely maximize her utility inthe coalition without considering the action of other SUsoutside the coalition.

In this paper, we propose a framework for jointlymodeling spectrum sensing and sharing in CRs as a CGin a characteristic form. The characteristic function cap-tures the essence of cooperation by quantifying what eachSU brings to the coalition by sensing the channels. Theresulting CGs have an inherent structure with desirableproperties such as balancedness and super-additivity.Balanced CGs have non-empty cores which make resourceallocation possible and stable. Super-additivity providesSUs with an incentive to form the grand coalition so thatSUs prefer to cooperate rather than compete with eachother. Since the core could be very large, one-pointsolutions that lie within the core are used to calculatesingleton solutions, i.e., SUs receive mutually agreeablepayoffs for cooperatively sensing the spectrum. Dependingupon their rate requirements, SUs bid using payoffs earnedfrom increasing awareness about state of spectrum occu-pancy to gain access to idle channels. A VCG auction isused to demonstrate that a socially optimal and feasiblescheduling mechanism exists that can allocate idle chan-nels to SUs based on their bids and data rate requirements.The proposed cooperative game-theoretic joint spectrumsensing and access model (CG-JSJA) provides the bestbalance between fairness, cooperation and performancein terms of data rates obtained by each SU as well as datasum rates of all SUs among other considered models suchas individual sensing and probabilistic access model (ISPA),joint sensing and probabilistic access model (JSPA), jointsensing and round robin access model (JSRR) and jointsensing and rate maximization access model (JSRM).Thecontributions of this paper are as follows:

�

A novel and comprehensive framework for jointlymodeling spectrum sensing and sharing in CRs as aCG in a characteristic form is proposed.

�

The characteristic function of the CG is derived based onthe worth of SUs, calculated according to the amount of

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–29 17

work done by a SU (from spectrum sensing) for thecoalition, which is measured in terms of improvement inawareness about state of spectrum and PU activity.

�

The resulting CG is shown to be balanced and super-additive in nature (ensuring that the resource alloca-tion is possible and that SUs have an incentive tocooperate) and payoffs to SUs are computed throughone-point solutions that lie within the core.

�

A VCG auction is proposed in CR context to demon-strate that a socially optimal scheduling mechanismexists that can allocate idle channels to SUs based ontheir data rate requirements for transmission.

�

tsx ttx

Fig. 1. Time slots are divided into sensing minislot and access slots.

The proposed CG model is compared with other com-mon allocation models to show that it achieves the bestbalance among fairness, cooperation and performancein terms of data rates obtained by SUs.

The rest of this paper is organized as follows. CRnetwork structure and model used for spectrum sensingand sharing is overviewed in Section 2. The proposed CGmodel and VCG auction is described in Section 3. Inparticular, spectrum sensing and sharing in CRs is modeledas a CG, characteristic function of the CG is derived, payoffsto SUs are computed, CG is characterized and VCG auctionis explained. Simulation examples that show how the CGmodel allocates resources in a fair and stable manner areillustrated in Section 4. The proposed CG model is com-pared against other common allocation models and theperformance in terms of data rate obtained by each SU isanalyzed. Section 5 concludes the paper. Analytical resultsthat prove the characteristics of CG such as non-negativity,monotonicity, balancedness and super-additivity are pro-vided in Appendix A.

2. System model

In this section, we describe the CR network structureand model used for spectrum sensing and sharing in CRs.The main idea of cooperative sensing is to enhance thesensing performance by exploiting the spatial diversity inthe observations of SUs. There are N spatially distributedSUs who wish to cooperate and make use of unusedlicensed spectrum. The available spectrum may consist ofcontiguous or scattered frequency bands and is assumed tobe divided into M sub-bands. There are one or more PUsoperating in these sub-bands. The FC manages the sensingand access policy of SUs and communicates with themthrough a common control channel. After sensing, decisionstatistics from SUs such as local binary decisions, LLRs orother sufficient statistic combined with side informationsuch as SNRs may be mapped to probability of detecting(Pd) PU at the FC. This is possible because, for most widelyused detectors, ROC curves and Pd as a function SNR havebeen established analytically [23]. The diversity gain incooperative sensing can be quantified by using the slope ofthe Pd versus SNR curve [24]. Based on detection prob-ability values sent by SUs, FC makes a decision about PUactivity. FC can choose any standard fusion rule such assoft linear combining or equal gain combining in order toarrive at this decision. FC may also employ special fusionrules such as selective down-weighted combining to

counter the effects of malicious or unreliable SUs. Thedetection performance and hence the diversity gain fromcooperative sensing depend on the fusion rule employedat the FC, but in general, spatial diversity improves thedetection performance and serves to counter the effects offading in wireless channels.

Once idle channels have been identified, the worth ofeach user and coalition is calculated by FC based on thequantity and the quality of work done for the coalition. Theworth of each user/coalition determines how the overallbenefit created by cooperative sensing is distributedamong SUs. Payoffs received by each SU is calculatedaccording to CG solution concepts. With this payoff, SUsbid for idle channels based on their data rate requirementsand power constraints or save the payoffs for later use. FCacts as the auctioneer and allocates idle channels to SUsthrough the VCG auction mechanism.

Every time slot t consists of a sensing (Sx) minislotfollowed by an access (Tx) slot as shown in Fig. 1. Sensingperiods are synchronized between SUs and simultaneoussensing, and transmission in the same channel is notpossible. In order to minimize the number of times eachSU communicates with FC during every time slot t,information exchange between them is modeled in thefollowing manner.

At the end of Sx minislot in t, each SU sends FC thefollowing:

�

Sensing results in terms of decision statistic (e.g., LLRs)and potential side information on interference levels(SU–PU channel SNRs) or channel quality for the justconcluded Sx minislot in t.

�

Estimated capacities of SU–SU channels in t. (Assumingside information is available to calculate instantaneousSINR.)

�

A sealed bid for accessing idle channels during theupcoming access slot in t depending upon SU's datarate requirements and power constraints.

�

Preference for number of channels to be sensed by SUduring Sx minislot in tþ1 depending upon powerconstraints of SU.

At the beginning of access slot in t, FC sends SUs thefollowing:

�

A Channel allocation map indicating which SU shouldtransmit over which idle channel during access slot in tbased upon SU bids and estimated channel capacities.

�

Normalized payoffs received by SUs for increasingawareness about state of spectrum occupancy up untilt slots with which SUs bid at the end of Sx minislot intþ1 for idle channels that will be available in tþ1.

�

A spectrum sensing allocation map according to whichSUs sense different sub-bands of spectrum during Sxminislot in tþ1.

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–2918

with each other more than once in t to optimize the

It is also possible that SUs and FC could communicate

bidding procedure. Instead of bidding based on their owndata rate requirements, SUs could first obtain informationabout idle channels and then make an informed bid basedon the estimated achievable data rates of idle channels. Forinstance, if estimated achievable data rates of idle channelsare not sufficiently high for SU's data rate requirements orif there are power constraints, then she can make a low bidand save her payoff for future bidding when the channelconditions are good. Even further complicated commu-nication systems can be considered, but in this work, weuse a simple and basic information exchange model asdescribed above to illustrate the basic idea.

3. Game modeling

In this section, we describe how the characteristicfunction of proposed game is modeled in a CR context,or in other words, how the worth of each SU and coalitionis calculated at FC and how their payoffs are computed. Wealso describe Vickrey–Clarke–Groves (VCG) auctions in aCG setting to demonstrate as an example that a feasiblescheduling mechanism exists to allocate idle channels toSUs based on their payoffs and data rate requirements.

In the absence of any prior information or model aboutPU activity, it is assumed to be random and is modeled as aBernoulli process. Thus, prior probability of deciding thatthe PU is active before sensing the channel is 0.5. Binaryentropy function H measures the amount of uncertaintyassociated with detection probabilities Pd, i.e., uncertaintyassociated with PU activity before sensing the channel is atits maximum with a value of Hð0:5Þ ¼ 1. The worth of a SUis the amount of work done and it can be measured interms of information that she brings from sensing thechannels. The key principle involved in calculating worthof each SU is based on amount of increase in awarenessabout spectrum state that she brings from sensing thechannels that may be quantified by reduction in uncer-tainty about PU activity. This is possible due to conditionalindependence assumption of the detectors (conditionedon the null and alternate hypotheses) and hence, each SU'scontribution to decision making process can be isolatedsince the joint probability or likelihood factors into theproduct of local (marginal) probability or likelihood.Reduction in uncertainty is calculated as

vðfigÞ ¼Hð0:5Þ�HðpijÞ ¼ 1�HðpijÞ; ð1Þ

where pij is the probability of detecting PU by SU i onchannel j. However, a SU is not rewarded if its detectionprobability value is not in agreement with the globaldecision taken at FC about PU activity as shown in Fig. 3.Moreover, there might be other users bringing in informa-tion to the FC about PU activity on that particular channel.Since FC values information from all entities are equal,information brought in by a SU will be appropriatelyweighted by the total number of entities sensing thatchannel. Thus, the worth of SU i on channel j is given by

v figð Þ ¼ 1�HðpijÞciðjÞ

; ð2Þ

where ci(j) is the total number of entities sensing channel jin addition to SU i. Aggregating the reduction in uncer-tainty from all sensed sub bands, the worth of SU is givenby

v figð Þ ¼ ∑M

j ¼ 1

1�HðpijÞciðjÞ

; ð3Þ

where M is the total number of sub-bands sensed.After sensing the channel, local decision statistics

(LLRs) and side information such as SNRs which improvethe awareness about state and occupancy of spectrum areused for detecting the presence of PU. SUs may themselvesmap decision statistics to detection probability pij based onROC curves for the employed detector and distribution ofdecision statistics. Thus, pij reported by a SU serves twopurposes. Firstly, it is used to estimate the amount of workdone by each SU for the coalition and secondly, it is alsoused by FC for detecting the presence of PU in the channel.Therefore, by reporting detection probability values, con-tribution of a SU to spectrum sensing is quantified bymeans of Eq. (3), a decision about PU activity is made and aconsiderable amount of communication overhead isreduced.

Calculating worth of a coalition is slightly more involved.The coalition is treated as if it were a single user. Thedetection probability value for a coalition that characterizesthe worth of the coalition is chosen from the variousdetection probabilities of the SUs within the coalition. Sincethe coalition is bound by FC's decision on spectrum occu-pancy, the detection probability for the coalition is deemedto be the probability that agrees as closely as possible withdecision taken at FC about PU activity, thereby maximizingthe worth of the coalition and payoffs obtained by itsmembers. It must be noted that this chosen detectionprobability among SUs is used only for the purpose ofcalculating the notional worth of a coalition if that particularcoalition were to materialize and is not used for detectingthe presence or the absence of PU in the channel. The worthof the coalition is calculated only after a decision about PUactivity has been made and idle channels have beenidentified.

The chosen detection probability for a coalition is givenby jmax8 iASðpijDjÞj, where Dj is the spectrum decision onchannel j taken at FC (þ1 when PU is present and �1when PU is absent). The main idea of cooperative sensingis to enhance the sensing performance by exploiting thespatial diversity in the observations of SUs [25]. Due todiversity gains in sensing, information obtained from acoalition is more reliable than information obtained from asingle SU belonging to that coalition. In order to accountfor this, the total reduction in uncertainty about PUactivity brought in by the coalition is appropriatelyweighted by number of SUs in the coalition.

The characteristic function is thus given by

v Sð Þ ¼ jSj ∑M

j ¼ 1

1�H jmax8 iA SðpijDjÞj� �

cSðjÞ

0@

1A; ð4Þ

where S is any coalition in f1;2;…;Ng, jSj representscardinality of set S, M is the number of channels, H is thebinary entropy function, pij is the probability of detecting

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–29 19

PU by SU i on channel j, Dj is the spectrum decision(þ1 when PU is present and �1 when PU is absent) onchannel j and cS(j) is the total number of entities sensingchannel j including coalition S.

The characteristic function formulated above measuresthe amount of work done by SUs both in terms of quality(amount of increase in spectrum awareness and reductionin uncertainty about PU activity) and quantity (number ofchannels sensed). If two SUs form a coalition, SU #2definitely benefits from the reduction in uncertainty aboutPU activity brought in by SU #1 and vice versa. Thus, theworth of the coalition is better than either of the individualusers’ worth due to sharing or partial transfer of increasedawareness about state of spectrum and improved decisionmaking performance amongst themselves. This mechanismexactly captures the essence of cooperation in spectrumsensing in CRs.

An allocation or a payoff a¼ ða1;…; aNÞ is a division ofthe overall value v(N) created, where ai is the valuereceived by SU i. An allocation is individually rational ifaiZvðfigÞ for every i, i.e., a SU will have an incentive to jointhe game if the allocation gives her greater than equal towhat she can get by herself. An allocation is efficient if∑N

i ¼ 1ai ¼ vðNÞ, i.e., all the value that is created is allocated.An allocation is said to be in the core of the game if it isindividually rational and efficient and for every subset(coalition) S of N, we have aðSÞZvðSÞ, where v(S) is theworth of the coalition and a(S) is the S-allocation or sum ofthe values allocated to each player i in the coalition S(aðSÞ ¼∑iASai). When it comes to sharing unused spec-trum, the total value v(N) created from sensing can bedistributed amongst SUs in any manner as long as theallocation a lies in the core. In other words, the worth ofthe coalition is unaffected irrespective of the division ofthe worth amongst its members. When this criterion issatisfied, the CG is considered to have transferable utility(TU) [9,10]. With TU, the cooperative possibilities of thegame are completely described by the characteristic func-tion v. Thus, it is justified to model spectrum sensing andsharing in CRs as a TU CG (N,v) in a characteristic form asdescribed in Eq. (3).

CGs modeled in this fashion have interesting proper-ties. If vðSÞZ0, for each non-empty SAN, the CG is non-negative. This implies that the worth of any coalition ispositive, meaning that by sensing channels, awarenessabout state of spectrum is increased and uncertainty aboutPU activity can only reduce and not increase. The CG ismonotonic if, for all non-empty S; TAN; S� T ; vðSÞrvðTÞ.Monotonicity ensures that addition of new user/s to acoalition can only result in further reduction in uncer-tainty about PU activity. The CG is balanced if, for eachnon-empty SAN,

∑SλSvðSÞrvðNÞ ð5Þ

for every balanced collection of weights λS. Balancedness isa necessary property for CGs because scheduling andresource allocation must be possible and stable, meaningthat SUs, after sensing the spectrum, must somehow beable to access idle channels without collisions as deter-mined by a scheduling rule. This condition is guaranteed

through the balanced nature of the CG by ensuring that thecore of the CG from which resource allocation is made isnot empty. If the characteristic function of CG were notbalanced, it is not possible for SUs to arrive at any solutionon how to share idle channels in a fair manner, therebyrendering cooperative sensing moot. The CG is super-additive if, for all non-empty S; TAN; S \ T ¼ϕ,

vðS [ TÞZvðSÞþvðTÞ: ð6ÞSuper-additivity ensures that SUs prefer to form a grandcoalition for cooperatively sensing and sharing the spec-trum. This is justified because the radio channel, propaga-tion environment and spectral resources are all localphenomena that vary as a function of time and frequency.As a result, it is beneficial for all SUs in a small neighbor-hood to form a coalition. Super-additivity property, thoughnot necessary, incentivizes the formation of a grand coali-tion amongst all SUs in a local neighborhood.

For a balanced CG, the core is a non-empty convexregion and every point in this region is a potential solutionto the CG. Existence of core ensures stability of allocation,but does not stipulate how to divide the value created bycooperating players. Common singleton solutions such asnucleolus aN , Shapley value aS and τ-value aT (each withits own fairness concept) are used to arrive at a payoff thatis mutually agreed by the SUs. The Shapley value for eachplayer i is the expected marginal contribution when itjoins the coalition, over all orders of a player i to the set ofplayers who precede her. In other words, the Shapley valueis an average measure of fairness:

aS ið Þ ¼ ∑S � N\fig

jSj!ðN�jSj�1Þ!N!

v S [ figð Þ�v Sð Þð Þ: ð7Þ

The excess of a coalition measures the dissatisfaction ofthe allocation within the coalition. The absolute value ofthe excess of a coalition is the measure of the amount overand above the worth of the coalition that it obtains when apayoff is allocated and is therefore an indirect measure ofthe fairness of the allocation. The nucleolus is the solutionthat minimizes the maximum excess in a non-increasingorder, or in other words, the nucleolus is the best alloca-tion under min–max fairness criterion:

aNðiÞ ¼mini

kijðvÞ; kijðvÞ ¼maxSAN

ðvðSÞ�xðSÞÞ; 8 iAS; jAN\S:

ð8ÞThe τ-value can be thought of as a trade-off betweenmarginal contribution and minimum right payoff of aplayer. In other words, τ-value is a compromise betweenutopian agreement and ultimate dissatisfaction:

aT ðiÞ ¼ αmiðvÞþð1�αÞMiðvÞ;αA ½0;1� s:t: ∑iAN

xT ðiÞ ¼ vðNÞ:

ð9ÞSingleton solutions provide payoffs to SUs which are

not directly used to assign idle channels, but is insteadused as currency in a bidding process to access idlechannels. Therefore, apart from minor numerical differ-ences in payoffs that is largely irrelevant in the biddingprocess, singleton solutions do not directly affect theallocation of idle channels to SUs. The nucleolus of acoalitional game is unique and exists as long as the core

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–2920

is non-empty. Therefore, it is safe to always use thenucleolus as a solution to a cooperative game in theabsence of any other specific fairness criterion.

The FC can proportionally allocate idle channels in theupcoming access slot to SUs for transmission based upontheir payoffs earned. However, it is possible that a numberof idle channels are quite limited in order to be distributedto all SUs. Also, SUs might not prefer to access idlechannels immediately after sensing but at a later time orthey may want to save power by not accessing a badchannel even though it is idle. Hence, instead of directlytranslating payoffs to spectrum resources, it would beadvantageous to have a scheduling mechanism wherepayoffs can be translated to some form of currency andbe used according to the channel conditions and wishesof SUs.

Payoffs obtained by SUs from cooperatively sensing thespectrum are normalized and directly translated to acommon currency unit. SUs use this currency to bidaccording to their data rate requirements for transmission.SUs with full buffers or an urgent need to transmit (for e.g.,delay intolerant data such as video calls) will bid almost allof their payoff to acquire best possible channel for trans-mission, while SUs having a less urgent need to transmit(for e.g., delay tolerant data such as file transfer) willjudiciously bid only a part of their payoff and reserve therest to bid when a greater demand to transmit arises in thefuture and SUs who do not wish to transmit need notmake a bid at all. Therefore, we consider a Vickrey–Clarke–Groves (VCG) auctioning procedure to prove that a feasiblescheduling mechanism exists that can allocate idle chan-nels to SUs based on their channel conditions, payoffs anddata rate requirements. VCG auction merely acts as ascheduling mechanism to enable SUs to use their payoffsearned from sensing for accessing idle channels and is nota part of the CG. A brief introduction to VCG auctions asapplied to spectrum sensing and sharing in CRs is given in[26]. Social optimality of VCG auctions has been proved in[11]. Thus, VCG auction mechanism provides SUs with adominant strategy to bid their true values and is thereforesocially optimal in the sense that it maximizes total utilityof all SUs, i.e., it allocates idle channels to SUs who value itthe most.

For a SU to convey to FC on how much she values achannel, she has to bid. The bid is a direct reflection of SU'schannel conditions and data rate requirement for trans-mission. The FC acting as VCG auctioneer arranges bidsfrom SUs and the highest bidding SU is allocated a channelthat is best for her (in terms of estimated channel capacity)and is charged a price equal to the second highest bid plus abid increment. The bid increment ensures that winner ofthe auction pays a higher price than the second highestbidder. This price is now subtracted from the bid of SUwho was just allocated the channel and bids from SUs arerearranged to allocate remaining idle channels in a similarmanner. This procedure continues until FC runs out of idlechannels to allocate. Balance payoffs of SUs after theauction are normalized again and are averaged withnormalized payoffs obtained from sensing the spectrumin previous time slots. Normalized averaging of payoffdeters malicious SUs by ensuring that no single SU can

collect a huge amount of payoff by either not bidding for along time or reporting false detection probability valuesand then suddenly hogging the auction by exploiting allavailable spectrum opportunities and denying other SUs afair chance of accessing the spectrum. This procedure ofSUs cooperatively sensing the spectrum, earning payoffsthrough a CG, bidding through a VCG auction based ontheir data rate requirements is repeated over all time slots.

Formally, a VCG auction mechanism can be described asfollows. Let g be the idle spectrum auctioned by the FC andlet biðgiÞ denote SU i's value for any non-negative vector gi.Each SU reports a value function b̂i to the FC which thencomputes a value-maximizing allocation given by

gnAarg maxgi

∑ib̂iðgiÞ subject to ∑

igirg: ð10Þ

The price paid by SU i is then given by

pi ¼ αi� ∑ma i

b̂mðgn

mÞ where αi ¼max ∑ma i

b̂mðgmÞj ∑ma i

gmrg

( ):

ð11ÞNote that αi depends only on the value reports of otherSUs and not on what SU i reports.

From the data rate point of view, by allocating to thehighest bidder the best channel as seen by her, VCGauction maximizes total utilities of all SUs and is henceconsidered to be socially optimal. Also, VCG auctionprovides SUs with a weakly dominant strategy to truth-fully bid according to its actual transmission requirements.If a SU experiences severe fading environment, it will notbid on bad channels and will save the rewards earned fromsensing for transmission in the future when the channel isgood. By allocating channels to SUs who value them mostand have worked qualitatively and quantitatively in termsof sensing the spectrum to achieve a larger payoff, theresulting CG is construed to be fair by all SUs whichprovides them with an incentive for future cooperation.

4. Simulation examples

In this section, we provide an example to illustrate theproposed CG approach to jointly modeling spectrum sen-sing and sharing in CRs. A multi-band, multi-user scenariois considered. We show that through the CG approach,allocation is fair, stable, provides SUs with an incentive tocooperate and is socially optimal. We also compare theproposed model to other common allocation models interms of data rates obtained by each SU and data sum rateof all SUs to show that the proposed method provides bestbalance among fairness, cooperation and performance.

Simulations were carried out in MATLAB with the helpof TUGlab toolbox [27]. PU signal used in simulations is aOFDM signal which is employed in many wireless com-munication systems such as 3GPP Long term evolution(LTE), IEEE 802.11 a/g/n Wireless local area networks(WLAN), IEEE 802.16 Wireless metropolitan area networks(WMAN), and Digital video broadcasting (DVB) standardsDVB-T and DVB-T2.

Let H0 be the null hypotheses, i.e., an OFDM based PU isabsent and H1 be the alternate hypotheses, i.e., an OFDMbased PU is active. The autocorrelation property of OFDM

Table 1Spectrum sensing map and SNR values from sensing the channel.

Channels

1 2 3

(a) Sensing mapSU 1 � – –

2 – � �3 � � –

(b) SNR valuesSNR matrixSU 1 �19.5949 – –

2 – �7.2246 �17.06423 �8.5656 �17.1763 –

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–29 21

systems using cyclic prefix (CP) is used for detection of PUs[28]. Let length of the useful symbol data Td¼32 andlength of the cyclic prefix Tcp ¼ Td=4¼ 8. SUs employ anautocorrelation based detector (any other widely useddetector can also be employed) and the detection periodis assumed to be 100 OFDM blocks. Therefore, the numberof samples for autocorrelation estimate at the SU detectoris 100ðTdþTcpÞ ¼ 4000 Samples. This corresponds to asensing time of approximately 0.6 ms for a OFDM systemwith 7 MHz VHF TV channel bandwidth.

The standard approach is to derive the distribution ofdecision statistics under both hypotheses and then basedon decision making strategy, a mapping between detectionprobability Pd and SNR may be established. Decisionmaking strategy may be chosen from among Neyman–Pearson, Bayes, Minimax, MAP, etc., and ROC curves for thedetector are plotted. For example, assuming that teststatistics r are Gaussian distributed [28] with zero meanand variance σ2 under H0 and Gaussian distributed withmean μ1 and variance σ2 under H1, for a fixed threshold η,probability of false alarm and probability of detection aregiven by

Pfa ¼ P r4η H0j Þ ¼ 12erfc

ηffiffiffi2

pσ

� �;

�

Pd ¼ P r4η H1j Þ ¼ 12erfc

η�μ1ffiffiffi2

pσ

� �:

�ð12Þ

SUs employ a decentralized cooperative detectionscheme where soft detection probability values from eachSU is combined at FC under conditional independenceassumption to detect PU activity based on a detection rule.In this case, assuming the parameters of OFDM based PUas mentioned above, SUs sense channels and translateSNRs to probability of detection Pd of PU using Neyman–Pearson detection strategy under a fixed probability offalse alarm Pfa ¼ 0:05 as shown in Fig. 2(a). SNR is definedas SNR¼10 log10 σ2

x=σ2n, where σ2x and σ2n are the trans-

mitted signal power and the noise variance respectively.Receiver operating characteristic (ROC) plots of SU detec-tor for various SNRs are shown in Fig. 2(b).

−25 −20 −15 −10 −5 00

0.2

0.4

0.6

0.8

1

SNR in dB

Pro

babi

lity

of d

etec

tion

Pd

Fig. 2. (a) SNR values are mapped to Pd of PU for the Neyman–Pearson detectiovalues of SNRs.

4.1. Demonstration of concept

In order to illustrate the proposed CG approach usingspectrum sensing and sharing in CRs as an example,consider the scenario in which there are 3 SUs sensing 3channels. SUs choose the number of channels they preferto sense according to their data rate requirements, powerconstraints, etc., and FC creates a spectrum sensing mapaccordingly as shown in Table 1(a). The � in the tablerepresent channels sensed by SUs. For channels that aresensed by SUs, SNR values are generated from a randomvariable that is uniformly distributed between �25 dB and�5 dB as shown in Table 1(b).

SNR values are translated to probability of detecting PUand SUs send the resulting soft detection probabilityvalues to FC where they are combined and idle channelsare decided as shown in Table 2. FC may choose anystandard fusion rule such as likelihood ratio test, softlinear combining or equal gain combining. In this example,we have used a simplistic assumption of equal gaincombining fusion rule at FC. The presence of PU is denotedby þ1, whereas the absence of PU is denoted by �1.

Reduction in uncertainty about PU activity is used toquantify the reward that a SU obtains for the informationthat she brings from sensing the channel. However, it must

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Probability of false alarm Pfa

Pro

babi

lity

of d

etec

tion

Pd

SNR = −25 dBSNR = −20 dBSNR = −15 dBSNR = −10 dBSNR = −5 dB

n under a constant false alarm rate Pfa ¼ 0:05. (b) ROC plots for different

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–2922

be noted that this reward is conditioned on the decisionabout channel availability made at FC as shown in Fig. 3.

Using (4), the characteristic function for this CG iscalculated to be

v¼ fvð1Þ; vð2Þ; vð3Þ; vð12Þ; vð13Þ; vð23Þ; vð123Þg¼ f0:3107;0:7819;0;2:1851;1:2427;2:0450;4:9316g:

ð13ÞCharacteristic function gives the worth of each SU and

all possible SU coalitions. Here, SU #2 has the highestworth vð2Þ for two reasons. Firstly, SU #2 sensed channel#3 that was not sensed by any other SU, thereby bringingvaluable information from that channel. Secondly, she wasresponsible for detecting PU activity on channel #2. Henceit is justified on the part of SU #2 to expect the highestpayoff. Even though she has sensed only one channel, SU#1 has the next highest individual worth vð1Þ as hersensing result was in direct agreement with the decisiontaken at FC about PU activity and expects to get secondhighest payoff. Though SU #3 sensed two channels, herresults were not in agreement with the decision taken atFC about PU activity and hence ends up with no worth.However, from channel sensing map, if coalition {23} wereto form, SU #3 is the only SU sensing channel #1 and ifcoalition {13} were to form, SU #3 is the only SU sensingchannel #2, thereby bringing valuable information fromthese channels. Since these contributions are valuable, itis fair on the part of SU #3 to expect a non-zero payoffeven though her individual worth is zero. From thecharacteristic function, different one-point solutions forthis CG based on Shapley value, τ-value and nucleolus arecomputed and shown in Table 3.

Table 2Identification of idle channels from detection probability values.

Pd matrix Channels

1 2 3

SU 1 0.0734 0.5 0.52 0.5 0.8837 0.09683 0.7054 0.0953 0.5

Decision �1 1 �1

0 0.5 10

0.2

0.4

0.6

0.8

1

Probability of detection (Pd)

Rew

ard

(1 −

H(P

d))

PU absent

Fig. 3. Reward calculated in terms of amount of reduction of uncertainty aboreporting Pd values that contradict the decision taken by FC are not rewarded.

From Table 3, it is clear that SU payoffs reflect theirexpectations. Differences in payoffs between various one-point solutions arise by the virtue of definition of thesesolutions. In this CG, singleton solutions provide payoffs toSUs which is not directly used to assign idle channels toSUs, but is instead used as a currency by SUs in a biddingprocess to access idle channels. Therefore, apart fromminor numerical differences in payoffs that is largelyirrelevant in bidding process, singleton solutions do notdirectly affect allocation of idle channels to SUs. For allpractical purposes, it is safe to use nucleolus, as it isguaranteed to lie in the core, ensuring that resourceallocation is stable and that SUs have an incentive tocooperate in the future.

Working with normalized nucleolus as payoffs, SUs bidfor access to channels based upon their transmissionrequirements which is modeled as a Gaussian randomwalk. The transmission requirements of the SUs can bequantified in terms of the size of the outgoing data buffer.Assuming additive white Gaussian noise (AWGN) channelbetween the SU and its receiver with SNR values drawnfrom a uniform distribution between �25 dB and �5 dB,data rates (in Mbps) for a 7 MHz bandwidth is estimatedusing Shannon's capacity formula as shown in Table 4. Itmust be noted that a different independent randomvariable is used to generate this distribution as comparedto the one used for obtaining spectrum sensing SNR valuesearlier.

VCG auction takes place as follows. From Table 4, SU #1has the highest bid and the best unoccupied channel forher in terms of obtainable data rate is channel #3. She getsaccess to channel #3 at the price of second highest bidder(SU #3) plus one bid increment totaling 22.3674. This priceis deducted from the bid of SU #1 and the new bid of SU

0 0.5 10

0.2

0.4

0.6

0.8

1

Probability of detection (Pd)

Rew

ard

(1 −

H(P

d))

PU present

ut PU activity from Pd values after decision on PU activity is made. SUs

Table 3Normalized payoffs for the CG based on various one-point solutions.

Normalized payoffs One-point solutions

Shapley values Tau values Nucleolus

SU 1 30.5526 30.6662 32.24842 43.4645 43.3531 41.80293 25.9830 25.9807 25.9487

Table 4Estimated channel capacities along with secondary user bids.

Channel capacity in Mbps Channels SU bids

1 2 3

SU 1 0.0547 0.0429 0.0974 24.79432 0.7187 0.0143 0.4765 6.99173 2.0485 0.9998 0.0318 22.3673

Table 5SU payoffs, bids and channel allocations based on VCG auction.

Allocation table Secondary users

1 2 3

Norm. payoff 32.2484 41.8029 25.9487Bids 24.7943 6.9917 22.3673Price Paid 22.3674 – –

Channel allocated #3 – –

Rate obtained (in Mbps) 0.0974 – –

Bids 2.4269 6.9917 22.3673Price paid – – 6.9918Channel allocated – – #1Rate obtained (in Mbps) – – 2.0485Balance payoff 9.8810 41.8029 18.9569Norm. balance payoff 13.9876 59.1767 26.8357

1 2 30

2

4

6

8

10

12

14

16x 106

Dat

a ra

te a

chie

ved

in M

bps

Secondary users

CG−JSJA vs JSPA, 3 SUs, 5 Channels

Fig. 4. Comparison between the proposed CG-JSJA and JSPA models. Thebox plot depicts minimum value, first quartile, median, mean, thirdquartile and maximum value of obtained data rates for both models foreach SU. It can be seen here that the proposed CG-JSJA model outper-forms the JSPA model in terms of data rates obtained by SUs.

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–29 23

#1 is 2.4269. Bids are sorted again and auction is con-tinued. SU #3 is the highest bidder now and the bestunoccupied channel for her is channel #1. As SU #2 is thesecond highest bidder now, SU #3 pays a price of 6.9918for channel #3. After completion of bidding and allocation,remaining payoffs of SUs are normalized and averagedwith normalized payoffs obtained from next time slot. Theallocation procedure is depicted in Table 5.

A few remarks are in place here. Though SU #3 had thelowest payoff, her need for transmission was greater thanSU #2 as was reflected in her bid and hence, she still getsaccess to a channel. It must be noted that price paid by aSU for accessing a channel is dependent only upon her bidand bids placed by other SUs (which reflect their data raterequirements) and is not on the estimated channel capa-city. This is why SU #1 pays a far higher price for a channelwith lower estimated capacity compared to SU #3. SU #2bids conservatively and reserves her payoff for future use.At the end of allocation, we see that SUs #1 and #3 have avery low normalized balance payoff compared to SU #2and will have to perform more sensing work for thecoalition to be able to bid competitively in the future.VCG auction does not guarantee data rate maximization,but allocates best possible channels to SUs according totheir bids and hence satisfies their data rate requirementsin a fair manner. Thus, the proposed CG model for jointlymodeling spectrum sensing and sharing scenario in CRsresults in an allocation that is fair, socially optimal andprovides SUs with an incentive to cooperate in the future.

In the following subsections, we compare the perfor-mance of proposed CG model for resource allocation with

other common allocation models in terms of data ratesobtained by each SU and data sum rate of all SUs. We showthat the proposed model provides the best balancebetween fairness, cooperation and performance amongall other models considered. Assuming a similar set-upas described in this section, we carry out simulations over1000 time slots with 3 SUs sensing 5 channels.

4.2. Comparison with joint sensing and probabilisticaccess model

The proposed CG joint sensing and joint access (CG-JSJA) model is compared with a joint sensing and prob-abilistic access model (JSPA). In JSPA, SUs jointly sense thechannel and collaboratively detect the PU at FC, but accessidle channels in a probabilistic fashion very similar tocarrier sense multiple access (CSMA) protocol. In order tocompare the two models in a fair manner, in JSPA, SUschoose the number of channels to access from the avail-able free channels according to their data rate require-ments for transmission which is proportional to their bidsin the CG-JSJA model. From the channel capacity estimatesthat they have, SUs choose channels with maximumcapacity for transmission. During sensing minislot, SUslisten to the channel and if found idle, transmit their datain the access slot. In the event of a collision, SUs back offfor a random amount of time before retransmitting theirdata. Data rates obtained by each SU in both models areshown as a box plot in Fig. 4. Each SU obtains a highermaximum, mean, median, lower quartile and upper quar-tile data rate in the CG approach when compared to theprobabilistic access model. Lower performance of the JSPAmodel can be attributed to collisions that occur among SUsdue to uncoordinated nature of their transmissions. More-over, collision leads to retransmission and consumes SUbattery energy. Thus, there is need for a joint accessmechanism for SUs to effectively making use of idlechannels and increase their individual data rate.

1 2 30

2

4

6

8

10

12

14

16x 106

Dat

a ra

te a

chie

ved

in M

bps

Secondary users

CG−JSJA vs JSRR, 3 SUs, 5 Channels

Fig. 6. Comparison between the proposed CG-JSJA and JSRR models. Thebox plot depicts minimum value, first quartile, median, mean, thirdquartile and maximum value of obtained data rates for both models foreach SU. It can be seen here that the proposed CG-JSJA model outper-forms the JSRR model in terms of data rates obtained by SUs.

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–2924

4.3. Comparison with independent sensingand probabilistic access model

The proposed CG-JSJA model is compared with aindividual sensing and probabilistic access model (ISPA).In ISPA, SUs sense the channels individually and detect thePU on their own and access idle channels in a probabilisticfashion very similar to CSMA protocol. As in JSPA model, inISPA too, SUs choose the number of channels to accessfrom the available free channels according to their datarate requirements which is proportional to their bids inthe CG-JSJA model. During sensing minislot, SUs listen tothe channel and if found idle, transmit their data in theaccess slot. In the event of a collision, the SUs back off for arandom amount of time before retransmitting their data.Data rates obtained by each SU in both models are shownas a box plot in Fig. 5. Due to independent sensing on 5channels, the 3 SUs failed to detect PU activity 2605 timesin 1000 time slots which could have been avoided bycooperatively sensing the channels. Lower performance ofISPA model is due to collisions occurring between SUsduring uncoordinated channel access, thereby demon-strating the necessity for both jointly sensing and acces-sing the channel cooperatively.

4.4. Comparison with joint sensing and round robin model

The proposed CG-JSJA model is compared with a jointsensing and round robin access model (JSRR). In JSRR, SUsjointly sense the channel and collaboratively detect the PUat FC, but FC allocates idle channels in a round robinfashion. Since SUs take turns to transmit their data, all SUsget an equal opportunity in terms of accessing idlechannels and there are no collisions between SUs. How-ever, in this model, FC does not take into account the datarate requirements of SUs or quality of the channel for theSU while making the allocation. Data rates obtained byeach SU in both models are shown as a box plot in Fig. 6.From the SU's perspective, JSRR model is fair enough to

1 2 30

2

4

6

8

10

12

14

16x 106

Dat

a ra

te a

chie

ved

in M

bps

Secondary users

CG−JSJA vs ISPA, 3 SUs, 5 Channels

Fig. 5. Comparison between the proposed CG-JSJA and ISPA models. Thebox plot depicts minimum value, first quartile, median, mean, thirdquartile and maximum value of obtained data rates for both models foreach SU. It can be seen here that the proposed CG-JSJA model outper-forms the ISPA model in terms of data rates obtained by SUs.

provide SUs with an incentive to cooperate, but thesocially optimal allocation in CG-JSJA model consistentlyachieves higher data rates over JSRR model for all SUs.

4.5. Comparison with joint sensing and ratemaximization model

The proposed CG-JSJA model is compared with a jointsensing and rate maximization access model (JSRM). InJSRM, SUs jointly sense the channel and collaborativelydetect PU, but FC allocates idle channels such that datasum rate of SUs is maximized. Hence, an idle channel isalways allocated to a SU whose capacity estimate on thatchannel is highest. This allocation may be highly unfair toSUs that are for example, in a cell edge, further away orexperience shadowing. However, in the CG-JSJA model,VCG mechanism chooses a SU based on their bids andallocates the best possible channel to her. While JSRMmodel maximizes the (data) sum rate obtained by SUs, theproposed CG-JSJA model maximizes the utility of SUs (interms of their data rate requirements). Data rates obtainedby each SU in both models are shown as a box plot inFig. 7. As expected, JSRM model consistently outperformsthe proposed CG-JSJA model, but is obviously less fair toSUs with lower channel quality.

4.6. Comparison among all allocation models

Cumulative data rates obtained by each SU in allmodels are shown in Fig. 8. It is interesting to note thatISPA model performs slightly better than JSPA model. Thisis because, in ISPA model, when SUs fail to detect PU bythemselves, they seemingly have more channels to choosefrom when randomly accessing channels. Since the num-ber of idle channels as perceived by each SU increases dueto missed detections of PU, there are fewer collisionsamongst SUs. However, increased data rate is achieved atthe cost of interfering with PU (2605 missed detections ofPU by 3 SUs sensing 5 channels over 1000 time slots) and

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3

3.5x 109

Time slots

Cum

ulat

ive

data

rate

Cumulative data rates achieved by SU1

JSRMCG−JSJAJSRRISPAJSPA

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3

3.5x 109

Time slots

Cum

ulat

ive

data

rate

Cumulative data rates achieved by SU3

JSRMCG−JSJAJSRRISPAJSPA

Fig. 8. Comparison among CG-JSJA, JSRM, JSRR, JSPA and ISPA models. The cumobtained by all SUs are shown for all models.

1 2 30

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2x 107

Dat

a ra

te a

chie

ved

in M

bps

Secondary users

CG−JSJA vs JSRM, 3 SUs, 5 Channels

Fig. 7. Comparison between the proposed CG-JSJA and JSRM models. Thebox plot depicts minimum value, first quartile, median, mean, thirdquartile and maximum value of obtained data rates for both models foreach SU. It can be seen here that the JSRM model outperforms theproposed CG-JSJA model in terms of data rates obtained by SUs.

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–29 25

thereby causing collisions and retransmissions. If thepenalty associated with such missed detections of PUs isextremely high, the marginal increase in data rate of ISPAmodel is not justified. As intended, JSRM model achieveshighest possible data rate, but does not take into accountthe data rate requirements of SUs. The JSRR model, whileproviding SUs with an incentive to cooperate, is not able tomatch data rates achieved by the CG-JSJA model.

The extent of fairness of these allocation models can bestudied in terms of number of channels accessed by eachSU based on the bid that she makes. Fig. 9 shows thescatter plot for number of channels allocated to SU #3 forthe normalized bid made by SU #3 in each time slot over1000 time slots when the number of detected idle chan-nels is equal to four for all models.

In CG-JSJA model, the number of channels allocated isdirectly dependent on the bid made by SU. The higher thebid made by SU, the higher the probability of beingallocated more channels. SUs make their bids based ontheir data rate requirements for transmission and hence,the number of channels allocated to a SU is directly

0 200 400 600 800 10000

0.5

1

1.5

2

2.5

3

3.5x 109

Time slots

Cum

ulat

ive

data

rate

Cumulative data rates achieved by SU2

JSRMCG−JSJAJSRRISPAJSPA

0 200 400 600 800 10000

2

4

6

8

10x 109

Time slots

Cum

ulat

ive

data

rate

Cumulative sum data rate achieved by all SUs

JSRMCG−JSJAJSRRISPAJSPA

ulative data rates obtained by each SU and the cumulative data sum rate

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Normalized bids

Num

ber o

f cha

nnel

s al

loca

ted

CG−JSJA

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Normalized bids

Num

ber o

f cha

nnel

s al

loca

ted

JSRM

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Normalized bids

Num

ber o

f cha

nnel

s al

loca

ted

JSRR

0 0.2 0.4 0.6 0.8 10

1

2

3

4

Normalized bids

Num

ber o

f cha

nnel

s al

loca

ted

JSPA

Fig. 9. Scatter plot depicting the number of channels allocated versus normalized bids. The plot shows the number of channels allocated to SU #3 for everynormalized bid made by SU #3 in each time slot over 1000 time slots when the number of detected idle channels is equal to four for CG-JSJA, JSRM, JSRRand JSPA models. In the CG-JSJA model, the tiered step-like structure indicates that the higher the bid made by the SU, the higher the probability of beingallocated more channels. Since bids are made by SUs from payoffs earned from sensing the channel, the amount of work done by SUs and their data raterequirements are fairly reflected in the number of channels allocated in CG-JSJA model. In comparison, other models are not construed by the SUs as fairdue the fact that the number of channels allocated to them is independent of the bids made by them.

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–2926

proportional to its data rate requirements for transmission.SUs cannot make high bids for the sake of being allocatedmore channels as bids are made with payoffs they earnfrom working for the coalition. In this case, SUs can makehigh bids only by earning higher payoffs in terms of thequality and the quantity of work done in sensing thechannels. Thus, CG-JSJA model is construed by SUs to befair as both data rate requirements of SUs and amount ofwork done by SUs is reflected in resource allocation with-out any room for manipulation by SUs.

The goal in JSRM model is to maximize data rate andhence, the number of channels allocated to a SU isdependent only on channel quality and not on bids madeby SUs. JSRM model chooses an idle channel and allocatesit to a SU that can obtain the maximum data rate on thatchannel, whereas CG-JSJA model chooses a SU from VCGauction and allocates her the best possible idle channel interms of available data rate. While the data sum rate of SUsis maximized in JSRM model, the utilities of SUs (in termsof their data rate requirements) are maximized in CG-JSJA

model. JSRM model gives access to SUs that experiencegood channel conditions, but the CG-JSJA model allocateschannels by taking into account the bids and data raterequirements of SUs and the sensing work done by themfor the coalition. Therefore, a SU experiencing somewhatbad channel conditions due to her location will not get anyaccess to the channel in JSRM model, but can get access inthe CG-JSJA model by sensing and earning a payoff to makea sufficiently high bid to win the VCG auction which leads toimproved fairness. Thus, higher data rates achieved by JSRMmodel over CG-JSJA model is achieved at the cost of fairness.Similarly, in JSRR model too, data rate requirements of SUsare not taken into account and no SU gets access to morethan ⌈number of idle channels=number of SUs⌉ channels.However, SUs are given an equal opportunity to access idlechannels in a round robin fashionwhich can be construed assomewhat fair despite the lack of consideration of SU's datarate requirements. In JSPA or ISPA model, there is noallocation per se, as channels are accessed in a probabilisticfashion and no SU is able to access all idle channels due to

Table 6Comparison between the proposed CG-JSJA and other allocation models.

Comparisontable

Model

CG-JSJA JSPA ISPA JSRR JSRM

PropertiesFairness Very high Low Low High LowCooperation Very high Moderate Low High HighObtained rate High Low Low Moderate Very high

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–29 27

collisions arising from uncoordinated channel access. Acomparison between allocation models is summarized inTable 6.

Thus, the proposed CG-JSJA model provides the bestbalance among fairness, cooperation and performance interms of data rates obtained by each SU as well as datasum rates of all SUs among the considered models.

5. Conclusion

In this paper, we have proposed a novel and compre-hensive framework for jointly modeling spectrum sensingand sharing problem in cognitive radios (CRs) as a coop-erative game (CG) in a characteristic form. Additionally,Vickrey–Clarke–Groves (VCG) auction is used to demon-strate the existence of a feasible scheduling mechanism toallocate unoccupied spectrum to secondary users (SUs).The proposed CG model provides a attractive trade-offamong design goals in fairness, stability, cooperation anddata rates. A CG is formulated to quantify and share thebenefits of cooperation to allocate idle spectrum to coop-erating SUs based on their sensing results (qualitatively interms of detection performance and quantitatively interms of effort put in for the coalition). The characteristicfunction that fully describes the CG is based on the worthof SUs measured by the amount of increase in awarenessabout the spectrum state that she brings from sensing thechannels which is quantified by the reduction in uncer-tainty about PU activity. CGs modeled in this fashion havedesirable properties such as monotonicity, balancednessand super-additivity ensuring that resource allocation isstable and providing SUs with an incentive to cooperate.One-point solutions that lie in the core of the CG are usedto compute singleton payoffs to SUs which is used to bidfor idle channels based on their data rate requirements.VCG auction provides SUs with a dominant strategy to bidtruthfully, maximizes total utility of SUs and allocatesresources to SUs who value them most, thereby resultingin a socially optimal allocation. The proposed CG model iscompared with other common allocation models to showthat it achieves the best balance among fairness, coopera-tion and performance in terms of data rates obtained bySUs. Thus, the proposed CG model allocates spectralresources in a fair and stable manner, ensuring that SUshave a very strong incentive in the future to cooperativelysense and share unoccupied bands of spectrum withoutcompromising on obtained data rates.

Acknowledgments

The authors would like to thank Dr. Jan Eriksson for hisvaluable comments during early stages of this work.

Appendix A. Properties of the characteristic function

The characteristic function of the CG is given in (4). Inthis section, we provide proofs for various properties ofthe characteristic function.

A.1. Non-negativity

Non-negativity ensures that the worth of any coalitionis positive, meaning that by sensing channels, awarenessabout state of spectrum is increased and uncertainty aboutPU activity can only reduce and not increase.

For each non-empty coalition SAN; vðSÞZ0: Since pij isthe probability of detection of PU by SU i on channel j,0rpijr1. Also, dj ¼ 71. Therefore, 0r jmax8 iA S ðpij�djÞjr1. Since H is the binary entropy function, we have0rHðjmax8 iA Sðpij � djÞjÞr1, which also implies that0r1�Hðjmax8 iA Sðpij � djÞjÞr1. Also, cSðjÞ40 and jSj40.Thus,

jSj ∑M

j ¼ 1

1�Hðjmax8 iA Sðpij � djÞjÞcSðjÞ

� �Z0: ðA:1Þ

Hence, vðSÞZ0.

A.2. Monotonicity

Monotonicity ensures that the addition of new user/s toa coalition can only result in further reduction in uncer-tainty about PU activity.

For all non-empty S; TAN with S� T ; vðSÞrvðTÞ: Letdj ¼ þ1. Since PU is present, only SUs with pijZ0:5 arerewarded. Therefore, jmax8 iA Sðpij � djÞjr jmax8 iAT ðpij � djÞj.For pijZ0:5, Hðjmax8 iA Sðpij � djÞjÞZHðjmax8 iAT ðpij � djÞjÞ.Let dj ¼ �1. Since PU is absent, only SUs with pijr0:5are rewarded. Therefore, jmax8 iA Sðpij � djÞjZ jmax8 iAT

ðpij � djÞj. For pijr0:5, Hðjmax8 iASðpij � djÞjÞZHðjmax8 iAT

ðpij � djÞjÞ. Thus, in both cases, 1�Hðjmax8 iASðpij � djÞjÞr1�Hðjmax8 iAT ðpij � djÞjÞ. Also, cSZcT for all S� T . SincejSjo jTj,

jSj ∑M

j ¼ 1

1�H jmax8 iA Sðpij � djÞj� �

cSðjÞ

0@

1A

r jT j ∑M

j ¼ 1

1�H jmax8 iAT ðpij � djÞj� �

cT ðjÞ

0@

1A: ðA:2Þ

Hence, vðSÞrvðTÞ.

A.3. Balancedness

Balancedness ensures that the core of the CG fromwhich resource allocation is made is non-empty. If thecharacteristic function of CG were not balanced, it is notpossible for SUs to arrive at any solution on how to share

J. Rajasekharan, V. Koivunen / Signal Processing 106 (2015) 15–2928

idle channels in a fair manner, thereby rendering coopera-tive sensing moot.

For each non-empty SAN; ∑SλSvðSÞrvðNÞ for everybalanced collection of weights λS: Let RS be S dimensionalEuclidean space in which dimensions are indexed bymembers of S. Characteristic vector 1SARN of coalition Sis given by

ð1SÞi ¼1 if iAS

0 otherwise:

(ðA:3Þ

A collection ðλSÞ of numbers in ½0;1� is a balancedcollection of weights, if for every player i, the sum of λSover all coalitions that contain i is 1, such that ∑SλS1S ¼ 1N .Since SDN, we have

∑M

j ¼ 1

1�H jmax8 iASðpij � djÞj� �

cSðjÞ

0@

1A

r ∑M

j ¼ 1

1�H jmax8 iANðpij � djÞj� �

cNðjÞ

0@

1A: ðA:4Þ

Therefore, ð1=jSjÞvðSÞrð1=jNjÞvðNÞ) vðSÞrðjSj=jNjÞvðNÞ.Multiplying both sides by λS and summing over all possiblecoalitions, ∑SλSvðSÞr ðvðNÞ=jNjÞ∑SλSjSj. Since weights arebalanced, ðvðNÞ=jNjÞ∑SλSjSj ¼ ðvðNÞjNjÞjNj ¼ vðNÞ. Hence,∑SλSvðSÞrvðNÞ.

A.4. Super-additivity

Super-additivity ensures that a SU will have an incen-tive to join the existing coalition and continue to cooperatein the future instead of selfishly trying to compete withother SUs for spectrum access.

For all non-empty S; TAN with S \ T ¼ϕ; vðS [ TÞZvðSÞþvðTÞ: Since S� S [ T , from the monotonicity prop-erty, we have

∑M

j ¼ 1

1�H jmax8 iAS[T ðpij � djÞj� �

cS[T ðjÞ

0@

1A

Z ∑M

j ¼ 1

1�H jmax8 iASðpij � djÞj� �

cSðjÞ

0@

1A: ðA:5Þ

Similarly, since T � S [ T , from the monotonicity prop-erty, we have

∑M

j ¼ 1

1�H jmax8 iAS[T ðpij � djÞj� �

cS[T ðjÞ

0@

1A

Z ∑M

j ¼ 1

1�H jmax8 iAT ðpij � djÞj� �

cT ðjÞ

0@

1A: ðA:6Þ

Multiplying (A.5) by jSj and (A.6) by jTj and addingtogether, we have

jSj ∑M

j ¼ 1

1�H jmax8 iAS[T ðpij � djÞj� �

cS[T ðjÞ

0@

1A

þjT j ∑M

j ¼ 1

1�H jmax8 iAS[T ðpij � djÞj� �

cS[T ðjÞ

0@

1A

Z jSj ∑M

j ¼ 1

1�H jmax8 iASðpij � djÞj� �

cSðjÞ

0@

1A

þjTj ∑M

j ¼ 1

1�H jmax8 iAT ðpij � djÞj� �

cT ðjÞ

0@

1A: ðA:7Þ

jS [ T j ∑M

j ¼ 1

1�H jmax8 iAS[T ðpij � djÞj� �

cS[T ðjÞ

0@

1A

Z jSj ∑M

j ¼ 1

1�H jmax8 iASðpij � djÞj� �

cSðjÞ

0@

1A

þjTj ∑M

j ¼ 1

1�H jmax8 iAT ðpij � djÞj� �

cT ðjÞ

0@

1A: ðA:8Þ

Hence, vðS [ TÞZvðSÞþvðTÞ.

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