This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts **for** publication **in** the IEEE SECON 2006 proceed**in**gs.**Belief**-**Assisted** **Pric ing**

secondary users (unlicensed users) coexist, primary usersattempt to sell unused spectrum resources to secondaryusers **for** monetary ga**in**s while secondary users try toacquire spectrum usage permissions from primary usersto achieve certa**in** communication goals, which generally**in**troduces reward payoffs **for** them. In order to solvethe above issues, we consider the spectrum shar**in**gas multistage dynamic games and propose a dynamicpric**in**g approach to optimize the overall spectrum efficiency,meanwhile, keep**in**g the participat**in**g **in**centivesof the users based on double-auction rules and cop**in**gwith the budget constra**in**ts by dynamic programm**in**g.The ma**in** contributions of this paper are multi-fold.First, by model**in**g the spectrum shar**in**g as a dynamicpric**in**g game, we are able to quickly and accuratelycoord**in**ate the spectrum allocation among primary andsecondary users through a trad**in**g process to maximizethe payoffs of both primary and secondary users. Further,we develop a belief system to assist greedy users updatetheir strategies adaptive to the spectrum demand and supplychanges, which not only approaches the theoreticaloptimal outcomes of the spectrum allocation problem butalso substantially decreases the pric**in**g overhead due tofrequent bid/ask updates and message exchange. Third,by consider**in**g the budget constra**in**ts of the secondaryusers, the proposed dynamic pric**in**g approach is able tofurther exploit the time diversity of spectrum resources.The rem**in**der of this paper is organized as follows:The system model of dynamic spectrum allocation isdescribed **in** Section II. In Section III, we **for**mulatethe spectrum allocation as pric**in**g games based on thesystem model. In Section IV, the belief-based dynamicpric**in**g approach is proposed **for** the optimal spectrumallocation. The simulation studies are provided **in** SectionV. F**in**ally, Section VI concludes this paper.II. SYSTEM MODELWe consider the wireless networks where multipleprimary users and secondary users operate simultaneously**in** a wireless network, which may represent variousnetwork scenarios. For **in**stance, the primary users can bethe spectrum broker connected to the core network andthe secondary users are the base stations equipped withcognitive radio technologies; or the primary users arethe access po**in**ts of a mesh network and the secondaryusers are the mobile devices. On one hand, every primaryuser has the license of us**in**g a certa**in** spectrum range,which can be divided **in**to non-overlapp**in**g orthogonalchannels. Consider**in**g that the authorized spectrum ofprimary users may not be fully utilized over time, theyprefer to lease the unused channels to the secondaryusers **for** monetary ga**in**s. On the other hand, s**in**ce theunlicensed spectrums become more and more crowded,the secondary users may try to lease some unusedchannels from primary users **for** more communicationga**in**s by provid**in**g leas**in**g payments.In our system model, we assume all users are selfishand rational, that is, their objectives are to maximizetheir own payoffs, not to cause damage to other users.However, users are allowed to cheat whenever theybelieve cheat**in**g behaviors can help them to **in**creasetheir payoffs. Generally speak**in**g, **in** order to acquire thespectrum licenses from regulatory bodies such as FCC,the primary users have certa**in** operat**in**g costs. Withregard to secondary users, **in** order to have the rewardsof achiev**in**g certa**in** communication goals, they want toutilize more spectrum resources. The selfishness of bothprimary and secondary users will prevent them from reveal**in**gtheir private **in****for**mation such as acquisition costsor reward payoffs, which makes traditional spectrumallocation approaches not applicable under this scenario.There**for**e, novel spectrum allocation approaches needto be developed which not only optimize the spectrumefficiency but also extract the private **in****for**mation fromthe selfish parties through certa**in** mechanisms to assistthe optimization of spectrum allocation.Specifically, we consider the collection of the availablespectrums from all primary users as a spectrum pool,which totally consists of N non-overlapp**in**g channels.Assume there are J primary users and K secondaryusers, **in**dicated by the set P = {p 1 ,p 2 , ..., p J } andS = {s 1 ,s 2 , ..., s K }, respectively. We represent thechannels authorized to primary user p i us**in**g a vectorA i = {a j i } j∈{1,2,...,n i}, wherea j irepresents the channel**in**dex **in** the spectrum pool and n i is the total number ofchannels which belong to user p i .Def**in**eA as the set ofall the channels **in** the spectrum pool. Moreover, denotethe acquisition costs of user p i ’s channels as the vectorC i = {c aj ii} j∈{1,2,...,ni}, where the jth element representsthe acquisition cost of the jth channel **in** A i . Forsimplicity, we write c aj iias c j i . As **for** secondary user s i,we def**in**e her/his payoff vector as V i = {v j i } j∈{1,2,...,N},where the jth element is the reward payoff if this usersuccessfully leases the jth channel **in** the spectrum pool.III. PRICING GAME MODELIn this paper, we model the dynamic spectrum allocationproblem as a pric**in**g game to study the **in**teractionsamong the players, i.e., the primary and secondary users.Based on the discussion **in** the previous section, we are120

able to have the payoff functions of the players **in** ourdynamic game. Specifically, if primary user p i reachesagreements of leas**in**g all or part of her/his channels tosecondary users, the payoff function of this primary usercan be written as follows.U pi (φ Ai ,α Aii)=∑n ij=1(φ aji− c j i )αaj ii, (1)where φ Ai = {φ aj} i j∈{1,2,...,ni} and φ aj is the paymentithat user p i obta**in**s from the secondary user by leas**in**gthe channel a j i**in** the spectrum pool. Note that α Aii={α aj ii} j∈{1,2,...,ni} and α aj ii∈ {0, 1} which **in**dicates ifthe jth channel of user p i has been allocated to asecondary user or not. For simplicity, we denote α aj iiasα j i . Similarly, the payoff function of secondary user s ican be modeled as follows.U si (φ A ,β A i )=N∑(v j i − φ j)β j i , (2)j=1where φ A = {φ j } j∈{1,2,...,N} , βiA = {β j i } j∈{1,2,...,N}.Note that β j i∈ {0, 1} illustrates if secondary user s isuccessfully leases the jth channel **in** the spectrum poolor not. Hence, the strategies of the primary users andsecondary users are actually def**in**ed by α Aiiand βi A,respectively.S**in**ce the players may have conflict **in**terests witheach other, our dynamic spectrum shar**in**g game canbe modeled as a multi-stage non-cooperation game. Tobe specific, from the primary users’ po**in**t of view,they want to earn the payments by leas**in**g the unusedchannels which not only cover their spectrum acquisitioncosts but also ga**in** as much extra payments as possible;from the secondary users’ po**in**t of view, they aim toaccomplish their communication goals by provid**in**g theleast possible payments to lease the channels; whilefrom the network designers’ po**in**t of view, they attemptto maximize the network per**for**mance, which **in** ourcase is the spectrum efficiency. There**for**e, the spectrumusers **in**volved **in** the spectrum shar**in**g process constructa non-cooperative pric**in**g game [11], [12]. S**in**ce theselfish users are their own authorities, they will notreveal their private **in****for**mation to others unless somemechanisms have been applied to guarantee that it is notharmful to disclose the private **in****for**mation. Generally,such non-cooperative game with **in**complete **in****for**mationis complex and difficult to study as the players do notknow the perfect strategy profile of others. But basedon our game sett**in**g, the well-developed auction theory[13] can be applied to **for**mulate and analyze the pric**in**ggame.In auction games [13], accord**in**g to an explicit setof rules, the pr**in**ciples (auctioneers) determ**in**e resourceallocation and prices on the basis of bids from the agents(bidders). In our spectrum allocation pric**in**g game, theprimary users can be viewed as the pr**in**ciples, whoattempts to sell the unused channels to the secondaryusers. The secondary users are the bidders who competewith each other to buy the permission of us**in**gprimary users’ channels, by which they may ga**in** extrapayoffs **for** future use. In our pric**in**g game, multiplesellers and buyers coexist, which **in**dicates the doubleauction scenario. It means that not only the secondaryusers but also the primary users need to compete witheach other to make the beneficial transactions possibleby elicit**in**g their will**in**gness of the payments **in** the**for**ms of bids or asks. Specifically, the double auctionis one of the most common exchange mechanisms, usedextensively **in** stock markets such as the New YorkStock Exchange (NYSE) or commodity markets suchas Chicago Merchandize Exchange (CME). The mostimportant property of double auction mechanism is itshigh efficiency, which is still not fully understood **in**economic theory. Moreover, it can respond quickly tochang**in**g conditions of auction participants. However, **in**order to achieve the full efficiency of the double auctionmechanism, a lot of messages need to be exchangedamong the auction participants, which can be easilyimplemented by powerful central authorities **in** stockor commodity markets. It is worth notic**in**g that **in** autonomouswireless networks either central authorities canbe pre-assumed or the bandwidth of control channels isvery limited. There**for**e, we aims to develop an efficientpric**in**g approach **for** spectrum allocation, which not onlyhas the prevalence of the double auction mechanismbut also uses simple message exchanges to quickly andaccurately coord**in**ate the spectrum shar**in**g.IV. BELIEF-ASSISTED DYNAMIC PRICING FOREFFICIENT SPECTRUM ALLOCATIONA. Static **Pric ing** Game and Competitive EquilibriumAssume that the available channels from the primaryusers are leased

considered as one stage **in** our pric**in**g game. We firststudy the **in**teractions of the players **in** static pric**in**ggames. Note that the users’ goals are to maximize theirown payoff functions. As **for** the primary users, theoptimization problem can be written as follows.O(p i )=max U pi (φ Ai ,α Aiφ Ai ,α A i), ∀i ∈{1, 2, ..., J} (3)iis.t. Uŝaji({φ −aj,φ i aji},β A i ) ≥ Uŝaji({φ −aj, ˜φi aj},βi A ),iŝ aj ≠0,a jii ∈ A i. (4)where ˜φ aj is any feasible payment and φi−aj is theipayment vector exclud**in**g the element of the payment**for** the channel a j i . Note that ŝ ais def**in**ed as follows.jiŝ aj =i{s k if β aj ik =1,0 if β aj ik =0, ∀k ∈{1, 2, ..., K}. (5)Thus, (4) is the **in**centive compatible constra**in**t [13]. Itmeans that the secondary users have **in**centives to providethe optimal payment because they cannot have extraga**in**s by cheat**in**g on the primary users. Similarly, theoptimization problem can be written **for** the secondaryusers as follows.O(s i )= maxφ A,β A iU si (φ A ,β A i ), ∀i ∈{1, 2, ..., K} (6)s.t. Uˆpj ({φ −j ,φ j },β A i ) ≥ Uˆpj ({φ −j , ˜φ j },β A i ),ˆp j ≠0,β j iwhere ˆp j is def**in**ed as{pk if β j ˆp j =i =1,j ∈ A k,α j k =10 otherwise, ∀k ∈{1, 2, ..., J}.=1. (7)(8)Similarly, (7) is the **in**centive compatible constra**in**t **for**the primary users, which guarantees that the primaryuser will give the usage permission of their channels tothe secondary users so that they can receive the optimalpayments.From (3) and (6), we can see that **in** order to obta**in**the optimal allocation and payments, a multi-objectiveoptimization problem needs to be solved, which becomesextremely complicated due to our game sett**in**g thatonly **in**volves **in**complete **in****for**mation. Thus, **in** orderto make this problem tangible, we analyze it from thegame theory po**in**t of view. Generally speak**in**g, gametheory provides well-developed equilibrium concepts oroptimality criteria to study the outcomes of games.For **in**stance, Nash Equilibrium [12] is an importantFig. 1: Illustration of supply and demand functions.concept to measure the outcome of a non-cooperationgame, which is a set of strategies, one **for** each player,such that no selfish player has **in**centive to unilaterallychange his/her action. In order to further measure theefficiency of game outcomes, Pareto Optimality [11]is def**in**ed such that a Pareto optimal outcome cannot beimproved upon without hurt**in**g at least one player. Often,a Nash equilibrium is not Pareto optimal while Paretooptimal outcomes may not be susta**in**ed consider**in**gthe selfishness of the players. Further, consider**in**g thedouble auction scenarios of our pric**in**g game, CompetitiveEquilibrium (CE) [14] is a well-known theoreticalprediction of the outcomes. It is the price at which thenumber of buyers will**in**g to buy is equal to the numberof sellers will**in**g to sell. Alternatively, CE can also be**in**terpreted as where the supply and demand match [13].The supply function can be def**in**ed as the relationshipbetween the acquisition costs of primary users and thenumber of correspond**in**g channels; the demand functioncan be def**in**ed as the relationship between the rewardpayoffs of secondary users and the number of correspond**in**gchannels. We describe the supply and demandfunctions **in** Figure 1. Note that CE is also proved tobe Pareto optimal **in** stationary double auction scenarios[15]. It is worth not**in**g that **in** order to achieve the CE thetraditional cont**in**uous bid/ask **in**teractions among playerswill **in**volve a great amount of message exchanges andrequire powerful centralized control, which may not beapplicable to wireless network**in**g scenarios due to thelimited bandwidth of control channels.B. **Belief**-**Assisted** **Dynamic** **Pric ing**Consider

certa**in** channels from primary users at different timestages. Specifically, s**in**ce the secondary users can bemobile devices, they may move out the access rangeof certa**in** channels and hence the correspond**in**g rewardpayoffs v j iare regarded as 0. Or, the secondary users mayface various channel fad**in**g conditions with**in** differentspectrum ranges or dur**in**g different time periods, whichchanges their payoff values v j iat different time stages.Moreover, the costs of primary users will also changeover time due to network dynamics. For **in**stance, if thelegacy users themselves have larger spectrum demands,some legacy channels may not be available **for** leas**in**ganymore, which actually **in**dicates an **in**f**in**ite leas**in**g costof those channels **in** our pric**in**g model. In brief, c j i and vj **in**eed to be considered as random variables **in** dynamicscenarios, which we assume to satisfy the probabilitydensity functions (PDF) f c (c) and f v (v), respectively.There**for**e, consider**in**g dynamic network conditions, wefurther model the spectrum shar**in**g as a multi-stagedynamic pric**in**g game. Let γ be the discount factor ofthe multi-stage game. Based on (3) and (6), the objectivefunctions **for** the primary users and secondary users canbe rewritten as follows.Õ(p i )=Õ(s i )=max Eφ Ai ,t,α A i cj [i ,vj ii,tmax E cj [φ A,t,βi,tA i ,vj i∞∑t=1γ t · U pi,t(φ Ai,t,α Aii,t)], (9)∞∑γ t · U si,t(φ A,t ,βi,t)], A (10)t=1where the subscript t **in**dicates the tth stage of themulti-stage game. Generally speak**in**g, there may existsome overall constra**in**ts of spectrum shar**in**g such aseach secondary user’s total budget **for** leas**in**g spectrumresources or each primary user’s total available spectrumsupply. Under these constra**in**ts, the above problem needto be further modeled as a dynamic programm**in**g process[16], [17] to obta**in** optimal sequential strategies byconsider**in**g some state parameters such as the numberof channels to be allocated at every stage or the residualmonetary budget. However, the major difficulty ofdynamic spectrum shar**in**g lies **in** that how to efficientlyand quickly update the spectrum shar**in**g strategies adaptto the chang**in**g network conditions only based on local**in****for**mation. There**for**e, **in** the follow**in**g parts, we firstfocus on develop**in**g a belief-assisted dynamic pric**in**gapproach, which can not only approach CE outcomesbut also responds quickly to network**in**g dynamics whileonly **in**troduc**in**g limited overhead. Then, the total budgetconstra**in**t is taken **in**to consideration and a dynamicprogramm**in**g approach is further proposed to obta**in** theoptimal sequential strategies.1) **Belief**-**Assisted** **Dynamic** **Pric ing**: S

Def**in**ition 1: Primary users’ beliefs: **for** each potentialask at x, def**in**e⎧⎪⎨ 1 x =0∑w≥xˆr p(x) =µ A(w)+ ∑ ∑ w≥x η(w)w≥x ⎪⎩µ A(w)+ ∑ x ∈ (0,M)w≥x η(w)+∑ w≤x µ R(w)0 x ≥ M(13)where µ R (w) is the number of asks at w that has beenrejected, M is a large enough value so that the asksgreater than M won’t be accepted. Also, it is **in**tuitivethat the ask at 0 will be def**in**itely accepted as no cost is**in**troduced.Def**in**ition 2: Secondary users’ beliefs: **for** each potentialbid at y, def**in**e⎧⎪⎨ˆr s(x) =⎪⎩0 y =0∑w≤y η A(w)+ ∑ w≤y µ(w)∑w≤y η A(w)+ ∑ w≤y µ(w)+∑ w≥y η R(w)y ∈ (0,M)1 y ≥ M(14)where η R (w) is the number of bids at w that has beenrejected. And, it is **in**tuitive that the bid at 0 will not beaccepted by any primary users.Not**in**g that it is too costly to build up beliefs on everypossible bid or ask price, we can update the beliefs onlyat some fixed prices and use **in**terpolation to obta**in** thebelief function over the price space. Then, it is worthdiscuss**in**g the effect of the available public **in****for**mationon the efficiency of the above belief system. First, **in** thescenario that only local **in****for**mation is available to eachuser, the user updates the belief based on her/his ownobserved past bid/ask **in****for**mation, which results **in** moremessage exchanges to achieve the equilibrium price.Second, consider**in**g the broadcast nature of wirelesschannels, the neighbors’ bid/ask **in****for**mation may beobserved by the users, which can also be utilized toupdate the beliefs. In this scenario, the users may havepart of the public **in****for**mation besides of their private**in****for**mation, which may accelerate their belief-updat**in**gpace and result **in** more efficient pric**in**g process. Moreover,if the users have the access to all the public**in****for**mation such as ask/bid **in**teractions through somecentralized po**in**t, the above belief function is able toquickly reflect current supply and demand relationships.Be**for**e us**in**g our def**in**ed belief functions to assist thestrategy decisions, we first look at the Spread ReductionRule (SRR) of double auction mechanisms. Generally,be**for**e the double auction pric**in**g game converges toCE, there may exist a gap between the highest bidand lowest ask, which is called the spread of doubleauction. The SRR states that any ask that is permissiblemust be lower than current lowest ask, i.e., outstand**in**gask [14], and then either each new ask results **in** anTABLE I: **Belief**-assisted dynamic spectrum allocation1. Initialize the users’ beliefs and bids/asks⋄ The primary users **in**itialize their asks as large values close to Mand their beliefs as small positive values less than 1;⋄ The secondary users **in**itialize their bids as small values close to 0and their beliefs as small positive values less than 1.2. **Belief** update based on local **in****for**mation:Update primary and secondary users’ beliefsus**in**g (13) and (14), respectively3. Optimal bid/ask update:⋄ Obta**in** the optimal ask **for** each primary user by solv**in**g (16);⋄ Obta**in** the optimal bid **for** each secondary user by solv**in**g (17).4. Update leas**in**g agreement and spectrum pool:⋄ If the outstand**in**g bid is greater than or equal to the outstand**in**g ask,the leas**in**g agreement will be signed between the correspond**in**g users;⋄ Update the spectrum pool by remov**in**g the assigned channel.5. Iteration:If the spectrum pool is not empty, go back to Step 2.agreed transaction or it becomes the new outstand**in**gask. A similar argument can be applied to bids. Bydef**in****in**g current outstand**in**g ask and bid as ox and oy,respectively, we let ¯r p (x) =ˆr p (x) · I [0,ox) (x) **for** eachx and ¯r s (y) =ˆr s (x) · I (oy,M](y) **for** each y, which aremodified belief function consider**in**g the SRR. Note thatI (a,b) (x) is def**in**ed as{ 1 if x ∈ (a, b);I (a,b) (x) =(15)0 otherwise.By us**in**g the belief function ¯r p (x), the payoff maximizationof sell**in**g the ith primary user’s jth channel can bewritten asmax E[U p i(x, j)], (16)x∈(oy,ox)where U pi (x, j) represents the payoff **in**troduced byallocat**in**g the jth channel when the ask is x, andthenE[U pi (x, j)] = (x − c j i ) · ¯r p(x). Similarly, as **for** thesecondary user s i , the payoff maximization of leas**in**gthe jth channel **in** the spectrum pool can be written asmax E[U s i(y, j)], (17)y∈(oy,ox)where U si (y, j) represents the payoff **in**troduced byleas**in**g the jth channel **in** the spectrum pool when the bidis y, andthenE[U si (y, j)] = (v j i − y) · ¯r s(y). There**for**e,by solv**in**g the optimization problem **for** each primaryand secondary user us**in**g (16) and (17), respectively,primary and secondary users can make the optimaldecision of spectrum allocation at every stage conditionalon dynamic spectrum demand and supply. Based onthe above discussions, we illustrate our belief-assisteddynamic pric**in**g algorithm **for** spectrum allocation **in**Table I.124

2) **Dynamic** **Pric ing** with Budget Constra

Fig. 2: Comparison of the total payoff **for** the proposed scheme andtheoretical Competitive Equilibrium.ends until |Q p+1s i( ˜ψ i ) − Q p s i( ˜ψ i )|≤ɛ, **for**all ˜ψi ∈S,where ɛ is the error bound **for** Q ∗ s i( ˜ψ i ).Intuitively, the basic idea beh**in**d our dynamic pric**in**gapproach **for** spectrum allocation with budget constra**in**tscan be expla**in**ed as follows: Consider**in**g the overallbudget constra**in**ts, the users make their spectrum shar**in**gdecisions not only based on their current payoffs butalso based on expected future payoffs. Specifically, ifthe competition **for** spectrum resources is high at currentstage, the users prefer to save their monetary budgets**for** future usage, which will yield higher overall payoffs**for** the users. There**for**e, by us**in**g our proposed dynamicpric**in**g approach, the spectrum allocation can be optimizednot only **in** the space and frequency doma**in**s butalso **in** the the time doma**in**.V. SIMULATION RESULTSIn this section, we evaluate the per**for**mance of theproposed belief-assisted dynamic spectrum shar**in**g approach**in** wireless networks. Consider**in**g a wirelessnetwork cover**in**g 100 × 100 area, we simulate J primaryusers by randomly plac**in**g them **in** the network.These primary users can be the base stations serv**in**g **for**different wireless network operators or different accesspo**in**ts **in** a mesh network. Here we assume the primaryusers’ locations are fixed and their unused channels areavailable to the secondary users with**in** the distance of50. Then, we randomly deploy K secondary users **in**the network, which are assumed to be mobile devices.The mobility of the secondary users is modeled us**in**ga simplified random waypo**in**t model [18], where weassume the “th**in**k**in**g time” at each waypo**in**t is closeto the effective duration of one channel-leas**in**g agreement,the waypo**in**ts are uni**for**mly distributed with**in** theFig. 3: Comparison of the overhead between the proposed schemeand cont**in**uous double auction scheme.distance of 10, and the travel**in**g time is much smallerthan the “th**in**k**in**g time”. Let the cost of an availablechannel **in** the spectrum pool be uni**for**mly distributed**in** [10, 30], the reward payoff of leas**in**g one channelbe uni**for**mly distributed **in** [20, 40]. If a channel is notavailable to some secondary users, let the correspond**in**greward payoffs of this channel be 0. Note that J =5and 10 3 pric**in**g stages have been simulated. Let n i =4, ∀i ∈{1, 2, ..., J} and γ =0.99.We first focus our simulation studies on dynamicspectrum shar**in**g without budget constra**in**ts, which canbe used to illustrate the efficiency of the proposedbelief-assisted pric**in**g algorithm **for** spectrum allocation.In our simulation, the local bid/ask **in****for**mation with**in**the transmission range of each node is used **for** beliefconstruction and update. In Figure 2, we compare thetotal payoff of all users of our proposed approach withthat of the theoretical CE outcomes **for** different numberof secondary users. It can be seen from this figure thatthe per**for**mance loss of our approach is very limitedcompared to that of the theoretical optimal solutions.Moreover, when the number of secondary users **in**creases,our approach is able to approach the optimal CE.It is because that the belief function reflects the spectrumdemand and supply more accurately when more users are**in**volved **in** spectrum shar**in**g.Now we study the overhead of our pric**in**g approach.Here we measure the pric**in**g overhead by show**in**gthe average number of bids and asks **for** each stage.In Figure 3, the overhead of our pric**in**g approach iscompared to that of the traditional cont**in**uous doubleauction when the same total payoff is achieved. Assumethe m**in**imal bid/ask step δ of the cont**in**uous doubleauction to be 0.01. It can be seen from the figure126

There are several avenues **for** future research. We**in**tend to further per**for**m the equilibrium analysis ofthe solution obta**in**ed by the proposed dynamic spectrumallocation scheme. Also, we would like to study howto adapt our scheme to mesh, cellular or comb**in**ednetworks under practical constra**in**ts caused by the natureof wireless network**in**g. Further, security issues needto be extensively **in**vestigated **for** dynamic spectrumallocation. Not only the security threats caused by thedesigned mechanism itself such as bidd**in**g r**in**g problems[13] need to be considered, but also the threats due tothe limitations of wireless systems and radio **in**terface.Fig. 4: Comparison of the total payoffs of the proposed schemewith those of the static scheme.that our approach substantially decreases the pric**in**gcommunication overhead. Note that when decreas**in**g theoverhead, our proposed approach may **in**troduce extracomplexity to update the beliefs.Then, we study the dynamic spectrum allocation wheneach secondary user is constra**in**ed by his/her monetarybudget. For comparison, we def**in**e a static scheme **in**which the secondary users make their spectrum-leas**in**gdecisions without consider**in**g their budget limits. Withoutloss of generality, we assume that the budget constra**in**ts**for** the secondary users are the same. In Figure 4,we compare the total payoffs of our proposed dynamicprogramm**in**g scheme with those of the static scheme**for** different budget constra**in**ts. It can be seen from thefigure that our proposed scheme achieves significant per**for**mancega**in**s over the static scheme when the budgetconstra**in**ts are taken **in**to consideration. Also, when thebudget limits **in**crease, the proposed scheme achieveshigher ga**in** by further exploit**in**g the time diversity.VI. CONCLUSIONS AND FUTURE WORKIn this paper, we have studied dynamic pric**in**g **for**efficient spectrum allocation **in** wireless networks withselfish users. We model the dynamic spectrum allocationas a multi-stage game and propose a belief-assisteddynamic pric**in**g approach to maximize the users’ payoffswhile provid**in**g them the participat**in**g **in**centives viadouble auction rules. Further, the dynamic pric**in**g underthe budget constra**in**ts of secondary users is analyzedus**in**g dynamic programm**in**g. Simulation results showthat the proposed scheme can approach the optimalper**for**mances by only us**in**g limited overhead. Moreover,the time diversity of spectrum resources can be fullyexploited when budget constra**in**ts exist.REFERENCES[1] FCC, “**Spectrum** policy task **for**ce report,” FCC Document ETDocket No. 02-135, November 2002.[2] FCC, “Facilitat**in**g opportunities **for** flexible, efficient, andreliable spectrum use employ**in**g cognitive radio technologies:notice of proposed rule mak**in**g and order,” FCC Document ETDocket No. 03-108, December 2003.[3] R. J. Berger, “Open spectrum: a path to ubiquitous connectivity,”FCC ACM Queue 1, 3, May 2003.[4] J. M. Peha, “Approaches to spectrum shar**in**g,” IEEE CommunicationsMagaz**in**e, vol. 43, pp. 10–12, Feburary 2005.[5] M. M. Buddhikot, “Dimsumnet: new directions **in** wirelessnetwork**in**g us**in**g coord**in**ated dynamic spectrum access,” **in**Proc. of IEEE WoWMoM’05, 2005.[6] C. Peng, H. Zheng, and B. Y. Zhao, “Utilization and fairness**in** spectrum assignment **for** opportunistic spectrum access,” toappear **in** Mobile Networks and Applications (MONET), 2006.[7] L. Cao and H. Zheng, “Distributied spectrum allocation vialocal barga**in****in**g,” **in** Proc. of IEEE DySpan, 2005.[8] R. Etk**in**, A. Parekh, and D. Tse, “**Spectrum** shar**in**g **for**unlicensed bands,” **in** Proc. of IEEE DySpan, 2005.[9] M. H. Halldorson, J. H. Halpern, L. Li, V. S. Mirrokni, “Onspectrum shar**in**g games,” **in** Proc. of ACM Symposium onPr**in**ciple of Distributed Comput**in**g (PODC), 2004.[10] M. Felegyhazi and J. P. Hubaux, “**Wireless** operators **in** a sharedspectrum,” **in** Proc. of IEEE INFOCOM’06, 2006.[11] D. Fudenberg and J. Tirole, Game Theory, The MIT Press,Cambridge, Massachusetts, 1991.[12] M. J. Osborne and A. Rub**in**ste**in**, A Course **in** Game Theory,The MIT Press, Cambridge, Massachusetts, 1994.[13] V. Krishna, Auction Theory, Academic Press, 2002.[14] S. Gjerstad and J. Dickhaut, “Price **for**mation **in** doubleauctions,” Games and Economic Behavior, vol. 22, pp. 1–29,1998.[15] L. Hurwicz, R. Radner, and S. Reiter, “A stochastic decentralizedresource allocation process: Part i,” Econometrica, vol. 43,pp. 363–393, 1975.[16] D. Bertsekas, **Dynamic** Programm**in**g and Optimal Control, vol.1,2, Athena Scientific, Belmont, MA, Second edition, 2001.[17] Z. Ji, W. Yu, and K. J. R. Liu, “An optimal dynamic pric**in**gframework **for** autonomous mobile ad hoc networks,” **in** Proc.of IEEE INFOCOM’06, 2006.[18] D. B. Johnson and D. A. Maltz, “**Dynamic** source rout**in**g **in** adhoc wireless networks, mobile comput**in**g,” IEEE Transactionson Mobile Comput**in**g, pp. 153–181, 2000.127