IEEE/ACM TRANSACTIONS ON NETWORKING 1 Pricing-Based Decentralized Spectrum...

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IEEE/ACM TRANSACTIONS ON NETWORKING 1

Pricing-Based Decentralized Spectrum AccessControl in Cognitive Radio Networks

Lei Yang, Student Member, IEEE, Hongseok Kim, Member, IEEE, Junshan Zhang, Fellow, IEEE,Mung Chiang, Fellow, IEEE, and Chee Wei Tan, Member, IEEE

Abstract—This paper investigates pricing-based spectrumaccess control in cognitive radio networks, where primaryusers (PUs) sell the temporarily unused spectrum and secondaryusers (SUs) compete via random access for such spectrum op-portunities. Compared to existing market-based approaches withcentralized scheduling, pricing-based spectrum managementwith random access provides a platform for SUs contending forspectrum access and is amenable to decentralized implementationdue to its low complexity. We focus on two market models, onewith a monopoly PU market and the other with a multiple-PUmarket. For the monopoly PU market model, we devise decen-tralized pricing-based spectrum access mechanisms that enableSUs to contend for channel usage. Specifically, we first considerSUs contending via slotted Aloha. Since the revenue maximizationproblem therein is nonconvex, we characterize the correspondingPareto-optimal region and obtain a Pareto-optimal solution thatmaximizes the SUs’ throughput subject to their budget con-straints. To mitigate the spectrum underutilization due to the“price of contention,” we revisit the problem where SUs contendvia CSMA, which results in more efficient spectrum utilizationand higher revenue. We then study the tradeoff between thePU’s utility and its revenue when the PU’s salable spectrum iscontrollable. Next, for the multiple-PU market model, we cast thecompetition among PUs as a three-stage Stackelberg game, whereeach SU selects a PU’s channel to maximize its throughput. Weexplore the existence and the uniqueness of Nash equilibrium,in terms of access prices and the spectrum offered to SUs, anddevelop an iterative algorithm for strategy adaptation to achievethe Nash equilibrium. Our findings reveal that there exists aunique Nash equilibrium when the number of PUs is less than athreshold determined by the budgets and elasticity of SUs.

Index Terms—Cognitive radio, nonconvex optimization, Paretooptimality, pricing, random access, spectrum access control.

Manuscript received June 21, 2011; revised February 13, 2012 and April30, 2012; accepted May 27, 2012; approved by IEEE/ACM TRANSACTIONS ONNETWORKING Editor S. Sarkar. This work was supported in part by the US NSFunder Grants CNS-0905603, CNS-0917087, and CNS-1117462; AFOSRMURIProject FA9550-09-1-0643, and the City University of HongKong under ProjectNo. 7002756. Part of this work was presented at the IEEE International Con-ference on Computer Communications (INFOCOM), Shanghai, China, April10–15, 2011.L. Yang and J. Zhang are with the School of Electrical, Computer and En-

ergy Engineering, Arizona State University, Tempe, AZ 85287 USA (e-mail:[email protected]; [email protected]).H. Kim is with the Department of Electronic Engineering, Sogang University,

Seoul 121-742, Korea.M. Chiang is with Department of Electrical Engineering, Princeton Univer-

sity, Princeton, NJ 08544 USA.C. W. Tan is with City University of Hong Kong, Kowloon, Hong Kong.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TNET.2012.2203827

I. INTRODUCTION

C OGNITIVE radio is expected to capture temporal andspatial “spectrum holes” in the spectrum white space

and to enable spectrum sharing for secondary users (SUs).One grand challenge is how SUs can discover spectrum holesand access them efficiently, without causing interference tothe primary users (PUs), especially when the demand foravailable spectrum nearly outstrips the supply. Market-basedmechanisms have been explored as a promising approachfor spectrum access, where PUs can dynamically trade unusedspectrum to SUs [2]–[17]. In particular, auction-based spectrumaccess mechanisms have been extensively studied, includingincentive compatibility [3]–[6], spectrum reuse [3], [4], [7], [8],auctioneer’s revenue maximization [4], social welfare maxi-mization [8], and power allocation for the SUs with interferenceprotection for the PU [9] (and the references therein). Theseworks focus on on-demand auctions where each SU requestsspectrum based on its traffic demand, and it is worth noting thatthe overhead can be significant in the auction procedure (e.g.,market setup time, bidding time, and pricing clearing time).Compared to auction-based spectrum access, pricing-based

spectrum access incurs lower overhead (see [10]–[16] andthe references therein). Notably, [10] studied pricing policiesfor a PU to sell unused spectrum to multiple SUs. Recentworks [11], [12] considered competition among multiple PUsthat sell spectrum, whereas [13] focused on competition amongmultiple SUs to access the PU’s channels. Reference [14] con-sidered spectrum trading across multiple PUs and multiple SUs.Reference [15] studied the investment and pricing decisionsof a network operator under spectrum supply uncertainty. Onecommon assumption used in these studies is that orthogonalmultiple access is used among SUs, either in time or frequencydomain, where a central controller is needed to handle SUs’admission control, to calculate the prices, and to charge theSUs. However, the computational complexity for dynamicspectrum access and the need of centralized controllers canoften be overwhelming or even prohibitive. To address theseproblems, a recent work [16] proposed a two-tier market modelbased on the decentralized bargain theory, where the spectrumis traded from a PU to multiple SUs on a larger timescale, andthen redistributed among SUs on a smaller timescale. Due tothe decentralized nature, coordination among SUs remains achallenge when SUs of different networks coexist [17], simplybecause contention between SUs is unavoidable.As a less-studied alternative in cognitive radio networks,

random access can serve as a platform for the contention amongSUs (e.g., in [18] and [19]) and can be employed for decentral-ized spectrum access to mitigate the overwhelming complexity.

1063-6692/$31.00 © 2012 IEEE


2 IEEE/ACM TRANSACTIONS ON NETWORKING

With this insight, we focus on pricing-based spectrum controlwith random access. In particular, we study the behaviors ofPUs and SUs in two spectrum-trading market models based onrandom access: one with a monopoly PU market and the otherwith a multiple-PU market.We first consider the monopoly PU market model where the

PU’s unused spectrum is fixed.We study pricing-based dynamicspectrum access based on slotted Aloha, aiming to characterizethe optimal pricing strategy maximizing the PU’s revenue. Dueto the nonconvexity of the optimization problem, the global op-timum is often unattainable. Instead, we first characterize thePareto-optimal region associated with the throughput vector ofSUs, based on the observation that the global optimum has tobe Pareto-optimal. Roughly speaking, for any Pareto-optimalsolution, the throughput of any individual SU cannot be im-provedwithout deteriorating some other SU’s throughput. Then,by maximizing the SUs’ throughput subject to the budget con-straints, we provide a Pareto-optimal solution that is near-op-timal. Furthermore, the structural properties of this Pareto-op-timal solution indicate that the access probabilities can be com-puted by the SUs locally. With this insight, we develop a de-centralized pricing-based spectrum access control algorithm ac-cordingly. To mitigate the spectrum underutilization due to the“price of contention,” we next turn to dynamic spectrum accessusing CSMA and quantify the improvements in spectrum uti-lization and PU’s revenue. We also consider the case when PU’ssalable spectrum is controllable, i.e., the PU can flexibly allo-cate the spectrum to its ongoing transmissions so as to balanceits own utility and revenue.Next, for the multiple-PU market model, we treat the compe-

tition among PUs as a three-stage Stackelberg game, where eachPU seeks to maximize its net utility and each SU selects a PU’schannel to maximize its own throughput. We explore optimalstrategies to adapt the prices and the offered spectrum for eachPU and show that the Nash equilibria of the game exist. We fur-ther prove that the Nash equilibrium is unique when the numberof PUs is less than a threshold, whose value is determined bythe budgets and elasticity of SUs. Intuitively, this threshold cri-terion can be used by PUs to decide whether to join in the com-petition or not, i.e., when the number of PUs grows larger thanthe threshold, the competition among PUs is too strong, indi-cating that it is unprofitable for a PU to sell spectrum to SUs.An iterative algorithm is devised to compute the Nash equilib-rium accordingly.The rest of this paper is organized as follows. In Section II, we

study the monopoly PU market and present the Pareto-optimalpricing strategy for the PU’s revenue maximization problem.We also characterize the tradeoff between the PU’s utility andits revenue. We study in Section III the competition amongPUs in the multiple-PU market, which is cast as a three-stageStackelberg game, and analyze the Nash equilibria of the game.Finally, we conclude the paper in Section IV.

II. MONOPOLY PU MARKET

A. System Model

We first consider a monopoly PU market with a set ofSUs, denoted by . The PU sells the available spectrum

Fig. 1. System model: (a) monopoly PU market; (b) multiple-PU market.

TABLE ISUMMARY OF THE KEY NOTATIONS AND DEFINITIONS

opportunity in each period,1 in terms of time-slots in a slottedwireless system based on random access, to SUs who arewilling to buy the spectrum opportunities [Fig. 1(a)]. When oneSU decides to buy the spectrum opportunity, it sends a requestmessage together with the budget information (to be elaboratedin the sequel) to the PU. Then, at the beginning of each period,the PU broadcasts to the SUs the salable spectrum opportunitiesand the prices to access them. Observe that message passinginvolved in this scheme is infrequent and minimum (insteadof sending the control message at each slot to manage thespectrum access).2

We study two cases: 1) the spectrum opportunity is fixed,and the PU desires to find the optimal prices (i.e., usage priceand flat price) to maximize its revenue; 2) the spectrum oppor-tunity is a control parameter that the PU can use to balance itsown utility and revenue. In both cases, each SU seeks to setits demand that maximizes its payoff, given the spectrum op-portunity , the usage price , and the flat price . For ease ofreference, the key notation in this paper is listed in Table I.3

B. Case With Fixed Spectrum Opportunity

We first study the case where the spectrum opportunity isfixed. We begin with the channel access model for SUs, as-suming a linear pricing strategy, i.e., the PU charges each SU

1The period refers to a time frame where the PU sells its unused part, denotedby . Please refer to Fig. 2 for an illustration of a period in the slotted wirelesssystem.2In cognitive radio networks, the control channel often does not exist [20].

In this study, assuming that no common control channel exists, we design algo-rithms that have minimal computational complexity and message passing over-head. The salable spectrum opportunity is a fraction of a period. When sellingthe spectrum at the beginning of each period, the PU does not know which slotswill be idle, as this depends on the PU’s traffic, and SUs would have to detectthe opportunities through spectrum sensing.3We use bold symbols (e.g., ) to denote vectors, and calligraphic symbols

(e.g., ) to denote sets.


YANG et al.: PRICING-BASED DECENTRALIZED SPECTRUM ACCESS CONTROL IN COGNITIVE RADIO NETWORKS 3

Fig. 2. Random access model for SUs.

a flat price and a usage price proportional to its successful trans-missions. We will show that the Pareto-optimal usage price isthe same for all SUs and is uniquely determined by the total de-mand of SUs and the available spectrum opportunity .1) Slotted Aloha Model for SUs’ Channel Access: Each SU

first carries out spectrum sensing to detect the PU’s activity.4

When the sensing result reveals that the PU is idle, SUs willcontend for channel access by random access; otherwise, SUswill remain silent as illustrated in Fig. 2. As in the standardslotted Aloha model [21], we assume that SUs under consid-eration are within the contention ranges of the others, and alltransmissions are slot-synchronized. We assume that SUs al-ways have packets to transmit, and traffic demands of SUs areelastic. Denote by the transmission probability of the th SU.The probability that the th SU’s packet is successfully receivedis . The expected number of successfullytransmitted packets of the th SU in one period can be writtenas , where . Accordingly, the th SU receivesa utility in one period equal to , where denotes theutility function of the th SU. The optimal demand is the so-lution to the following optimization problem:

maximize

subject to

variables (1)

As is standard, we define the demand function that capturesthe successful transmissions given the price as

ifotherwise.

(2)

Assuming -fair utility functions, the utility and the demandfunctions of each SU can be written, respectively, as

(3)

ifotherwise

(4)

where , the multiplicative constant in the -fair utility func-tion, denotes the utility level5 of the th SU, which reflects thebudget6 of the th SU (see [22] and the references therein).

4In this study, we assume that sensing errors are negligible.5The PU collects the budget information of SUs from the SUs’ request mes-

sage at the beginning of each period, and this allows the PU to infer .6Given the prices, the SUs with larger budget, i.e., larger , would buy more

spectrum from the PU based on (4).

Note that we mainly consider the case of becauseis the elasticity of the demand seen

by the PU, and it has to be strictly larger than 1 so that the mo-nopoly price is finite [23]. Clearly, the -fairness boils down tothe weighted proportional fairness when and to selectingthe SU with the highest budget when .2) PU’s Pricing Strategy: We have the following revenue

maximization problem for the monopoly PU market:

maximize

subject to

variables (5)

The constraint ensures that SUs havenonnegative utility under the prices and ; otherwise, SUsmay not transmit. Let denote the optimal solu-tion to (5). It is clear that ;otherwise, the PU can always increase its revenue by increasingto make the SU’s net utility equal to zero.Lemma 2.1: The optimal prices for (5) are given by

(6)

Based on Lemma 2.1, (5) can be rewritten as

maximize

subject to

variables (7)

Since the utility function is increasing, the optimal solu-tion to (7) is achieved at the point when

. Also, the objective function of (7) can be writtenas . Since is aconstant, maximizing is equivalentto maximizing , i.e., solving (5) is equivalent tosolving the following problem without considering flat prices

maximize

subject to

variables (8)

In general, (8) is nonconvex, and therefore it is difficult tofind the global optimum. Observing that the global optimumof (8) is Pareto-optimal, we shall confine our attention to thePareto-optimal region, i.e., the set consisting of Pareto-optimalsolutions to (8).Definition 2.1: A feasible allocation is Pareto-optimal if

there is no other feasible allocation such that for alland for some .

Lemma 2.2: The Pareto-optimal region corresponding to (8)has the following properties.1) The global optimum is in the Pareto-optimal region.



Fig. 3. Sketch of the Pareto-optimal region: the case with two SUs.

2) The solution to (8) is Pareto-optimal if and only if.

The proof of Lemma 2.2 is given in Appendix A. ByLemma 2.2, for any Pareto-optimal allocation ,we have . Let

be the Pareto-optimal region. Therefore, it suf-fices to search for the points in that can maximize (8). Inlight of Lemma 2.2, instead of tackling the original problemgiven by (8), hereafter we focus on obtaining a Pareto-optimalsolution to (8) that maximizes the sum of SUs’ throughputgiven the SUs’ budget constraints by confining the search spaceto the hyperplane , where de-notes the spectrum utilization percentage under the allocation

.We now consider this “constrained” version of (8) for finding

the maximum feasible spectrum utilization , i.e., the tangentpoint of the hyperplane and , as illustrated in Fig. 3

maximize

subject to

variables (9)

We note that the solution to (9) is in general suboptimal for(8). However, by exploring the connections between the pricingstrategy and the spectrum utilization, we derive a closed-formsolution to (9) that is also a near-optimal solution to (8).Proposition 2.1: For , the optimal solution to (9)

is given by

(10)

where , , and is theunique solution of

Proof Outline: In what follows, we sketch a outline forproving the above proposition (see Appendixes B and C for theproof details). First, the following result establishes the relation-ship between the optimal pricing strategy of (9) and when

.Lemma 2.3: Given , the optimal pricing strategy of (9) for

is given by

(11)

Next, a key step is to find . By Lemma 2.3, we have. Utilizing those constraints, we

can find by solving the following problem:

maximize

subject to

variables (12)

Still, (12) is nonconvex, but by first taking logarithms ofboth the objective function and the constraints and then letting

, we can transform (12) into the following convexproblem:

maximize

subject to

variables (13)

Thus, the optimal solution to (12) can be summarized by thefollowing lemma.Lemma 2.4: The optimal solution to (12) is given by

(14)

where is the unique solution of

(15)

Furthermore, when the number of SUs in the network is large,i.e., , we can approximate by .Based on Lemmas 2.3 and 2.4, the optimal solution to (9)

is given by Proposition 2.1, which is a Pareto-optimal solutionto (5).Remarks: The “constrained” revenue maximization

problem (9) is nonconvex. However, (9) exhibits a hiddenconvexity property, i.e., by utilizing Lemma 2.3, (9) can betransformed into a convex problem (13). Intuitively speaking,this property ensures that the line, , asillustrated in Fig. 3, can touch the Pareto-optimal region withone unique point.So far, we have focused on the case with . Next, we

consider the special cases when and 1. Interestingly, wewill see that the Pareto-optimal solutions are also global-optimalin those special cases.



Corollary 2.1: When , the Pareto-optimal solutionto (5) is also the global optimal solution, which is given by

(16)

for all , where .Corollary 2.1 implies that when , the PU selects the SU

with the highest budget and only allows that SU to access thechannel with probability 1 to maximize its revenue.Remarks: The Pareto-optimal solution in (10) converges to

the globally optimal solution as goes to zero, i.e.,

When , (7) can be transformed into a convex problemby taking logarithms of the constraints, and the global optimalaccess probabilities of SUs in this case are

i.e., the random access probability is proportional to SU’s utilitylevel, where the revenue is dominated by the flat rate. This issimilar to the observation made in [22]. As approaches 1, therevenue computed by (10) also converges to the global optimalsolution, since the Pareto-optimal flat rates in (10) converge tothe global optimal ones.3) Decentralized Implementation: Based on the above study,

we next develop decentralized implementation of the pricing-based spectrum access control. Based on the structural prop-erties of the Pareto-optimal solution given in Proposition 2.1,we develop a decentralized pricing-based spectrum access con-trol algorithm (Algorithm 1). In particular, the PU only needs tocompute and broadcast the common parameters , , and ,based on which each SU can compute its access probability lo-cally. It is clear that Algorithm 1 significantly reduces the com-plexity and the amount of the message passing, which wouldotherwise require a centralized coordination for the PU.

Algorithm 1: Decentralized Pricing-based Spectrum AccessControl

Initialization:1) The PU collects the budget information of SUs, i.e., .2) The PU computes , , , and by (10) and broadcasts

, , and to SUs.3) Each SU computes by (10) based on and andinfers from its own utility and by (10).

Repeat at the beginning of each period:

If New SUs join the system or SUs leave the system then

The PU updates the budget information of SUs. Thenrun Steps 2 to 3.

Endif

Fig. 4. Pareto optimum versus global optimum.

C. Numerical Example: Pareto Optimum vs. Global Optimum

To reduce the computational complexity in solving the globaloptimum of (5), we first solve (9). To examine the efficiencyof this Pareto-optimal solution, we exhaustively search for theglobal optimum of (5) to compare to the Pareto optimum, in asmall network with three SUs so as to efficiently generate thetrue global optimum as the benchmark. In this example, is setto 5, and each SU’s is generated uniformly in the interval[0, 4] and fixed for different for the sake of comparison. Asshown in Fig. 4, the Pareto-optimal solution is close to the globaloptimal solution. Furthermore, the gap between the objectivevalue evaluated at the Pareto-optimal solution and the globaloptimal value diminishes as approaches 1. In addition, thegap goes to zero as goes to zero, corroborating Corollary 2.1.

D. CSMA Model for SUs’ Channel Access

Needless to say, the contention among SUs leads to spectrumunderutilization. For the slotted Aloha model, the spectrum uti-lization approaches , when the network size grows large,indicating that the unused spectrum is .Definition 2.2: We define the unused spectrum, , as the

“price of contention.”Obviously, the “price of contention” using slotted Aloha is

high, compared to orthogonal access that requires centralizedcontrol. It is well known that spectrum utilization can be en-hanced by using CSMA. Thus motivated, next we consider aCSMA-based random access for the SUs’ channel access.When CSMA is employed, a SU can successfully access the

channel after an idle period if no other SUs attempt to accessthe channel at the same time. Let denote the idle time of thechannel. For a given , the network service throughput for alarge network can be approximated by [21]

(17)

and the successful channel access probability of the th SU canbe approximated by

(18)



where denotes the rate at which the SUsattempt to access the channel at the end of an idle period.Since is maximized when , the PU can

always adjust the prices to make the SUs’ channel access ratessatisfy . In this case, can be uti-lized by the SUs per period, i.e., the spectrum utilization underCSMA is . Similar to the approach for solving (5),by confining the search space to , the “con-strained” revenue maximization problem under CSMA is givenby7

maximize

subject to

variables (19)

The above problem can be solved by following the same ap-proach to (5), and the optimal solution to (19) is given by thefollowing proposition.Proposition 2.2: For , the optimal solution to (19)

is given by

(20)

Remarks: Note that (17) and (18) offer good approximationsfor large networks, in which case we can compare the perfor-mance of slotted Aloha and CSMA, e.g., revenue and spectrumutilization. As expected, the spectrum utilization under CSMAis higher, resulting in higher revenue from the SUs. We cautionthat when the network size is small, such comparisons may notbe accurate since the system capacity under CSMA is unknown.Under slotted Aloha, the results hold for an arbitrary number ofSUs. In the remainder of the paper, we focus on the system underslotted Aloha only.

E. Case With Controllable Spectrum Opportunity

When the spectrum opportunity is a control parameter, thereexists a tradeoff between the PU’s utility and its revenue. Forease of exposition, we use the logarithmic utility function toquantify the PU’s satisfaction

(21)

where denotes the total length of a period, and the utilitylevel is a positive constant depending on the application

7The solution to (19) is a Pareto-optimal solution to the original revenue max-imization problem under CSMA without the constraint .

Fig. 5. PU’s profit under different .

type [14]. Given the PU’s utility function, the net utility (orprofit) of the PU for can be expressed as

(22)

Based on Proposition 2.1, the Pareto-optimal price is a functionof the spectrum opportunity . Then, the optimal can be foundby solving the following problem:

maximize

subject to

variable (23)

where is a positive constant.Note that is constrained to be an integer in the slotted system

noted above. However, the objective function of (23) has theunimodal property, and therefore (23) can be solved efficiently(e.g., by the Fibonacci search algorithm [24]).Lemma 2.5: The objective function of (23) is unimodal for

.The unimodal property of the objective function of (23) di-

rectly follows from that of its continuous version, whose op-timal solution can be determined by the first order conditionand the boundary conditions, which is the solution to

(24)

When the length of each period, , is reasonably large,can be approximated by the solution to (24). In what follows,we adopt this continuous approximation of .As an illustration, we plot two possible curves of (22) for

different in Fig. 5. In this example, we set , ,, and . Each SU’s is generated uniformly

in the interval . For the two realizations of, the optimal tradeoff decision, corresponding to the highest

point of each curve within , increases with . Intuitivelyspeaking, the PU would allocate more spectrum opportunity tothose SUs who would pay more.



Fig. 6. Three-stage Stackelberg game.

III. MULTIPLE-PU MARKET

A. System Model

When there are multiple PUs in a cognitive radio network,they compete with each other in terms of prices and spectrumopportunities in order to maximize their net utilities. It can beseen from Lemma 2.1 that PU’s flat prices depend on usageprices and that each SU wishes to choose the PU with the lowestusage price for its transmission.8 Thus, we focus on usage pricesin what follows, and the corresponding flat prices can be ob-tained accordingly.We assume that both PUs and SUs are selfish and yet ra-

tional. As the PUs are spectrum providers, they have the rightto decide the prices and spectrum opportunities, so as to maxi-mize their net utilities. Based on PU’s decisions, each SU thenchooses a PU’s channel to maximize its transmission rate. Ob-serve that it is a typical leader–follower game that can be an-alyzed by using the Stackelberg game framework. Specifically,we cast the competition among the PUs as a three-stage Stackel-berg game, as summarized in Fig. 6, where the PUs and the SUsadapt their decisions dynamically to reach an equilibrium point.The PUs first simultaneously determine in Stage I their availablespectrum opportunities, and then in Stage II simultaneously an-nounce the prices to the SUs. Finally, each SU accesses only onePU’s channel to maximize its throughput in Stage III. Here, weconsider a set of PUs. We assume that all SUs are within theintersection of those PU’s coverage areas shown in Fig. 1(b).In the sequel, we focus on the game for and use

the index for SUs and the index for PUs.

B. Backward Induction for the Three-Stage Game

We analyze the subgame perfect equilibrium of the gameby using the backward induction method [23], which is a pop-ular technique for determining the subgame perfect equilibrium.First, we start with Stage III and analyze SU’s behaviors, undergiven PU’s spectrum opportunities and prices. Then, we turnour focus to Stage II and analyze how PUs determine pricesgiven spectrum opportunities and the possible reactions of SUsin Stage III. Finally, we study how PUs determine spectrum op-portunities given the possible reactions in Stages II and III.1) Channel Selection in Stage III: In this stage, each SU

determines which PU’s channel to access based on the set ofprices . The admission of SUs also depends on the availablespectrum opportunities in Stage I. Since (4) decreases with

8Based on Lemma 2.1, each SU ends up with zero profit in any PU’s channel.However, the SU canmaximize its transmission rate by choosing the PUwith thelowest usage price. Furthermore, the access probability of each SU is determinedby (10).

price , the th SU would choose the th PU’s channel if.

Given the set of prices , the total demand of SUs in the thPU’s channel can be written as

where denotes the set of SUs choosing the th PU, anddenotes the set of prices of PUs other than the th PU. Bothand depend on prices and are independent of . Therefore,the demand function can be written as

where is defined in Proposition 2.1, anddenotes the set of PUs with the smallest

price in . In this paper, we assume that the SUs randomly pickone PU in with equal probability.Given the size of available spectrum opportunities , theth PU always adjusts its price to make the demand of SUsequal to the supply so as to maximize its revenue (based onProposition 2.1). It follows that at the Nash equilibrium point

(25)

where denotes the maximum feasible spectrum utilization ofthe th PU’s channel and is given by Lemma 2.4. Since de-pends on the budgets of SUs, it is difficult for each PU to decidewhether or not to admit new SUs based on the current demand.Note that, in the multiple-PU market, the available spectrumis much larger than the single PU case, which can accommo-date a large number of SUs. Since approaches when thenumber of SUs is reasonably large based on Lemma 2.4, wewill approximate using this asymptote. Accordingly, (25) canbe rewritten as

(26)

2) Pricing Competition in Stage II: In this stage, the PUs de-termine their pricing strategies while considering the demandsof SUs in Stage III, given the available spectrum opportunitiesin Stage I. The profit of the th PU can be expressed as

Since is fixed at this stage, the th PU is only interestedin maximizing the revenue . Obviously, if theth PU has no available spectrum to sell, i.e., , it wouldnot compete with other PUs by price reduction to attract the SUs.For convenience, define as the setof PUs with positive spectrum opportunity.Game at Stage II: The competition among PUs in this stage

can be modeled as the following game:• Players: the PUs in the set ;• Strategy: each PU can choose a price from the feasibleset ;

• Objective function: ;where denotes the minimum price that each PU can chooseand is determined by (10) at .



Proposition 3.1: Anecessary and sufficient condition for PUsto achieve a Nash equilibrium price is . More-over, when , there exists a unique Nash equi-librium price, and the Nash equilibrium price is given by

, where

(27)

The proof of Proposition 3.1 is given in Appendix D.Proposition 3.1 shows that no PU would announce a pricehigher than its competitors to avoid losing most or all of itsdemand to its competitors, and the optimal strategy is to makethe same decision as its competitors. Since ,the equilibrium price (27) can be rewritten as

(28)

where .3) Spectrum Opportunity Allocation in Stage I: In this stage,

the PUs need to decide the optimal spectrum opportunities tomaximize their profits. Based on Proposition 3.1, the th PU’sprofit can be written as

where . For convenience, define. Since

is a constant, maximizing isequivalent to maximizing .Game at Stage I: The competition among the PUs in this

stage can be modeled as the following game:• Players: the PUs in the set ;• Strategy: the PUs will choose from the feasible set

;

• Objective function: .We first examine the existence of the Nash equilibrium of

the game at this stage. Based on [25], the existence of theNash equilibrium can be obtained by showing the concavity of

in terms of .Proposition 3.2: There exists a Nash equilibrium in the spec-

trum opportunity allocation game, which satisfies the followingset of equations:

(29)In general, the Nash equilibrium that satisfies (29) is not nec-

essarily unique, as illustrated by the following example. Sup-pose a market with two heterogeneous PUs, with and

. Let and . Then,and are two possible Nash equilibria thatsatisfy (29).In what follows, we provide a necessary and sufficient condi-

tion for the uniqueness of Nash equilibrium in the market withhomogeneous PUs (i.e., ).

To find the Nash equilibrium of the game at this stage,we first examine the strategy of the th PU given otherPU’s decisions. By checking the first order condition

and the boundary conditions, wecan obtain the best response strategy of the th PU. As ex-pected, the best response strategy for the th PU dependson and its competitors’ decision . Let andbe the thresholds for PU’s decision making associated with

and , respectively. They are givenexplicitly byand (derivation in Appendix E). We nowestablish the response strategy for the PUs.Proposition 3.3: The best response strategy for the th PU in

the above game is outlined as follows.1) The case with :If , then is the solution to

(30)If , then .

2) The case with :If , then is the solution to

(31)If , then .

The proof of Proposition 3.3 is given in Appendix E. As ex-pected, the Nash equilibrium of the spectrum opportunity allo-cation game depends on , and the number of PUs. We havethe following necessary and sufficient condition for the unique-ness of Nash equilibrium.Proposition 3.4: The Nash equilibrium of the spec-

trum opportunity allocation game is unique if and only if, and at the spectrum opportunity equilib-

rium, , where is the solution to

(32)

The proof is given in Appendix F. Note that there exists athreshold for the number of PUs, denoted by , in the casewith , where is given by .9

Accordingly, Proposition 3.4 can be treated as a criterion for thePUs to decide whether to join in the competition or not becauseeach PU can calculate the pricing equilibrium when it gathersthe necessary information based on Proposition 3.4. In the casewith , it needs to check whether the condition

holds. This is because if , the pricingequilibrium will be , which indicates that it is unprofitableto sell spectrum to SUs due to the strong competition.

C. Algorithm for Computing Nash Equilibria

To achieve the Nash equilibrium of the dynamic game,we present an iterative algorithm for each PU. Based onProposition 3.1, if , the spectrum allocation isinefficient, i.e., there always exists some PU whose supply islarger than the demand. Thus, each PU first updates its spectrumopportunity based on the demand to fully utilize its spectrum.

9 denotes the largest integer not greater than .



After the necessary condition is satisfied, eachPU can update its spectrum opportunity in the Stage I based onProposition 3.3. We assume that the “total budget” of SUsis available to each PU. The proposed algorithm for computingthe market equilibrium is summarized in Algorithm 2. Based onPropositions 3.1, 3.3, and 3.4, Algorithm 2 provably convergesto the equilibrium of the game, as also verified by simulation.

Algorithm 2: Computing the Nash equilibrium of themultiple-PU market

Initialization: Each PU collects the budget information ofSUs, i.e., .

At the beginning of each period1) If then

Each PU sets , and broadcasts .Else

Each PU sets based on Proposition 3.3, andbroadcasts .

Endif2) Each PU sets , andbroadcasts .

3) Each SU randomly chooses a PU’s channel from the setof PUs with the lowest price in with equal probability.

4) Each PU admits new SUs when .

Remarks: Algorithm 2 is applicable to the scenarios wherethe PUs can vary their spectrum opportunities. When the spec-trum opportunities are fixed, the three-stage game reduces to atwo-stage game without the stage of spectrum opportunity al-location. In this case, the equilibrium of the game is given byProposition 3.1.

D. Numerical Examples: Equilibria for Competitive PUs

In this section, we examine the Nash equilibrium of the three-stage game in the market with homogeneous PUs. First, we il-lustrate the existence and the uniqueness of the Nash equilib-rium for two PUs, in the case with . Then, we con-sider a more general system model with four PUs and examinethe convergence performance of Algorithm 2. In the end, wedemonstrate how the equilibrium price evolves under differentelasticities of SUs and different numbers of PUs. In each exper-iment, is equal to 20, and each SU’s budget is generateduniformly in the interval [0, 4] and is fixed for different forthe sake of comparison.The existence and the uniqueness of the Nash equilibrium,

corresponding to the competitive spectrum opportunity of twoPUs, is illustrated in Fig. 7. Based on Proposition 3.4, when

, there exists a unique Nash equilibrium, as verifiedin Fig. 7. In particular, we change the inverse elasticity (i.e.,) of SUs from 0.3 to 0.4 in order to show how the spectrumopportunity equilibrium evolves. We observe that the spectrumequilibrium lies on the line with slope one, due to the symmetryof the best response functions, and increases with since theSUs become more insensitive to prices, which motivates thePUs to offer more spectrum to the SUs.

Fig. 7. Illustrating the existence and uniqueness of spectrum opportunityequilibrium.

Fig. 8. Convergence of spectrum opportunity equilibrium.

Next, we examine the convergence performance ofAlgorithm 2 in the case of four PUs. Here, we chooseand such that and . Asillustrated in Fig. 8, the sum of initial normalized spectrumopportunities10 is greater than 1, i.e., , in whichcase there is no equilibrium point based on Proposition 3.1.Each PU then updates its offered spectrum opportunity basedon its current demand (this process corresponds to the iterationsfrom 1 to 3 in Fig. 8). Once the total supply is within the feasibleregion, each PU adjusts its supply based on Proposition 3.3.According to Proposition 3.4, there exists a unique Nash equi-librium, which is further verified in Fig. 8.Fig. 9 depicts how the equilibrium price evolves under dif-

ferent elasticities of SUs and different numbers of PUs. Specif-ically, we choose and . As expected, theequilibrium price increases with . For each , the equilibriumprice decreases with the number of PUs due to more competitionamong the PUs. In other words, the SUs can benefit from thecompetition among the PUs. Moreover, the equilibrium priceapproaches as the number of PUs increases.

10The spectrum opportunity of each PU is normalized by .



Fig. 9. Price equilibrium corresponding to different elasticities of SUs and dif-ferent numbers of PUs.

IV. CONCLUSION

This paper studied pricing-based decentralized spectrumaccess in cognitive radio networks, where SUs compete viarandom access for available spectrum opportunities. We de-veloped two models: one with the monopoly PU market andthe other with the multiple-PU market. For the monopoly PUmarket model, we applied the revenue maximization approachto characterize the appropriate choice of flat and usage prices,and derived a Pareto-optimal solution, which was shown tobe near-optimal. More importantly, this Pareto-optimal solu-tion exhibits a decentralized structure, i.e., the Pareto-optimalpricing strategy and access probabilities can be computed bythe PU and the SUs locally. We also analyzed a PU profitmaximization problem by examining the tradeoff between thePU’s utility and its revenue.We then studied the multiple-PUmarket model by casting the

competition among PUs as a three-stage Stackelberg game, interms of access prices and the offered spectrum opportunities.We showed the existence of the Nash equilibrium and derived anecessary and sufficient condition for the uniqueness of Nashequilibrium for the case with homogeneous PUs. Intuitively,this condition can be used by PUs to decide whether to joinin the competition or not, i.e., when the number of PUs growslarger than a certain threshold, the competition among PUs is toostrong, indicating that it is unprofitable for a PU to sell spectrumto SUs. Then, we developed an iterative algorithm for strategyadaption to achieve the Nash equilibrium.It remains open to characterize the condition for the unique-

ness of Nash equilibrium for the case with heterogeneous PUs.Another interesting direction is to investigate transient behav-iors corresponding to dynamic spectrum access in the presenceof spectrum hole dynamics.

APPENDIX APROOF OF LEMMA 2.2

The proof of the first property is contained in that for thesecond one, which can be derived from the Lagrangian of (8) by

utilizing the Karush–Kuhn–Tucker (KKT) conditions. Specifi-cally, the Lagrangian of (8) is given by

By the KKT conditions, for optimal and , the system mustsatisfy the following equations:

Based on [26, Theorem 1], the solution to the above equationsis

(33)

It follows that

To prove the converse for the second property, note that theset of given in (33) is a stationary point for the function .It is straightforward to see that the set of given in (33) cannotachieve a minimum point of the function . Hence, the set ofgiven in (33) must maximize .

APPENDIX BPROOF OF LEMMA 2.3

The Lagrangian of (9) is given by

By the KKT conditions, the optimal of the system must sat-isfy the following equations:

The above equations can be written as

where is the elasticity of SUs.Therefore, the optimal pricesare the same. Further based on the constraint of (9),

, the optimal price can be derived as (11)by substituting the demand function for (4).

APPENDIX CPROOF OF LEMMA 2.4

Since (13) is strictly convex, we can solve it by first consid-ering its dual problem. The Lagrangian of (13) is given by



By the KKT conditions, we have and .Thus, the solution of the dual problem satisfies the Pareto-op-timal condition, i.e., , given by Lemma 2.2.The dual problem can be written as

minimize

subject to

variables (34)

By manipulating the summations and utilizing the constraint,the above problem can be written as

minimize

subject to

variables (35)

Since (35) is strictly convex, therefore the optimal solutioncan be derived by the KKT conditions, which imply that

, where is given by (15).When the number of SUs is large, let , where

. By (15), we have

Then, we have . For any , we have

as

Thus, , as . Since , wehave . Therefore, Lemma 2.4 is proved.

APPENDIX DPROOF OF PROPOSITION 3.1

We first show that is the sufficient condi-tion for the existence of the Nash equilibrium price, and thatunder that condition, the equilibrium price is uniquely deter-mined by (27).First, we consider the case where there are only two PUs. We

show that there does not exist a pricing equilibrium such that. Suppose that PU 1 and PU 2 have positive spectrum

opportunities and . Without loss of generality, we assumethat . The optimal prices for PU 1 and PU 2 are shownin Fig. 10, where the curve represents the optimalrevenue that each PU can earn when there is no competition.

Fig. 10. PU’s Revenue at different prices and spectrum opportunities.(a) . (b) .

Since it is only when the demand equals to the supply that eachPU can achieve its Pareto-optimal revenue, the optimal pricefor each PU is at the intersection point of the curve and the lineshown in Fig. 10. Obviously, when . From theanalysis of Stage I, we know that SUs will choose PU 1, whichmakes PU 2 have no revenue. In this case, PU 2 will decreaseits price at least less than or equal to PU 1 to get some revenue.Also, the price reduction will not end until both PUs announcethe same price.Next, we show that the equilibrium price is at the point as

shown in Fig. 10. Suppose that the equilibrium price is notat the point . In this case, the total demand is not equal to thetotal supply. Thus, at least one PU’s demand is not equal to itssupply. For the case , the total demand is less than thetotal supply, which means at least one PU’s demand is less thanits supply. Without loss of generality, we assume that the PU 1’sdemand is less than its supply. Based on Lemma 2.2, PU 1 willdecrease its price to make its demand equal to its supply, whichwill alsomake PU 2 decrease its price to achieve pricing equilib-rium. Thus, when , both PUs will decrease their prices.For the other case , the total demand is larger than thetotal supply, which means at least one PU’s demand is largerthan its supply. Without loss of generality, we assume that thePU 1’s demand is larger than its supply. Based on (26), PU 1will increase its price to make its demand equal to its supply. Inthis case, the price of PU 1 will be larger than the price of PU 2,which makes all SUs choose PU 2. Therefore, the demand ofPU 2will be larger than its supply, which forces PU 2 to increaseits price to achieve more revenue. Thus, when , bothPUs will increase their prices. Hence, the equilibrium price is at, which can be determined by (27). Since , the

total supply is less than or equal to .



Thus far, we have discussed the case for two PUs. The aboveresults can be easily generalized to the case with more than twoPUs by following similar steps. Therefore, the equilibrium priceis uniquely determined by (27), when .To show that is the necessary condition for

the existence of the Nash equilibrium price, it suffices to showthat no equilibrium exists when . By the defini-tion ofGame at Stage II, each PU can choose a price from thefeasible set , where is determined by (10)at . When , the equilibrium price would besmaller than , which means that the total supply is greaterthan the total demand. In other words, some PU’s spectrum op-portunities are unused. Thus, those PUs can always decrease thesupplied spectrum to improve their own utilities and make thedemand equal to their supply in the end. Intuitively speaking,since the revenue curve has no intersection point with the linewith slope as shown in Fig. 10, this meansthat in this region there is no equilibrium point. Therefore, thenecessary condition for PUs to achieve the Nash equilibrium is

.

APPENDIX EPROOF OF PROPOSITION 3.3

Due to the concavity of , we can obtain the bestresponse function by checking the first-order condition andboundary conditions. The first-order condition is

and the boundary conditions can be written as

(36)

(37)

both of which depend on its competitors spectrum opportunitiesand the parameter . For different and , the th PU’s bestresponse strategy can be written as follows.1) Case :

i.e., the th PU would sell spectrum opportunity to theSUs. Then, the optimal spectrum opportunity dependson the boundary condition (37). From Fig. 11, we knowthat there exists a decision threshold

for PU , which is the solu-tion to

Fig. 11. Decision threshold .

Based on its competitors’ spectrum opportunities, the thPU’s decision is outlined as follows.a) :

The best response strategy of the th PU is determinedby its first-order condition (30).

b) :

The best response strategy of the th PU is to sellas much spectrum opportunity as possible, i.e.,

.2) Case :From Fig. 11, we know

The optimal spectrum opportunity depends on theboundary condition (36), from which we know there existsa decision threshold . Based on andits competitors’ spectrum opportunities, the decisions ofthe th PU are as follows.a) :

The best response strategy of the th PU is determinedby its first-order condition (31).

b) :

The best response strategy of the th PU is not to sellany spectrum opportunity, i.e., .



APPENDIX FPROOF OF PROPOSITION 3.4

Due to the concavity of in , the existence ofequilibrium can be readily shown based on [25]. In what fol-lows, we will derive the necessary and sufficient condition forthe uniqueness of Nash equilibrium, based on the best responsestrategy.1) Case :Defineand as the sets ofPUs choosing decision as the solution to (30) and

, respectively, where . Letand denote the spectrum op-

portunity equilibrium for PUs in the set and , respec-tively. Assume that both sets and are nonempty.Based on Proposition 3.3, we know , andis the solution of

Thus, . Since, we can get .

Utilizing the condition(i.e., ), needs to satisfy

Therefore, we can summarize the spectrum opportunityequilibria as follows.a) Case :In this case, there exist infinitely many spectrum op-portunity equilibria that satisfy

b) Case :In this case, we have either , or, . For the case , , we have

and , by whichwe have , i.e.,

, which yieldsdue to . Obviously, this con-tradicts the fact . Hence, the only possible caseis , . Due to the homogeneity of thebest response function, there exists a unique spectrumopportunity equilibrium [27], i.e., ,where is the solution to

2) Case :Define and

as the sets of PUs choosingdecision as the solution to (31) and , respectively,where . Let anddenote the spectrum opportunity equilibrium for PUs in

the set and , respectively. Based on Proposition 3.3,we know that , and we can use (31) to calculate .Then, , where is the solution to

After some algebra, we have .Since for each PU in , , i.e.,

, the best response for PUs in is thedecision (31), which shows that all PUs will choose thedecision (31). Due to the homogeneity of best responsefunction, there exists a unique spectrum opportunity equi-librium [27], which can be determined by (32).

When , we have based on the proof ofProposition 3.3. Therefore, . In summary,when , there exists a unique Nash equi-librium. The other direction follows directly based on the abovediscussion of spectrum opportunity equilibria.

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Lei Yang (S’10) received the B.S. and M.S. degreesin electrical engineering from Southeast University,Nanjing, China, in 2005 and 2008, respectively, andis currently pursuing the Ph.D. degree in electricalengineering at Arizona State University, Tempe.His research interests include wireless network op-

timization/control, cognitive radio, and smart grid.

Hongseok Kim (S’06–M’10) received the B.S. andM.S. degrees in electrical engineering from SeoulNational University, Seoul, Korea, in 1998 and 2000,respectively, and the Ph.D. degree in electrical andcomputer engineering from the University of Texasat Austin in 2009.He is an Assistant Professor with the Department

of Electronic Engineering, Sogang University,Seoul, Korea. He was a Member of Technical Staffwith Korea Telecom (KT) Labs, Daejeon, Korea,from 2000 to 2005. He worked as a Post-Doctoral

Research Associate with the Department of Electrical Engineering, PrincetonUniversity, Princeton, NJ, from 2009 to 2010, and worked as a Member ofTechnical Staff with Bell Labs, Murray Hill, NJ, from 2010 to 2011. Hisresearch interests are network resource allocation and optimization includingcross-layer design of wireless communication systems, green IT/wireless,MIMO and OFDMA, network economics, and smart grid.Dr. Kim is the recipient of a Korea Government Overseas Scholarship in

2005–2008.

Junshan Zhang (S’98–M’00–SM’06–F’12) re-ceived the Ph.D. degree in electrical and computerengineering from Purdue University, West Lafayette,IN, in 2000.He joined the Electrical Engineering Department,

Arizona State University, Tempe, in August 2000,where he has been a Professor since 2010. Hisresearch interests include communications networks,cyber-physical systems with applications to smartgrid, stochastic modeling and analysis, and wirelesscommunications. His current research focuses on

fundamental problems in information networks and network science, includingnetwork optimization/control, smart grid, cognitive radio, and network infor-mation theory.Prof. Zhang was an Associate Editor for the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS. He is currently an Editor for Computer Net-works and the IEEE Wireless Communication Magazine He is TPC Co-Chairfor INFOCOM 2012. He served as TPC Co-Chair for WICON 2008 andIPCCC 2006, TPC Vice Chair for ICCCN 2006, and a member of the technicalprogram committees of INFOCOM, SECON, GLOBECOM, ICC, MOBIHOC,BROADNETS, and SPIE ITCOM. He was the General Chair for the IEEECommunication Theory Workshop 2007. He is a recipient of the ONR YoungInvestigator Award in 2005 and the NSF CAREER Award in 2003. He receivedthe Outstanding Research Award from the IEEE Phoenix Section in 2003. Hecoauthored a paper that won the IEEE ICC 2008 Best Paper Award, and one ofhis papers was selected as the INFOCOM 2009 Best Paper Award Runner-up.

Mung Chiang (S’00–M’03–SM’08–F’12) receivedthe B.S. degree (Honors) in electrical engineering andmathematics andM.S. and Ph.D. degrees in electricalengineering from Stanford University, Stanford, CA,in 1999, 2000, and 2003, respectively.He is a Professor of electrical engineering with

Princeton University, Princeton, NJ, and an affiliatedfaculty in Applied and Computational Mathematicsand in Computer Science. He was an Assistant Pro-fessor from 2003 to 2008, and an Associate Professorfrom 2008 to 2011 with Princeton University. He

wrote an undergraduate textbook Networked Life: 20 Questions and Answers(Cambridge Univ. Press, 2012)Prof. Chiang served as an IEEE Communications Society Distinguished Lec-

turer in 2012 to 2013. His research on networking received the IEEE KiyoTomiyasu Award in 2012, a US Presidential Early Career Award for Scien-tists and Engineers in 2008, several young investigator awards, and a few paperawards including the IEEE INFOCOM Best Paper Award in 2012. His inven-tions resulted in a few technology transfers to commercial adoption, and he re-ceived a Technology Review TR35 Award in 2007 and founded the PrincetonEDGE Lab in 2009.

Chee Wei Tan (M’08) received the M.A. and Ph.D.degrees in electrical engineering from PrincetonUniversity, Princeton, NJ, in 2006 and 2008,respectively.He is an Assistant Professor with the City Univer-

sity of Hong Kong, Hong Kong. Previously, he was aPostdoctoral Scholar with the California Institute ofTechnology (Caltech), Pasadena. He was a VisitingFaculty with Qualcomm R&D, San Diego, CA, in2011. His research interests are in wireless and broad-band communications, signal processing, and non-

linear optimization.Dr. Tan currently serves as an Editor for the IEEE TRANSACTIONS ON

COMMUNICATIONS. He was the recipient of the 2008 Princeton UniversityWu Prize for Excellence and the 2011 IEEE Communications Society APOutstanding Young Researcher Award.

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