Duopoly Competition in Dynamic Spectrum Leasing and Pricing

Duopoly Competition in DynamicSpectrum Leasing and Pricing

Lingjie Duan, Member, IEEE, Jianwei Huang, Senior Member, IEEE, and Biying Shou

Abstract—This paper presents a comprehensive analytical study of two competitive secondary operators’ investment (i.e., spectrum

leasing) and pricing strategies, taking into account operators’ heterogeneity in leasing costs and users’ heterogeneity in transmission

power and channel conditions. We model the interactions between operators and users as a three-stage dynamic game, where

operators simultaneously make spectrum leasing decisions in Stage I, and pricing decisions in Stage II, and then users make purchase

decisions in Stage III. Using backward induction, we are able to completely characterize the dynamic game’s equilibria. We show that

both operators’ investment and pricing equilibrium decisions process interesting threshold properties. For example, when the two

operators’ leasing costs are close, both operators will lease positive spectrum. Otherwise, one operator will choose not to lease and the

other operator becomes the monopolist. For pricing, a positive pure strategy equilibrium exists only when the total spectrum investment

of both operators is less than a threshold. Moreover, two operators always choose the same equilibrium price despite their

heterogeneity in leasing costs. Each user fairly achieves the same service quality in terms of signal-to-noise ratio (SNR) at the

equilibrium, and the obtained predictable payoff is linear in its transmission power and channel gain. We also compare the duopoly

equilibrium with the coordinated case where two operators cooperate to maximize their total profit. We show that the maximum loss of

total profit due to operators’ competition is no larger than 25 percent. The users, however, always benefit from operators’ competition in

terms of their payoffs. We show that most of these insights are robust in the general SNR regime.

Index Terms—Cognitive radio, spectrum trading, dynamic spectrum leasing, spectrum pricing, multistage dynamic game, subgame

perfect equilibrium

Ç

1 INTRODUCTION

WIRELESS spectrum is often considered as a scarceresource, and thus has been tightly controlled by

the governments through static license-based allocations.However, several recent field measurements show thatmany spectrum bands are often underutilized even indensely populated urban areas [2]. To achieve more efficientspectrum utilization, secondary users may be allowed toshare the spectrum with the licensed primary users. Variousdynamic spectrum access mechanisms have been proposedalong this direction. One of the proposed mechanisms isdynamic spectrum leasing,1 where a spectrum ownerdynamically transfers and trades the usage right oftemporarily unused part of its licensed spectrum tosecondary network operators or users in exchange formonetary compensation [3], [4], [5], [6], [7]. In this paper, we

study the competition of two secondary operators under thedynamic spectrum leasing mechanism.

Our study is motivated by the successful operations ofmobile virtual network operators (MVNOs) in manycountries today.2 An MVNO does not own wirelessspectrum or even the physical infrastructure. It providesservices to end-users by long-term spectrum leasing agree-ments with a spectrum owner. MVNOs are similar to the“switchless resellers” of the traditional landline telephonemarket. Switchless resellers buy minutes wholesale fromthe large long distance companies and resell them to theircustomers. It is shown by Lev-Ram [9] and Dewenter andHaucap [10] that it can be more efficient for the spectrumowner to hire an MVNO as intermediary to retail itsspectrum resource, as MVNO has a better understanding oflocal user population and local users’ demand. However,an MVNO is often stuck in a long-term leasing contractwith a spectrum owner and cannot make flexible spectrumleasing and pricing decisions to match the dynamicdemands of the users. The secondary operators consideredin this paper do not own wireless spectrum either.Compared with a traditional MVNO, the secondaryoperators can dynamically adjust their spectrum leasingand pricing decisions to match the users’ demands thatchange with users’ channel conditions.

This paper studies the competition between secondaryoperators in spectrum acquisition and pricing to serve acommon pool of secondary users. To abstract the interac-tions among operators, we focus on two operator case

1706 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. 11, NO. 11, NOVEMBER 2012

. L. Duan is with the Engineering Systems and Design Pillar, SingaporeUniversity of Technology and Design, 20 Dover Drive, Singapore 138682.E-mail: [email protected].

. J. Huang is with the Network Communications and Economics Laboratory,Department of Information Engineering, The Chinese University of HongKong, Ho-Sin Hang Engineering Building, Shatin, N.T., Hong Kong999077. E-mail: [email protected].

. B. Shou is with the Department of Management Sciences, City Universityof Hong Kong, Room 7519, Academia Building Tat Chee Avenue,Kowloon, Hong Kong 99907. E-mail: [email protected].

Manuscript received 2 Nov. 2010; revised 29 Aug. 2011; accepted 16 Sept.2011; published online 10 Oct. 2011.For information on obtaining reprints of this article, please send e-mail to:[email protected], and reference IEEECS Log Number TMC-2010-11-0500.Digital Object Identifier no. 10.1109/TMC.2011.213.

1. “Dynamic” here means that spectrum leasing can be flexibly done at ashort time scale between any two involved parties.

2. Since the late 1990s, there have been over 400 mobile virtual networkoperators owned by over 360 companies worldwide as of February 2009 [8].

1536-1233/12/$31.00 � 2012 IEEE Published by the IEEE CS, CASS, ComSoc, IES, & SPS

(i.e., duopoly) and will study multiple operator case (i.e.,oligopoly) in our future work. The secondary operatorswill dynamically lease spectrum from spectrum owners,and then compete to sell the resource to the secondaryusers to maximize their individual profits. We would liketo understand how the operators make the equilibriuminvestment (leasing) and pricing (selling) decisions, consideringoperators’ heterogeneity in leasing costs and wireless users’heterogeneity in transmission power and channel conditions.

We adopt a three-stage dynamic game model to studythe (secondary) operators’ investment and pricing decisionsas well as the interactions between the operators and the(secondary) users.3 From here on, we will simply use“operator” to denote “secondary operator” and “users” todenote “secondary users.” In Stage I, the two operatorssimultaneously lease spectrum (bandwidth) from thespectrum owners with different leasing costs. In Stage II,the two operators simultaneously announce their spectrumretail prices to the users. In Stage III, each user determineshow much resource to purchase from which operator. Eachoperator wants to maximize its profit, which is thedifference between the revenue collected from its usersand the cost paid to the spectrum owner.

Key results and contributions of this paper include

. An appropriate wireless spectrum sharing model. Weassume that heterogeneous users share the spectrumusing orthogonal frequency division multiplexing(OFDM) technology. Then, a user’s achievable rateand thus its spectrum demand depend on itsallocated bandwidth, maximum transmissionpower, and channel condition. This model is moresuitable to our problem than the generic economicmodels used in related literature [6], [11], [12], [13]. Itcan also provide more engineering insights on howdifferent wireless network parameters in the spec-trum sharing model (e.g., users’ various wirelesscharacteristics) impact the operators’ leasing andpricing decisions.

. Symmetric pricing structure. We show the twooperators always choose the same equilibrium price,even when they have different leasing costs andmake different investment decisions. Moreover, thisprice is independent of users’ transmission powersand channel conditions.4

. Threshold structures of investment and pricing equili-brium. We show that both operators’ investment andpricing equilibrium decisions process interestingthreshold properties. For example, when the twooperators’ leasing costs are close, both operators willlease positive spectrum. Otherwise, one operatorwill choose not to lease and the other operatorbecomes the monopolist. For pricing, a positive purestrategy equilibrium exists only when the totalspectrum investment of both operators is less thana threshold.

. Fair service quality achieved by users. We show thateach user achieves the same signal-to-noise ratio(SNR) that is independent of the users’ populationand wireless characteristics.

. Impact of competition. We show that the operators’competition leads to a maximum 25 percent loss oftheir total profit compared with a coordinated case.The users, however, always benefit from theoperators’ competition by achieving better payoffs.

Next, we briefly discuss the related literature. In Section 2,we describe the network model and game formulation. InSection 3, we analyze the dynamic game through backwardinduction and calculate the duopoly leasing/pricing equili-brium. We discuss various insights obtained from theequilibrium analysis in Section 4. In Section 6, we showthe impact of duopoly competition on the total operators’profit and the users’ payoffs. We conclude in Section 7together with some future research directions.

1.1 Related Work

Recently, researchers have started to study the economicaspect of dynamic spectrum access, such as the secondaryoperators’ strategies of spectrum acquisition from spectrumowners and service provision to the users. For example,several auction mechanisms have been proposed for thespectrum owner to allocate spectrum [14], [15], [16], [17].Auction is a good choice when a spectrum owner does nothave a good estimation of how much the spectrum is worthto the users, and thus relies on the bids of the users todetermine the pricing and resource allocation. When aspectrum owner has the complete network information,however, he can simply price the spectrum accordingly andlease to the users.

For operators’ service provision, most related resultslooked at the pricing interactions between network opera-tors and the secondary users [6], [11], [12], [13], [17], [18],[19]. There are few papers that jointly studied resourceinvestment and service pricing decisions for intermediarysecondary operator(s) [6], [20], [21] as this paper. Such jointoptimization is very important since different investmentamounts at the beginning weigh heavily on later pricingand service accommodation capabilities of operators. Jiaand Zhang [6] only used a generic economic model for totalusers’ spectrum demand without much wireless details.Niyato et al. [20] did not explicitly analyze the jointoptimization over leasing and pricing resource and ex-tensive simulations are mainly used instead. Also, Duanet al. [21] only considered a single operator case withoutdiscussing operator competition. In multiple competitiveoperator case, the operators’ strategy spaces are oftencoupled and is hard to analyze over multiple stages.

The key difference of this work here is that we present acomprehensive analytical study that characterizes both thecompetitive operators’ equilibrium investment and pricingdecisions, with heterogeneous leasing costs for the operatorsand an appropriate wireless spectrum sharing model forthe users.

2 NETWORK AND GAME MODEL

We consider two operators (i; j 2 f1; 2g and i 6¼ j) and a setK ¼ f1; . . . ; Kg of users in an ad hoc network as shown in

DUAN ET AL.: DUOPOLY COMPETITION IN DYNAMIC SPECTRUM LEASING AND PRICING 1707

3. Here, the word “dynamic” in “dynamic game” is different from that infootnote 1. It refers to the interactions between operators and users overtime.

4. Such independency is good for the development of spectrum market,since a user does not need to worry about how variations of user populationand wireless characteristics change its performance in spectrum trading.

Fig. 1. The operators obtain wireless spectrum fromdifferent spectrum owners with different leasing costs,and compete to serve the same set K of users. Each user hasa transmitter-receiver pair. We assume that users areequipped with software-defined radios and can transmitin a wide range of frequencies as instructed by theoperators, but do not have the capability of spectrumsensing in cognitive radios.5 Such a network structure putsmost of the implementation complexity for dynamicspectrum leasing and allocation on the operators, and thusis easier to implement than a “full” cognitive radio networkespecially for a large number of users. A user may switchamong different operators’ services (e.g., WiMAX, 3G)depending on operators’ prices. It is important to study thecompetition among multiple operators as operators arenormally not cooperative.

The interactions between the two operators and the userscan be modeled as a three-stage dynamic game, as shown inFig. 2. Operators i and j first simultaneously determinetheir leasing bandwidths in Stage I, and then simulta-neously announce the prices to the users in Stage II. Finally,each user chooses to purchase bandwidth from only oneoperator to maximize its payoff in Stage III.

The key notations of the paper are listed in Table 1.Some are explained as follows:

. Leasing decisions Bi and Bj: leasing bandwidths ofoperators i and j in Stage I, respectively.

. Costs Ci and Cj: the fixed positive leasing costs perunit bandwidth for operators i and j, respectively.These costs are determined by the negotiationbetween the operators and their own spectrumsuppliers.

. Pricing decisions pi and pj: prices per unit band-width charged by operators i and j to the users inStage II, respectively.

. The User k’s demand wki or wkj: the bandwidthdemand of a user k 2 K from operator i or j. A usercan only purchase bandwidth from one operator.

2.1 Users’ and Operators’ Models

OFDM has been proposed as a promising physical layerchoice for dynamic spectrum sharing [22], [23]. We assumethat the users share the spectrum using OFDM to avoidmutual interferences. The main analysis in this paperassumes that users are located close-by, and thus no twousers will transmit over the same channel (also calledsubcarriers in the OFDM literatures [24], [25]). We willrelax this assumption later on (Appendix E, which can befound on the Computer Society Digital Library at http://doi.ieeecomputersociety.org/10.1109/TMC.2011.213) andshow that our results can be extended to the case withspectrum spatial reuse.

If a user k 2 K obtains bandwidth wki from operator i,then it achieves a data rate (in nats) of [26]

rkðwkiÞ ¼ wki ln 1þ Pmaxk hkn0wki

� �; ð1Þ

where Pmaxk is user k’s maximum transmission power, n0 is

the noise power density, hk is the channel gain betweenuser k’s transmitter and receiver. The channel gain hk isindependent of the operator, as the operator only sellsbandwidth and does not provide a physical infrastruc-ture.6 Here, we assume that user k spreads its power Pmax

k

across the entire allocated bandwidth wki. To simplify laterdiscussions, we let

gk ¼ Pmaxk hk=n0;


Fig. 1. Network model for the secondary network operators.

Fig. 2. Three-stage dynamic game: the duopoly’s leasing and pricing,and the users’ resource allocation.

TABLE 1Key Notations

5. Spectrum sensing is the most important functionality of cognitiveradios, which enables users to actively monitor the external radioenvironments to communicate efficiently without interfering primary users.The capability of spectrum sensing includes comprehensive monitoring offrequency spectrum, user behavior, and network state over time.

6. We also assume that the channel condition is independent oftransmission frequencies, such as in the current 802.11d/e standard [27]where the channels are formed by interleaving over the tones. In otherwords, each user experiences a flat fading over the entire spectrum.

thus gk=wki is the user k’s SNR. The rate in (1) is calculatedbased on the Shannon capacity.

To obtain closed-form solutions, we first focus on thehigh SNR regime where SNR� 1. This will be the casewhere a user has limited choices of modulation andcoding schemes, and thus cannot decode a transmission ifthe SNR is below some threshold. In the high SNR regime,the rate in (1) can be approximated as

rkðwkiÞ ¼ wki lngkwki

� �: ð2Þ

Although the analytical solutions in Section 3 are derivedbased on (2), we will show later in Section 5 that all majorengineering insights remain unchanged in the general SNR regime.

If a user k purchases bandwidth wki from operator i, itreceives a payoff of

ukðpi; wkiÞ ¼ wki lngkwki

� �� piwki; ð3Þ

which is the difference between the data rate and thepayment. The payment is proportional to price pi an-nounced by operator i. This linear pricing scheme has beenwidely used in the literature [28], [29].

For an operator i, its profit is the difference between therevenue and the total cost, i.e.,

�iðBi;Bj; pi; pjÞ ¼ piQiðBi;Bj; pi; pjÞ �BiCi; ð4Þ

where QiðBi;Bj; pi; pjÞ and QjðBi;Bj; pi; pjÞ are realizeddemands of operators i and j. The concept of realizeddemand will be defined later in Definition 4.

3 BACKWARD INDUCTION OF THE THREE-STAGE

GAME

A common approach of analyzing dynamic game isbackward induction to find the subgame perfect equili-brium (SPE) [30]. Subgame perfect equilibrium (or simply,equilibrium) represents a Nash equilibrium of everysubgame of the original game. In this paper, we start withStage III and analyze the users’ behaviors given theoperators’ investment and pricing decisions. Then, we lookat Stage II and analyze how operators make the pricingdecisions taking the users’ demands in Stage III intoconsideration. Finally, we look at the operators’ leasingdecisions in Stage I knowing the results in Stages II and III.Throughout the paper, we will use “bandwidth,” “spec-trum,” and “resource” interchangeably.

In the following analysis, we only focus on pure strategySPE and rule out mixed SPE in the multistage game.7 Such amethodology has been widely used in the literature [31],[32]. Following the definition in [31], we use conditionallySPE to denote an SPE with pure strategies only, where thenetwork’s pure strategies constitute a Nash equilibrium inevery subgame. The concept of conditionally SPE ismotivated by the concept of SPE but rules out mixedstrategies. In Section 3.2, we will show that a conditionally

SPE will not include any investment decisions ðBi;BjÞ inthe medium investment regime in Stage I. Otherwise, thereis no pure strategy Nash equilibrium for pricing in Stage II,and it will not be a conditionally SPE.8

Following very similar statements in [31], we list severalreasons to focus on conditionally SPE in this paper withoutconsidering mixed strategies

. First, we want to emphasize the result that a purestrategy pricing equilibrium may not exist inStage II, as this result highlights the very importantEdgeworth paradox for the medium investmentregime (which will be introduced in Section 3.2).Such result reveals the special structure of ourproblem and leads to important engineering insightsfor practical network design.

. Second, a standard criticism of mixed strategyequilibrium is that they impose very large informa-tional burdens on users [30]. If operators chooseprices according to mixed strategies, users need toconsider price distributions (from which the finalprices will be drawn by operators) when they choosewhich operator to purchase from. When the opera-tors’ leasing costs change over time, the leasingamounts and the corresponding mixed pricingstrategies can also be time varying. Given all thesecomplexities, it is unlikely that end users will havethe computational capacities and willingness tocalculate the “equilibrium choices” in real spectrummarket. In other words, the analysis results whenallowing mixed strategies may not be very relevantfor engineering practice.

. Third, two operators need to run the randomizationprocedure in the pricing stage of each time slot ifthey adopt mixed pricing strategies. However, suchrandomization over time may be too complicatedto implement in practice in a short time scale [34].

In the following analysis, we derive the conditionallySPE, which is also referred to as equilibrium for simplicity.

3.1 Spectrum Allocation in Stage III

In Stage III, each user needs to make the following twodecisions based on the prices pi and pj announced by theoperators in Stage II:

1. Which operator to choose?2. How much to purchase?

If a user k 2 K obtains bandwidth wki from operator i,then its payoff ukðpi; wkiÞ is given in (3). Since this payoff isconcave in wki, the unique demand that maximizes thepayoff is

w�kiðpiÞ ¼ arg maxwki�0

ukðpi; wkiÞ ¼ gk expð�ð1þ piÞÞ: ð5Þ

Demand w�kiðpiÞ is always positive, linear in gk, anddecreasing in price pi. Since gk is linear in channel gain hkand transmission power Pmax

k , then a user with a better


7. For interested readers, we have provided some preliminary analysis ofmixed strategy SPE in Appendix F, available in the online supplementalmaterial.

8. If we do not focus on the concept of conditionally SPE, there may be anSPE with mixed strategies. For example, in the pricing subgame in Stage II,mixed pricing strategy Nash equilibrium can exist in the mediuminvestment regime, which is supported by our analysis in Appendix F,available in the online supplemental material, and [33].

channel condition or a larger transmission power has alarger demand. It is clear that w�kiðpiÞ is upper bounded bygk expð�1Þ for any choice of price pi � 0. In other words,even if operator i announces a zero price, user k will notpurchase infinite amount of resource since it cannot decodethe transmission if SNRk ¼ gk=wki is low.

Equation (5) shows that every user purchasing band-width from operator i obtains the same SNR

SNRk ¼gk

w�kiðpiÞ¼ expð1þ piÞ;

and obtains a payoff linear in gk

ukðpi; w�kiðpiÞÞ ¼ gk expð�ð1þ piÞÞ:

3.1.1 Which Operator to Choose?

Next, we explain how each user decides which operatorto purchase from. The following definitions help thediscussions.

Definition 1. The Preferred User Set KPi includes the userswho prefer to purchase from operator i.

Definition 2. The Preferred Demand Di is the total demandfrom users in the preferred user set KPi , i.e.,

Diðpi; pjÞ ¼X

k2KPi ðpi;pjÞgk expð�ð1þ piÞÞ: ð6Þ

The notations in (6) imply that both set KPi and demandDi only depend on prices ðpi; pjÞ in Stage II and areindependent of operators’ leasing decisions ðBi;BjÞ inStage I. Such dependance can be discussed in two cases:

1. Different prices (pi < pj). Every user k 2 K prefers topurchase from operator i since

ukðpi; w�kiðpiÞÞ > ukðpj; w�kjðpjÞÞ:

We have KPi ¼ K and KPj ¼ ;, and

Diðpi; pjÞ ¼ G expð�ð1þ piÞÞ and Djðpi; pjÞ ¼ 0;

where G ¼P

k2K gk represents the aggregate wire-less characteristics of the users. This notation will beused heavily later in the paper.

2. Same prices (pi ¼ pj ¼ p). Every user k 2 K is indif-ferent between the operators and randomly choosesone with equal probability. In this case,

Diðp; pÞ ¼ Djðp; pÞ ¼ G expð�ð1þ pÞÞ=2:

Now, let us discuss how much demand an operator canactually satisfy, which depends on the bandwidth invest-ment decisions ðBi;BjÞ in Stage I. It is useful to define thefollowing terms:

Definition 3. The Realized User Set KRi includes the userswhose demands are satisfied by operator i.

Definition 4. The Realized Demand Qi is the total demand ofusers in the Realized User Set KRi , i.e.,

Qi Bi; Bj; pi; pj� �

¼X

k2KRi Bi;Bj;pi;pjð Þgk expð�ð1þ piÞÞ:

Notice that both KRi and Qi depend on prices ðpi; pjÞ inStage II and leasing decisions ðBi;BjÞ in Stage I. Calculatingthe Realized Demands also requires considering twodifferent pricing cases:

1. Different prices (pi < pj). The Preferred Demands areDiðpi; pjÞ ¼ G expð�ð1þ piÞÞ and Djðpi; pjÞ ¼ 0.

a. If Operator i has enough resource ði:e:; Bi �Diðpi; pjÞÞ. All Preferred Demand will be satis-fied by operator i. The Realized Demands are

Qi ¼ minðBi;Diðpi; pjÞÞ ¼ G expð�ð1þ piÞÞ;Qj ¼ 0:

b. If Operator i has limited resource ði:e:; Bi <

Diðpi; pjÞÞ. Since operator i cannot satisfy the

Preferred Demand, some demand will be

satisfied by operator j if it has enough resource.

Since the realized demand QiðBi;Bj; pi; pjÞ ¼Bi ¼

Pk2KRi gk expð�ð1þ piÞÞ, then

Pk2KRi gk ¼

Bi expð1þ piÞ.9 The remaining users want to

purchase bandwidth from operator j with a total

demand of ðG�Bi expð1þ piÞÞ expð�ð1þ pjÞÞ.Thus, the Realized Demands are

Qi ¼ minðBi;Diðpi; pjÞÞ ¼ Bi;

Qj ¼ min Bj;G�Bi expð1þ piÞ

exp 1þ pj� �

!:

2. Same prices (pi ¼ pj ¼ p). Both operators will attractthe same Preferred Demand G expð�ð1þ pÞÞ=2. TheRealized Demands are

Qi ¼ min Bi;Diðp; pÞ þmax Djðp; pÞ �Bj; 0� ��

¼ min

�Bi;

G

2 expð1þ pÞ

þmaxG

2 expð1þ pÞ �Bj; 0

� ��;

Qj ¼ min Bj;Djðp; pÞ þmax Diðp; pÞ �Bi; 0ð Þ� �

¼ min

�Bj;

G

2 expð1þ pÞ

þmaxG

2 expð1þ pÞ �Bi; 0

� ��:

3.2 Operators’ Pricing Competition in Stage II

In Stage II, the two operators simultaneously determinetheir prices ðpi; pjÞ considering the users’ preferred de-mands in Stage III, given the investment decisions ðBi;BjÞin Stage I.

An operator i’s profit is defined earlier in (4). Since thepayment BiCi is fixed at this stage, operator i’s profitmaximization problem is equivalent of maximization of its


9. In this paper, we consider a large number of users and each user isnonatomic (infinitesimal). Thus, an individual user’s demand is infinitesi-mal to an operator’s supply and we can claim equality holds for Qi ¼ Bi.

revenue piQi. Note that users’ total demand Qi to operator idepends on the received power of each user (product of itstransmission power and channel gain). We assume that anoperator i knows users’ transmission powers and channelconditions. This can be achieved similarly as in today’scellular networks, where users need to register with theoperator when they enter the network and frequentlyfeedback the channel conditions. Thus, we assume that anoperator knows the user population and user demand.

Game 1 (Pricing game). The competition between the twooperators in Stage II can be modeled as the following game:

. Players: two operators i and j.

. Strategy space: operator i can choose price pi from thefeasible set Pi ¼ ½0;1Þ. Similarly for operator j.

. Payoff function: operator i wants to maximize therevenue piQiðBi;Bj; pi; pjÞ. Similarly for operator j.

At an equilibrium of the pricing game, ðp�i ; p�j Þ, eachoperator maximizes its payoff assuming that the otheroperator chooses the equilibrium price, i.e.,

p�i ¼ arg maxpi2Pi

piQiðBi;Bj; pi; p�j Þ; i ¼ 1; 2; i 6¼ j:

In other words, no operator wants to unilaterally change itspricing decision at an equilibrium.

Next, we will investigate the existence and uniquenessof the pricing equilibrium. First, we show that it issufficient to only consider symmetric pricing equilibriumfor Game 1.

Proposition 1. Assume both operators lease positive bandwidthin Stage I, i.e., minðBi;BjÞ > 0. If pricing equilibrium exists,it must be symmetric p�i ¼ p�j .

The proof of Proposition 1 is given in our onlinetechnical report [35]. The intuition is that no operator willannounce a price higher than its competitor to avoid losingits Preferred Demand. This property significantly simpli-fies the search for all possible equilibria.

Next, we show that the symmetric pricing equilibrium isa function of (Bi;Bj) as shown in Fig. 3.

Theorem 1. The equilibria of the pricing game are as follows:

. Low investment regime (Bi þBj � G expð�2Þ as inregion (L) of Fig. 3). There exists a unique nonzeropricing equilibrium

p�i ðBi;BjÞ ¼ p�j ðBi;BjÞ ¼ lnG

Bi þBj

� �� 1: ð7Þ

The operators’ profits in Stage II are

�II;iðBi;BjÞ ¼ Bi lnG

Bi þBj

� �� 1� Ci

� �; ð8Þ

�II;jðBi;BjÞ ¼ Bj lnG

Bi þBj

� �� 1� Cj

� �: ð9Þ

. Medium investment regime (Bi þBj > G expð�2Þand minðBi;BjÞ < G expð�1Þ as in regions (M1)-(M3) of Fig. 3). There is no pricing equilibrium.

. High investment regime (minðBi;BjÞ � G expð�1Þas in region (H) of Fig. 3). There exists a unique zeropricing equilibrium

p�i ðBi;BjÞ ¼ p�j ðBi;BjÞ ¼ 0;

and the operators’ profits are negative for any positivevalues of Bi and Bj.

Proof of Theorem 1 is given in Appendix A, available inthe online supplemental material. Intuitively, higher invest-ments in Stage I will lead to lower equilibrium prices inStage II. Theorem 1 shows that the only interesting case isthe low investment regime where both operators’ totalinvestment is no larger than G expð�2Þ, in which case thereexists a unique positive symmetric pricing equilibrium.Notice that the same prices at equilibrium do not imply thesame profits, as the operators can have different costs (Ciand Cj) and thus can make different investment decisions(Bi and Bj) as shown next.

Note that our equilibrium results in medium investmentregime are consistent with the well-known Edgeworthparadox [36] in economics. Edgeworth paradox describes asituation where two players cannot reach a state ofequilibrium with pure strategies. Each operator facescapacity constraints when determining pricing decisionsin Stage II. The choice of both operators charging zero pricesis not an equilibrium in the medium investment regime,since at least one operator can raise its price and obtainnonzero revenue. Nor is the case where one operatorcharges less the other an equilibrium, since the lower priceoperator can profitably raise its price toward the other. Noris the case where both operators charge the same positiveprice, since at least one operator can lower its price slightlyand increase its profit.

The above nonequilibrium cases will not happen in thelow investment regime where operators have very limitedresources. This is because that in the low investmentregime no operator can satisfy the whole demand alone,and thus it is possible for the two operators to share themarket at the equilibrium.

Also, these nonequilibrium cases will not happen in thehigh investment regime, where both two operators havemore resources than users’ total demand even the price iszero. In this regime, we can ignore the resource constraints(similar to the Bertrand competition) and the zero priceequilibrium is the same as the Bertrand paradox [37]. Inthe Bertrand paradox, either operator deviating from zeroprice cannot attract any demand from its competitor whocan already serve all users.


Fig. 3. Pricing equilibrium types in different (Bi; Bj).

3.3 Operators’ Leasing Strategies in Stage I

In Stage I, the operators need to decide the leasingamounts ðBi;BjÞ to maximize their profits. Based onTheorem 1, we only need to consider the case where thetotal bandwidth of both the operators is no larger thanG expð�2Þ. We emphasize that the analysis of Stage I is notlimited to the case of low investment regime; we actuallyalso consider the medium investment regime and the highinvestment regime. The key observation is that an SPE willnot include any investment decisions (Bi;Bj) in themedium investment regime, as it will not lead to a pricingequilibrium in Stage II. Moreover, any investment deci-sions in the high investment regime lead to zero operatorrevenues and are strictly dominated by any decisions inlow investment regime. After the above analysis, theoperators only need to consider possible equlibria in thelow investment regime in Stage I.

Game 2 (Investment Game). The competition between the two

operators in Stage I can be modeled as the following game:

. Players. Two operators i and j.

. Strategy space. The operators will choose ðBi;BjÞfrom the set B ¼ fðBi; BjÞ : Bi þBj � G expð�2Þg.Notice that the strategy space is coupled across theoperators, but the operators do not cooperate witheach other.

. Payoff function. The operators want to maximize theirprofits in (8) and (9), respectively.

At an equilibrium of the investment game, ðB�i ; B�j Þ, eachoperator has maximized its payoff assuming that the otheroperator chooses the equilibrium investment, i.e.,

B�i ¼ arg max0�Bi� G

expð2Þ�B�j

�II;iðBi;B�j Þ; i ¼ 1; 2; i 6¼ j:

To calculate the investment equilibria of Game 2, we canfirst calculate operator i’s best response given operator j’s(not necessarily equilibrium) investment decision, i.e.,

B�i ðBjÞ ¼ arg max0�Bi� G

expð2Þ�Bj

�II;iðBi;BjÞ; i ¼ 1; 2; i 6¼ j:

By looking at operator i’s profit in (8), we can see that alarger investment decision Bi will lead to a smaller price.The best choice of Bi will achieve the best tradeoff betweena large bandwidth and a small price.

After obtaining best investment responses of duopoly,we can then calculate the investment equilibria, givendifferent costs Ci and Cj.

Theorem 2. The duopoly investment (leasing) equilibria in

Stage I are summarized as follows:

. Low costs regime (0 < Ci þ Cj � 1, as region (L) inFig. 4). There exists infinitely many investmentequilibria characterized by

B�i ¼ �G expð�2Þ; B�j ¼ ð1� �ÞG expð�2Þ; ð10Þ

where � can be any value that satisfies

Cj � � � 1� Ci: ð11Þ

The operators’ profits are

�LI;i ¼ B�i ð1� CiÞ;

�LI;j ¼ B�j ð1� CjÞ;

where “L” denotes the low costs regime.. High comparable costs regime (Ci þ Cj > 1 andjCj � Cij � 1, as region (HC) in Fig. 4). There existsa unique investment equilibrium

B�i ¼ð1þ Cj � CiÞG

2exp �Ci þ Cj þ 3

2

� �; ð12Þ

B�j ¼ð1þ Ci � CjÞG

2exp �Ci þ Cj þ 3

2

� �: ð13Þ

The operators’ profits are

�HCI;i ¼1þ Cj � Ci

2

� �2

G exp � Ci þ Cj þ 3

2

� �� ;

�HCI;j ¼1þ Ci � Cj

2

� �2

G exp � Ci þ Cj þ 3

2

� �� ;

where “HC” denotes the high comparable costs regime.. High incomparable costs regime (Cj > 1þ Ci or

Ci > 1þ Cj, as regions (HI) and (HI 0) in Fig. 4).For the case of Cj > 1þ Ci, there exists a uniqueinvestment equilibrium with

B�i ¼ G expð�ð2þ CiÞÞ; B�j ¼ 0;

i.e., operator i acts as the monopolist and operator j isout of the market. The operators’ profits are

�HII;i ¼ G expð�ð2þ CiÞÞ; �HII;j ¼ 0;

where “HI” denotes the high incomparable costs. The

case of Ci > 1þ Cj can be analyzed similarly.

The proof of Theorem 2 is given in Appendix B, which isavailable in the online supplemental material. Let us furtherdiscuss the properties of the investment equilibrium inthree different costs regimes.

3.3.1 Low Costs Regime (0 < Ci þ Cj � 1)

In this case, both the operators have very low costs. It is thebest response for each operator to lease as much as possible.However, since the strategy set in the Investment Game is


Fig. 4. Leasing equilibrium types in different (Ci; Cj).

coupled across the operators (i.e., B ¼ fðBi;BjÞ : Bi þBj � G expð�2Þg), there exist infinitely many ways for the

operators to achieve the maximum total leasing amount

G expð�2Þ. We can further identify the focal point, i.e., the

equilibrium that the operators will agree on without prior

communications [30]. The details can be found in our online

technical report [35].

3.3.2 High Comparable Costs Regime (Ci þ Cj > 1 and

jCj � Cij � 1)

First, the high costs discourage the operators from leasing

aggressively; thus, the total investment is less thanG expð�2Þ.Second, the operators’ costs are comparable, and thus the

operator with the slightly lower cost does not have sufficient

power to drive the other operator out of the market.

3.3.3 High Incomparable Costs Regime (Cj > 1þ Ci or

Ci > 1þ Cj)First, the costs are high and thus the total investment of two

operators is less than G expð�2Þ. Second, the costs of the

two operators are so different that the operator with the

much higher cost is driven out of the market. As a result,

the remaining operator thus acts as a monopolist.

4 EQUILIBRIUM SUMMARY

Based on the discussions in Section 3, we summarize the

equilibria of the three-stage game in Table 2, which includes

the operators’ investment decisions, pricing decisions, and

the resource allocation to the users. Without loss of

generality, we assume Ci � Cj in Table 2. The equilibrium

for Ci > Cj can be described similarly.Several interesting observations are as follows:

Observation 1. The operators’ equilibrium investment

decisions B�i and B�j are linear in the users’ aggregate

wireless characteristics

G ¼Xk2K

gk ¼Xk2K

Pmaxk hk

�n0

!:

This shows that the operators’ total investment increaseswith the user population, users’ channel gains, and users’transmission powers.

Observation 2. The symmetric equilibrium price p�i ¼ p�jdoes not depend on users’ wireless characteristics.

Observations 1 and 2 are closely related. Since the totalinvestment is linearly proportional to the users’ aggregatecharacteristics, the “average” equilibrium resource alloca-tion per user is “constant” and does not depend on the userpopulation. Since resource allocation is determined by theprice, this means that the price is also independent of theuser population and wireless characteristics.

Observation 3. The operators can determine differentequilibrium leasing and pricing decisions by observingsome linear thresholds in Figs. 3 and 4.

For equilibrium investment decisions in Stage I, thefeasible set of investment costs can be divided into threeregions by simple linear thresholds as in Fig. 4. As leasingcosts increase, operators invest less aggressively; as theleasing cost difference increases, the operator with a lowercost gradually dominates the spectrum market. For theequilibrium pricing decisions, the feasible set of leasingbandwidths is also divided into three regions by simplelinear thresholds as well. A meaningful pricing equilibriumexists only when the total available bandwidth from the twooperators is no larger than a threshold (see Fig. 3).

Observation 4. Each user k’s equilibrium demand ispositive, linear in its wireless characteristic gk, anddecreasing in the price. Each user k achieves the sameSNR independent of gk, and obtains a payoff linear in gk.

Observation 4 shows that the users receive fair resourceallocation and service quality. Such allocation does notdepend on the wireless characteristics of the other users.

Observation 5. In the High Incomparable Costs Regime,users’ equilibrium SNR increases with the costs Ci andCj, and the equilibrium payoffs decrease with the costs.


TABLE 2Operators’ and Users’ Behaviors at the Equilibria (Assuming Ci � Cj)

As the costs Ci and Cj increase, the pricing equilibrium(p�i ¼ p�j ) increases to compensate the loss of the operators’profits due to increased costs. As a result, each user willpurchase less bandwidth from the operators. Since a userspreads its total power across the entire allocated band-width, a smaller bandwidth means a higher SNR but asmaller payoff.

Finally, we observe that the users achieve a high SNR atthe equilibrium. The minimum equilibrium SNR that usersachieve among the three costs regime is expð2Þ. In this case,the ratio between the high SNR approximation andShannon capacity, lnðSNRÞ= lnð1þ SNRÞ, is larger than 94 per-cent. This validates our assumption on the high SNRregime. The next section, on the other hand, shows thatmost of the insights remain valid in the general SNR regime.

4.1 How Network Dynamics Affect EquilibriumDecisions

Our analysis so far has not considered network dynamics,as we have focused on a single time slot where an operatorknows users’ channel conditions through proper feedbackmechanisms. In this section, we will look at how theequilibrium results in Table 2 change over multiple timeslots with the network dynamics. Note that operators stillhave the complete network information in each time slot.Users are myopic in the sense that they do not take intoaccount the effect of time-varying network parameters onfuture prices when they determine bandwidth demand inthe current time slot.

First, we consider the case where the spectrumavailable for leasing changes over time. Intuitively, whena spectrum owner faces a strong demand from its ownprimary users, it will have less spectrum resource for thevirtual operator and will set a higher leasing cost. Here,we look at the case where operators’ leasing costs Ci andCj change over time according to some Markov decisionprocesses. We write two costs as CiðtÞ and CjðtÞ toemphasize their dependancies in time. For the illustrationpurpose, we consider three possible values for both CiðtÞand CjðtÞ: 0.4, 0.8, and 2, and the transition probabilities(same for two operators) are shown in Fig. 5.

Fig. 6 shows how costs CiðtÞ and CjðtÞ, equilibriumleasing decisions B�i and B�j , and pricing decisions p�i and p�jchange over time. Here, we represent a price N=A inTable 2 as a zero price. This means that whenever we see azero price in the figure, the corresponding operator doesnot participate in the game and the other operator becomesthe monopoly in the market. We observe that as anoperator’s leasing cost increases, its leasing amount

decreases. The operator with a lower cost will lease moreand will become the monopolist if its cost is much lowerthan its competitor (i.e., with jCjðtÞ � CiðtÞj > 1 in the highincomparable costs regime). In this case, its competitordecides not to lease. As costs increase, operators’ symmetricprices tend to increase to compensate the costs. When twocosts are low (with CiðtÞ þ CjðtÞ � 1), both operatorsannounce the same high price.

Second, we can consider the dynamics of users’ channelgains and their population over time. Users’ channel gainsmay follow, for example, different Rayleigh distributions.Also, there can be users departing or entering the networkover different time slots. As a result, users’ aggregatewireless characteristics G will change over time. Table 2 hasclearly shown that operators’ leasing amounts and profitswill change proportionally to G. But the equilibrium priceswill not be affected, since operators will balance theirleasing amounts with users’ demands. For the sake of space,we will not show additional plots for this case.

5 EQUILIBRIUM ANALYSIS UNDER THE GENERAL

SNR REGIME

In Sections 3 and 4, we computed the equilibria of thethree-stage game based on the high SNR assumption in(2), and obtained five important observations (Observa-tions 1-5). The high SNR assumption enables us to obtainclosed-form solutions of the equilibria analysis and clearengineering insights.

In this section, we further consider the more generalSNR regime where a user’s rate is computed according to(1) instead of (2). We will follow a similar backwardinduction analysis, and extend Observations 1, 2, 4, 5, andpricing threshold structure of Observation 3 to the generalSNR regime.

We first examine the pricing equilibrium in Stage II.

Theorem 3. Define Bth :¼ 0:462G. The pricing equilibria in the

general SNR regime are as follows:

. Low investment regime (Bi þBj � Bth as in region(L) of Fig. 7). There exists a unique pricingequilibrium


Fig. 5. Transition matrix of CiðtÞ and CjðtÞ over time slots.

Fig. 6. Costs, equilibrium bandwidth and pricing decisions as functionsof time slots. Here, we fix � to be 0.5.

p�i ðBi;BjÞ ¼ p�j ðBi;BjÞ

¼ ln 1þ G

Bi þBj

� �� G

Bi þBj þG:

The operators’ profits in Stage II are

�iðBi;BjÞ ¼

Bi ln 1þ G

Bi þBj

� �� G

Bi þBj þG� Ci

� �;ð15Þ

�jðBi;BjÞ ¼

Bj ln 1þ G

Bi þBj

� �� G

Bi þBj þG� Cj

� �:ð16Þ

. High investment regime (Bi þBj > Bth as in region(H) of Fig. 7). There is no pricing equilibrium.

Proof of Theorem 3 is given in Appendix C, available inthe online supplemental material. This result is similar toTheorem 1 in the high SNR regime, and shows that thepricing equilibrium in the general SNR regime still has athreshold structure in Observation 3.

Unlike Theorem 1, here we only have two investmentregimes. The “high investment regime” in Theorem 1 isgone, and the “medium investment regime” in Theorem 1corresponds to the high investment regime here. Intuitively,the high SNR assumption in Section 3 requires each user todemand relatively small amount of bandwidth to spread itstransmission power efficiently; thus, the users’ totaldemand is not elastic to prices and is always upperbounded by G expð�1Þ in Fig. 3. But in the general SNRcase, users’ demands are elastic to prices and is no longerupper bounded. Hence, we only have two regimes here. Formore details, please refer to Appendix C, which is availablein the online supplemental material.

Based on Theorem 3, we are ready to prove Observa-tions 1, 2, 4, and 5 in the general SNR regime.

Theorem 4. Observations 1, 2, 4, and 5 in Section 4 still hold forthe general SNR regime.

Proof of Theorem 4 is given in Appendix D, available inthe online supplemental material.

6 IMPACT OF OPERATOR COMPETITION

We are interested in understanding the impact of operatorcompetition on the operators’ profits and the users’payoffs. As a benchmark, we will consider the coordinatedcase where both operators jointly make the investment and

pricing decisions to maximize their total profit. In this case,there does not exist competition between the two operators.However, it is still a Stackelberg game between a singledecision maker (representing both operators) and the users.Then, we will compare the equilibrium of this Stackelberggame with that of the duopoly game as in Section 4.

6.1 Maximum Profit in the Coordinated Case

We analyze the coordinated case following a three-stagemodel as shown in Fig. 8. Compared with Fig. 2, the keydifference here is that a single decision maker makes thedecisions for two operators in both Stages I and II. In otherwords, the two operators coordinate with each other.

Again, we use backward induction to analyze the three-stage game. The analysis of Stage III in terms of thespectrum allocation among the users is the same as inSection 3.1 (still assuming the high SNR regime), and wefocus on Stages II and I. Without loss of generality, weassume that Ci � Cj.

In Stage II, the decision maker maximizes the followingtotal profit T� by determining prices pi and pj:

T�ðBi;Bj; pi; pjÞ ¼ �iðBi;Bj; pi; pjÞ þ �jðBi;Bj; pi; pjÞ;

where �iðBi;Bj; pi; pjÞ is given in (4) and �jðBi;Bj; pi; pjÞ canbe obtained similarly.

Theorem 5. In Stage II, the optimal pricing decisions for thecoordinated operators are as follows:

. If Bi > 0 and Bj ¼ 0, then operator i is the monopolistand announces a price

pcoi ðBi; 0Þ ¼ lnG

Bi

� �� 1: ð17Þ

Similar result can be obtained if Bi ¼ 0 and Bj > 0.. If minðBi;BjÞ > 0, then both operators i and j

announce the same price

pcoi ðBi;BjÞ ¼ pcoj ðBi;BjÞ ¼ lnG

Bi þBj

� �� 1:

Proof of Theorem 5 can be found in our online technicalreport [35]. Theorem 5 shows that both operators will acttogether as a monopolist in the pricing stage.

Now, let us consider Stage I, where the decision makerdetermines the leasing amounts Bi and Bj to maximize thetotal profit

maxBi;Bj�0

T�ðBi;BjÞ ¼

maxBi;Bj�0

Biðpcoi ðBi;BjÞ � CiÞ þBjðpcoj ðBi;BjÞ � CjÞ;


Fig. 8. The three-stage Stackelberg game for the coordinated operators.Fig. 7. Pricing equilibrium types in different (Bi;Bj) regions for generalSNR regime.

where pcoi ðBi;BjÞ and pcoj ðBi;BjÞ are given in Theorem 5. Inthis case, operator j will not lease (i.e., Bco

j ¼ 0) as operator ican lease with a lower cost. Thus, the optimization problemin (18) degenerates to

maxBi�0

T�ðBiÞ ¼ maxBi�0

Biðpcoi ðBi; 0Þ � CiÞ:

This leads to the following result:

Theorem 6. In Stage I, the optimal investment decisions for thecoordinated operators are

Bcoi ðCi; CjÞ ¼ G expð�ð2þ CiÞÞ; Bco

j ðCi; CjÞ ¼ 0; ð19Þ

and the total profit is

Tco� ðCi; CjÞ ¼ G expð�ð2þ CiÞÞ:

6.2 Impact of Competition on the Operators’ Profits

Let us compare the total profit obtained in the competitiveduopoly case (Theorem 2) and the coordinated case(Theorem 6).

6.2.1 Low Costs Regime (0 < Ci þ Cj � 1)

First, the total equilibrium leasing amount in the duopolycase is B�i þB�j ¼ G expð�2Þ, which is larger than the totalleasing amount G expð�ð2þ CiÞÞ in the coordinated case. Inother words, operator competition leads to a moreaggressive overall investment. Second, the total profit atthe duopoly equilibria is

TL� ðCi; Cj; �Þ ¼ ½�ð1� CiÞ þ ð1� �Þð1� CjÞ�G expð�2Þ;

where � can be any real value in the set of ½Cj; 1� Ci�. Eachchoice of � corresponds to an investment equilibrium andthere are infinitely many equilibria in this case as shown inTheorem 2. The minimum profit ratio between the duopolycase and the coordinated case optimized over � is

T�RLðCi; CjÞ ¼4 min

�2½Cj;1�Ci�

TL� ðCi; Cj; �ÞTco� ðCi; CjÞ

:

Since TL� ðCi; Cj; �Þ is increasing in �, the minimum profitratio is achieved at

�� ¼ Cj:

This means

T�RLðCi; CjÞ ¼ ½Cjð1� CiÞ þ ð1� CjÞ2� expðCiÞ: ð20Þ

Although (20) is a nonconvex function of Ci and Cj, we cannumerically compute the minimum value over all possiblevalues of costs in this regime

minðCi;CjÞ:0<CiþCj�1

T�RLðCi; CjÞ ¼ lim

�!0T�R

Lð�; 0:5þ �Þ ¼ 0:75:

This means that the total profit achieved at the duopolyequilibrium is at least 75 percent of the total profit achievedin the coordinated case under any choice of cost parametersin the Low Costs Regime.

6.2.2 High Comparable Costs Regime (Ci þ Cj > 1 and

Cj � Ci � 1)

First, the total duopoly equilibrium leasing amount is

B�i þB�j ¼ G exp � Ci þ Cj þ 3

2

� ��

which is greater than G expð�ð2þ CiÞÞ of the coordinatedcase. Again, competition leads to a more aggressive overallinvestment. Second, the total profit of duopoly is

THC� ðCi; CjÞ ¼1þ ðCj � CiÞ2

2G exp �Ci þ Cj þ 3

2

� �:

And the profit ratio is

T�RHCðCi; CjÞ ¼4

THC� ðCi; CjÞTco� ðCi; CjÞ

¼ 1þ ðCj � CiÞ2

2exp

1� ðCj � CiÞ2

� �;

which is a function of the cost difference Cj � Ci. Let uswrite it as T�R

HCðCj � CiÞ. We can show that it is a convexfunction and achieves its minimum at

minðCi;CjÞ:CiþCj>1;0�Cj�Ci�1

T�RHCðCj � CiÞ

¼ T�RHCð2�

ffiffiffi3pÞ ¼ 0:773:

6.2.3 High Incomparable Costs Regime (Cj � Ci > 1)

In this case, only one operator leases a positive amount atthe duopoly equilibrium and achieves the same profit as inthe coordinated case. The profit ratio is 1.

We summarize the above results as follows:

Theorem 7 (Operators’ Profit Loss). Comparing with thecoordinated case, the operator competition leads to a maximumtotal profit loss of 25 percent in the low costs regime.

Since low leasing costs lead to aggressive leasingdecisions and thus intensive competitions among operators,it is not surprising to see that the maximum profit losshappens in the low cost regime. For detailed discussions on

the relationship between the profit loss and the costs, seeour online technical report [35].

6.3 Impact of Competition on the Users’ Payoffs

Theorem 8. Comparing with the coordinated case, users obtainthe same or higher payoffs under the operators’ competition.

By substituting (19) into (17), we obtain the optimal pricein the coordinated case as 1þ Ci. This means that user k’spayoff equals to gk expð�ð2þ CiÞÞ in all three costs regimes.According to Table 2, users in the duopoly competition casehave the same payoffs as in coordinated case in the highincomparable costs regime. The payoffs are larger in theother two costs regimes with the competitor competition.

The intuition is that operator competition in those tworegimes leads to aggressive investments, which results inlower prices and higher user payoffs.

7 CONCLUSION AND FUTURE WORK

Dynamic spectrum leasing enables the secondary networkoperators to quickly obtain the unused resources from the


primary spectrum owner and provide flexible services tosecondary end-users. This paper studies the competitionbetween two secondary operators and examines theoperators’ equilibrium investment and pricing decisionsas well as the users’ corresponding achieved service qualityand payoffs.

We model the economic interactions between theoperators and the users as a three-stage dynamic game.Our appropriate OFDM-based spectrum sharing modelcaptures the wireless heterogeneity of the users in terms ofmaximum transmission power levels and channel gains.The two operators engage in investment and pricingcompetitions with asymmetric costs. We have discoveredseveral interesting features of the game’s equilibria. Forexample, the operators can determine different equilibriumleasing and pricing decisions by observing some linearthresholds. We also study the impact of operator competi-tion on operators’ total profit loss and the users’ payoffincreases. Compared with the coordinated case where thetwo operators cooperate to maximize their total profit, weshow that at the maximum profit loss due to competition isno larger than 25 percent. We also show that the usersalways benefit from competition by achieving the same orbetter payoffs. Although we have focused on the high SNRregime when obtaining closed-form solutions, we show thatmost engineering insights summarized in Section 4 stillhold in the general SNR regime. Due to the page limit, moredetailed discussions and all proofs can be found in theonline technical report [35].

There are several ways to extend the results here

. First, we can consider the case where the operatorscan also obtain resource through spectrum sensing[21], [38]. Compared with leasing, sensing is cheaperbut the amount of useful spectrum is less predictabledue to the primary users’ stochastic traffic. With thepossibility of sensing, we need to consider a four-stage dynamic game model.

. Second, we can consider the case where usersmight experience different channel conditions whenthey choose different providers, e.g., when theyneed to communicate with the base stations of theoperators. Competition under such channel hetero-geneity has been partially considered by Gajic et al.[39] without considering the cost of spectrumacquisition.

. Third, we can consider a more general and realisticmodel for user’s transmission. For example, a user’spayoff and demand are affected by the receivedsignal strength indicator (RSSI) to its receiver andwhether the transmitter is able to transmit the datato the receiver with the allocated bandwidth andtime given. Also, we can consider detailed signalfading and its effect on users’ changing demand, anddifferent allocated frequencies may have differentpower requirements.

. Fourth, we can study the multiple-operator (oligo-poly) case, where the analysis becomes much morecomplicated without closed-form solutions. Forexample, when we do backward induction analysisat Stage II, all possible combinations of multiple

operators’ leasing decisions in Stage I need to beconsidered. Nevertheless, we can still infer someintuitions about the oligopoly case based on ourduopoly analysis. For example, operators’ competi-tion will be more severe and their equilibriumsymmetric prices will be closer to leasing costs witholigopoly. Also, operators will be more conservativein leasing decisions in Stage I, since each operator isexpected to serve fewer users in Stage III. Finally,each operator’s profit will decrease in the number ofoperators. It will be useful to verify these intuitionsand discover additional new insights in the oligo-poly case.

. Finally, we can consider the case where the operatorimproves its profit by price differentiation,i.e., charging different users different prices basedon their channel conditions and transmission power.The key issue is to achieve the best tradeoff betweenpricing complexity and profit improvement. Asimilar tradeoff has been studied for a monopolynetwork service provider in Li et al. [40].

8 NOTICE FOR APPENDICES OF PROOFS

In accordance with TMC’s publication guidelines, theappendices for this paper are provided in a separatesupplemental material file that can be found on the ComputerSociety Digital Library at http://doi.ieeecomputersociety.org/10.1109/TMC.2011.213. Interested readers can also findthe proofs in our online technical report [35].

ACKNOWLEDGMENTS

This work was supported by the General Research Funds(Project Numbers CUHK 412710, CUHK 412511, and CityU144209) established under the University Grant Committeeof the Hong Kong Special Administrative Region, China.This work was also partially supported by grants from theCity University of Hong Kong (Project Numbers 7002517and 7008116). Part of the results appeared in IEEE DySPAN,Singapore, April 2010 [1]. Jianwei Huang was the corre-sponding author for this paper. Lingjie Duan was pre-viously with the Network Communications and EconomicsLaboratory, Department of Information Engineering, TheChinese University of Hong Kong.

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Lingjie Duan received the BE degree inelectrical engineering from the Harbin Instituteof Technology, China, in 2008, and the PhDdegree in information engineering from TheChinese University of Hong Kong in 2012. Heis an assistant professor in the EngineeringSystems and Design Pillar at Singapore Uni-versity of Technology and Design. During 2011,he was a visiting scholar in the Department ofElectrical Engineering and Computer Sciences

at the University of California at Berkeley. His research interests are inthe areas of resource allocation and game theoretical analysis ofcommunication networks, with current emphasis on cognitive radionetworks and small cell networks. He has served on the technicalprogram committees (TPC) for multiple top conferences (e.g., IEEEVTC, IEEE PIMRC, IEEE WCNC, and ACM MobiArch). He is a memberof the IEEE.


Jianwei Huang received the BS degree inelectrical engineering from the Southeast Uni-versity, Nanjing, Jiangsu, China, in 2000, andthe MS and PhD degrees in electrical andcomputer engineering from the NorthwesternUniversity, Evanston, Illinois, in 2003 and 2005,respectively. He is an assistant professor in theDepartment of Information Engineering at theChinese University of Hong Kong. He worked asa postdoctoral research associate in the Depart-

ment of Electrical Engineering at Princeton University from 2005-2007and as a summer intern at Motorola, Arlington Heights, Illinois, in 2004and 2005. He currently leads the Network Communications andEconomics Laboratory (ncel.ie.cuhk.edu.hk), with a main research focuson nonlinear optimization and game theoretical analysis of communica-tion networks, especially on network economics, cognitive radionetworks, and smart grid. He is the recipient of the IEEE Marconi PrizePaper Award in Wireless Communications in 2011, the IEEE GlobeComBest Paper Award in 2010, the IEEE ComSoc Asia-Pacific OutstandingYoung Researcher Award in 2009, the Asia-Pacific Conference onCommunications Best Paper Award in 2009, and the Walter P. MurphyFellowship at Northwestern University in 2001. He has served as aneditor of the IEEE Transactions on Wireless Communications, the IEEEJournal on Selected Areas in Communication - Cognitive Radio Series, aguest editor of the IEEE Journal on Selected Areas in Communicationsspecial issue on economics of networks and wireless systems, the leadguest editor of the IEEE Journal of Selected Areas in Communicationsspecial issue on game theory in communication systems, and guesteditor of several other journals including Wireless Communications andMobile Computing (Wiley), the Journal of Advances in Multimedia, andthe Journal of Communications. Dr. Huang served as the chair of theIEEE ComSoc Multimedia Communications Technical Committee(MMTC) from 2012-2014, steering committee member of the IEEETransactions on Multimedia and IEEE ICME from 2012-2014, vice-chairof the IEEE MMTC from 2010-2012, director of the IEEE MMTC e-letterin 2010, and chair of meetings and conference committees of the IEEEComSoc Asia Pacific Board from 2012-2013. He also served as a TPCcochair of the IEEE GlobeCom Selected Areas of CommunicationsSymposium (Game Theory for Communications Track) 2013, a publicitycochair of the IEEE Communications Theory Workshop 2012, a TPCcochair of the International Symposium on Modeling and Optimization inMobile, Ad Hoc, and Wireless Networks (IEEE WiOpt) 2012, a TPCcochair of the IEEE ICCC Communication Theory and SecuritySymposium 2012, a student activities cochair of IEEE WiOpt 2011, aTPC cochair of the IEEE GlobeCom Wireless CommunicationsSymposium 2010, a TPC cochair of the International WirelessCommunications and Mobile Computing (IWCMC) Mobile ComputingSymposium 2010, and a TPC cochair of the International Conference onGame Theory for Networks (GameNets) 2009. He is a TPC member ofleading conferences such as INFOCOM (2009-2013), MobiHoc (2009,2012), ICC, GlobeCom, DySPAN, WiOpt, NetEcon, and WCNC. He is asenior member of the IEEE.

Biying Shou received the BE degree fromTsinghua University and the MS and PhDdegrees from the Northwestern University. Sheis an assistant professor of managementsciences at the City University of Hong Kong.Her main research interests include operationsand supply chain management, game theory,and economics of wireless networks. She haspublished in leading journals including Opera-tions Research, Production and Operations

Management, Naval Research Logistics, and the IEEE Transactionson Mobile Computing.

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Duopoly Competition in Dynamic Spectrum Leasing and Pricing

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