1

Price and Inventory Competition in Oligopoly TVWhite Space Markets

Yuan Luo, Lin Gao, and Jianwei Huang

Abstract—In this paper, we investigate an oligopoly competitiveTV white space market, where multiple secondary networkoperators compete to serve a common pool of secondary end-users by using TV white space purchased from a white spacedatabase. We first study the competitive interactions amongsecondary operators. Specifically, we formulate the interactionsas a non-cooperative Price-Inventory competition game, whereoperators determine the spectrum inventory (purchased fromthe database) and the service price (charged to end-users)simultaneously. We prove the existence and uniqueness of theNash equilibrium (NE) using the super-modular game theory.

Then we study the impact of the database manager’s wholesalepricing strategy on the market equilibrium. Specifically, weanalytically show how the wholesale prices affect the operators’equilibrium inventory and pricing decisions. Based on this analy-sis, we further propose two different spectrum wholesale pricingstrategies that maximize the database manager’s profit and thetotal network profit, respectively. Our simulations evaluate theperformance difference between these two wholesale pricingstrategies.

Index Terms—TV White Space, Price and Inventory, OligopolyCompetition, Game Theory

I. INTRODUCTION

A. Background and Motivation

Wireless spectrum is becoming more congested and scarcewith the explosive development of wireless services andnetworks. Dynamic spectrum sharing can effectively improvespectrum efficiency and alleviate spectrum scarcity, by al-lowing unlicensed devices to access the licensed spectrumopportunistically. TV white space network is one of the mostpromising commercial realizations of dynamic spectrum shar-ing [1]–[6],1 where unlicensed devices (called white space de-vices, WSDs) explore and exploit the under-utilized broadcasttelevision spectrum (called TV white spaces, TVWS2) via awhite space database (called geo-location database [7]). In aTV white space network, a WSD is usually an infrastructure-based device (e.g., a base station) owned by a secondarynetwork operator, who serves a set of secondary end-users.A geo-location database is usually operated by a third-partydatabase manager, who is authorized (by spectrum regulatorsuch as FCC and Ofcom) to get TVWS spectrum information

This work is supported by the General Research Funds (Project NumberCUHK 412713 and CUHK 412511) established under the University GrantCommittee of the Hong Kong Special Administrative Region, China.

Yuan Luo, Lin Gao, and Jianwei Huang (corresponding author) are withNetwork Communications and Economics Lab (NCEL), Department of In-formation Engineering, The Chinese University of Hong Kong, HK, E-mail:{ly011, lgao, jwhuang}@ie.cuhk.edu.hk.

1Supporters of TV white space network include regulators (e.g., FCC inthe USA [1] and Ofcom in the UK [2]), standards bodies (e.g., IEEE 802.22Group [3]), and industrial organizations (e.g., Whitespace Alliance [4]).

2For convenience, we will call TVWS as spectrum in this paper.

Database(Google, Microsoft, etc)

Licensees(TV, PMSE, etc)

Step 0Step 3

Step 1

Step 2 End-users WSD(Secondary Operator)

Fig. 1. Spectrum Access in Database-Assistant TV White Space Network.In Step 0, the white space database updates the TV licensee informationperiodically. In Step 1, the WSD reports its location as well as spectrumdemand information to the database (via Internet). In Step 2, the databasecomputes and allocates the available TVWS to the WSD. In Step 3, the WSDserves end-users using the allocated TVWS.

at various times and locations, and to allocate spectrum tounlicensed WSDs. Figure 1 illustrates the detailed spectrumaccess process in such a database-assistant TV white spacenetwork. Obviously, the successful operation of such a networkrequires the coordination of all involved network entities(e.g., TV spectrum licensees, white space database managers,secondary operators, and end-users), which form the so-calledWhite Space EcoSystem (WSES) [8].

While most of the existing studies focused on the technicalissues in TV white space networks such as the networkdeployment and optimization [9]–[13], in this work we focuson the economic issues in TV white space networks from theperspective of the whole white space ecosystem. In particular,we consider such a competitive network scenario, wheremultiple secondary operators compete to serve a pool of end-users by using TV white spaces requested from a third-partywhite space database. We will study the economic interactionsamong all involved network entities.

Some recent works also studied the economic issues (e.g.,spectrum reservation [14] and pricing [15]) in TV white spacenetworks, but did not consider the competition among differentsecondary operators. Such competition is important in thewhite space ecosystem, as secondary operators not only needto compete for the subscriptions of secondary end-users, butalso need to share TV white space spectrum resources ina public manner. This is quite different from the currentcellular networks, where different cellular network operatorsown exclusive spectrum licences and do not use overlappingspectrum bands. Because of this, when a commercial entity(such as an existing cellular network operator) wants to enterthe TV white space network and serve as a secondary networkoperator, it needs to understand the complicated interactionsin this scenario and evaluate the profitability of providingservices in such a new type of network. Without a goodunderstanding of these issues, it is difficult to envision strongcommercializations of the new technology.

B. Contributions

In this paper, we consider an oligopoly competitive TVwhite space network, where multiple secondary operators

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The Database Manager: Determining the spectrum

wholesale price

Secondary Operators: Determining the initial inventory

and the service price

Stage I Stage IIStage III

End-users: Choosing an operaor, and demanding service from that operator

Secondary Operators: Replenishing inventory by the shared spectrum (if

needed), and serving end-users

Fig. 2. Three-Stage Hierarchical Model

(each operating one or multiple WSDs) compete for a commonpool of secondary end-users. Each operator can serve end-users by using two types of spectrum resources: the dedicatedspectrum and the shared spectrum [1], [2] (both requestedfrom a geo-location database). The dedicated spectrum arethose licensed to specific TV spectrum licensees. A dedicatedspectrum can be exclusively used by a secondary operator fora particular time period with the agreement of the spectrumlicensee (and thus has no co-channel interference from otheroperators). The shared spectrum are those unlicensed to anyTV spectrum licensee. A shared spectrum can be used by mul-tiple secondary operators concurrently (and thus co-channelinterference exists). Moreover, a dedicated spectrum must beordered well ahead of time, while a shared spectrum can berequested in real-time.

Specifically, we propose and analyze such a business model:secondary operators purchase spectrum (both dedicated andshared) from a white space database to serve their subscribedend-users. In this case, the database manager acts as a spec-trum broker, purchasing the dedicated spectrum from TVlicensees or requesting the shared spectrum from regulators,and then selling spectrum to secondary operators.3 This leadsto a three-stage hierarchical model illustrated in Figure 2. InStage I, the database manager determines the wholesale pricesof dedicated spectrum and shared spectrum to each secondaryoperator.4 In Stage II, each secondary operator determines theorder quantity of dedicated spectrum (called initial inventory,or inventory for short), and the service price to end-users. InStage III, each end-user demands service from one of thesesecondary operators based on his quality-of-service (QoS)in different secondary operators’ networks and the serviceprices of secondary operators; as a result, each secondaryoperator purchases the shared spectrum (called replenishedinventory) to replenish the excess demand if the total end-userdemand exceeds his initial inventory (of dedicated spectrum).Through such a three-stage hierarchical model, we will studythe following problems systematically:• How should the database manager determine the whole-

sale prices (of dedicated and shared spectrum) for eachsecondary operator (in Stage I) for different purposes(e.g., individual or social optimization)?

• How should each secondary operator determine the ini-tial inventory and service price (in Stage II) to maximizehis profit, considering the competition of other operators?

More specifically, we will consider two types of differentdatabases in this work: social-planning database and profit-

3Such a business model is supported by the widely adopted assumption thatin a TV white space network, WSDs can only interact with an authorized geo-location database for secondary spectrum access, and cannot interact directlywith TV licensees [14]–[16].

4Note that we allow the database manager to charge different secondaryoperators different wholesale prices.

seeking database. For a social-planning database (e.g., thosemanaged by non-profit organizations such as governmentdepartments), the database manager’s objective is to maximizethe total network profit, i.e., the aggregate profit of secondaryoperators and the database manager. For a profit-seekingdatabase (e.g., those managed by third-party businesses suchas Google [17] and SpectrumBridge [18]), the database man-ager’s objective is to maximize his profit.

As far as we know, this is the first work that systematicallystudies the competition among secondary operators and theimpact of the database manager’s wholesale pricing decisionon such a competition. In summary, the key contributions ofthis paper are summarized as follows.• Practical meaning and significance: We propose a three-

stage hierarchical business model for competitive TVwhite space network, which takes the heterogeneity ofspectrum and the uncertainty of end-user demand intoconsideration. Such a holistic study that involves theinteractions of the database manager, secondary operatorsand end-users has not been considered before.

• Competition among secondary operators: We formulatethe competition among secondary operators as a non-cooperative game (called the Price-Inventory competitiongame), and study the existence and uniqueness of theNash equilibrium systematically.

• Impact of wholesale price: We study the impact of thedatabase manager’s wholesale pricing decision on theoperators’ equilibrium inventory and pricing decisions.Based on this, we propose two different wholesale pricingstrategies that maximize the database manager’s profit(for profit-seeking database) and the total network profit(for social-planning database), respectively.

• Performance analysis: We evaluate the system perfor-mance (e.g., the database manager’s profit, the opera-tors’ profit, and the total network profit) under differentwholesale pricing schemes via extensive simulations. Ourresults show that (i) a profit-seeking database managerwill charge larger wholesale prices to the higher QoSsecondary operators (to draw more profit from thoseoperators), and accordingly the higher QoS operators willchoose higher equilibrium service prices; and (ii) a social-planning database manager will charge smaller wholesaleprices to the higher QoS secondary operators (to drivemore end-users to those operators), and all operators willchoose the same equilibrium service prices.

The rest of this paper is organized as follows. In SectionII, we review the related literature. In Section III, we providethe system model. In Section IV, we formulate and analyzethe operators’ competition game. In Section V, we study thedatabase manager’s best wholesale pricing strategies underdifferent objectives. We provide numerical results in SectionVI, and finally conclude in Section VII.

II. RELATED WORK

Most of the existing studies in TV white space networkfocused on the technical issue such as the white spaceexploration, database deployments and network optimization

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[9]–[13]. In [9], Gurney et al. discussed the computation ofthe protection areas for TV stations. In [10], Murty et al.presented and evaluated a database driven white space networkusing a more accurate propagation model with terrain data.In [11], Goncalves et al. compared the geo-location databaseapproach with sensing approach in terms of the technical andbusiness values. In [12], Feng et al. presented the design andimplementation of a multi-cell infrastructure-based TV whitespace network. In [13], Chen et al. considered the joint channelselection and access point association problem. However, weconsider in this work the economic issues (such as pricing andcompetition) in TV white space networks.

Only a few of recent works in TV white space networkconsidered the economic issues. For example, in [14], Luo etal. studied the (dedicated) TVWS reservation problem for asingle secondary operator. In [15], Feng et al. studied the hy-brid pricing scheme for the database manager. However, theseworks did not consider the competition between operators. Ourpreliminary work [16] studied the competition of secondaryoperators, but did not consider the economic interaction ofthe database manager.

Our proposed price and inventory competition game isrelated to the newsvendor model widely studied in operationsmanagement and marketing science literature. In [20], Porteuset al. gave an excellent literature review on the newsvendormodel where the price is exogenously given. In [21]–[23], theauthors studied a more general newsvendor model where theprice is endogenous. These papers, however, considered themonopoly scenario with a single retailer (which correspondsto the secondary operator in our model). Bernstein et al. in[24] (and many references cited therein) studied a competitiveoligopoly newsvendor model, considering stochastic demandsand endogenously determined prices, which are similar toour model. The key differences between our work and [24]are as follows. First, the underlying models in [24] and inour work are different. Specifically, [24] considered a modelwith buy-back policy (under overstock) and without inventoryreplenishment (under understock), while our paper considersa model with inventory replenishment (under understock) andwithout buy-back policy (under overstock). Second, the re-search focuses in [24] and our paper are different. Specifically,[24] focused only on the social-planning supplier, whoseobjective is to maximize the aggregate profit of the supplierand the competitive retailers. In our paper, we focuses onboth the social-planning supplier (database), whose objectiveis to maximize the aggregate profit of the database and thesecondary operators, and the profit-seeking supplier (database),whose objective is to maximize its own profit. Moreover,we systematically compare the wholesale pricing decisions ofsocial-planning and profit-seeking databases, and the equilibriaof the operators’ competition game under different wholesalepricing strategies.

III. SYSTEM MODEL

A. System Overview

We consider a competitive TV white space network, wherea set M = {1, 2, ...,M} of secondary network operators

Database

WSD 3(Secondary Operator 3)

WSD 2(Secondary Operator 2)

WSD 1(Secondary Operator 1)

Internet

End-usersEnd-users

Fig. 3. Illustration of a TV white space network with 3 competitive operatorsand 9 end-users.

compete to serve the same pool of end-users. Each operatorowns one or multiple infrastructure-based WSDs (e.g., basestations), which transmit over the TV white spaces (alsocalled spectrum, for convenience) allocated by a geo-locationdatabase.5 Figure 3 illustrates such a white space networkwith 3 competitive operators (each operating one WSD) and9 end-users. Each operator can serve some end-users by usingeither the dedicated spectrum (which is exclusively allocatedto one operator for a particular time period) or the sharedspectrum (which can be allocated to and used by multipleoperators simultaneously using CDMA or CSMA techniques).Notice that a dedicated spectrum must be ordered in advance(i.e., before the end-user demand is realized), while a sharedspectrum can be requested in real-time (i.e., after the end-userdemand is realized) [1], [2].

We consider a broker-based business model for the abovedatabase-assistant TV white space network, where the geo-location database manager acts as a spectrum broker, purchas-ing the dedicated spectrum from TV licensees or requestingthe shared spectrum from regulators, and then selling spectrumto operators. Moreover, in this work, we consider a time periodof T time slots. In Stage I, the database manager determinesthe wholesale prices, aiming at maximizing the expected profitor the expected network profit in the whole time period. InStage II, the operators determine the initial inventories andservice prices concurrently for the entire time period, aimingat maximizing their expected profits in the whole time period.In each time slot of Stage III, end-users reveal their demandsin that time slot, and operators purchase the shared spectrumfor that time slot if needed. We illustrate the detailed flowchartof the above three-stage model in Algorithm 1.

B. Database Manager ModelingAs mentioned previously, the database manager purchases

the dedicated spectrum from TV licensees or requests theshared spectrum from regulators, and then sells the spectrumto secondary operators. Let c and cs denote the prices ofdedicated and shared spectrum when the database managerpurchases spectrum from spectrum licensees or requests spec-trum from regulators. We will also refer c and cs as the costsof dedicated and shared spectrum for the database manager.As we do not consider the strategic interactions of spectrumlicensees or regulators, we assume that c and cs are pre-definedsystem parameters, and fixed in the entire time period. Letwm and wsm denote the wholesale prices of dedicated andshared spectrum when the database manager sells spectrum tooperator m. Note that wm and wsm are decision variables of the

5For convenience, in the rest of this paper we will use “WSD”, “secondarynetwork operator”, and “operator” interchangeably.

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Algorithm 1: Algorithmic statement for the three-stagehierarchical model.

Initialization PhaseStep 0: The database manager announces the wholesale price

wm and wsm for each operator m ∈M;

Step 1: Each operator m ∈M determines the order bm ofdedicated spectrum (initial inventory);

Step 2: Each operator m ∈M announces the service pricepm to end-users;for each time slot t = 1, ..., T do

Step 3: Each end-user chooses the best operator. Thus, thedemand dm to each operator m ∈M is realized;for each operator m = 1, ...,M do

if dm > bm thenStep 4: operator m requests (dm − bm)+ units ofshared spectrum;

endStep 5: operator m serves end-users using dedicatedspectrum min(dm, bm) and shared spectrum(dm − bm)+;

endend

database manager, and determined by the database manager inStage I for the entire time period. Given c, cs, wm and wsm,the database manager’s expected profit collected from operatorm is

Vm = (wm − c) · bm + (wsm − cs) · Edm [dm − bm]+, (1)

where bm is the operator m’s order of dedicated spectrum(initial inventory) for the entire time period, dm is the totalend-user demand to operator m, and [dm − bm]

+ is the amountof shared spectrum (replenished inventory) that operator mrequires. Note that dm is a random variable, which variesrandomly in different time slots (see Section III-D for de-tails). Edm [·] denotes the expectation with respect to dm,and thus Edm [dm − bm]

+ is the expected amount of sharedspectrum (replenished inventory) that operator m requires inthe whole time period. Therefore, the first term of (1) denotesthe database manager’s profit achieved from the dedicatedspectrum, and the second term of (1) denotes the databasemanager’s expected profit achieved from the shared spectrum.Moreover, the total profit of the database manager is just thesummation of the profits collected from all operators, denotedby V =

∑m∈M Vm.

The database manager determines the wholesale prices fordifferent objectives. In this work, we consider two types oftypical databases: the profit-seeking database and the social-planning database. For the profit-seeking database manager,he will choose wholesale prices to maximize his own profit.For the social-planning database manager, he will choosewholesale prices to maximize the total network profit (whichwill be defined in the next subsection).

C. Secondary operator Modeling

Each operator m decides the initial inventory bm (of ded-icated spectrum) and the service price pm in Stage II forthe entire time period. Notice that in each time slot of thefollowing Stage III, if the realized end-user demand dm islarger than the initial inventory bm, operator m will further

purchase the shared spectrum (replenished inventory) to meetthe excess demand. As mentioned before, the QoS of end-usersmay degrade on the shared spectrum due to the co-channelinterferences. Let δm denote the average payoff loss of anend-user (due to the QoS degradation) on the shared spectrumoffered by operator m.6 To compensate such a loss, we assumethat operator m will offer a price discount δm to end-users whouse the shared spectrum. Hence, the service price for end-userswho use the shared spectrum is psm = pm−δm. By doing this,the dedicated spectrum and shared spectrum are indifferent toend-users.7

Given the initial inventory bm and service price pm, opera-tor m’s expected profit is

Um = pm · Edm [min {dm, bm}]− wm · bm+ (pm − δm − wsm) · Edm [dm − bm]

+,

(2)

where [x]+ = 0 if x < 0, and [x]+ = x if x ≥ 0. The firstterm in (2) is the operator’s total revenue from the dedicatedspectrum; the second term is the operator’s total paymentfor the dedicated spectrum; and the last term is the profitfrom the shared spectrum. Later we will show that the totalend-user demand dm to operator m is a random variable,depending not only on operator m’s service price pm, but alsoon other operators’ service prices. Thus, operators’ decisionsare coupled with each other.

As we focus on the interactions of the database managerand secondary operators, the network profit can be defined asthe aggregate profit of the database manager and all secondaryoperators, denoted by

W =

M∑m=1

(Um + Vm)

=

M∑m=1

(pm · Edm [min {dm, bm}]− c · bm

+ (pm − δm − cs) · Edm [dm − bm]+).

(3)

It is easy to see that the network welfare depends on the serviceprice pm and the initial inventory bm, ∀m = 1, ...,M , whilenot on the wholesale prices wm and wsm, ∀m = 1, ...,M .

D. End-user Modeling

We assume that all secondary operators’ coverage areasare completely overlapping with each other, and end-usersmove randomly in this area. Each end-user becomes activerandomly, and requests service from a secondary operator’snetwork. We assume that each active end-user always requiresone unit of spectrum (e.g., one TV channel) for his service.Let d denote the total spectrum demand (i.e., the total numberof active end-users) in a particular time slot. Due to theuncertainty of end-user activity, d is a random variable, and

6Here we assume that all end-users suffer the same average payoff loss onany shared spectrum of operator m, and thus δm is fixed in the entire period.This is reasonable when the total number of devices is large in the network.The more general case will be left for our future work.

7This is possible given the assumption that the quality discount of theshared spectrum is fixed. With the fixed quality discount, a secondary operatoris able to perfectly compensate the average performance loss for end-userson the shared spectrum with a fixed price discount. We will consider thisgeneralization in our future work.

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changes randomly in different time slots (but keeps unchangedwithin a particular time slot). Let G(d) denote the cumulativedistribution function (CDF) of d.

Since there are multiple secondary operators (each associ-ated with a secondary network), an end-user may have multiplenetwork choices. We assume that each end-user can onlyconnect to one network at a particular time. We further assumethat each end-user is rational and will always choose the bestnetwork that maximizes his payoff. The payoff of an end-user is defined as the difference between the benefit (achievedfrom data transmission via a network) and the cost (i.e., thepayment to the operator) [14], [15]. Formally, when choosingthe network of operator m, the payoff of an end-user is definedas

U EUm = Rm + εm − pm, (4)

where Rm denotes the average benefit that an end-user canachieve from operator m, and εm ∈ [εm, εm] denotes the ran-dom fluctuation of the realized benefit to the mean value Rm.8

Intuitively, we can also view Rm as the expected Quality-of-Service (QoS) of an end-user on the operator m’s network,and εm as the random fluctuation of the realized QoS to themean value. It is notable that Rm is a constant, and fixed in thewhole time period, while εm is a random variable, changingrandomly in different time slots (but keeping unchanged withinevery time slot). Moreover, we assume that εm, m = 1, ...,M,are independent and identically distributed.

Since each end-user will always choose the best networkthat maximizes his payoff, the average probability of an end-user choosing an operator m, by (4), is9

θm = PR{U EUm ≥ 0 and U EU

m ≥ maxi∈M

U EUi

}. (5)

We denote θm,m ∈ M as the end-user’s network selectionprobabilities. We further denote θ0 = 1 −

∑m∈M θm as the

end-user’s dropping probability (i.e., not choosing any opera-tor). Obviously, θm depends not only on operator m’s serviceprice pm, but also on other operators’ prices pj ,∀j 6= m.Thus, we will also write θm as θm(p1, ..., pM ). Moreover, θmdepends on the distributions of εm,m ∈M.

Given the total end-user demand d and the end-user’snetwork selection probability θm, the total end-user demandto operator m, denoted by dm, can be calculated by

dm(p1, ..., pM ) = d · θm(p1, ..., pM ), (6)

which is a random variable related to all operators’ prices.For the analytic convenience, we introduce the following

assumptions for the above end-user demand.

Assumption 1. The demand dm to each operator m isdifferentiable, and

∂dm∂pm≤ 0, ∂dm

∂pj≥ 0, ∀j 6= m ∈M. (7)

8In our model, different end-users’ benefits of choosing operator m arestatistically the same, but the realized values may be different.

9It is important to note that such a probabilistic selection does not meanthat an end-user will frequently change its choice. In fact, in a particular timeslot, an end-user will select one operator and stick to that operator based onhis location, service type, and service price. The value θm is mainly used tocharacterize the behavior of end-users from the system perspective.

Assumption 2. The log-demand log(dm) has increasing dif-ferences in (pm, pj), ∀j 6= m ∈M.10

The first assumption implies that the demand to operator mdecreases with its own service price, while increases with itscompetitor’s price. The second assumption implies that whenoperator m increases his service price pm, it becomes moreprofitable for an other operator j to increase his price pj aswell. Note that many commonly-used distribution functionsfor εm (such as uniform distribution, exponential distribution,and normal distribution) satisfy the above assumptions.

To better illustrate our idea, in this work we assume a morespecific distribution form for the above end-user demand εm:the Gumbel distribution (also known as the double exponentialdistribution [26]), which is widely used to model the distri-bution of “the maximum of several samples of a distribution”(e.g., the maximum level of a river in a particular year).11

Thus, it is reasonable to assume the Gumbel distribution inthe following scenario: each operator will always assign thebest spectrum (among all spectrums he owns) to end-users.With this distribution, by [26] we can formally write the end-user’s network selection probability as follows:

θm(p1, ..., pM ) = eRm−pm

1+∑

i∈M eRi−pi, ∀m ∈M. (8)

Obviously, θm decreases with the operator m’s own price pm,but increases with other operators’ prices pi, i 6 m.

IV. SECONDARY OPERATORS’ PRICE AND QUANTITYCOMPETITION

In this section, we study the interactions among M sec-ondary operators in Stage II, given the wholesale prices speci-fied by the database manager in Stage I. In the next section, wewill study the database manager’s optimal wholesale pricingdecision in Stage I. Specifically, in this section we will for-mulate the interactions among operators as a non-cooperativegame, called the Price-Inventory competition game, and studythe Nash equilibrium systematically.

A. Price-Inventory Competition Game

Each operator’s strategy is to determine the order quantity ofdedicated spectrum (initial inventory), and the service price toend-users, considering the potential competition of other oper-ators. Thus, we formulate the interaction among operators as anon-cooperative Price-Inventory competition game (PI-game),denoted by Γ = (M, {(bm, pm)}m∈M, {Um}m∈M), where• M is the set of game players (operators);• (bm, pm) is the strategy of operator m, where bm ≥ 0

and pm ≥ 0;• Um is the payoff of operator m defined in (2).

For notational convenience, we denote p = (p1, ..., pM )as the vector of all operators’ service prices, and b =(b1, ..., bM ) as the vector of all operators’ inventories. Besides,

10A function f(x1, x2) has increasing differences in (x1, x2) if forall x1 ≥ x′1, the difference f(x1, x2) − f(x′1, x2) is nondecreasing inx2. If the function f is twice differentiable, the property is equivalent to∂2f/∂x1∂x2 ≥ 0.

11We use the Gumbel distribution to obtain the analytical form of ourresults. Note that our solutions also hold for other distributions such as thetruncated Normal distribution (see the simulations in Section VI).

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we denote p−m = (p1, ...pm−1, pm+1, ...pM ) and b−m =(b1, ...bm−1, bm+1, ...bM ) as the price and inventory vectorsof all operators except m.

In the competitive scenario, an operator m’s received de-mand dm depends not only on its own pm, but also on otheroperators’ prices p−m. Based on Section III-D, we have

dm(pm,p−m) = d · θm(pm,p−m) = d · eRm−pm

1+∑

i∈M eRi−pi, (9)

where d is the total demand (i.e., the number of active end-users) with the CDF G(d). Then, under any price vector p,the CDF of dm can be written as

Hm(dm|p) = G(

dmθm(pm,p−m)

). (10)

We assume that G(·) is a strictly increasing function (e.g.,the CDF of uniform distribution, exponential distribution, andnormal distribution). Then, Hm(·) is a strictly increasingfunction, which implies that the inverse function of Hm(·),denoted by H−1m (·), is also a strictly increasing function.

Substituting (9) and (10) into (2), we can easily find thatevery operator m’s profit Um depends not only on its ownstrategy, but also on the strategies of other operators. Thus, wecan write Um as Um(b,p) or Um(bm, pm, b−m,p−m). Givenall other operators’ strategies p−m and b−m, the best choiceof operator m (i.e., the best response correspondence) isBm(b−m,p−m) = arg max

{bm≥0,pm≥0}Um(bm, pm, b−m,p−m).

(11)The Nash equilibrium (NE) of the proposed PI-game is

defined as follows.

Definition 1 (Nash Equilibrium - NE). A Nash equilibriumof the PI-game is a strategy profile {b∗,p∗} that satisfies thefollowing conditions:

(b∗m, p∗m) ∈ Bm(p∗−m, b

∗−m), ∀m ∈M.

That is, every operator’s strategy is in its best responsecorrespondence to others.

Note that it is hard to jointly solve the operator’s best inven-tory and price in (11). However, we notice that operator m’sexpected profit does not depend on its competitors’ inventorydecision b−m. This observation can help us to decouple thepricing decision and inventory decision of an operator. Inwhat follows, we will first solve the best inventory for everyoperator, given the prices of all operators. Then, we will solvethe equilibrium prices of all operator.

B. Optimal Initial Inventory

We first consider every operator’s optimal inventory (ofdedicated spectrum), given the service price vector p of alloperators. Notice that Um is strictly concave in bm. Then,given any price p, the optimal inventory b∗m can be derivedby the first-order condition. Formally,

Proposition 1 (Optimal initial inventory). Given prices p, theoptimal inventory of operator m is

b∗m(p) = H−1m(1− wm

δm+wsm|p), if δm +wsm > wm, (12)

and b∗m(p) = 0 if δm + wsm ≤ wm.

Intuitively, we can view δm+wsm as the total cost of one unitof shared spectrum for operator m, including the discount δm

to end-users and the payment wsm to the database manager.Obviously, if δm + wsm ≤ wm, there is no incentive foroperators to order any dedicated spectrum, since it can alwaysget a higher profit by using the shared spectrum than using thededicated spectrum. In the rest of the paper, we will skip thistrivial case, and focus on the case of δm +wsm > wm, that is,for each operator m, the unit cost of dedicated spectrum isalways no larger than the unit cost of shared spectrum. In thisnon-trivial case, the operator needs to consider the tradeoffbetween using dedicated and shared spectrum.

By Proposition 1, the operator m’s optimal inventory b∗mis a function of p. Substituting (12) into (2), we can rewriteoperator m’s profit as a function of p, denoted by

Um(p) = pm · µm − wm ·H−1m (τm|p)

− (δm + wsm) · Edm[dm −H−1m (τm|p)

]+,

(13)

where τm = 1 − wm

δm+wsm

, and µm = Edm [dm] = θm(pm) ·Ed[d] = θm(pm) · µ is the expected demand to operator m.Here µ = Ed[d] is the expected total demand. By (10), we haveH−1m (x|p) = θm(p) · G−1(x). Thus, we can further rewrite(13) as

Um(p) = (pm · µ− wm) · θm(p), (14)

where wm can be viewed as a virtual wholesale price ofspectrum to operator m, and defined by

wm = wm ·G−1(τm) + (δm + wsm) · Ed[d−G−1(τm)

]+.

(15)Obviously, wm is independent of the service prices p, and isconcavely increasing with the spectrum wholesale prices wmand wsm, and the service price discount δm.

Hence, we can transform the original Price-Inventory com-petition game into a Reduced Price competition game, whereoperators decide prices p to maximize their respective profitsdefined in (14). Formally, we denote this reduced price compe-tition game as Γ = (M, {pm}m∈M, {Um}m∈M), where Umis the operator m’s profit and is defined in (14).

The following proposition shows that once we find the NEof the reduced game Γ, then the NE of the original game Γ canbe characterized immediately. This implies that the reducedgame is equivalent to the original game.

Proposition 2. If p∗ is a NE of the reduced game Γ, then{b∗(p∗),p∗} is a NE of the original game Γ, where b∗(p∗)is the optimal inventory given in Proposition 1.

C. Existence and Uniqueness of Nash Equilibrium

We study the NE by using the supermodular game theory[27]. Specifically, we transform the reduced game to a super-modular game, based on which we can prove the existenceand uniqueness of equilibria of the reduced game and originalPI-game. Notice that there are several appealing properties fora supermodular game. First, it guarantees at least one NE.Second, if the NE is unique, a simple best response basedalgorithm globally converges to the NE. Our key results aboutthe existence and uniqueness of NE is below.

Theorem 1 (Existence and Uniqueness). There exists a uniqueNE p∗ for the reduce price competition game, and thus aunique NE (b∗(p∗),p∗) for the original PI-game.

7

Proof: The proof sketch for the existence of NE is as fol-lows. First, we define the log-transformed game of the reducedgame Γ, denoted by Γ = (M, {pm}m∈M, {log Um}m∈M).We can prove that under Assumptions 1 and 2, the log-transformed game Γ is a supermodular game, and thus hasat least one NE. Second, by the property of monotone trans-formation [28], an NE of the log-transformed game is also anNE of the original game. Hence, we can show the existence ofNE for the reduced game Γ. By Proposition 2, we can furthershow the existence of NE for the original game Γ. The proofsketch for the uniqueness of NE is as follows. We first showthat under Assumptions 1 and 2, the following conditions hold:

−∂2log Um(p)∂pm2 ≥

∑j 6=m

∂2log Um(p)∂pm∂pj

, ∀m ∈M. (16)

This implies that the log-transformed game Γ has a unique NE[29]. Then, by the property of monotone transformation [30],the reduced game Γ has a unique NE, so does the originalPI-game Γ.

We now study the impact of wholesale prices (wm, wsm) onthe equilibrium, which is important for analyzing the databasemanager’s best wholesale pricing decision in Section V.

Proposition 3 (Parameter Impact). The equilibrium pricep∗m of each operator m is non-decreasing in the spectrumwholesale prices wm and wsm, and non-decreasing in the pricediscount δm, ∀m ∈M.

This proposition states that when the wholesale price ofspectrum (dedicated or shared) charged by the database man-ager increases, operator m will increase his service price aswell in order to compensate for the increase of cost. Moreover,when providing a larger price discount to end-users, operatorm will also increase his service price due to a similar reason.Notice that increasing an operator’s service price will decreasehis attractiveness to the end-users in the end-user market, andreduce the end-user’s probability of choosing that operator.This implies that the database manager can choose differentwholesale prices to different operators, either to alleviate orto intensify the competition among operators, and thereforedrive the outcome to a desirable NE (which will be furtherdiscussed in Section V).

D. Algorithm

For a supermodular game with a unique NE, several com-monly used updating rules are guaranteed to converged to theNE [27]. In this work, we propose the following best responserule: (i) start with an arbitrary price vector p(0), where 0 isthe round index; (ii) at every round k + 1, each operator mupdates its price based on its best response to other operators’prices in the previous round k, that is,

pm(k + 1) = arg maxpm≥0 Um(pm,p−m(k)); (17)

and (iii) repeat the above procedure until the NE is reached.

Theorem 2 (Convergence). The best response update in (17)converges to the unique NE.

Due to space limit, we present the detailed algorithm andthe proof of the above theorem in [31].

V. DATABASE MANAGER’S WHOLESALE PRICINGSTRATEGY

In the previous section, we have studied the competitionamong secondary operators in Stage II. In this section, we willstudy the impact of the database manager’s wholesale pricingdecision in Stage I on such a competition. That is, how thesewholesale prices affect the operations’ equilibrium invento-ries and pricing decisions. We further analyze the databasemanager’s optimal wholesale pricing strategy that maximizesthe network profit (for the social-planning database) or thedatabase manager’s profit (for the profit-seeking database).

Due to the difficulty in characterizing the NE in Stage II,it is hard to derive the close-form of the optimal wholesalepricing strategies. Thus, we will focus on the conditions for theoptimal wholesale prices. Later we will illustrate the optimalwholesale prices by simulations in Section VI.

A. Network Profit Maximization Wholesale Pricing

We first consider a social-planning database (e.g., thosemanaged by government departments), whose objective is tomaximize the network profit, i.e., the aggregate profit of thedatabase manager and all operators. We will refer to the bestwholesale pricing strategy of a social-planning database asthe coordinated wholesale pricing. Note that this coordinatedwholesale pricing will serve as a benchmark to other whole-sale pricing strategies. It may also be actually performed bythose databases managed by non-profit organizations such asgovernment regulators.

Since the coordinated wholesale pricing aims at maximizingthe network profit, we first look at what is the maximumnetwork network in a centralized system (where the databasemanager and operators act as an integrated player). By (3),the network profit W depends only on the service prices pand inventory b, while not on the wholesale prices w and ws.Thus, the network profit maximization problem is

(p◦, b◦) = arg maxp≥0,b≥0

W (p, b). (18)

We further notice that Proposition 1 is still applicable in thiscase (by simply replacing wm and wsm with c and cs). Thatis, given prices p, the optimal inventory that maximizes thenetwork profit is

b◦m(p) = H−1m(1− cm

δm+csm|p), if δm + csm > cm, (19)

and b◦m(p) = 0 if δm + csm ≤ cm for each operator m. Then,the network welfare can be written as a function of p, and thenetwork profit maximization problem can be written as

p◦ = arg maxp≥0

W (p) =

M∑m=1

(pmµ− cm

)· θm(p), (20)

where cm is virtual cost of operator m, and defined by

cm = cm ·G−1(τm) + (δm + csm) · Ed[d−G−1(τm)

]+.

By using the similar approach in Section IV, we can easilyshow that there exists a unique p◦ that maximizes the networkprofit W (p). Moreover, the optimal inventory b◦ is given by(19).

Now we study the coordinated wholesale price that leads tothe network profit maximization. That is, we want to find thedatabase manager’s wholesale pricing strategy (w,ws) that

8

drives the unique equilibrium prices p∗ and inventories b∗ ofoperators in Stage II to the network profit-maximization pricesp◦ and inventories b◦, i.e., p∗ = p◦ and b∗ = b◦.

We first consider the wholesale prices for price coordina-tion, i.e., p∗ = p◦. Recall from Section IV that each operatorm will chooses a price p∗m that maximizes its expected profit,i.e.,

p∗m = arg maxpm≥0 Um(pm,p∗−m),

where Um(p) = (pm · µ − wm) · θm(p) is the operatorm’s expected profit under the optimal inventory defined in(14). Notice that Um is a concave function of pm, and thus∂Um(pm,p

∗−m)

∂pm|pm=p∗m

= 0. Hence, the coordinated wholesaleprices wm and wsm, if exist, must satisfy∂Um(pm,p−m|wm)

∂pm

∣∣∣∣pm=p◦m, p−m=p◦−m

= 0, ∀m = 1, . . . ,M,

(21)where wm = wm ·G−1(τm)+(δm+wsm)·Ed

[d−G−1(τm)

]+is the virtual wholesale price defined in (15). This impliesthat if we can find a set of virtual wholesale prices wm,m =1, ...,M, such that the conditions in (21) are satisfied for allm ∈ M, then wm,m = 1, ...,M, are the coordinated virtualwholesale prices. However, it is difficult to characterize theclose-form of the coordinated virtual wholesale prices due tothe difficulty in characterizing the close-form of the networkprofit-maximization service prices p◦. Therefore, in whatfollows, we will only give the conditions for the coordinatedwholesale pricing.

Lemma 1 (Service Price Coordination). The service price canbe coordinated, i.e., p∗ = p◦, if the virtual wholesale priceswm,m = 1, ...,M , satisfy the conditions in (21).

Lemma 1 characterizes the virtual wholesale priceswm,m = 1, ...,M , for service price coordination. Since wmis a function of wm and wsm as shown in (15), Lemma 1 alsocharacterizes the necessary conditions on wm and wsm for pricecoordination. That is, any feasible wholesale prices wm andwsm satisfying (21) coordinate the equilibrium prices p∗ andthe network profit-maximization prices p◦.

Next we study the wholesale prices for inventory coordi-nation, i.e., b∗ = b◦. By Proposition 1, we can see that theequilibrium inventory in the PI-game is

b∗m(p∗) = H−1m(1− wm

δm+wsm|p∗), ∀m ∈M. (22)

By (19), we can find that the network profit-maximizationinventory in the centralized system is12

b◦m(p◦) = H−1m(1− cm

δm+csm|p◦), ∀m ∈M. (23)

Suppose the price is already coordinated, i.e., p∗ = p◦.13

Then, to achieve inventory coordination (i.e., b∗m(p∗) =b◦m(p◦), ∀m ∈M), the following conditions must hold:

wm

δm+wsm

= cmδm+csm

, ∀m ∈M. (24)

Based on above, we immediately have the following inven-tory coordination conditions.

12The detailed derivation for b◦m(p◦) is similar to that for b∗m(p∗). Seeour technical report [31] for details.

13By Lemma 1, the wholesale prices wm and wsm must satisfy (21) for all

m = 1, ...,M .

Lemma 2 (Inventory Coordination). Suppose that the priceis coordinated, i.e., p∗ = p◦. Then, the inventory can becoordinated, i.e., b∗ = b◦, if the wholesale prices satisfy theconditions in (24).

Combining Lemma 1 and Lemma 2, we have the followingcoordinated wholesale pricing.

Theorem 3 (Coordinated Wholesale Pricing). If the wholesaleprices w and ws are set according to (21) and (24), thenboth price and inventory are coordinated, i.e., p∗ = p◦ andb∗ = b◦.

Now we provide some useful properties for the coordinatedwholesale prices w and ws. By (21), together with the mono-tonicity of ∂Um(p)/∂pm (with respect to wm), we can easilyfind that the coordinated virtual wholesale price wm must belarger than the virtual cost cm, which implies that at least oneof the following conditions hold: (i) the wholesale price ofdedicated spectrum is larger than the database manager’s costfor dedicated spectrum, i.e., wm > cm, and (ii) the wholesaleprice of shared spectrum is larger than the database manager’scost for shared spectrum, i.e., wsm > csm. Furthermore, by(24), we can find that wm > cm if and only if wsm > csm,and vise versa. Therefore, we can find that the coordinatedwholesale prices w and ws are both larger than the costsc and cs. Notice that this is one of the key observations inthe competitive network scenario. In the traditional monopolynetwork scenario, however, the coordinated wholesale pricesw and ws equal to the costs c and cs (see [31] for details).

B. Database Profit Maximization Wholesale Pricing

In the previous subsection, we have studied the coordinatedwholesale pricing for a social-planning database, whose objec-tive is to maximize the total network profit. In practice, how-ever, the geo-location database may be operated by third-partybusinesses such as Google and Microsoft, whose objectivesare to maximize their own profits. In this subsection, we willstudy the best wholesale pricing that maximizes the databasemanager’s own profit for such a profit-seeking database. Wewill also show the social welfare loss under this best wholesalepricing via simulations in Section VI.

For the analytic convenience, we assume that the wholesaleprice wsm for the shared spectrum is fixed at cs, ∀m =1, ...,M . This implies that the database manager can gain zeroprofit from the shared spectrum. This assumption coincideswith the opinion of most current regulators, who suggestthat the unlicensed TV spectrum band is the public resource,and cannot be traded by the database. This type of spectrumresource corresponds to the shared spectrum in our model.On the other hand, regulators also support that the databasemanager can interact with certain spectrum licensees for thelicensed TV spectrum band (as one of the current databasemanager Spectrum Bridge does [18]). This type of spectrumresource can be traded by the database for exclusive usage,and thus corresponds to the dedicated spectrum in our model.Based on this assumption, the database manager’s expectedprofit defined in (1) can be written as

V (w) =∑Mm=1 Vm =

∑Mm=1(wm − cm) · b∗m. (25)

9

4.0 4.5 5.0 5.5 6.00

1

2

3

4

5 x 104

Network

Welfare

QoS of Operator 2 – R2

Total Operator Profit (NPM)Total Operator Profit (DPM)Database Profit (NPM)Database Profit (DPM)

4.0 4.5 5.0 5.5 6.00

0.5

1

1.5

2

2.5 x 104

QoS of Operator 2 – R2

OperatorProfit

Operator 1 (NPM)Operator 2 (NPM)Operator 1 (DPM)Operator 2 (DPM)

Fig. 4. (a) Network profit and (b) Operator profit vs R2 under NPM and DPM wholesale pricing schemes. The QoS of operator 1 is fixed at R1 = 4.

Notice that b∗m is operator m’s equilibrium inventory in the PI-game, which, by Proposition 1, is a function of the operators’equilibrium prices p∗ and the database manager’s wholesaleprice wm. Moreover, each equilibrium price in p∗ is a functionof all wholesale prices w. Therefore, b∗m is a function of w.

The database manager’s decision is to find the best whole-sale prices (denoted by w†) for the dedicated spectrum tomaximize his expected profit, that is,

w† = arg maxw≥0

V (w) (26)

Note that it is difficult to characterize the close-form ofthe database manager’s profit-maximization wholesale prices,since it is difficult to characterize the close-form expression ofthe NE of the PI-game. This difficulty is also highlighted bythe non-convexity of the optimization problem (26) (see [31]for details). Therefore, we will focus on using the numericalmethod to obtain the optimal solution of (26).

We notice that the best wholesale price wm to each operatorm is bounded in [cm, Rm + εm]. On one hand, if wm < cm,the database manager will achieve a negative profit selling thededicated spectrum to operator m, and thus it has no incentiveto set a wholesale price wm less than cm. On the other hand,if wm > Rm + εm, operator m has to set a service pricepm > wm > Rm + εm, otherwise he will achieve a negativeprofit from serving end-users. However, if pm > Rm + εm,all end-users will achieve negative utilities on operator m’snetwork, and thus no end-user will choose operator m. Thisimplies that the total demand to operator m will always be 0,and thus the operator m’s equilibrium inventory will also be0. Accordingly, the database manager will always get a zeroprofit from operator m if wm > Rm + εm.

Based on above, we can transform (26) into an equivalentoptimization problem on a closed and bounded feasible set∏m∈M[cm, Rm+ εm], where

∏denote the Cartesian product

of multiple sets. Thus, we can show the existence of theoptimal solution of of (26) in the following lemma.

Lemma 3. There must exist at least one optimal solution forthe optimization problem (26). In addition, the optimal solutioncan be effectively found through numerical methods such asthe exhaustive search.

VI. SIMULATION RESULTS

In this section, we provide simulations to illustrate the NEof the operators’ Price-Inventory competition game, and toevaluate the system performance (e.g., the network profit, thedatabase manager’s profit, and the operators’ profit) achievedat the NE, under both the network profit maximization (NPM)and database profit maximization (DPM) wholesale pricingstrategies.

A. Performance Evaluation

We first show the network profit, the database manager’sprofit, and the operators’ total profit achieved at the NE. Inthis simulation, we consider a network with 2 operators. Thetotal end-user demand d is uniformly distributed in [80, 120].14

The random QoS fluctuation εm follows the truncated Normaldistribution. We fix the QoS provided by operator 1 as R1 =4, while changing the QoS R2 provided by operator 2 fromR2 = 4 to R2 = 6.

Figure 4.a shows the database manager’s profit, the op-erators’ total profit, and the network profit achieved at theNE, under both NPM and DPM wholesale pricing schemes.The first bar (blue) in each bar group denotes the networkprofit (hollow + solid bar) under the NPM scheme, where thedatabase manager’s profit is denoted by the solid bar, and theoperators’ total profit is denoted by the hollow bar. Similarly,the second bar (red) in each bar group denotes those valuesunder the DPM scheme. We can see that under both NPMand DPM schemes, the network profit achieved at the NEincreases with R2 (i.e., the QoS provided by operator 2). Thisis because a higher QoS will attract more end-users, that is,end-users will choose a higher QoS operator with a relativelyhigher probability. Moreover, the network profit under theDPM scheme is less than that under the NPM scheme. Thisimplies that maximizing the database manager’s profit maylead to certain network profit loss from the system perspective.

We can further see from Figure 4.a that under the DPMscheme, the database manager’s profit (solid red bar) increases

14We can obtain similar engineering insights when using other commonly-used distributions such as the truncated normal distribution.

10

4.0 4.5 5.0 5.5 6.00

1

2

3

4

5

6

QoS of Operator 2 – R2

WholesalePrice

Operator 1 (NPM)Operator 2 (NPM)Operator 1 (DPM)Operator 2 (DPM)

4.0 4.5 5.0 5.5 6.03.5

4

4.5

5

5.5

6

6.5

QoS of Operator 2 – R2

NE

ServicePrice

Operator 1 (NPM)Operator 2 (NPM)Operator 1 (DPM)Operator 2 (DPM)

Fig. 5. (a) Database’s wholesale prices vs R2 and (b) Operators’ NE service prices vs R2 under NPM and DPM wholesale pricing schemes. The QoS ofoperator 1 is fixed at R1 = 4.

with R2, while under the NPM scheme, the database man-ager’s profit (solid blue bar) first increases and then decreaseswith R2. This reason is as follows. With the increase of R2,operator 2 becomes more competitive in the market, and canattract more end-users and thus generate more profit. Underthe DPM scheme, the database manager can charge a higherwholesale price to operator 2, so as to draw more profit fromoperator 2. Under the NPM scheme, however, the databasemanager will charge a lower wholesale price to operator 2,so as to give operator 2 even higher competitive power. Thiscan drive more end-users to the higher QoS operator, and thusincrease the total network profit (we will further discuss thisin the next subsection). However, this may reduce the databasemanager’s own profit.

Figures 4.b further shows every operator’s profit achieved atthe NE under NPM and DPM wholesale pricing schemes. Thefirst two bars in each bar group denote the profits of operators1 and 2 under the NPM wholesale pricing scheme, and the lasttwo bars denote those values under the DPM wholesale pricingscheme. We can see that under both NPM and DPM schemes,operator 1’s profit decreases with R2, while operator 2’s profitincreases with R2. The profit gap between operators increaseswith their QoS gap (i.e., R2 − R1). Moreover, the profit gapincreases much faster under the NPM scheme. As mentionedpreviously, the reason is that under the NPM scheme, thedatabase manager will charge a lower wholesale price to thehigher QoS operator (to drive more end-users to this operatorand thus generate higher network profit), while under DPMscheme, the database manager will charge a higher wholesaleprice to the higher QoS operator (to draw more profit fromthis operator). This can be shown by Figure 5 in the nextsubsection.

B. Wholesale Price and NE Service Price

Now we show the best wholesale pricing strategy of thedatabase manager and the equilibrium service prices of opera-tors. We use the same parameter setting as that in the previoussubsection.

Figure 5 shows the wholesale pricing strategy for thedatabase manager (left figure) and the NE service prices foroperators (right figure), under both NPM and DPM wholesalepricing schemes. From Figure 5.a, we can see that under the

NPM scheme, optimal wholesale price to operator 1 increaseswith R2, while the optimal wholesale price to operator 2decreases with R2. Thus, the wholesale price to operator 1is larger than that to operator 2. The purpose is to drive moreend-users to the higher QoS operator 2, and thus to increasethe total network profit. Under the DPM scheme, however,the best wholesale price to operator 1 increases with R2

slightly, while the best wholesale price to operator 2 increaseswith R2 fastly. Thus, the wholesale price to operator 1 issmaller than that to operator 2. The purpose is to draw moreprofit from the higher QoS operator 2, who can generate ahigher profit. To summarize, under the NPM wholesale pricingscheme, the higher QoS operator becomes even stronger (dueto the lower wholesale price) in the competition; while usingthe DPM wholesale pricing scheme, the higher QoS operatorbecomes weaker (due to the higher wholesale price) in thecompetition.

From Figure 5.b, we can see that under the NPM scheme,the NE service prices of two operators are identical, andincrease with R2. This implies that end-users will choose oper-ator just based on their realized QoS (i.e., Rm+εm,m = 1, 2)on different operators (since the service prices are identical).Thus, from the system perspective, the network profit ismaximized. Under the DPM scheme, both NE service pricesincrease with R2 as well. However, the NE service priceof operator 2 increases much faster than that of operator 1does. This is not only because operator 2 offers a higher QoS(and thus he can charge a higher service price to increasehis potential profit), but also operator 2 is charged a largerwholesale price (and thus he has to charge a higher serviceprice to compensate such a cost).

VII. CONCLUSION

In this paper, we consider the competition of secondarynetwork operators in the white space ecosystem, and examinethe operators’ equilibrium inventory and pricing decisionsfrom a game-theoretic perspective. We also study the impactof the database manager’s wholesale pricing strategy on theoperators’ decisions as well as the resulted equilibrium, basedon which we further propose two wholesale pricing strategiesfor the network profit maximization and the database profitmaximization, respectively. We discover several interesting

11

features of the operators’ Price-Inventory competition game(and its equilibrium) and the database optimal manager’swholesale pricing strategy. For example, the database managerwill charge a higher wholesale price to the higher QoS operatorfor its profit maximization, while a lower price to the higherQoS operator for the network welfare maximization. Theseresults can help us to better understand the complex interac-tions among the database manager, secondary operators, andend-users in a competitive scenario, which is essential for thedevelopment and optimization of TV white space networks.

In addition to the secondary network operators, there areother important players in a TV white space network, suchas spectrum licensees (who offer spectrum resource), andtraditional network operators (who may compete with sec-ondary operators). How to properly involve these differentplayers in an unified optimization framework is importantand meaningful for a TV white space network, and deservesour future effort. For example, a spectrum licensee may beinvolved by deciding the amount of spectrum for secondarydedicated utilization and shared utilization.

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Yuan Luo is a Ph.D. student in the Department ofInformation Engineering at the Chinese Universityof Hong Kong. She received the B.S. degree fromTianjin University (China) and M.S. degree fromBeijing University of Posts and Telecommunications(China) in 2008 and 2011, respectively. Her researchinterests lie in the field of wireless communicationsand network economics, with current emphasis onTV white space networks. She is the recipient of theBest Paper Award in the IEEE International Sympo-sium on Modeling and Optimization in Mobile, Ad

Hoc and Wireless Networks (WiOpt).

Lin Gao is a Postdoctoral Researcher in the De-partment of Information Engineering at the ChineseUniversity of Hong Kong. He received the M.S.and Ph.D. degrees in Electronic Engineering fromShanghai Jiao Tong University (China) in 2006 and2010, respectively. His research interests lie in thefield of wireless communications and networkingwith emphasis on the economic incentives in var-ious communication and network scenarios, includ-ing cooperative communications, dynamic spectrumaccess, cognitive radio networks, TV white space

networks, cellular-WiFi internetworks, and user-provided networks.

Jianwei Huang (S’01-M’06-SM’11) is an AssociateProfessor in the Department of Information Engi-neering at the Chinese University of Hong Kong. Heis the recipient of 7 Best Paper Awards in leadinginternational journal and conferences, including the2011 IEEE Marconi Prize Paper Award in Wire-less Communications. He is the co-author of threerecent monographs: “Wireless Network Pricing”,“Monotonic Optimization in Communication andNetworking Systems”, and “Cognitive Mobile Vir-tual Network Operator Games”. He is the Editor of

IEEE Journal on Selected Areas in Communications–Cognitive Radio Seriesand IEEE Transactions on Wireless Communications, and Chair of IEEECommunications Society Multimedia Communications Technical Committee.