spectrum reservation contract design in tv white space...

IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 2, JUNE 2015 147

Spectrum Reservation Contract Design in TV WhiteSpace Networks

Yuan Luo, Student Member, IEEE, Lin Gao, Member, IEEE, and Jianwei Huang, Fellow, IEEE

(Invited Paper)

Abstract—In this paper, we study a broker-based TV whitespace market, where unlicensed white space devices (WSDs) pur-chase white space spectrum from TV licensees via a third-partygeo-location database (DB), which serves as a spectrum bro-ker, reserving spectrum from TV licensees and then resellingthe reserved spectrum to WSDs. We propose a contract-theoreticframework for the database’s spectrum reservation under demandstochasticity and information asymmetry. In such a framework,the database offers a set of contract items in the form of reser-vation amount and the corresponding payment, and each WSDchooses the best contract item based on its private information. Wesystematically study the optimal reservation contract design (thatmaximizes the database’s expected profit) under two differentrisk-bearing schemes: DB-bearing-risk and WSD-bearing-risk,depending on who (the database or the WSDs) will bear the riskof over reservation. Counter-intuitively, we show that the opti-mal contract under DB-bearing-risk leads to a higher profit forthe database and a higher total network profit. Our numericalresults show that the proposed optimal spectrum reservation con-tract improves the total network profit up to 5%, compared withthe scheme without information sharing.

Index Terms—TV white space networks, spectrum reservation,contract theory, game theory.

I. INTRODUCTION

A. Background and Motivations

R ADIO spectrum is becoming increasingly more con-gested and scarce with the explosive development of

wireless services and networks. Dynamic spectrum sharing caneffectively improve the spectrum efficiency and alleviate thespectrum scarcity, by allowing unlicensed secondary devicesto access the licensed spectrum in an opportunistic manner.TV white space network is one of the promising paradigms ofdynamic spectrum sharing [2], [3], where unlicensed devices(called white space devices, WSDs) exploit the un-used or

Manuscript received April 6, 2015; revised August 28, 2015; acceptedOctober 27, 2015. Date of publication November 10, 2015; date of currentversion February 25, 2016. This work was supported by the General ResearchFunds (Project Number CUHK 412713 and 14202814) established under theUniversity Grant Committee of the Hong Kong Special Administrative Region,China. Part of the results have appeared in IEEE GLOBECOM 2012 [1]. Theassociate editor coordinating the review of this paper and approving it forpublication was L. DaSilva. (Corresponding author: Jianwei Huang.)

Y. Luo and J. Huang are with the Network Communications andEconomics Lab (NCEL), Department of Information Engineering, The ChineseUniversity of Hong Kong, Hong Kong (e-mail: [email protected];[email protected]).

L. Gao is with the Shenzhen Graduate School, Harbin Institute of Technology(HIT), Shenzhen, China (e-mail: [email protected]).

Digital Object Identifier 10.1109/TCCN.2015.2499198

under-utilized broadcast television spectrum (called TV whitespaces, TVWS1) opportunistically.

In order to fully utilize TVWS while not harming licenseddevices, regulatory bodies (e.g., FCC in the US and Ofcom inthe UK) have advocated a database-assisted spectrum accesssolution, which relies on a third-party geo-location white spacedatabase [2], [3].2 In this solution, WSDs obtain the avail-able spectrum information through querying the geo-locationdatabase, instead of performing spectrum sensing. More specif-ically, WSDs periodically report their location informationand other optional information (e.g., spectrum demand) to ageo-location database, and the database returns the availablespectrum in the respective locations and time periods to WSDs.

In general, there are two types of TV white space spectrumresources. The first type of spectrum is not registered to any TVlicensee or Programme Making and Special Events (PMSE) ata particular location. This type of spectrum is usually for theopen and shared usage among unlicensed WSDs, according tothe regulators’ policies [2]. The second type of spectrum isalready registered to some TV licensees and PMSE, but notfully utilized by those licensees. Hence, the licensees can tem-porarily lease these idle spectrum to unlicensed WSDs for theirexclusive usage. In such a secondary spectrum market, the geo-location database can act as an intermediary (e.g., a broker)between the licensees (sellers) and the WSDs (buyers), due toits proximity to both sides of the market.3

In this work, we focus on the secondary sharing and trad-ing of the second type spectrum resource, i.e., those registeredbut under-utilized spectrum. Such spectrum can be exclusivelyused by a WSD (with the permission of the licensees), henceare particularly suitable for supporting applications that requirea high QoS.

B. Market Model and Problem

Specifically, we study a broker-based secondary spectrummarket, where TV licensees lease their idle spectrum to unli-censed WSDs via a spectrum broker acted by a geo-locationdatabase. As a broker, the database purchases spectrum from

1For convenience, we simply use spectrum to represent TVWS in this paper.2Based on the database-assisted solution proposed by the regulators, IEEE

802.22 (http://www.ieee802.org/22/), CEPT ECC (http://www.erodocdb.dk/default.aspx), and ETSI (http://www.etsi.org/standards) proposed correspond-ing standards for WSDs operating in a database-assisted TVWS network.

3This model is used by real-world geo-location database operators suchas SpectrumBridge (https://spectrumbridge.com/) in the US and COGEU(http://www.ict-cogeu.eu/) in Europe.

2332-7731 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

148 IEEE TRANSACTIONS ON COGNITIVE COMMUNICATIONS AND NETWORKING, VOL. 1, NO. 2, JUNE 2015

Fig. 1. Broker-based spectrum reservation market. In Step 0, the geo-locationdatabase (broker) reserves spectrum from TV licensees and PMSE for everyreservation period (e.g., one day). In Step 1, each WSD (master) reports itslocation and demand in every access period (e.g., one hour). In Step 2, thedatabase sells the corresponding spectrum to the WSD in every access period.In Step 3, the WSD serves end-users (slaves) in every access period using theobtained spectrum. Notice that Steps 1-3 will occur repeatedly within everyreservation period, as one reservation period consists of many access periods.

TV licensees in advance, and then resells the leased spectrum toWSDs. Figure 1 illustrates such a broker-based spectrum reser-vation market model. As the TV towers have fixed locationsand TV programs have well planned schedules, the reserva-tion period of TV spectrum can be relative long [5]. Thus, wemodel and analyze a spectrum reservation market, where thedatabase reserves spectrum from TV licensees in advance for arelatively large time period (e.g., more than one day), called thereservation period. Then, within each reservation period, thedatabase sells the reserved spectrum to WSDs periodically witha relatively small time period (e.g., one hour), called the accessperiod. Namely, the spectrum reservation decision is made atthe beginning of the reservation period, which consists of manyaccess periods.

In such a spectrum reservation market, the database needs toreserve spectrum in advance, without knowing the actual futuredemands from WSDs. Therefore, an important problem for thedatabase in this market is:

• How much bandwidth should the database reserve foreach WSD, aiming at maximizing the database’s profit?

The problem is challenging due to the demand stochasticity andthe information asymmetry.

(i) Demand Stochasticity. Due to the stochastic nature ofend-users’ activities and requirements, each WSD’s spec-trum demand (for serving its end-users) is a random vari-able, and cannot be precisely predicted by the WSD or thedatabase in advance. Therefore, there is inevitably a riskof reservation mismatch, e.g., spectrum over-reservationor under-reservation. Accordingly, the database’s spec-trum reservation decision depends on the risk-bearingscheme, namely, who will bear the risk of spectrum over-reservation: the database (called DB-bearing-risk) or theWSD (called WSD-bearing-risk)? In the former case, theWSD only pays for the spectrum it actually purchases inevery access period; while in the latter case, the WSD hasto pay for the reserved spectrum (even if it is more thanactually needed) in every access period.

(ii) Information Asymmetry. The above mentioned demandinformation is asymmetric between the database andWSDs. Due to the proximity to end-users, the WSD usu-ally has more information (i.e., with less uncertainty)about the spectrum demand than the database. Thisimplies that the database can potentially make a betterreservation decision, if it is able to know the WSD’sprivate information regarding the demand. However,without proper incentives, the WSD may not be willing

to share its private information with the database. As willbe shown in Section 5, the WSD may even report a falseinformation to the database intentionally, as long as sucha misreport can increase the WSD profit.

C. Results and Contributions

We propose a contract-theoretic spectrum reservation frame-work, in which the database offers a list of contract items inthe form of reservation amount and the corresponding pay-ment, and each WSD chooses the best contract item basedon its private demand information (from its served end-users).We first study the incentive compatible contract design, underwhich each WSD will disclose its private demand informationcredibly. With the incentive compatibility, we further derivethe optimal spectrum reservation contracts that maximize thedatabase expected profit under both DB-bearing-risk and WSD-bearing-risk schemes. For clarity, we summarize the key resultsregarding the optimal contract design in Table 1.

As far as we know, this is the first paper that systemati-cally studies the contract-based spectrum reservations underdifferent risk-bearing schemes for TV white space markets. Theproposed market model, together with the derived spectrumreservation solutions, can offer the proper economic incentivesfor the database operator, and support the practical and com-mercial deployment of TV white space networks. The maincontributions of this paper are summarized as follows.

• Novel modeling and solution techniques: We studya generic spectrum reservation market under demandstochasticity and information asymmetry, and propose acontract-theoretic reservation framework, which ensuresthat WSDs disclose their private information truthfully,and meanwhile maximizes the database profit.

• Optimal contract design: We analytically derive the opti-mal spectrum reservation contract design under DB-bearing-risk and WSD-bearing-risk schemes, and numer-ically compare their performances. Through these numer-ical comparisons, we characterize the impacts of risk-bearing scheme, demand stochasticity, and informationasymmetry on the spectrum reservation solutions.

• Numerical results and insights: Our numerical resultsshow that the optimal contract under the DB-bearing-risk scheme can achieve a higher database profit and ahigher total network profit, compared to the optimal con-tract under the WSD-bearing-risk scheme. The intuitionis that the WSD is more risk-averse than the database.

The rest of this paper is organized as follows. In Section II,we review the related literature. In Section III, we present thesystem model. In Sections IV, we provide the integrated opti-mal reservation solution as a benchmark. In Sections V and VI,we study the decentralized spectrum reservations without infor-mation sharing and with information sharing (via contract),respectively. We provide numerical results in Section VII, andfinally conclude in Section VIII.

II. RELATED WORK

A proper business model is very important for the commer-cialization of TV white space network. Some recent works have

LUO et al.: CONTRACT-BASED SPECTRUM RESERVATION IN COGNITIVE RADIO NETWORKS 149

TABLE IKEY RESULTS IN THIS PAPER

studied the business modeling and economic issues in TVWS[8]–[14]. In [8], Feng et al. studied the hybrid pricing schemefor the database manager. In [9], Luo et al. studied the pric-ing strategy of oligopoly competitive WSDs. In [10]–[14], Luoet al. proposed an alternative information market model forTVWS. However, none of the existing work considered thebandwidth reservation problem under demand uncertainty andinformation asymmetry.

We proposed a contract-theoretic reservation framework inthis work. The concept of contract has recently been introducedinto the spectrum trading model (e.g. [15]–[17]). In [15], Gaoet al. proposed a quality-price contract for the spectrum tradingin a monopoly spectrum market. In [16], Duan et al. proposeda contract-based cooperative spectrum sharing mechanism topromote the cooperation of a primary user and a secondaryuser. In [17], Sheng et al. proposed a contract for a primarylicense holder to sell its excess spectrum capacity to poten-tial secondary users. In this paper, we propose a contract-basedmechanism for the spectrum reservation problem. In our model,the demand of a WSD consists of two parts: one is unknown byboth the database and the WSD, and the other is only known bythe WSD (hence is the WSD’s private information). Thus, theoptimal contract design needs to consider not only the truthfulinformation disclosure of the WSD, but also the uncertainty ofdemand for both the database and the WSD. This makes ourcontract design much more challenging than existing contractdesigns.

III. SYSTEM MODEL

A. System Overview

We consider a TV white space network where unlicensedWSDs exploit the under-utilized broadcast television spec-trum (called TV white space, or spectrum for simplicity) viaa geo-location database. Each WSD is an infrastructure-baseddevice (e.g., a base station), and serves a set of unlicensedend-users/devices called “slave” devices. We assume that thenumber of unlicensed WSDs is large enough, so that the spec-trum demand of a particular WSD does not affect other WSDs’demand. This allows us to concentrate on the interactionbetween the database and each WSD.

We focus on the secondary sharing and trading of the under-utilized licensed spectrum of TV licensees. In particular, we

model a broker-based secondary spectrum market, where thegeo-location database acts as a spectrum broker, reserving spec-trum from TV licensees in advance and then reselling thereserved spectrum to unlicensed WSDs.

B. Broker-Based Spectrum Reservation Market

Now we discuss the proposed spectrum reservation marketmore detailedly. Let c denote the unit price (cost) at which thedatabase reserves spectrum from TV licensees. Let w denotethe unit price (wholesale price) at which the database sellsspectrum to the WSD. Let r and s denote the unit price(market price) at which the WSD serves the subscribed andun-subscribed end-users, respectively.4 In order to concentrateon the spectrum reservation problem, we consider a fixed spec-trum trading model, that is, the trading prices c, w, r, and sare fixed system parameters.5 This implies that our proposedspectrum reservation framework does not need to alter the spec-trum trading process, and thus is compatible with many existingspectrum market mechanism designs. Moreover, to make thetrading model meaningful, we assume that min{r, s} > w > c,i.e., both the database and the mater will benefit from thetrading process.

We illustrate the detailed spectrum reservation and trad-ing/access processes in Figure 2 and Algorithm 1. It is notablethat the spectrum reservation process (Step 0) is performed ata relatively large time period (e.g, oncen every day or everyweek), called the reservation period (denoted by T ); while thespectrum trading/access processes (Steps 1-3) are performed ata relatively small time period (e.g., once per hour), called theaccess period (denoted by t).

We focus on the following database’s spectrum reservationproblem: how to determine the proper spectrum reservationamount k to maximize the database profit? The problem ischallenging due to the demand stochasticity (see Section III-C) as well as the information asymmetry (see Section III-D).Moreover, the spectrum reservation decision also depends onthe risk-bearing scheme (see Section III-E), namely, who (i.e.,the database or the WSD) will bear the risk of spectrumover-reservation. This further complicates the problem.

4In Section III-C, We will discuss the two types of users in details.5Our model allows the possibility of changing the prices over a longer time

horizon. Detailed disucssion can be found in [20].


Fig. 2. Spectrum reservation and access processes. Step 0: the databasereserves spectrum for every reservation period T ; Step 1: the WSD reports therealized demand in every access period t ; Step 2: the database returns spectrumto the WSD in every access period t ; Step 3: the WSD serves end-users in everyaccess period t .

Algorithm 1. Algorithmic statement for the three-stage hierar-chical model

for each reservation period T = 1, 2, . . . doStep 0: The database reserves k unit of spectrum from TVlicensees at a unit price c, for each reservation period;for each access period t = 1, . . . , T do

Step 1: The WSD collects the realized end-userdemand d, and requests d units of spectrum from thedatabase in each access period;Step 2: The database sells min{k, d};Step 3: The WSD serves end-users using the receivedspectrum at a market price r or s in each access period.

endend

C. Demand Stochasticity

In each access period, a WSD n ∈ N uses the purchasedspectrum to serve its end-users. We consider two types of end-users for each WSD: registered end-users (called subscribers)and unregistered end-users (called random access users orrandom users). Let Jn and In denote the sets of WSD n’ssubscribers and random users, respectively.

Specifically, subscribers characterize the residents in theWSD’s serving area, and these users can sign a service con-tract with the WSD in advanced. Because of this, the WSD hasa good knowledge regarding the demand of these users basedon the long-term interactions. The random end-users character-ize the travelers to the WSD’s serving area, and these users donot have any prior contractual relationship with the WSD. It isdifficult for the WSD to predict the demand from these users.Naturally, we assume that subscribers have a higher priority inobtaining service than random users. That is, when the spec-trum received by the WSD (from the database) is not enoughto meet all end-users’ demand, the WSD will satisfy the sub-scribers’s demand first, and then serve the random users usingthe remaining spectrum. Recall that r and s are the unit prices(of spectrum) for serving subscribers and random users, respec-tively. Due to the high priority of subscribers, it is reasonable toassume that r > s.

Let ξn, j and εn,i denote the spectrum demands of a sub-scriber j ∈ Jn and a random user i ∈ In (to WSD n) in one

access period, respectively. We assume that (i) ξn, j keepsunchanged within each reservation period T (but may varyacross T ), which implies that each contract’s validity is largerthan one access period; and (ii) εn,i keeps unchanged withineach access period t (but may vary across t), which impliesthat each random user’s average QoS and wireless character-istic remain constant in each access period. The total demand(of all subscribers and random users) of WSD n in one accessperiod is:

dn =∑j∈Jn

ξn, j +∑i∈In

εn,i � ξn + εn, (1)

where ξn �∑

j∈Jnξn, j is total subscriber demand, and εn �∑

i∈Inεn,i � ξn is total random user demand. For convenience,

we refer to ξn as the scheduled demand of WSD n (as it isknown at the beginning of each reservation period, and keepsunchanged during the whole reservation period), and refer toεn as the bursty demand of WSD n (as it is known only atthe beginning of each access period, and changes randomly indifferent access periods).

Based on the assumptions mentioned above, the scheduleddemand ξn is a random variable changing each reservationperiod T , and the bursty demand εn is a random variable chang-ing each access period t . For simplicity, we assume that ξn

and εn are independent and identically distributed (i.i.d) in dif-ferent reservation periods and access periods, respectively. Letf (ξ) and F(ξ) denote the probability density function (pdf) andcumulative distribution function (cdf) of ξ , and g(ε) and G(ε)

denote the pdf and cdf of ε, respectively. As in many mech-anism design literature (see, e.g., [15]–[17]), we assume thatsuch distribution information are public information to boththe database and the WSD. In practice, they can be obtainedthrough machine learning in a sufficiently long time period. Asmentioned previously, the number of WSDs is large enoughso that one WSD’s strategy is independent of others. Hence,we can concentrate on the interaction between the databaseand one WSD. We provide the detailed discussion of multipleWSDs in [20].

Since the total demand d changes randomly in each accessperiod t , while the spectrum reservation is performed at thebeginning of each reservation period T , the database or theWSD faces a spectrum reservation problem under demandstochasticity. Obviously, a higher reservation can serve moredemand potentially, but may also lead to a higher risk of spec-trum over-reservation. A lower reservation, however, may leadto a higher loss due to the spectrum under-reservation.

Next we draw some useful properties of the scheduleddemand ξ and the bursty demand ε. First, we notice that therandom users’ bursty demand usually depends on the real-timemarket price s and end-users’ wireless characteristics. As anexample commonly used in the literature (e.g., [1]–[18]), a ran-dom user i’s utility πi can be defined as the difference betweenthe achiavable data rate (e.g., the Shannon capacity assuminghigh SNR [19]) and the payment, e.g.,

πi = β · εi · ln

(Pi |hi |2εi n0

)− s · εi ,


where |hi | is the channel gain, Pi is the transmission power, n0is the noise power per unit bandwidth, and β denotes the mon-etary income per unit of data rate. Based on the above utilitydefinition, the optimal bursty demand for a random user i thatmaximizes its payoff πi is

εi = Pi · e−(1+s/β) · |hi |2n0

.

Notice that the channel coefficient hi satisfies: (i) hi ∼ C(0, 1),the complex normal distribution (when the channel experiencesthe Rayleigh fading), and (ii) hi is i.i.d for different users i ∈In . Therefore, both εi and ε follow the chi-square distribution[19] (with different degrees of freedom). Note, however, thatour analysis also holds for other demand distributions such asthe normal distribution.

Second, the subscribers’ scheduled demand ξ is a long-termaverage demand (changing every reservation period), and usu-ally independent of short-term wireless characteristics. Ouranalysis holds for arbitrary ξ distribution with the increasingfailure rate (IFR), i.e., f (ξ)

1−F(ξ)is increasing in ξ .6

D. Information Asymmetry

Due to the different proximities to end-users, the databaseand the WSD usually have different knowledge about the sched-uled demand ξ and the bursty demand ε at the beginning of eachreservation period T (when making the reservation decision).Specifically,

• Bursty demand ε of random users: Notice that ε changesrandomly every access period. Thus, neither the WSD northe database knows the exact value of ε at the beginningof the reservation period. That is, both the WSD and thedatabase only know the distribution of ε.

• Scheduled demand ξ of subscribers: Notice that ξ keepsunchanged within each reservation period. Thus, theWSD is able to know the exact value of ξ (e.g., throughbilateral agreements signed with subscribers) at the begin-ning of the reservation period. The database, however,does not know the exact value of ξ unless the WSD sharessuch information. That is, the database only knows thedistribution of ξ .

We refer to the difference between the database’s knowledgeand the WSD’s knowledge regarding demand information asinformation asymmetry. The co-existence of these two typesof end-users and the information asymmetry provide incentivesfor the WSD to misreport its private information.7

E. Risk-Bearing Scheme

Due to the demand stochasticity, there is a risk of spec-trum over-reservation.8 Thus, the spectrum reservation decisiondepends greatly on the risk-bearing scheme. Namely, who will

6Many commonly used distributions, such as the uniform distribution,exponential distribution, and normal distribution, satisfy the IFR constraint.

7We will discuss it in more detailed in Section V.8Note that spectrum under-reservation will hurt the profits of both the

database and the WSD directly, and thus there is no need to discuss the risksharing under spectrum under-reservation. Under spectrum over-reservation,however, the database and the WSD must decide who will pay for theover-reserved spectrum.

bear the risk of spectrum over-reservation, i.e., the databaseor WSDs? We refer to the former scheme as DB-bearing-risk(Scheme I) and the latter scheme as WSD-bearing-risk (SchemeII). Specifically,

• DB-bearing-risk (Scheme I): In this case, the WSD onlypays for the spectrum it actually purchases in each accessperiod, and thus the database bears all the risk of spec-trum over-reservation. That is, in each access period, theWSD will only pay for min{k, d} units of spectrum that itconsumes.

• WSD-bearing-risk (Scheme II): In this case, the WSDpays for all the spectrum reserved, and thus the WSDbears all the risk of spectrum over-reservation. That is,in each access period, the WSD will pay for all k units ofreserved spectrum, even if the total demand d is smallerthan k.

In this paper, we will study the spectrum reservation problemunder both risk-bearing schemes systematically.

In the following sections, we first study the central-ized/integrated spectrum reservation solution as a (central-ized) benchmark (Section IV). Then we study the decentralizedreservation solution without information sharing as another(decentralized) benchmark (Section V), and show that it maylead to a poor performance (in terms of database profit and net-work profit) due to the asymmetry of information. To this end,we study the decentralized reservation solution with contract-based credible information sharing (Section VI). To facilitatethe understanding, we have listed the key results of this work inTable I.

IV. INTEGRATED SPECTRUM RESERVATION SOLUTION

In this section, we consider an integrated system, where thedatabase and the WSD act as an integrated decision maker tomaximize their aggregate profit (called network profit, denotedby �). We will study this integrated/centralized optimal reser-vation as the centralized benchmark.

Obviously, in this case the integrated player (database andWSD) knows the precise value of ξ and the distribution of ε.Moreover, there is no difference between the DB-bearing-riskscheme and the WSD-bearing-risk scheme. Specifically, givenany spectrum reservation k, the expected network profit is

�(k, ξ) = r · min {k, ξ} + s · Eε

[min

{ε, (k − ξ)+

}] − c · k,

(2)

where (x)+ = max{x, 0}. This formula implies that the WSDwill satisfy the subscribers’ scheduled demand first (1st term),and then satisfy the random users’ bursty demand using theremaining spectrum (2nd term).

Next we study the centralized optimal reservation k◦ thatmaximizes the network profit defined in (2). Notice that whenk ≤ ξ , we have ∂�(k,ξ)

∂k = r − c > 0, which implies that theoptimal k cannot be smaller than ξ ; when k ≥ ξ , we have

(i) ∂�(k,ξ)∂k = s [1 − G(k − ξ)] − c, and (ii) ∂2�(k,ξ)

∂k2 = −s ·g (k − ξ) ≤ 0. Thus, the centralized optimal reservation k◦is given by the first-order condition ∂�(k,ξ)

∂k = 0, and moreformally,

k◦ = ξ + G−1(

s − c

s

). (3)


Intuitively, k◦ consists of two parts: (i) the scheduled demandξ , and (ii) the best response to the bursty demand ε. Note thatthe centralized optimal reservation k◦ is a function of ξ , but nota function of ε. This is because the integrated player knows theprecise value of ξ , but not the value of ε.

V. DECENTRALIZED SPECTRUM RESERVATION—NO

INFORMATION SHARING

Now we consider a general decentralized system, where thedatabase and the WSD make decisions independently, aim-ing at maximizing their individual profits. In this section,we will study the decentralized spectrum reservation solutionunder information symmetry and under information asymmetrywithout information sharing as the decentralized benchmarks.

A. Scheme I: DB-Bearing-Risk

Under the DB-bearing-risk scheme, the WSD only pays forthe spectrum it actually uses, and thus the database bears all therisk of spectrum over-reservation. That is, in each access period,the WSD will only purchase min{k, d} units of spectrum.

1) Information Symmetry: We first study the database’soptimal spectrum reservation solution under information sym-metry, where the database is assumed to know the precise valueof ξ . Specifically, for any reservation k, the WSD’s and thedatabase’s (ex-ante) expected profits are, respectively,

πMS(k, ξ) = (r − w) · min{k, ξ}+ (s − w) · Eε

[min

{ε, (k − ξ)+

}], (4)

πDB(k, ξ) = w · Eε [min{ε + ξ, k}] − c · k. (5)

The optimal reservation for the database (i.e., that maximizesits profit defined in (5)) is

kSYM(I) = ξ + G−1

(w − c

w

). (6)

Similar to the centralized optimal reservation k◦, the abovedecentralized optimal reservation kASY

(I) (I)SYM under informa-tion symmetry is also a function of ξ .

2) Information Asymmetry: As discussed in Section III-D,the demand information is asymmetric between the databaseand the WSD. Now we study the database’s optimal spectrumreservation solution under information asymmetry, where thedatabase does not know the precise value of ξ .

We first show that the reservation solution kSYM(I) in (6) under

information symmetry may not be the database’s optimal solu-tion in this case, as it cannot ensure that the WSD shares itsprivate information ξ with the database credibly. Notice that(i) the WSD profit πMS(k, ξ) in (4) increases with the spectrumreservation k, and (ii) the database’s optimal spectrum reserva-tion kSYM

(I) in (6) is linear to ξ . This implies that the WSD hasan incentive to inflate its private information ξ . The key reasonbehind this phenomenon is that the database bears all the riskof over-reservation.

As a consequence, the database will not trust the information(i.e., the value of ξ ) informed by the WSD, and therefore will

act based on its own prior distribution information of ξ and ε.That is, it will maximize the following expected profit:

π̄DB(k) � Eξ [πDB(k, ξ)] = w · Eξ,ε [min{ε + ξ, k}] − c · k,

(7)

where the expectation Eξ,ε is taken over the distribution of ξ

and ε. The optimal reservation for the database that maximizesits expected profit defined in (7) is

kASY(I) = (F × G)−1

(w − c

w

), (8)

where F × G is the joint c.d.f. of ξ + ε.Note that kASY

(I) is not a function of ξ , which is different from(3) and (6). This implies that the database cannot adjust itsspectrum reservation decision to account for the WSD’s privateinformation. Therefore, both parties’s profits may reduce due tothe ignorance of information ξ (that WSD has) in the spectrumreservation. To solve this problem, we will propose a spectrumreservation contract to achieve the credible information sharingbetween the database and the WSD in Section VI-A.

B. Scheme II: WSD-Bearing-Risk

Under the WSD-bearing-risk scheme, the WSD pays for allthe spectrum reserved, and thus the WSD bears all the risk ofspectrum over-reservation. That is, in each access period, theWSD will pay for all k units of reserved spectrum, even if thetotal demand d is smaller than k.

1) Information Symmetry: Similarly, we first study theWSD’s optimal spectrum reservation decision under informa-tion symmetry. Specifically, for any reservation k, the WSD’sand the database’s (ex-ante) expected profits are, respectively,

πMS(k, ξ) = r · min{k, ξ} + s · Eε

[min

{ε, (k − ξ)+

}] − wk,

(9)

πDB(k, ξ) = (w − c) · k. (10)

Note that if the WSD bears the risk, then the WSD will deter-mine the spectrum reservation amount. Otherwise, the databasewill always choose a very large reservation as it does not bearthe risk of over-reservation. Accordingly, the optimal reser-vation for the WSD (i.e., that maximizes its profit defined in(9)) is

kSYM(II) = ξ + G−1

(s − w

s

), (11)

which is also a function of ξ .2) Information Asymmetry: Since the WSD itself holds the

private information under information asymmetry, the WSD’sexpected profit under information asymmetry is exactly sameas (9). Thus, the optimal reservation for the WSD underinformation asymmetry is same as that under informationsymmetry, i.e.,

kASY(II) = kSYM

(II) = ξ + G−1(

s − w

s

). (12)


Fig. 3. (a) Spectrum Reservation vs Scheduled Demand ξ , and (b) Network Profit vs Wholesale Price w. Here, σξ denotes the variance of ξ .

Notice that the database profit πDB(k, ξ) defined in (10) isincreasing in the spectrum reservation k. This implies that itis possible for the database to improve its profit by incen-tivizing the WSD to increase the spectrum reservation k. InSection VI-B, we will propose a spectrum reservation contractto maximize the database profit.

C. Comparison

Now we compare the above decentralized optimal reser-vations (without information sharing). It is easy to see thatthese decentralized solutions deviate from the integrated opti-mal solution (3), due to the “double marginalization” effectas well as the lack of information on the database side underinformation asymmetry.

1) Performance Under Information Symmetry: We firstcompare two spectrum reservation solutions under informationsymmetry, i.e., kSYM

(I) and kSYM(II) .

Lemma 1: There exists a critical wholesale price w∗ = √sc

such that1) when w < w∗, then k◦ > kSYM

(II) > kSYM(I) ;

2) when w > w∗, then k◦ > kSYM(I) > kSYM

(II) .We illustrate the spectrum reservation solutions vs scheduled

demand ξ in Figure 3.a, where s = 0.8, w = 0.5, c = 0.2, andobviously, w >

√sc = 0.4. It is easy to see that kSYM

(I) underDB-bearing-risk (the blue triangle curve) is always larger thankSYM

(II) under WSD-bearing-risk (the red square curve). This isbecause with a large wholesale price (e.g., w >

√sc), the risk

of over-reservation that the WSD bears under WSD-bearing-risk is higher than that the database bears under DB-bearing-risk, and thus the WSD will reserve less spectrum than thedatabase. We can further see that kSYM

(I) and kSYM(II) are smaller

than k◦ in the integrated system (the green circle curve). Thegap between kSYM

(I) (or kSYM(II) ) and k◦ is caused by the double

marginalization effect.Lemma 2: Under information symmetry, there exists a criti-

cal wholesale price w∗ = √sc such that

1) when w < w∗, the optimal network profit under WSD-bearing-risk (i.e., under kSYM

(II) ) is larger than that under

DB-bearing-risk (i.e., under kSYM(I) );

2) when w > w∗, the optimal network profit under WSD-bearing-risk (i.e., under kSYM

(II) ) is smaller than that under

DB-bearing-risk (i.e., under kSYM(I) )

Lemma 2 can be obtained by Lemma 1, together with thefact that the network profit increases with k when k ≤ k◦. Forclarity, we illustrate the network profit under different reser-vation solutions vs wholesale price w in Figure 3.b. We cansee that (i) the centralized optimal network profit (the greencircle curve) does not depend on the wholesale price w, and(ii) the decentralized optimal network profit under DB-bearing-risk (the blue triangle curve) increases with the wholesaleprice w, while the decentralized optimal network profit underWSD-bearing-risk (the red square curve) decreases with thewholesale price. This is because with a larger wholesale price,the database will reserve more spectrum under DB-bearing-risk (hence, the network profit increases), while the WSDwill reserve less spectrum under WSD-bearing-risk (hence, thenetwork profit decreases).

2) Performance Under Information Asymmetry: We nowcompare two spectrum reservation solutions under informationasymmetry, i.e., kASY

(I) and kASY(II) .

From Figure 3.a, we can see that kASY(I) (blue dashed curve

with mark “x”) under DB-bearing-risk is independent of ξ ,while kASY

(II) (red dashed curve with mark “+”) under WSD-

bearing-risk increases linearly with ξ . Obviously, kASY(I) > kASY

(II)

when ξ is small (e.g., ξ < 14), while kASY(I) < kASY

(II) when ξ islarge (e.g., ξ > 14). This is because the database makes thereservation decision kASY

(I) without knowing the exact value ofξ , and thus it is likely to over-reserve spectrum when ξ is small,while under-reserve spectrum when ξ is large.

Similarly, from Figure 3.b, we can see that (i) the decentral-ized optimal network profits under DB-bearing-risk (the bluedash curves with mark “x”) increases with the wholesale pricew, while the decentralized optimal network profit under WSD-bearing-risk (the red dash curve with mark “+”, overlappingwith the red square curve) decreases with w. The reason issimilar to that under information symmetry, i.e., a larger whole-sale price will increase the database’s reservation kASY

(I) under

DB-bearing-risk, but reduce the WSD’s reservation kASY(II) under

WSD-bearing-risk. Moreover, we can see that the decentralizedoptimal network profit under DB-bearing-risk (the blue dashcurves with mark “x”) decreases with the variance of scheduled


demand ξ (denoted by σξ ). This is because the database’s spec-trum reservation kASY

(I) under DB-bearing-risk does not considerthe exact value of ξ ; hence, a larger variance of ξ will lead to alarger network profit loss.

D. Observation

By the above comparison, we can see that performancesof the decentralized optimal solution under information asym-metry (i.e., kASY

(I) in (8) and kASY(II) in (12) depend on the

wholesale price w and the variance of scheduled demand ξ .Moreover, both of these solutions may lead to low profits forboth the database and the WSD (comparing with the centralizedbenchmark), due to the lack of information and/or the doublemarginalization effect.

VI. DECENTRALIZED SPECTRUM

RESERVATION—CONTRACT-THEORETIC APPROACH

In the previous section, we have shown that lacking of infor-mation and/or the double marginalization effect may result inprofit losses for both the database and the WSD. In this section,we will propose a contract-theoretic approach to achieve cred-ible information sharing and hedge double marginalization inspectrum reservation.

A. Contract Under DB-Bearing-Risk

As shown in (8), under the DB-bearing-risk scheme, theprofit loss under information asymmetry is mainly due to thelack of information ξ (when the database makes the spec-trum reservation decision). Therefore, we propose a SpectrumReservation Contract to achieve the credible information shar-ing between the database and the WSD. We derive the optimalcontract that maximizes the database profit under informationasymmetry analytically. Simulations demonstrate that with theoptimal contract, the total network profit can also be improved,comparing with that (under information asymmetry) withoutcredible information sharing.

1) Contract Design: The key idea of a spectrum reserva-tion contract is as follows. To motivate the WSD credibly revealits private information ξ , the database put an additional chargeon the WSD for spectrum reservation (on top of the wholesalecharge of w · min [k, ξ ]). This forces the WSD to share thecost of over-reservation, such that the WSD has no incentiveto inflate the value of ξ .

Based on this idea, we design the following contract: (I) �{〈k(ξ), p(ξ)〉}∀ξ , which consists of a menu of contract items,〈k(ξ), p(ξ)〉, each intending for a possible scheduled demandξ . Here, k(ξ) and p(ξ) denote the spectrum reservation and theWSD’s payment to the database, respectively, when the sched-uled demand is ξ .9 The detailed spectrum reservation processis as follows.

1) Before reserving spectrum, the database announces thecontract (I) = {〈k(ξ), p(ξ)〉}∀ξ ;

9Note that p(ξ) is the WSD’s payment for reserving spectrum via thedatabase, and is not the total cost of using spectrum.

2) The WSD selects the contract item 〈k(ξ̂ ), p(ξ̂ )〉 thatmaximizes its expected profit, based on its private infor-mation ξ ;

3) The database reserves spectrum k(ξ̂ ) for one reservationperiod, and charges the WSD a reservation fee p(ξ̂ ) (Step0 in Figure 2);

4) The database sells spectrum to the WSD in each accessperiod (Steps 1-3 in Figure 2).

When the WSD with information ξ chooses a contract item〈k(ξ̂ ), p(ξ̂ )〉 (i.e., that intended for information ξ̂ ), the WSDprofit, the database profit, and the aggregate profits (networkprofit) are, respectively,

πMS(k(ξ̂ ), p(ξ̂ ), ξ) = (r − w) · min{k(ξ̂ ), ξ}+ (s − w) · Eε

[min

{ε, (k(ξ̂ ) − ξ)+

}]− p(ξ̂ ), (13)

πDB(k(ξ̂ ), p(ξ̂ ), ξ) = w · Eε

[min{ε + ξ, k(ξ̂ )}

]

− c · k(ξ̂ ) + p(ξ̂ ), (14)

�(k(ξ̂ ), p(ξ̂ ), ξ) = r · min {k(ξ̂ ), ξ}+ s · Eε

[min

{ε, (k(ξ̂ ) − ξ)+

}]− c · k(ξ̂ ). (15)

We define a feasible contract as follows.Definition 1 (Feasible Contract): A contract is feasible, if

and only if• Incentive Compatibility (IC): The WSD with any

information ξ prefers the contract item 〈k(ξ), p(ξ)〉(that is intended for ξ ) than all other contract items〈k(ξ̂ ), p(ξ̂ )〉,∀ξ̂ �= ξ . Formally, we have

πMS(k(ξ), p(ξ), ξ) ≥ πMS(k(ξ̂ ), p(ξ̂ ), ξ), ∀ξ̂ , ξ. (16)

• Individual Rationality (IR): The WSD can achieve a mini-mum acceptance profit πmin

MS when choosing 〈k(ξ), p(ξ)〉.Formally, we have

πMS(k(ξ), p(ξ), ξ) ≥ πminMS , ∀ξ. (17)

Moreover, we define an optimal contract, denoted by ∗(I) =

{〈k∗(I)(ξ), p∗

(I)(ξ)〉}∀ξ , as follows.Definition 2 (Optimal Contract): The contract ∗

(I) ={〈k∗

(I)(ξ), p∗(I)(ξ)〉}∀ξ is optimal if this contract is feasible and

maximizes the database expected profit. Formally, the optimalcontract is given by

max〈k(ξ),p(ξ)〉,∀ξ

Eξ

[πDB(k(ξ), p(ξ), ξ)

],

subject to: IC and IR in (16) and (17). (18)

In the following, we first provide the necessary and sufficientconditions for a feasible contract. Then, we derive the opti-mal contract systematically. For clarity, we present all of thedetailed proofs in [20].

2) Feasibility: Suppose that a contract (I) ={〈k(ξ), p(ξ)〉}∀ξ is feasible. Then, the following necessaryconditions hold.


Proposition 1 (Necessary Condition I for Feasibility):

k(ξ1) > k(ξ2), if and only if p(ξ1) > p(ξ2).

Proposition 2 (Necessary Condition II for Feasibility):

k(ξ1) ≥ k(ξ2), ∀ξ1 > ξ2.

Proposition 1 implies that in a feasible contract, a larger spec-trum reservation k(·) must correspond to a larger reservationfee p(·). This is quite intuitive, as the WSD’s profit is increas-ing in k(·) but decreasing in p(·). Proposition 2 implies that thespectrum reservation k(·) increases with the value of scheduleddemand ξ .

For convenience, we denote πMS(ξ) � πMS(k(ξ), p(ξ), ξ)

as the WSD profit when choosing the contract item intendedfor its true private information ξ . Given any feasible k(ξ) (i.e.,those non-decreasing with ξ ), we have the following necessaryconditions for the feasible p(ξ), or equivalently, for the WSDprofit πMS(ξ).

Proposition 3 (Necessary Condition III for Feasibility):

πMS(ξ1) ≥ πMS(ξ2), ∀ξ1 > ξ2.

Proposition 4 (Necessary Condition IV for Feasibility):

πMS(ξ) = πMS(ξ) + (r − s) · (ξ − ξ)

+∫ ξ

ξ

(s − w) · G (k(x) − x) dx .

Proposition 3 implies that in a feasible contract, the WSDprofit increases with the value of ξ . Proposition 4 further givesthe detailed form of the WSD profit in a feasible contract, givenany feasible k(ξ). Note that the third term on the r Here, ξ is theminimum achievable value of scheduled demand ξ , i.e., g(ξ) =0 if ξ < ξ .

By Proposition 4, we can get the following feasible reserva-tion fee p(ξ) directly:

p(ξ) = − πMS(ξ) + (r − w) · min{k(ξ), ξ}+ (s − w) · Eε

[min

{ε, (k(ξ) − ξ)+

}],

(19)

where πMS(ξ) is given in Proposition 4.We have shown the necessary conditions for a feasible

contract through Propositions 1-4. Next we show that theseconditions are also sufficient for a contract to be feasible.

Proposition 5 (Sufficient Conditions for Feasibility): A con-tract (I) = {〈k(ξ), p(ξ)〉}∀ξ is feasible, if the following condi-tions hold:

• k(ξ) is non-decreasing in ξ (i.e., Necessary Condition IIin Proposition 2),

• p(ξ) is given by (19) (i.e., Necessary Condition IV inProposition 4),

• πMS(ξ) ≥ πminMS (i.e., IR Condition).

Intuitively, the first two conditions guarantee the IC conditionfor the contract, and the last condition guarantees the IR condi-tion for the contract. Therefore, the conditions in Proposition 5are sufficient.

3) Optimality: Now we study the database’s optimal con-tract characterized by (18). By (13) and (14), we notice that thetotal profit can be freely transferred between the database andthe WSD through the reservation fee p(ξ). Therefore, to maxi-mize the database profit, we need to shrink the WSD’s profitas much as possible. This leads to the following optimalitycondition immediately.

Proposition 6 (Optimality Condition I):

πMS(ξ) = πminMS .

Proposition 6 implies that in the optimal contract, thedatabase will assign the minimal acceptable profit to the WSD.Intuitively, if the WSD profit πMS(ξ) = X > πmin

MS , then thedatabase can immediately improve its profit by increasing thereservation fee p(ξ) by a constant (X − πmin

MS ) for all ξ .Denote πDB(ξ) � πDB(k(ξ), p(ξ), ξ) and �(ξ) �

�(k(ξ), p(ξ), ξ). By (13)-(15), we can write the database’sprofit as πDB(ξ) = �(ξ) − πMS(ξ). Together withProposition 4 and Proposition 6, we can rewrite the databaseprofit maximization problem (18) as follows.

maxk(ξ),∀ξ

Eξ [πDB(ξ)] �∫ ξ̄

ξ

φ(I) (k(ξ), ξ) · f (ξ)dξ − πminMS ,

subject to: k(ξ) is non-decreasing in ξ, (20)

where

φ(I) (k(ξ), ξ) � �(ξ)

− 1 − F(ξ)

f (ξ)[r − s + (s − w) · G (k(ξ) − ξ)] .

We first notice that φ(I)(k(ξ), ξ) is related to a particularξ only, and is independent of other ξ̂ �= ξ . Thus, the optimalsolution of (20) can be obtained by maximizing φ(I) (k(ξ), ξ)

for each ξ independently (as long as the non-decreasing con-dition is not violated). However, due to the non-convexity ofG(·), φ(I)(k(ξ), ξ) is non-convex in k(ξ), and thus the clas-sic Karush-Kuhn-Tucker (KKT) analysis cannot be directlyapplied here.10

Next we can show that φ(I)(k(ξ), ξ) has the nice prop-erty of piecewise convexity. Based on this, the maximizer ofφ(I)(k(ξ), ξ) is unique, and it satisfies the first-order condition:∂φ(I)(k,ξ)

∂k = 0. Formally, the optimal k(ξ),∀ξ , is given by

∂φ(I)(k, ξ)

∂k= s · [1 − G(k(ξ) − ξ)] − c

− 1 − F(ξ)

f (ξ)· (s − w) · g(k(ξ) − ξ) = 0. (21)

We can further check that optimal k(ξ) given by (21) is indeednon-decreasing in ξ , due to the IFR assumption for F(·), i.e.,1−F(ξ)

f (ξ)decreases with ξ . Therefore, we have the following

optimal contract under DB-bearing-risk.Theorem 1: Under DB-bearing-risk, the database’s optimal

contract ∗(I) = {〈k∗

(I)(ξ), p∗(I)(ξ)〉}∀ξ is given by: ∀ξ ∈ [ξ, ξ̄ ],

• k∗(I)(ξ) is given by (21), and

• p∗(I)(ξ) is given by (19) with πMS(ξ) = πmin

MS .

10As an example mentioned in Section III-C, the bursty demand ε’s distribu-tion G(·) is the chi-square distribution, which is non-convex.


Now we provide some useful properties for the optimalcontract ∗

(I) = {〈k∗(I)(ξ), p∗

(I)(ξ)〉}∀ξ . Specifically,

dp∗(I)

dk∗(I)

= dp∗(I)/dξ

dk∗(I)/dξ

= (s − w) ·[1 − G(k∗

(I) − ξ)]

≥ 0, (22)

d2 p∗(I)

d k∗(I)

2=

d

(dp∗

(I)dk∗

(I)

)/dξ

dk∗(I)/dξ

=−(s − w) · g(k∗

(I) − ξ) ·(

dk∗(I)/dξ − 1

)dk∗

(I)/dξ≤ 0. (23)

The above properties show that p∗(I) is concavely increasing

in k∗(I) (which can be seen from Figure 4.a). This implies that

the database’s reservation fee charge for each additional unitof spectrum reservation will decrease with the total amount ofspectrum reservation.

B. Contract Under WSD-Bearing-Risk

Comparing (3) and (12), we can see that under WSD-bearing-risk, the gap between the centralized optimal reserva-tion k◦ and the decentralized optimal reservation kASY

(II) (underinformation asymmetry without information sharing) is mainlydue to the double marginalization effect, which further leadsto some loss in both the database profit and the total networkprofit. The perfect coordination of the WSD’s optimal solution(12) and the centralized optimal solution (3) requires the whole-sale price to be as low as the cost (i.e., w = c). This is obviouslyundesirable for a profit-maximizing database. To this end, wepropose a Spectrum Reservation Contract to mitigate the dou-ble marginalization effect in this case. Similarly, we analyticallyderive the optimal contract that maximizes the database profitunder information asymmetry. Simulations demonstrate thatwith the optimal contract, the total network profit can also beimproved, comparing with that (under information asymmetry)without information sharing.

1) Contract Design: The detailed contract formulationunder WSD-bearing-risk is similar to that under DB-bearing-risk (in Section VI-A). Specifically, to motivate the WSD toorder spectrum according to the database’s profit-maximizingobjective, the database charges the WSD for the spectrum reser-vation (in addition of the wholesale charge of w · k).11 Thisforces the database to share the cost of over-reservation,such that the WSD operates as the database desired.

Similarly, we design the following contract: (II) �{〈k(ξ), p(ξ)〉}∀ξ , where each contract item 〈k(ξ), p(ξ)〉 spec-ifies a spectrum reservation level k(ξ) and the correspondingWSD’s payment p(ξ). The detailed spectrum reservation pro-cess is the same as that in Section VI-A. However, the defini-tions for the database’s and the WSD profits are different, dueto the different risk-bearing schemes.

Specifically, when the WSD with information ξ chooses acontract item 〈k(ξ̂ ), p(ξ̂ )〉 (i.e., that intended for ξ̂ ), the WSD’s

11Note that this wholesale charge is different from that under DB-bearing-risk. The latter is w · min [k, ξ ], as the WSD only needs to pay for the spectrumit actually purchases.

profit, the database profit, and the aggregate profits (networkprofit) are, respectively,

πMS(k(ξ̂ ), p(ξ̂ ), ξ) = r · min{k(ξ̂ ), ξ} − w · k(ξ̂ )

+ s · Eε

[min

{ε, (k(ξ̂ ) − ξ)+

}]− p(ξ̂ ), (24)

πDB(k(ξ̂ ), p(ξ̂ ), ξ) = (w − c) · k(ξ̂ ) + p(ξ̂ ), (25)

�(k(ξ̂ ), p(ξ̂ ), ξ) = r · min {k(ξ̂ ), ξ}+ s · Eε

[min

{ε, (k(ξ̂ ) − ξ)+

}]− c · k(ξ̂ ). (26)

Obviously, the aggregate profit in (26) is same as that in (15),that is, the network profit does not depend on the choice of therisk-bearing scheme.

Similar as in Definition 1 and Definition 2, we first define thecontract feasibility and optimality.

Definition 3 (Feasible Contract under WSD-risk-bearing):The contract (II) = {〈k(ξ), p(ξ)〉}∀ξ is feasible, if and only ifit satisfies the following conditions.

IC: πMS(k(ξ), p(ξ), ξ) ≥ πMS(k(ξ̂ ), p(ξ̂ ), ξ), ∀ξ̂ , ξ ; (27)

IR: πMS(k(ξ), p(ξ), ξ) ≥ πminMS , ∀ξ. (28)

We denote the optimal contract by ∗(II) =

{〈k∗(II)(ξ), p∗

(II)(ξ)〉}∀ξ , which is defined below.Definition 4 (Optimal Contract): The contract ∗

(II) ={〈k∗

(II)(ξ), p∗(II)(ξ)〉}∀ξ is optimal if this contract is feasible and

maximizes the database expected profit. Formally, the optimalcontract is given by

max〈k(ξ),p(ξ)〉,∀ξ

Eξ

[πDB(k(ξ), p(ξ), ξ)

],

subject to: IC and IR in (27) and (28). (29)

2) Feasibility: It is easy to check that the necessary condi-tions II and III in Propositions 2-3 also hold for the feasiblecontract under WSD-bearing-risk. However, the necessary con-dition IV in Proposition 4 is a bit different. Specifically,

Proposition 7 (Necessary Condition IV for Feasibilityunder WSD-bearing-risk): Given a feasible k(ξ), the WSD’sexpected profit is

πMS(ξ) = πMS(ξ) + (r − s) · (ξ − ξ) +∫ ξ

ξ

s · G(k(x) − x) dx .

Accordingly, the feasible reservation fee p(ξ) is

p(ξ) = − πMS(ξ) + r · min{k(ξ), ξ}+ s · Eε

[min

{ε, (k(ξ) − ξ)+

}] − w · k(ξ),(30)

where πMS(ξ) is given in Proposition 7.3) Optimality: Notice that the optimality condition in

Proposition 6 also holds for the WSD-bearing-risk scheme.


Fig. 4. (a) Illustration of Optimal Contracts, (b) Contract-based Spectrum Reservations vs Scheduled Demand ξ . Here, σξ denotes the variance of ξ .

Thus, we can similarly rewrite the database profit maximizationproblem (29) as

maxk(ξ),∀ξ

Eξ [πDB(ξ)] �∫ ξ̄

ξ

φ(II) (k(ξ), ξ) · f (ξ)dξ − πminMS ,

subject to: k(ξ) is non-decreasing in ξ,

(31)where

φ(II)(k(ξ), ξ) � �(ξ) − 1−F(ξ)

f (ξ)· [r − s + s · G(k(ξ) − ξ)] .

Using a similar analysis as in Section VI-A, we can showthat the optimal solution of (31) can be obtained by maximizingφ(II) (k(ξ), ξ) for each ξ independently. Moreover, the optimal

k(ξ) satisfies the first-order condition: ∂φ(II)(k,ξ)

∂k = 0. Formally,

∂φ(II)(k(ξ), ξ)

∂k= s · [1 − G(k(ξ) − ξ)] − c

− 1 − F(ξ)

f (ξ)· s · g(k(ξ) − ξ) = 0. (32)

Therefore, the optimal contract under the WSD-bearing-riskscheme is given in the following theorem.

Theorem 2: Under WSD-bearing-risk, the optimal contract∗

(II) = {〈k∗(II)(ξ), p∗

(II)(ξ)〉}∀ξ is given by: ∀ξ ∈ [ξ, ξ̄ ],• k∗

(II)(ξ) is given by (32), and

• p∗(II)(ξ) is given by (30) with πMS(ξ) = πmin

MS .We provide some useful properties for the optimal contract

∗(II) = {〈k∗

(II)(ξ), p∗(II)(ξ)〉}∀ξ . Specifically,

dp∗(II)

dk∗(II)

= dp∗(II)/dξ

dk∗(II)/dξ

= s ·[1 − G(k∗

(II) − ξ)]

− w, (33)

d2 p∗(II)

d k∗(II)

2=

d

(dp∗

(II)dk∗

(II)

)/dξ

dk∗(II)/dξ

=−s · g(k∗

(II) − ξ) ·(

dk∗(II)/dξ − 1

)dk∗

(II)/dξ≤ 0. (34)

The second property shows that p∗(II) is concave in k∗

(II), and thefirst property shows that p∗

(II) is non-monotonous in k∗(II). More

precisely, p∗(II) first increases with k∗

(II) and then decreases withk∗

(II), as illustrated in Figure 4.a.

C. Comparison

Now we compare the optimal contract ∗(I) = {〈k∗

(I)(ξ),

p∗(I)(ξ)〉}∀ξ under the DB-bearing-risk scheme (in Theorem 1)

and the optimal contract ∗(II) = {〈k∗

(II)(ξ), p∗(II)(ξ)〉}∀ξ under

the WSD-bearing-risk scheme (in Theorem 2).Figure 4.a compares the structures of both contracts, by

showing the relationships of reservation and reservation feeunder both optimal contracts.

• For the optimal contract ∗(I) under DB-bearing-risk,

we can see that the reservation fee p∗ monotonicallyincreases with the spectrum reservation k∗. This isbecause the WSD always benefits from a larger spectrumreservation level (as it does not need to bear the risk);hence, the database can charge a higher reservation feefor a higher reservation level.

• For the optimal contract ∗(II) under WSD-bearing-risk,

we can see that the reservation fee p∗ first increases andthen decreases with the spectrum reservation k∗. This isbecause the WSD’s profit first increases with the reser-vation level, and then decreases with the reservation level(due to the high risk of over-reservation); hence, the reser-vation fee first increases with the reservation level, andthen decreases with the reservation level.

We can further see that under the same reservation levelk∗, the reservation fee under DB-Bear-Risk is larger than thatunder WSD-Bear-Risk, hence charges a higher reservation feeto compensate its expected cost due to over-reservation.

Then we compare the spectrum reservations under bothcontracts. By Proposition 2, both k∗

(II)(ξ) and k∗(I)(ξ) are increas-

ing in ξ . By (21) and (32), we further have the followingobservation.

Lemma 3 (Contract-based spectrum reservation):

k∗(II)(ξ) ≤ k∗

(I)(ξ) ≤ k◦(ξ), ∀ξ ∈ [ξ, ξ̄ ],

and k∗(II)(ξ) = k∗

(I)(ξ) = k◦(ξ) only when ξ = ξ̄ .That is, only when the realized scheduled demand ξ reaches

its maximum value (i.e., ξ = ξ̄ ), the spectrum reservationsunder both optimal contracts are identical, and are equal theintegrated optimal spectrum reservation. Under other values ofξ , the spectrum reservation in the contract ∗

(II) (under WSD-bearing-risk) is smaller than that in the contract ∗

(I) (under


Fig. 5. (a) Database Profit vs Wholesale Price, and (b) Network Profit vs Wholesale Price.

DB-bearing-risk), which is further smaller than the integratedoptimal spectrum reservation.

We illustrate the result of Lemma 3 in Figure 4.b. Intuitively,When the database bears the risk, it has an incentive to chargea high reservation fee in order to force the WSD to shouldersome of the potential cost. When the WSD bears the risk, how-ever, the database has less incentive to charge a high reservationfee. Hence, for the same ξ , we find that p∗

(I)(ξ) > p∗(II)(ξ).

Combined with Proposition 1, we have k∗(II)(ξ) < k∗

(I)(ξ).

VII. NUMERICAL RESULTS

In this section, we provide numerical results to compare theperformances of the proposed contract-based spectrum reserva-tion mechanisms. Practically speaking, the database’s contractchoice depends on many factors, among which the spectrumreservation decision and the resulting (expected) profit are themost important ones. Hence, we will present the expected prof-its (of the database, WSD, and the aggregated one) under dif-ferent contracts associated with different risk-bearing schemes.Unless specified otherwise, we assume the following spectrumtrading parameters: r = 1, s = 0.8, w = 0.5, and c = 0.2. Wefurther assume that the scheduled demand ξ follows the normaldistribution, and the bursty demand ε follows the chi-squaredistribution.12

A. Profit vs Wholesale Price

Figure 5 illustrates (a) the database profit and (b) the networkprofit (aggregate profit) achieved in different spectrum reserva-tion solutions (associated with information asymmetric underdifferent wholesale prices w). In this simulation, we assumethat ξ follows the normal distribution with mean μξ = 30 andvariance σ 2

ξ = 64, and ε follows the chi-square distribution

with mean με = 30 and variance σ 2ε = 60.

From Figure 5.a, we have the following observations regard-ing the database profit.

• Under both risking bear-schemes, the contract-basedspectrum reservation leads to a much higher profit forthe database, compared to the reservation solution withoutinformation sharing.

12The parameter setting is for an illustrative purpose; similar insights can beobtained using other parameter settings.

• The database can achieve a higher profit with the optimalspectrum reservation contract under DB-bearing-risk (theblue triangle curve) than that under WSD-bearing-risk(the red square curve).

This is quite counter-intuitive. The reason is that the WSD ismore risk-averse than the database.

From Figure 5.b, we have the following observations. (i) Theoptimal network profit achieved in the centralized reserva-tion solution is independent of the wholesale price w, andserves as an upper-bound of the network profit under anyother reservation solution; (ii) Information sharing based onthe optimal spectrum reservation contract proposed in thispaper improves the total network profit up to 5%, under DB-Bear-Risk; (iii) Different from the DB-Bearing-Risk scheme,we can see that only when the wholesale price w is large(e.g., w > 0.62 in this example), the performance under theoptimal spectrum reservation contract is better than that withoutinformation sharing when WSD bears risk. This is because thepurpose of contract under the WSD-bearing-risk is to reducethe double marginalization effect. Hence, the network profitunder WSD-Bearing-Risk contract is independent of the whole-sale price. However, as the objective of contract is maximizingthe database profit, the database would charge an equiva-lent high “wholesale price” to the WSD. As shown by theFigure 5.b, such equivalent “wholesale price” lies between 0.6and 0.7. This high equivalent high wholesale price decreasesthe performance of social welfare.

Our results provide the following important insight for a gen-eral reservation problem: it is not only individually better, butalso socially better to leave the over-reservation risk to the lessrisk-averse decision maker.

B. Profit vs Scheduled Demand Variance

Figure 6 illustrates (a) the database profit and (b) the net-work profit achieved in different spectrum reservation solu-tions (associated with information asymmetry), under differentscheduled demand variance σ 2

ξ . Notice that σ 2ξ reflects the

degree of information asymmetry. That is, a higher σ 2ξ implies

a larger variance of ξ , and thus a higher uncertainty of thedatabase regarding ξ . In this simulation, we assume that ξ fol-lows the normal distribution with mean μξ = 30 (and with


Fig. 6. (a) Database Profit vs Scheduled Demand Variance, and (b) Network Profit vs Scheduled Demand Variance.

different variances), and ε follows the chi-square distributionwith mean με = 30 and variance σ 2

ε = 60.From Figure 6.a, we can further see that under both risk-

bearing schemes, the optimal contracts (∗(I) and ∗

(II)) cangreatly improve the database profit. Moreover, the database canachieve a slightly higher profit with the optimal contract ∗

(I)under DB-bearing-risk, than the optimal contract ∗

(II) underWSD-bearing-risk.

Figure 6.b leads to a similar observation as Figure 5.b.Specifically, under DB-bearing-risk, the optimal contract ∗

(I)can always increase with the total network profit; while underWSD-bearing-risk, the optimal contract ∗

(II) can only increase

the total network profit when σ 2ξ is small (i.e., when the degree

of information asymmetry is low). We can further see that theprofits under both optimal contracts decrease with σ 2

ξ . This is

because with a larger σ 2ξ , the scheduled demand ξ varies more

dramatically. As the scheduled demand ξ is the WSD’s privateinformation, the larger variance of ξ means that the databaseneeds to pay a higher information rent to the WSDs.

VIII. CONCLUSION

We propose a broker-based spectrum reservation marketmodel for TV white space network, under stochastic demandand information asymmetry. To solve the problem, we pro-pose a contract-based spectrum reservation framework, whichensures WSDs share their private information credibly. Ouranalysis and extensive simulations indicate that the optimalcontract under DB-bearing-risk leads to a higher database profitand higher network profit than that under WSD-bearing-risk.Our work serves as a first step to give theoretical insights intothe problem of risk-bearing between the database and the WSD,and promote the economic study of such a network.

In this work, we have focused on the TV white space net-work, where the primary users are the TV broadcasters. As theTV towers have fixed locations and TV programs have wellplanned schedules, the database has full information regard-ing the primary usage of TV spectrum ahead of time. Thisallows us to focus on the demand uncertainty from unlicensedusers in this paper. On the other hand, the issue of primaryusage uncertainly becomes much more important, if we con-sider the Licensed Shared Access (LSA) and Authorised Shared

Access (ASA) models, where unlicensed users may access spe-cific non-TV band (e.g., 3.5 GHz band in the United States and2.3 GHz band in Europe). Our model can be directly extendedto analyze the LSA/ASA systems, if there is no penalty to thedatabase and the WSD for not being able to serve all demands.However, when the expected payoffs of the database and theWSD depend on both the demand randomness and the avail-able spectrum randomness, it would be much more challengingto obtain theoretical results by solving the contract design prob-lem. We will consider the issue of two-sided uncertainty and theinteractions among the licensee, the database, and the WSDs inour future work.

REFERENCES

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[4] S. Jones and T. Phillips, “Initial evaluation of the performance ofprototype TV-band white space devices,” Federal CommunicationCommission, Washington, DC, USA, Tech. Rep. FCC/OET 07-RT-1006,2007.

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[6] D. Niyato and E. Hossain, “Spectrum trading in cognitive radio net-works: A market-equilibrium-based approach,” IEEE Wireless Commun.,vol. 15, no. 6, pp. 71–80, Dec. 2008.

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[8] X. Feng, Q. Zhang, and J. Zhang, “Hybrid pricing for TV white spacedatabase,” in Proc. IEEE INFOCOM, 2013, pp. 1995–2003.

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[19] A. Goldsmith, Wireless Communications. Cambridge, U.K.: CambridgeUniv. Press, 2005.

[20] Y. Luo, L. Gao, and J. Huang, “MINE GOLD to deliver greencognitive communications,” Chin. Univ. Hong Kong, HongKong, China, Tech. Rep., arXiv: 1511.00544 [Online]. Available:http://arxiv.org/abs/1511.00544

Yuan Luo (S’10) received the B.S. degree fromTianjin University, Tianjin, China, and the M.S.degree from Beijing University of Posts andTelecommunications, Beijing, China, in 2008 and2011, respectively. She is currently pursuing thePh.D. degree at the Department of InformationEngineering, Chinese University of Hong Kong,Hong Kong. Her research interests include the fieldof wireless communications and network economics,with current emphasis on TV white space networksand crowdsourcing. She was the recipient of the

Best Paper Award in the IEEE International Symposium on Modeling andOptimization in Mobile, Ad Hoc, and Wireless Networks (WiOpt).

Lin Gao (S’08–M’10) received the M.S. and Ph.D.degrees in electronic engineering from ShanghaiJiao Tong University Shanghai, China, in 2006 and2010, respectively. He is an Associate Professor withthe Shenzhen Graduate School, Harbin Institute ofTechnology (HIT), Shenzhen, China. He worked as aPostdoc Research Fellow with the Chinese Universityof Hong Kong, Hong Kong, from 2010 to 2015. Hisresearch interests include the area of game theoryand network economics, with applications in wirelesscommunications, networks, and Internet of Things.

He was the recipient of the Best (Student) Paper Awards from the IEEEInternational Symposium on Modeling and Optimization in Mobile, Ad Hoc,and Wireless Networks (WiOpt) in 2013, 2014, and 2015.

Jianwei Huang (S’01–M’06–SM’11–F’16)is an Associate Professor and Director of theNetwork Communications and Economics Lab(ncel.ie.cuhk.edu.hk), Department of InformationEngineering, The Chinese University of Hong Kong,Hong Kong. He received the Ph.D. degree fromNorthwestern University, Evanston, IL, USA, in2005, and worked as a Postdoc Research Associateat Princeton University during, Princeton, NJ, USA,2005–2007. His main research interests are inthe area of network economics and games, with

applications in wireless communications, networking, and smart grid. Heis a Fellow of IEEE (Class of 2016), and a Distinguished Lecturer of IEEECommunications Society (2015–2016).

Dr. Huang is the co-recipient of eight Best Paper Awards, including IEEEMarconi Prize Paper Award in Wireless Communications in 2011, and Best(Student) Paper Awards from IEEE WiOpt 2015, IEEE WiOpt 2014, IEEEWiOpt 2013, IEEE SmartGridComm 2012, WiCON 2011, IEEE GLOBECOM2010, and APCC 2009. He received the CUHK Young Researcher Award in2014 and IEEE ComSoc Asia-Pacific Outstanding Young Researcher Award in2009. He has co-authored four books: Wireless Network Pricing, MonotonicOptimization in Communication and Networking Systems, Cognitive MobileVirtual Network Operator Games, and Social Cognitive Radio Networks. Heis a co-author of five ESI Highly Cited Papers.

Dr. Huang has served as an Editor of the IEEE TRANSACTIONS ON

COGNITIVE COMMUNICATIONS AND NETWORKING (2015–present), theIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS (2010–2015), theIEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS-COGNITIVE

RADIO SERIES (2011–2014), and Editor and Associate Editor-in-Chief ofthe IEEE COMMUNICATIONS SOCIETY TECHNOLOGY NEWS (2012–2014).He has served as a Guest Editor of IEEE TRANSACTIONS ON SMART

GRID special issue on Big Data Analytics for Grid Modernization (2016),the IEEE NETWORK special issue on Smart Data Pricing (2016), IEEEJOURNAL ON SELECTED AREAS IN COMMUNICATIONS special issues onGame Theory for Networks (2017), Economics of Communication Networksand Systems (2012), and Game Theory in Communication Systems (2008),and the IEEE Communications Magazine feature topic on CommunicationsNetwork Economics (2012). He has served as Vice Chair (2015–2016) ofIEEE Communications Society Cognitive Network Technical Committee, ViceChair (2010–2012) and Chair (2012–2014) of IEEE Communications SocietyMultimedia Communications Technical Committee, a Steering CommitteeMember of IEEE Transactions on Multimedia (2012–2014) and IEEEInternational Conference on Multimedia & Expo (2012–2014), Chair ofMeeting and Conference Committee (2012–2013) and Vice Chair of TechnicalAffairs Committee (2014–2015) of IEEE ComSoc Asia-Pacific Board. Hehas served as the TPC Co-Chair of IEEE WiOpt 2017, IEEE SDP 2016,IEEE ICCC 2015 (Wireless Communications System Symposium), IEEE SDP2015, NetGCoop 2014, IEEE SmartGridComm 2014 (Demand Response andDynamic Pricing Symposium), IEEE GLOBECOM 2013 (Selected Areasof Communications Symposium), IEEE WiOpt 2012, IEEE ICCC 2012(Communication Theory and Security Symposium), IEEE GlOBECOM 2010(Wireless Communications Symposium), IWCMC 2010 (Mobile ComputingSymposium), and GameNets 2009. He is a frequent TPC member of lead-ing networking conferences such as INFOCOM and MobiHoc. He is therecipient of IEEE ComSoc Multimedia Communications Technical CommitteeDistinguished Service Award in 2015 and IEEE GLOBECOM OutstandingService Award in 2010.

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