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1

Balancing Income and User Utility inSpectrum Allocation

Yanjiao Chen, Student Member, IEEE, Lingjie Duan, Member, IEEE, Jianwei Huang, SeniorMember, IEEE, and Qian Zhang, Fellow, IEEE

Abstract—To match wireless users’ soaring traffic demand, spectrum regulators are considering allocating additional spectrumto the wireless market. There are two major directions for the spectrum allocation: licensed (e.g., 4G cellular service) andunlicensed services (e.g., Super Wi-Fi service). The 4G service provides a ubiquitous coverage, has a higher spectrum efficiency,and often charges users a high service price. The Super Wi-Fi service has a limited coverage, a lower spectrum efficiency, butoften charges users a low service price. The spectrum regulator now simply allocates the spectrum to maximize its income,but such an income-centric allocation does not ensure the best spectrum utilization by the users. This motivates us to designa new spectrum allocation scheme which jointly considers the spectrum regulator’s income and the users’ aggregate utility byinvestigating three market tiers: the spectrum regulator, 4G and Super Wi-Fi operator coalitions, and all the wireless users. Weformulate it as a three-stage game and derive the unique subgame perfect equilibrium. Compared with the traditional income-centric allocation, we prove that the proposed scheme significantly improves users’ aggregate utility with a limited spectrumregulator’s income loss.

Index Terms—Spectrum allocation, user utility improvement, three-tier dynamic game

✦

1 INTRODUCTION

The number of customers using wireless broadbandservices has been increasing dramatically during re-cent years. Such a demand will surpass the capacityof allocated wireless spectrum for mobile broadbandservices by as soon as 2013 [1]. To provide more spec-trum resources to support mobile broadband services,the Federal Communications Commission (FCC) hasdecided to make 500 MHz of new wireless spectrumavailable within 10 years for licensed and unlicenseduse [2]. In July 2012, the President’s Council of Ad-visors on Science and Technology (PCAST) of theU.S. [3] further proposed to identify 1,000 MHz ofFederal spectrum for shared-use among commercialusers. The remaining key question is: how to allocatethese spectrum bands to different operators and howto make the best use of these spectrum to improve

• This work was supported by grants from 973 project 2013CB329006,China NSFC under Grant 61173156, RGC under the contracts CERG622613, 16212714, HKUST6/CRF/12R, and M-HKUST609/13, thegrant from Huawei-HKUST joint lab, the General Research Funds(Project Number CUHK 412713 and CUHK 14202814) establishedunder the University Grant Committee of the Hong Kong SpecialAdministrative Region, China, the Competitive Earmarked ResearchGrants, and the SUTD-MIT International Design Centre (IDC) Grant(Project No.: IDSF1200106OH).

• Y. Chen and Q. Zhang are with the Department of Computer Scienceand Engineering, Hong Kong University of Science and Technology.E-mail: {chenyanjiao, qianzh}@ust.hk.

• L. Duan is with the Engineering Systems and Design Pil-lar, Singapore University of Technology and Design. E-mail:lingjie [email protected].

• J. Huang is with the Department of Information Engineering, TheChinese University of Hong Kong. E-mail: [email protected]

end-users’ utility? It is not only a technical issue, butalso a complicated policy and socio-economic issue[4].

There are two mainstream systems of providingmobile Internet access: licensed cellular networks andunlicensed wireless local area networks (WLANs).The representative next generation technologies ofthese two systems are 4G and Super Wi-Fi. However,these two representative technologies are quite differ-ent.

• The 4G cellular system is based on OFDM andMIMO technologies, and can achieve a high spec-trum efficiency due to careful network planningand efficient interference mitigation. Moreover, asa cellular network technology, 4G can provideubiquitous Internet access. However, a 4G cellu-lar operator usually charges a high price to theend users to compensate the high deploymentand operational cost.

• Super Wi-Fi has a relatively low spectrum ef-ficiency, as it operates in the unlicensed modeand different networks or systems may share thesame spectrum without a centralized interferencemanagement [5]. Furthermore, it only has limitedoverall coverage due to a limited number of ac-cess points and regulatory power limitation1. ButSuper Wi-Fi can be easier to deploy and cheaperto maintain than 4G [6], resulting in a low serviceprice. For example, China Mobile Hong Kong

1. Even with the Super Wi-Fi operating on the TV white space(which has a better propagation property than the traditional2.4GHz spectrum bands), the ad hoc deployment of access pointsmay still make it hard to achieve the full coverage.



2

Spectrum

Allocation

Spectrum

Regulator

Wi-Fi Operator

4G Operator

Service

Pricing

Wireless

Users

Fig. 1: Illustration of Three-Stage Wireless Allocationand Service Model.

charges a monthly fee of HK$278 for 2 GBytesof 4G data, but only a monthly fee of HK$38 forunlimited Wi-Fi Service [7].

Users with different coverage requirements and pricesensitivities thus have different preferences over thesetwo services.

The FCC’s current practice is to reserve certainamount of unlicensed spectrum for Wi-Fi service,while auctioning the remaining spectrum to cellularoperators who bid the highest price [8]. However, it isnot clear what is the optimal amount of spectrum thatshould be reserved for unlicensed use. Recent study(e.g., [9]) has shown that an inappropriate allocationof unlicensed spectrum may hurt the total surplusof both operators and users. On the other hand,the spectrum auction mechanisms [10]–[13], thoughaiming at maximizing the total utility of winningoperators, does not guarantee that these operators willuse the spectrum to provide desirable and affordableservices for all wireless users. Hence from the socialpersecutive, the FCC should not only interact with theoperators, but also consider the impact on the users.Such multi-level interactions can be modeled by athree-tier model. Existing works on three-tier models(the regulator, the operators, and the end users, e.g.,[14]–[16]) mainly concerned about how the operatorsshould lease spectrum to maximize their profits in themarket, without concerning about how the regulatorshould allocate the spectrum to maximize a weightedsum of its income and the end-users’ utility. The mainpurpose of this paper is to fill in this important policygap.

In this paper, we formulate the interactions amongthe FCC (the regulator), the 4G and the Super Wi-Fi operators, and the wireless users as a three-stagedynamic game as illustrated in Fig. 1. We assumethat all licensed 4G operators form a coalition wheninteracting with the FCC; similarly, the Super Wi-Ficoalition represents all the Super Wi-Fi operators, whoneed spectrum for unlicensed use. In the rest of thepaper, we will use the “4G operator” to refer to the4G coalition, and the “Wi-Fi operator” to refer to theSuper Wi-Fi coalition. 2 In Stage I, the FCC decidesthe spectrum allocations to both operators. In StageII, two operators optimally price their services tomaximize their profits based on their limited spectrum

2. We focus on the competition and interaction between licensedand unlicensed services. We will further study the competitionwithin each service in a future work.

amounts. Finally in Stage III, the users choose be-tween the two services to maximize their own utilitiesbased on the service prices.

Our key results and major contributions are asfollows.

• Spectrum allocation with consideration of end-userutility. The proposed three-tier dynamic gamemodel, which practically characterizes all wire-less users’ utility and service choices, is helpfulfor the FCC’s supervision of spectrum utilizationin end-market.

• Significant user utility improvement with limitedincome loss. The proposed spectrum allocationscheme, shown by simulation results, signifi-cantly improves user utility; while, by theoreticalanalysis, have bounded income loss (comparedwith the income-centric spectrum allocation).

• Price competition under different spectrum conditions.We comprehensively analyze the price competi-tion between different operators, given the spec-trum allocation. We find that an operator will self-ishly reserve some capacity unused and charge ahigh price if assigned relatively large amount ofspectrum.

• Users’ service choices: We analyze how the prices(decided by the operators) and capacity (decidedby the FCC’s spectrum allocation) jointly influ-ence the users’ service choice. The proposed spec-trum allocation scheme will lead to lower serviceprices and serve more wireless users.

The rest of the paper is organized as follows. Webriefly review the related work in Section 2. The sys-tem model and key assumptions are given in Section3. We describe the three-stage game framework inSection 4. Then we use backward induction to analyzethe game, starting from the market response of StageIII in Section 5, to the service competition game ofStage II in Section 6, finally to the spectrum allocationsof Stage I in Section 7. In Section 8, we derive thebound of the regulator’s income loss and analyze theimpacts of critical parameters. Finally, we summarizeour work in Section 9. Due to page limitations, allproofs are given in our online technical report [17].

2 RELATED WORK

Multi-tier game models for wireless markets havebeen widely studied. Lehr and McKnight in [6] sur-veyed the competitive and complementary relation-ship of Wi-Fi and 3G technology. Niyato and Hossainin [18] built a 2-tier pricing competition model be-tween Wi-Fi and WiMAX operators. However, theydid not consider the regulator’s spectrum allocationin a higher tier. Several recent results studied three-tier models that involve the spectrum owner, theoperators, and the users [14], [15]. These prior resultsfocused on how much spectrum each operator shouldlease or buy from the spectrum owners to maximize



3

their profits, while our work focuses on how muchspectrum the regulator should allocate to Wi-Fi (forunlicensed use) and 4G (for licensed use) consideringthe utility of end-users.

Spectrum auction is a way of distributing spectrumto different operators for licensed access [11], [13],[19]. Usually the spectrum is auctioned for exclu-sive licensed use by the cellular operators. This willnot help users who want to use unlicensed wirelessservices. In addition, spectrum auction usually con-siders the utility of operators rather than the users’aggregate utility. However, the operators who valuethe spectrum the most may not choose to maximizethe aggregate utility of all users, especially when theoperators have enough market power and the marketentry barrier is significant.

Spectrum is also reserved for unlicensed access.Nguyen et al. in [9] studied the influence of additionalunlicensed spectrum on social welfare, but the userdemand function is a bit over-simplified. Further-more, they did not make suggestions regarding howspectrum should be allocated between licensed andunlicensed ones. The follow-up work [20] establisheda pricing model among unlicensed operators. How-ever, the model did not consider the competition fromlicensed operators, nor the spectrum allocation issueof the regulator in a higher tier.

Our paper is different from the previous results:1) we aim at addressing the spectrum allocation is-sue for the FCC, considering allocating additionalspectrum available for licensed and unlicensed use,respectively; 2) we build a three-tier model, in whichthe FCC not only cares about its own income, butalso the users’ aggregate utility.

3 SYSTEM MODEL

3.1 Spectrum Allocation

We consider a regulator (referred to as the FCC forthe illustration purpose), who possesses S units ofspectrum and intends to assign the spectrum to wire-less operators to satisfy their users’ increasing wire-less data demands. In this paper, we only considertwo wireless operator coalitions: one cellular coalitionproviding 4G service, and one wireless local areanetwork (WLAN) coalition providing Wi-Fi service3.Also, we focus on the additional capacity yielded bythe spectrum to be allocated to the two operators bythe FCC, without considering their existing capacity4.Let Sw and Sg denote the spectrum allocated to Wi-Fi and 4G, respectively. The FCC has no intention toreserve the spectrum for other purposes. Therefore,Sg + Sw = S.

3. In the following texts, we use Wi-Fi to represent Super Wi-Fi,for simplicity.

4. We assume that the additional spectrum to be distributed tothe two operators will not affect how they manage their existingspectrum and capacity. We will consider the impact of new spec-trum allocation on old services in the future work.

We assume that the FCC charges the Wi-Fi and the4G operators according to the quantity of spectrumallocated to them. The FCC charges the 4G operator atotal amount of φg(Sg), and charges the Wi-Fi operatora total amount of φw(Sw). Both pricing functionsare non-decreasing in the spectrum allocation. TheFCC can design different pricing schemes based onthe estimation for the profitability of 4G and Wi-Fiservices5.

3.2 Service Model

3.2.1 Network Capacity

We assume that the expected data rate required bya user is δ67. The capacity of 4G network is f(Sg),under an exclusive spectrum license and an efficientinterference management, in which f(·) is a non-decreasing function. The capacity of Wi-Fi networkis f(ηSw), where η < 1 is the interference parameterand characterizes the low spectrum efficiency in shar-ing the unlicensed band with other Wi-Fi networksand other unlicensed services (e.g., ZigBee devices,Bluetooth devices, and cordless phones) [21]. So themaximum number of concurrent in-service users thatcan be supported by 4G service and Wi-Fi serviceis f(Sg)/δ and f(ηSw)/δ, respectively. We considerthe requirement of data rate instead of bandwidth,circumventing the complicated issue of interferencemanagement. Thus, our setup can be easily general-ized and applied to different types of technologies, aslong as we are able to characterize the relationshipbetween bandwidth and capacity.

3.2.2 Network CoverageThe transmission range of a cellular base station isin the order of kilometers, being able to serve usersover a large contiguous area. 4G operators generallydeploy enough base stations to realize full coverage.Hence, we assume that the future 4G network’s cov-erage is 1 (full coverage) in the new spectrum8.

Though improved greatly on existing Wi-Fi, theSuper Wi-Fi is still limited in coverage. For example,the Super Wi-Fi base station launched by Altai hasa signal coverage of 500m in radius [22]. According

5. In this paper, we consider the case that φg(·) and φw(·) aredetermined by the FCC, unlike the spectrum auction where theoperators’ bids determine the price. In the future work, we willstudy the situation where the operators have the right to (partly)decide the spectrum fee.

6. The main results will still hold if we allow a user to havedifferent data rate requirements under the Wi-Fi and 4G services.

7. The 4G service can satisfy a user’s fixed δ requirement, aslicensed spectrum access can ensure QoS. Thanks to the betterpropagation characteristics of the White Space than the ISM band,future Super Wi-Fi network also intends to provide QoS-guaranteedservice to satisfy users’ fixed δ requirement [14]. In the futurework, we plan to consider the case where users have heterogeneousdemands.

8. At the regions where 4G base stations haven’t been deployedyet, subscribers can still enjoy cellular service through the existing3G network [7].



4

to [23], approximately 700 traditional Wi-Fi accesspoints would be needed to cover the same area asone cellular base station does under the current tech-nology, but such a dense deployment to achieve fullcoverage will have a formidably high cost. Even if us-ing the whitespace spectrum with better propagationcharacteristics (due to low frequency), the existing adhoc deployment of Wi-Fi may not be able to achievea full coverage. Hence, we assume Wi-Fi network’scoverage θ < 1.

3.2.3 Service Price

We assume that the Wi-Fi and 4G operators charge pwand pg per subscriber, respectively, as long as the datarate requirement is fulfilled [24]. We consider N wire-less users who are potential Wi-Fi or 4G subscribers.The numbers of users who eventually choose the twoservices (called subscribers) are denoted by Nw andNg, respectively. Naturally we have Nw+Ng ≤ N . Foreach network, the operator needs to perform admis-sion control if its subscribers’ total demand exceedsthe network capacity, hence we have Nw ≤ f(Sg)/δand Ng ≤ f(ηSw)/δ at the equilibrium.

3.3 User Preference

Users are different in their preferences for the cov-erage of Internet access. We characterize such het-erogeneity by a type parameter α ∈ [0, A], whichrepresents users’ sensitivity due to their mobility [25].A user with a higher mobility prefers a higher net-work coverage, while a user who always stays at onelocation is less sensitive to network coverage in otherplaces. Therefore, a higher mobility user will havea higher preference for coverage than for price. Forsimplification, we assume that α follows a uniformdistribution in [0, A] in the following analysis9.

For a type α user, he achieves a utility uαg = α− pg

if choosing 4G10, and a utility of uαw = αθ − pw if

choosing Wi-Fi. All users have the same reservationutility u0 ≥ 0, which means that a type α user willchoose neither 4G nor Wi-Fi if max{uα

g , uαw} < u0.

To summarize, a type α user’s service choice is asfollows11:

4G service, if uαg ≥ uα

w and uαg ≥ u0

Wi-Fi service, if uαw > uα

g and uαw ≥ u0

No service, otherwise.(1)

9. Uniform distribution of the QoS sensitivity is commonly usedfor tractable analysis and changing to another continuous distribu-tion is unlikely to change the key results (see [24] and [26]).

10. If choosing 4G, the user’s utility is uαg = α · 1− pg, as the 4G

coverage is 1.11. In this paper, we assume that each user can choose at most

one service. In future work, we will study the case when a usermay choose both Wi-Fi and 4G services at the same time.

Stage I – Spectrum allocation Player: the FCC

Decisions: spectrum allocation Sw and Sg Objective: joint income and user aggregate utility maximization

Stage II – Service competition game Players: 4G operator and Wi-Fi operator

Decisions: service prices pw and pg Objective: profit maximization

Stage III – Market response Players: end users

Decision: choose 4G, Wi-Fi or neither Objective: utility maximization

Fig. 2: Three stages of the dynamic game.

4 THREE-STAGE GAME FRAMEWORK

The three-stage game model is illustrated in Fig. 2,showing the game players, what decisions they makeand their objectives.

4.1 Stage I - Spectrum Allocation

The FCC decides the spectrum allocation Sw and Sg tomaximize its utility, Uf , which is the weighted sum ofthe end users’ aggregate utility Uuser and the FCC’sincome,

Uβf = βUuser + φg(Sg) + φw(Sw). (2)

Here the weight β ≥ 0, representing how much theFCC values the user utility over its income. The FCCcan adjust β to tailor for different spectrum allocationpurposes. In the benchmark case, where the FCC onlycares about its income, β = 0. If the FCC cares moreabout user utility than its income, then β can be setlarger than 1. We define end users’ aggregate utilityas the weighted sum of Wi-Fi subscribers’ enjoyedcoverage and 4G subscribers’ enjoyed coverage:

Uuser = ω1

∫

Wi-Fi subscribers

θ

Adα+ ω2

∫

4G subscribers

1

Adα

=ω1θNw

N+

ω2Ng

N.

(3)

where the first term is Wi-Fi subscribers’ aggregatedvaluation of coverage, and the second term is 4Gsubscribers’ aggregated valuation of coverage. The pa-rameters ω1 and ω2 represent the relative importanceof Wi-Fi and 4G network from the FCC’s point ofview. End users’ aggregate utility is determined bythe numbers of subscribers to the Wi-Fi service and 4Gservice (Nw and Ng), which are directly affected by theservice prices pw and pg decided by the two operators,and indirectly affected by the spectrum allocation Sw

and Sg decided by the FCC.

4.2 Stage II - Service Competition Game

After observing the spectrum allocation in Stage I,two operators play a pricing game in Stage II, where



5

they determine the prices of their own services tomaximize the profits. The profit of the 4G operator isthe difference between the revenue and the spectrumpayment:

Ug = Ngpg − φg(Sg). (4)

The profit of the Wi-Fi operator is the differencebetween the revenue and the spectrum payment:

Uw = Nwpw − φw(Sw) (5)

Apart from the cost of paying for spectrum, thecost for the Wi-Fi and 4G operators will also includethe operating expense (OPEX) and the capital expense(CAPEX). If the OPEX and CAPEX are fixed anddo not depend on the subscriber numbers or theamount of spectrum, our analytical results are directlyapplicable as long as those costs can be compensatedby the profits made by the two operators. On the otherhand, if the OPEX and CAPEX are functions of eithersubscriber number or the amount of spectrum, theywill affect the computation of the equilibrium. Forexample, if the OPEX is an increasing function of thesubscriber number, then the service prices of both Wi-Fi and 4G in Stage II will be higher. The influence onthe spectrum allocation is more complicated, since wehave to compare the OPEX and CAPEX of Wi-Fi and4G. We will leave the detailed study of the impact ofOPEX and CAPEX in a future work.

4.3 Stage III - Market Response

After observing the prices of 4G and Wi-Fi services,each user compares and decides which service tosubscribe (or not to subscribe to any service at all).A user may be rejected by a network if the users’demand exceeds the network’s capacity.

4.4 Nash Equilibrium and Subgame Perfect Equi-librium

We introduce the concept of Nash Equilibrium andSubgame Perfect Equilibrium as follows [27]:

Definition 1: Nash Equilibrium: Consider a game{I, (Si)i∈I , (ui)i∈I}, where I is the set of players, Si

is the strategy set of player i ∈ I, and ui is the utilityof player i ∈ I. Let s = (si, s−i) denote the strategyprofile of all users, where s−i includes the strategychoices of all players other than i. Denote S = ΠiSi

as the set of all strategy profiles. A strategy profiles∗ ∈ S is a (pure) Nash Equilibrium if and only ifui(s

∗i , s

∗−i) ≥ ui(si, s

∗−i) is true for all i ∈ I.

The game that we consider is a dynamic game,where players act sequentially in multiple stages. Asubgame is part of a dynamic game, and we havethree subgames here. Stage III is a subgame, Stages IIand III together is another subgame, and the wholegame with three stages is also a subgame.

Definition 2: Subgame Perfect Equilibrium (SPE): Astrategy profile of the three-stage game is an SPE if thechoices of the FCC, the 4G and the Wi-Fi operators,and the end users constitute a Nash Equilibrium in

each of the subgame of the whole game. In otherwords, no player at SPE will deviate unilaterally fromhis equilibrium strategy.

In Sections 5 to 7, we will derive the SPE of thethree-stage game using backward induction, that is,first to find the SPE of Stage III, then to find the SPEof Stage II and III together, finally to find the SPE ofthe whole game with three stages.

5 STAGE III: MARKET RESPONSE

In Stage III, users can observe the service prices givenin Stage II: pw and pg. Recall that the subscribernumbers of the Wi-Fi and 4G services are Nw and Ng,respectively. The following Proposition 1 gives the Nw

in response to pw; and Proposition 2 gives the Ng inresponse to pg .

Proposition 1: Given Wi-Fi capacity f(ηSw), Wi-Fisubscriber number Nw in response to Wi-Fi price pwand 4G price pg is as follows12:

Nw = min

(Nw,

f(ηSw)

δ

), (6)

whereNw ={

NAθ(1−θ) [θpg − pw − (1− θ)u0], if pw > pg − (1− θ)A,NAθ

[−pw + θA− u0], otherwise.(7)

Proposition 2: Given the 4G capacity f(Sg), 4G sub-scriber number Ng in response to 4G price pg andWi-Fi price pw is as follows:

Ng = min

(Ng,

f(Sg)

δ

), (8)

whereNg ={

NA(1−θ) [−pg + pw + (1 − θ)A], if pg > 1

θ[pw + (1− θ)u0],

NA[−pg +A− u0], otherwise.

(9)

The number of Wi-Fi subscribers corresponding toProposition 1 is shown in Fig. 3, which shows how thenumber of Wi-Fi subscribers Nw changes with the Wi-Fi price pw, given different capacity constraints andcompetitor’s service price13. We show two differentcapacity constraints f(ηS′

w)/δ and f(ηSw)/δ, in whichS′w > Sw (see the two subfigures (a) and (b)); and two

curves (solid line shows the overlapped area; dottedline and dot dash line show the difference) under twodifferent competitor’s prices p′g < pg. We have thefollowing observations.

• Influence of the service’s own price: For all curves,the higher the Wi-Fi service price, the smaller the

12. The numbers of subscribers to both services are expected val-ues, since the user type α follows a uniform distribution. Therefore,the resultant utility of the two operators and the FCC are alsoexpected values. To have a clean presentation, we omit notationE[·] on these variables.

13. The analysis of Proposition 2 is similar to that of Proposition1, therefore we ignore it here due to page limitation.



6

( )w

w

f S

!

wp

wN

Given pg’

Given pg

(a) Low capacity constraint

wp

wN

( ' )w

w

f S

!

Given pg’Given pg

(b) High capacity constraint

Fig. 3: Number of Wi-Fi subscribers in Stage III, undertwo different cellular prices p′g < pg.

Wi-Fi subscriber number. The slope of each curvechanges at the point when pw = pg − (1− θ)A (orp′g if the 4G price is p′g). The reason is that, whenpw < pg − (1− θ)A, we can prove that users of alltypes prefer Wi-Fi service (to 4G service) becauseof the Wi-Fi’s low price. Without competitionfrom 4G, the increase in the Wi-Fi price inducesa mild decline of the Wi-Fi subscriber number.In contrast, when pw > pg − (1 − θ)A, Wi-Fifaces competition from 4G, therefore Wi-Fi losessubscribers faster with the increase of the Wi-Fiprice pw.

• Influence of the competitor’s price: The 4G pricemainly horizontally “shifts” the curves in Fig. 3without changing the slopes. When 4G has a lowprice (p′g), more users will be attracted to the 4Gservice, leading to fewer Wi-Fi subscribers.

• Influence of the capacity constraint: When the spec-trum level for Wi-Fi is high (e.g., S′

w in Fig. 3.(b)),the capacity constraint f(ηS′

w)/δw does not affectthe subscriber number, because the W-Fi networkhas enough capacity even if all users subscribeto Wi-Fi; However, when the spectrum level forWi-Fi is low (e.g., Sw in Fig. 3.(a)), the capacitylimits the number of subscribers that the Wi-Fican support when the Wi-Fi price is low. In thiscase, the further decrease in Wi-Fi price cannotfurther increase the Wi-Fi subscriber number.

6 STAGE II - SERVICE COMPETITION GAME

In Stage II, given the spectrum allocations from theFCC, two operators need to optimize their pricesbased on the analysis of user behavior in Stage III.

Definition 3 (Best Response): Given the 4G serviceprice pg, the Wi-Fi operator’s best responseprice is p∗w(pg), such that Wi-Fi operator’s profitUw(p

∗w(pg), pg) ≥ Uw(pw, pg) for any pw ≥ 0. The

4G service’s best response p∗g(pw) can be definedsimilarly.

According to the analysis in Section 5, the sub-scriber number of a service is affected by the spectrumallocation of that service, in particular, whether thespectrum level is low (that constrains the subscriber

number, i.e., min(Nw,

f(ηSw)δ

)= f(ηSw)

δin (6) or

min(gw,

f(Sg)δ

)=

f(Sg)δ

in (8)) or high (that has no

influence on the subscriber number). Intuitively, thespectrum allocations will also affect the two serviceprices. Let’s define two thresholds14:

Sw =1

ηf−1

(Nδ(θA− u0)

θ(2− θ)A

),

Sg = f−1

(Nδ(A− u0)

θ(2 − θ)A

).

(10)

which distinguish whether the spectrum level of Wi-Fi

or 4G service is low or high. If Sw ≥ Sw, the capacityof Wi-Fi is able to support all the potential users who

want to subscribe to Wi-Fi service; if Sw < Sw, thenumber of Wi-Fi subscribers will be limited by theWi-Fi capacity, and some of the users who want tosubscribe to Wi-Fi service have to be rejected. Themeaning of Sg can be understood similarly for the4G service. If the total user number N or the userrequirement δ is high, then both thresholds are high.For the Wi-Fi service, if the spectrum efficiency η islow, the threshold becomes high, since more spectrumare needed to achieve the same capacity.

In the following, we will first exploit the best re-sponses of the Wi-Fi operator and the 4G operator,respectively. Then, we will examine the fixed point ofthe two operators’ best response functions in order todetermine the Nash Equilibrium (NE).

6.1 The Wi-Fi Operator’s Best Response

Proposition 3: Define pw = [θpg−(1−θ)u0]/2. Giventhe 4G operator’s service price as pg, the best responseof the Wi-Fi operator p∗w(pg) is as follows.

• Low Wi-Fi Spectrum Level. If Sw ≤ Sw, the bestresponse price for the Wi-Fi operator is

p∗w(pg) =

pw, if (1θ− 1)u0 ≤ pg ≤

(1θ− 1)u0 +

2A(1−θ)f(ηSw)Nδ

2pw − Aθ(1−θ)f(ηSw)Nδ

, if (1θ− 1)u0 +

2A(1−θ)f(ηSw)Nδ

< pg ≤ A− u0 −Aθf(ηSw)

Nδ

pg − (1− θ)A, if A− u0 −Aθf(ηSw)

Nδ<

pg ≤ A− u0

(11)

14. Refer to the online technical reports [17] for the derivation ofthe two thresholds.



7

0(1 ) /u q q-0A u- gp

0A uq -

* ( )w gp p

(a) High Wi-Fi spectrum level

0A uq -

* ( )w gp p

gp0(1 ) /u q q-0A u-

(b) Low Wi-Fi spectrum level

Fig. 4: Wi-Fi operator’s best response p∗w(pg)

• High Wi-Fi Spectrum Level. If Sw>Sw, the bestresponse price for the Wi-Fi operator is

p∗w ={

pw, if (1θ− 1)u0 ≤ pg ≤ 1−θ

2−θ(2A− u0)

pg − (1 − θ)A, if 1−θ2−θ

(2A− u0) < pg ≤ A− u0

(12)

Fig. 4 shows Wi-Fi operator’s best response in caseof high and low Wi-Fi spectrum levels. When pg isextremely low, p∗w(pg) has to remain zero in orderto attract users (the leftmost flat line at each subfig-ure). Then, p∗w(pg) increases with pg . When the Wi-Fispectrum level is high (subfigure (a)), the subscribernumber is not affected by the capacity. The slope ofp∗w(pg) is first small, then becomes large when pgbecomes too high and no users choose 4G service.Hence the Wi-Fi operator can increase its price moreaggressively in the high 4G price regime withoutlosing many potential subscribers. However, whenWi-Fi spectrum level is low (subfigure (b)), the slopeof p∗w(pg) has a medium value between the small andlarge values, where the subscriber number is boundedby the capacity, and the Wi-Fi operator increases itsprice at a rate that balances the user demand and thecapacity constraint (see the online technical report [17]for more details).

6.2 The 4G Operator’s Best Response

Proposition 4: Define pg = [pw + (1 − θ)A]/2. Giventhe Wi-Fi operator’s service price as pw, the bestresponse of the 4G operator p∗g(pw) is as follows.

• Low 4G Spectrum Level. If Sg ≤ Sg , the best

response price for the 4G operator is

p∗g(pw) =

pg, if 0 ≤ pw

≤2A(1−θ)f(Sg)

Nδ− (1− θ)A

2pg −A(1−θ)f(Sg)

Nδ, if 2A(1−θ)f(Sg)

Nδ− (1− θ)A

< pw ≤ θA− u0 −Aθf(Sg)

Nδ1θ[pw + (1− θ)u0], if θA− u0 −

Aθf(Sg)Nδ

< pw ≤ θA− u0

(13)• High 4G Spectrum Level. If Sg ≥ Sg, the best

response price for the 4G operator is

p∗g =

pg, if 0 ≤ pw ≤ 1−θ2−θ

(θA− 2u0)1θ[pw + (1− θ)u0], if 1−θ

2−θ(θA − 2u0) <

pw ≤ θA− u0

(14)6.3 Equilibrium Prices

To derive the equilibrium Wi-Fi and 4G prices, wehave to jointly consider the best responses of Wi-Fiand 4G operators. As analyzed in Sections 6.1 and 6.2,when the spectrum level for Wi-Fi and 4G operatorsare low or high (compared with the thresholds Sw andSg defined in (10)), the best responses of Wi-Fi and4G operators are different. However, to characterizethe spectrum conditions for the equilibrium prices,we need two different thresholds Sw and Sg. Thesetwo thresholds decide the high and low regimes ofspectrum allocation for Wi-Fi and 4G services. Whenthe FCC’s spectrum allocation at Stage I falls intodifferent regimes, the two operators’ best responsefunctions will be determined by Sw and Sg (Eq. (11)or (12) for Wi-Fi operator and Eq. (13) or (14) for4G operator), then the intersection points of the twopiecewise best response functions will be determinedby Sw and Sg.

Sw =1

ηf−1

(Nδ(θA− 2u0)

θ(4− θ)A

),

Sg = f−1

(Nδ(2A− u0)

(4− θ)A

).

(15)

Theorem 1: In Stage II, a unique equilibrium exists.The equilibrium prices of the Wi-Fi and 4G servicesare summarized in Table 1. The four spectrum condi-tions are characterized as

• Case LwLg (low Wi-Fi Spectrum, low 4G Spec-

trum): Sw ≤ Sw, Sg ≤ Sg ;• Case HwLg (high Wi-Fi Spectrum, low 4G Spec-

trum): Sw > Sw, Sg ≤ Sg ;• Case LwHg (low Wi-Fi Spectrum, high 4G Spec-

trum): Sw ≤ Sw, Sg > Sg ;• Case HwHg (high Wi-Fi Spectrum, high 4G Spec-

trum): Sw > Sw, Sg > Sg .

There exists a unique equilibrium because the bestresponses of two operators intersect only once (inter-ested readers can check Figures 4∼12 in the technical



8

TABLE 1: Equilibrium price in Stage II

4G service price pg Wi-Fi service price pw

LwLg A−u0−A[θf(ηSw)+f(Sg)]

NδθA−u0−

Aθ[f(ηSw)+f(Sg)]

Nδ

HwLg(1−θ)[2A−u0−

2Af(Sg)

Nδ]

2−θ

(1−θ)[θA−u0−θAf(Sg )

Nδ]

2−θ

LwHg(1−θ)[A−u0−

θAf(ηSw)Nδ

]

2−θ

(1−θ)[θA−2u0−2θAf(ηSw )

Nδ]

2−θ

HwHg1−θ4−θ

(2A− u0)1−θ4−θ

(θA− 2u0)

TABLE 2: SPE subscriber number in Stage III

Wi-Fi subscriber number 4G subscriber number

LwLgf(Sg)

δ

f(ηSw)δ

HwLgN(θA−u0)(2−θ)Aθ

− 12−θ

f(Sg)

δ

f(Sg)

δ

LwHgf(ηSw)

δ

N(A−u0)(2−θ)A

− θ2−θ

f(ηSw)δ

HwHgN(θA−2u0)(4−θ)Aθ

N(2A−u0)(4−θ)A

report [17]) for more details. Note that we have dif-ferent thresholds for low or high spectrum levels inTheorem 1 and Propositions 3 and 4. We can easily

prove that Sw < Sw and Sg < Sg . One of the reasonsthat we have different spectrum thresholds is that, thecapacity constraint may affect the best response of Wi-Fi operator as shown in Fig. 4, but may not affect theequilibrium price if the intersection of the two bestresponses does not fall into the transition phase inFig. 4. In order for the intersection of the two bestresponses (and the resulting equilibrium prices) to bewithin the transition phase (which reflects the impactof capacity constraint), the spectrum thresholds Sw

and Sg need to be even smaller. Detailed derivationof Sw and Sg is in the online technical report [17].

An interesting observation is that the equilibriumprices are only affected by the spectrum level of theservice that has low spectrum level. More specifically:1) when both the Wi-Fi and 4G spectrum levels arehigh (the HwHg case), the equilibrium prices do notrely on the spectrum allocation; 2) when the 4Gspectrum level is low but the Wi-Fi spectrum levelis high (the HwLg case), both the Wi-Fi and 4G pricewill decrease with Sg; we have the similar observationwhen the Wi-Fi spectrum level is low but the 4Gspectrum level is high (the LwHg case); 3) when boththe Wi-Fi and 4G spectrum levels are low (the LwLg

case), the equilibrium prices decrease in both Sg andSw. This is because the spectrum level only affects aservice’s best response price when it is low, in whichcase it will further affect the equilibrium prices.

6.4 Subgame Perfect Equilibrium

Once we have the equilibrium prices in Stage II, wecan determine the user subscriber numbers in StageIII at the SPE according to the analysis in Section 5.

Corollary 1: The equilibrium prices in Stage II resultin such subscriber numers in Stage III that Nw = Nw

and Ng = Ng, in which Nw and Ng are defined in (7)and (9), respectively.

The subscriber numbers at the SPE (Stage II andIII together) under different spectrum allocations issummarized in Table 2. We can see that if one service

has low spectrum level, the subscriber number of thatservice equals its capacity. However, if one servicehas a high spectrum level, the subscriber numberof that service is less than its capacity. This meansthat the network will reserve some capacity in thisregime in order to maximize the profit. Understandingthe influence of spectrum allocation on the subscribernumber can help the FCC to achieve a better balancebetween user benefits and income.

7 STAGE I - SPECTRUM ALLOCATION

In Stage I, the FCC optimizes its spectrum allocationbased on the prediction of the equilibrium prices inStage II (Table 1) and the subscriber numbers in StageIII (Proposition 1 and 2).

We first analyze the end users’ aggregate utilityUuser in (3). As an example, we assign ω1 = ω2 = 1,i.e., we view Wi-Fi and 4G networks as equally im-portant15. According to Corollary 1, we have Nw =Nw, Ng = Ng , in which Nw and Ng are defined in (7)and (9), respectively. Thus we have

Uuser =1

N(θNw +Ng) = −

1

A(pg −A+ u0), (16)

which depends on 4G price pg, which further dependson the spectrum allocations Sw and Sg according toTable 1. In the following analysis, we focus on thederivation of Sw, by keeping in mind that Sg = S−Sw.To derive clean insights for FCC’s spectrum allocationto 4G and Wi-Fi services, we assume that A = 1 andu0 = 0.7.1 Income-centric Spectrum Allocation Bench-mark

We first derive the spectrum allocation when the FCConly cares about its income, a special case with β =0 that serves as a benchmark to compare with ourproposed spectrum allocation with a positive β.

FCC’s income-only utility is U0f = φg(Sg) +

φw(Sw)= φg(S − Sw) + φw(Sw). The first order partial

derivative with respect to Sw is∂U0

f

∂Sw= −φ′

g(S −Sw) + φ′

w(Sw). The second order partial derivative

is∂2U0

f

∂S2w

= φ′′g(S − Sw) + φ′′

w(Sw). We assume that

φ′′g < 0 and φ′′

w < 0, which means that the unitprice (charged by the FCC to the network operators)will decrease as the spectrum increases16. This meansthat U0

f is a strictly concave function of Sw, and wecan obtain the income maximizing S0∗

w by solving∂U0

f /∂Sw = 017. The spectrum allocation to the 4G

15. Our results can also be extended to different weights of ω1

and ω2 when the FCC has different preference towards servingusers, e.g., if he intends to focus on high-end users, he would paymore attention to 4G service with ω1 < ω2.

16. This is reasonable because, according to diminishing marginalreturns [28] in economics, the marginal improvement in transmis-sion QoS will decrease as the quantity of spectrum increases whilekeeping other factors constant. So the FCC should charge a smallerunit price as the quantity of spectrum increases.

17. Let S0,optw denote the root of ∂U0

f /∂Sw = 0: if S0,optw <

0, S0∗w = 0; if S0,opt

w > S,S0∗w = S; otherwise, S0∗

w = S0,optw .



9

TABLE 3: Equilibrium spectrum allocation in Stage ISpectrum allocation Sopt

w satisfies Supperw Slower

w

LwLgβ[θηf ′(ηSopt

w )−f ′(S−Soptw )]

Nδ− φ′

g(S − Soptw ) + φ′

w(Soptw ) = 0 Sw S − Sg

HwLg −2β(1−θ)f ′(S−Sopt

w )

(2−θ)Nδ− φ′


w(Soptw ) = 0 S max(Sw , S − Sg)

LwHgβθ(1−θ)ηf ′(ηSopt

w )

(2−θ)Nδ− φ′


w(Soptw ) = 0 min(S − Sg, Sw) 0

HwHg −φ′


w(Soptw ) = 0 S − Sg Sw

network is then S − S0∗w . Such a spectrum allocation

scheme only depends on how the FCC designs thepayment function φg and φw, without consideringend users’ aggregate utility. The analysis in Section7 is applicable to general forms of Φg and Φw. InSection 8, when we perform numerical study of thebound of income loss and aggregate user utility, wewill consider polynomial functions of Φg and Φw. Thestudy of the optimal design of Φg and Φw as well astheir influences on the spectrum allocation will be afuture work.

7.2 Proposed Spectrum Allocation

When jointly considering end users’ utility and theFCC’s income, the FCC maximizes its utility as in(2). We assume that the second derivative of capacityfunction f ′′(·) < 0, which means that capacity perunit spectrum will decrease due to limitations such asinterferences. Therefore, (2) is still a concave functionof Sw.

Theorem 2: In Stage I, equilibrium spectrum alloca-tion for Wi-Fi service is

S∗w =

Slowerw , if Sopt

w < Slowerw

Soptw , if Slower

w ≤ Soptw ≤ Supper

w

Supperw , if Sopt

w > Supperw

(17)

in which Soptw , Supper

w , Slowerw are summarized in Table

3, and the equilibrium spectrum allocation for 4Gservice is S∗

g = S − S∗w.

Corollary 2: More spectrum allocation to the servicewith a low spectrum level. If one operator receivesa low spectrum allocation and the other operatorreceives a high spectrum allocation in the income-centric benchmark (with β = 0), then our proposedscheme (with β > 0) will allocate more spectrum tothe operator with the low spectrum allocation (henceless spectrum to the other operator).

Corollary 2 shows that if the income-centric ap-proach leads to a very imbalanced spectrum allocationacross two operators (one low and one high), makingthe allocation more balance can improve the users’aggregate utility.

Corollary 3: More spectrum allocation to 4G servicewhen both service have low spectrum level. If both opera-tors receive low spectrum allocations (based on theirown thresholds, respectively) in the income-centricbenchmark (with β = 0), then our proposed scheme(with β > 0) will allocate more spectrum to the 4Goperator (hence less spectrum to the Wi-Fi operator).

Corollary 3 shows that the FCC tends to prioritizealleviating the spectrum shortage of 4G network first,

partly due to the 4G service’s wider coverage andhigh spectrum efficiency.

According to Table 3, we also have the followinginsights:

• When both services have a high spectrum levelin the income-centric benchmark, the further con-sideration of users’ utility does not change thespectrum allocation. In the case of HwHg in Table3, the spectrum allocation results are the sameas the income-centric spectrum allocation. Thisshows that when spectrum resource is abundant,maximizing the FCC’s income does not hurtusers’ utility.

• The equilibrium spectrum allocation is closelyrelated to the Wi-Fi network coverage θ andspectrum efficiency η. For example, in the caseof LwLg in Table 3, if θ or η increases, thespectrum allocation to Wi-Fi Sw increases (sinceφ′w(Sw)− φ′

g(S − Sw) decreases). This means thatthe FCC will allocate more spectrum to Wi-Fi ifthe Wi-Fi’s network coverage or spectrum effi-ciency are improved (hence become similar as the4G network).

7.3 Subgame Perfect Equilibrium of the EntireGame

By substituting Sw and Sg in Table 1 and 2 withS∗w and S − S∗

w, we get the SPE service prices andsubscriber numbers. Now we analyze how the consid-eration of user utility affects the equilibrium prices inStage II and subscriber number in Stage III, comparedwith the income-centric spectrum allocation. We focuson the first three cases, as our scheme is the same asthe income-centric benchmark in the HwHg case.

7.3.1 Case LwLg

In the SPE, the FCC’s utility becomes

Uf = φg(S∗g ) + φw(S

∗w)︸︷︷︸

FCC′s income

+βθf(ηS∗

w) + f(S∗g )

δN︸︷︷︸user aggregate utility

(18)

As some spectrum is moved from Wi-Fi to 4G(compared with the income-centric benchmark), thevalue of f(ηSw) + f(Sg) will increase. According toTable 1, both service prices pg and pw decrease. It iseasy to understand for the 4G operator, which reducesprice pg to attract more users because the 4G capacityincreases. Counter-intuitively, the Wi-Fi operator alsoreduces price pw, even though the Wi-Fi capacitydecreases. The reason is that the 4G service, with ahigher spectrum efficiency, is able to support more



10

( )gf S

d

gp

gN

0*

gp*

gp

(a) 4G subscribers

( )w

w

f Sh

d

wp

wN

0*

wp*

wp

(b) Wi-Fi subscribers

Fig. 5: Change of subscriber number in case of lowWi-Fi spectrum and low 4G spectrum.

users with the same amount of spectrum comparedwith the Wi-Fi. Therefore, the subscriber churn fromWi-Fi to 4G reduces the need for Wi-Fi spectrum morethan the amount of spectrum that has been moved to4G. Hence the Wi-Fi operator needs to reduce pricepw to attract more low-end subscribers, to utilize theavailable Wi-Fi spectrum due to subscriber churn.

The change of subscriber numbers is shown inFig. 5. The solid lines are the income-centric bench-mark (with the corresponding equilibrium prices p0∗gand p0∗w ). The dashed lines are the allocation underour proposed scheme (with the corresponding equi-librium prices p∗g and p∗w). The 4G subscriber numberNg increases due to the following reasons: 1) The4G capacity increases, and can serve more users;2) The 4G price pg decreases more than the Wi-Fiprice pw, making some previous price sensitive Wi-Fi subscribers churn to 4G service.

The Wi-Fi subscriber number Nw decreases due tothe decrease of Wi-Fi capacity (according to Table 2,in the case of LwLg, the Wi-Fi subscriber number isdetermined by the Wi-Fi capacity).

If we consider two operators’ subscribers together,then the total number of 4G and Wi-Fi subscribersincreases, as the decrease in Wi-Fi service price at-tracts lower-end non-subscribers who previously didnot choose any services.

The users’ aggregate utility increases because: 1)Some Wi-Fi subscribers churn to 4G service, enjoyinga better network coverage; 2) Some previous non-subscribers now choose the Wi-Fi service, enjoying abetter network coverage.7.3.2 Case HwLg


( )gf S

d

gp

gN

0*

gp*

gp

(a) 4G subscribers

( )w

w

f Sh

d

wp

wN

0*

wp*

wp

(b) Wi-Fi subscribers

Fig. 6: Change of subscriber number in case of highWi-Fi spectrum and low 4G spectrum.


∗w)︸︷︷︸

FCC′s income

+β

((1 − θ)

(2− θ)N

f(S∗g )

δ+

θ

2− θ

)

︸︷︷︸user aggregate utility

(19)Compared with the income-centric benchmark, the

proposed spectrum allocation has a higher optimal 4Gspectrum allocation S∗

g (than S0∗g ), so both prices pg

and pw are lower according to Table 1. The 4G opera-tor reduces price pg for a similar reason as in the caseof LwLg . The Wi-Fi operator reduces price pw also dueto the subscriber churn from Wi-Fi to 4G. Notice thatin the case of HwLg, the decrease in Wi-Fi capacityno longer affects the Wi-Fi operator’s decision, sincethe Wi-Fi capacity is excessive compared with its userdemand.

The change of subscriber numbers is shown inFig. 6. The major difference between Fig. 5 and Fig. 6is: Although Nw decreases in both cases (as the spec-trum is reallocated from Wi-Fi to 4G), Nw equals to theWi-Fi capacity in the LwLg case, but is always lowerthan the Wi-Fi capacity in the HwLg case.

7.3.3 Case LwHg



∗w)︸︷︷︸

FCC′s income

+β

((1− θ)θ

(2− θ)N

f(ηS∗w)

δ+

1

2− θ

)

︸︷︷︸user aggregate utility

(20)Compared with the income-centric benchmark, S∗

w

is higher than S0∗w , so pg and pw both decrease accord-

ing to Table 1.



11

0 A

No service 4G service Wi-Fi service

0 A

No service 4G service Wi-Fi service

Fig. 7: User flow in the case of HwLg (upper plot) andLwHg (lower plot).

Intuitively, HwLg and LwHg should have sym-metrical interpretations. However, they are differentin terms of user flow as shown in Fig. 7. In thecase of HwLg, the increase of 4G users comes fromprevious Wi-Fi subscribers, and Wi-Fi attracts non-subscribers who previous choose no wireless service.The users’ aggregate utility increases, because eachof the churning subscribers (from Wi-Fi to 4G) andthe new subscribers (from nothing to Wi-Fi) enjoy ahigher utility. In the case of LwHg , the increase of Wi-Fi subscribers comes from both previous 4G users andnew users who previously choose nothing. Althoughthe churning subscribers’ service quality (from 4G toWi-Fi) decreases, such loss is compensated by theincreased utility of new subscribers (from choosingnothing to Wi-Fi).

In summary, in all of the above three cases, whenuser utility is considered (with β > 0), the proposedspectrum allocation will reduce the service price ofboth services (compared with income-centric bench-mark with β = 0). Some lower-end non-subscriberswho previously did not choose any services will nowchoose Wi-Fi service. The total number of 4G and Wi-Fi subscribers and their aggregate utility increase.

8 BOUNDED ALLOCATION INCOME LOSS

In this section, we analyze the loss of FCC’s income(compared with the income-centric benchmark), as-suming that the spectrum charge functions (φg(·) andφw(·)) and the capacity function (f(·)) are polyno-mial18. We show that the income loss can be analyti-cally upper bounded, and we examine how such lossdepends on several critical parameters including theweight for user utility β, the Wi-Fi spectrum efficiencyη, and the Wi-Fi network coverage θ. We furthercharacterize the increase of the users’ aggregate utilitythat accompanies the loss of FCC’s income. We againfocus on the three cases of LwLg, HwLg, and LwHg.

8.1 Analysis of income loss and aggregate userutility bounds

We define the user utility ratio as the ratio betweenthe users’ aggregate utility of our proposed spectrumallocation and that of the income-centric benchmark,

18. Here, we study polynomial functions for simplicity. In thefuture, we will consider more complex functions and their influenceon the FCC’s income and users’ utility.

denoted by Rutility . We also define the FCC’s incomeratio in a similar fashion, denoted by Rincome. In-tuitively, Rutility ≥ 1 and 0 < Rincome ≤ 1. Weare interested in characterizing the upper-bound ofRutility and the lower-bound of Rincome.

The FCC charges the Wi-Fi and 4G operators basedon functions φw(·) and φg(·) respectively. We assumethat 1) the operators are willing to pay more formore spectrum, i.e., φw(·) and φg(·) are increasingfunctions; 2) the operators’ willingness to pay for eachadditional spectrum decreases with the total spectrumallocation (“diminishing returns”), i.e., φ′

w(·) and φ′g(·)

(the first order derivatives) are decreasing functions.As a concrete example, we assume that φw and φg arenegative quadratic functions which satisfy the abovetwo properties. For simplicity, we assume that thenetwork capacity function f(·) is a linear function. Inparticular, we have

φg(Sg) = −aS2g + bSg, φw(Sw) = −cS2

w + dSw,

f(Sg) = eSg, f(ηSw) = eηSw

(21)

where a, b, c, d, e are positive parameters. We furtherassume that S < min{b/(2a), d/(2c)}, which guaran-tees that φ′

g > 0, φ′′g < 0, φ′

w > 0, and φ′′w < 0.

Under the income-centric benchmark, we have

Wi-Fi Spectrum allocation : S0∗w =

2aS + d− b

2(a+ c)

FCC’s Income: − aS2 + bS +(2aS + d− b)2

4(a+ c)

8.1.1 The case of LwLg:According to Table 3, we have

S∗w =

1

2(a+ c)

[2aS + d− b− (1− θη)

eβ

δN

]. (22)

As β increases, the FCC cares more about userutility. According to Corollary 3, the FCC reallocatesmore spectrum from Wi-Fi to 4G, so S∗

w decreases.We can prove that the minimum FCC’s income is−aS2 + bS > 0 when S∗

w = 0 (with a large enoughβ). Therefore, the income ratio is lower-bounded by

Rincome ≥−aS2 + bS

−aS2 + bS + (2aS+d−b)2

4(a+c)

(23)

Correspondingly, the user utility ratio is the highestwhen S∗

w = 0 with the following upper-bound:

Rutility =S − (1− θη)S∗

w

S − (1− θη)S0∗w

≤S

S − (1 − θη)S0∗w

(24)

The basic reason for the existence of such an upperbound is due to the limited spectrum that FCC canallocate. If the Wi-Fi spectrum efficiency η increases,the upper-bound of Rutility decreases, because thespectrum efficiency difference between Wi-Fi and 4Gis smaller, and reallocating spectrum to 4G no longersignificantly increases the overall network capacity(hence the users’ utility). If the Wi-Fi coverage θ



12

increases, the upper-bound of Rutility also decreases,because the coverage difference between Wi-Fi and 4Ggets smaller, and the Wi-Fi subscribers who churn to4G obtain a smaller utility improvement.

8.1.2 The case of HwLg:

According to Table 3, we have

S∗w =

1

2(a+ c)

[2aS + d− b−

eβ(1− θ)

δN(2− θ)

].

The income ratio is lower-bounded by the samevalue as in (23). However, the user utility ratio hasa different upper-bound as follows:

Rutility ≤(1− θ)eS +Nδθ

(1− θ)e(S − S0∗w ) +Nδθ

(25)

The upper-bound of Rutility decreases in the Wi-Ficoverage θ for the same reason as in the case of LwLg.However, the upper-bound of Rutility is independentof the Wi-Fi spectrum efficiency η. This is becausein this case of HwLg, the Wi-Fi subscriber numberis lower than the Wi-Fi capacity (determined by η)according to Table 2, so neither the Wi-Fi subscribernumber change nor the aggregate utility improvementdepend on η. The upper-bound of Rutility decreases inthe total number of users N , because the equilibriumprice gap pg − pw increases with N according toTable 1. And the increasing price gap deters Wi-Fisubscribers from churning to 4G for better QoS, sothe utility improvement decreases, that is, the upper-bound of Rutility decreases.

8.1.3 The case of LwHg:

According to Table 3, we have

S∗w =

1

2(a+ c)

[2aS + d− b+

eβ(1− θ)θη

δN(2− θ)

].

As β increases, according to Corollary 2, the FCCallocates more spectrum to Wi-Fi and S∗

w increases. Wecan prove that the minimum FCC’s income is −cS2+dS when S∗

w = S.Therefore, the income ratio is lower-bounded by

Rincome ≥−cS2 + dS

−aS2 + bS + (2aS+d−b)2

4(a+c)

(26)

The user utility ratio is the highest when S∗w = S:

Rutility =(1− θ)θηeS∗

w +Nδ

(1 − θ)θηeS0∗w +Nδ

≤(1− θ)θηeS +Nδ

(1 − θ)θηeS0∗w +Nδ

(27)If the Wi-Fi spectrum efficiency η increases, the upper-bound of Rutility increases, because the spectrum effi-ciency gap between Wi-Fi and 4G is smaller, and real-locating spectrum to Wi-Fi does not decrease much ofthe capacity. The influence of a higher Wi-Fi coverageθ is two-fold. On one hand, more non-subscribers willbecome Wi-Fi subscribers, enjoying a higher networkcoverage. On the other hand, more 4G subscriberswill become Wi-Fi subscribers, hence having a lower

network coverage. Which of the two factors domi-nates depends on the value of θ. When θ > 1/2, theupper-bound of Rutility decreases in θ, because thesubscriber churn from 4G to Wi-Fi is the dominatingfactor. When θ < 1/2, the upper-bound of Rutility

increases in θ, because non-subscribers switching toWi-Fi subscribers becomes the dominant factor. Theupper-bound of Rutility decreases in N , because theequilibrium price pw increases with N according toTable 1. Non-subscribers are less likely to choose Wi-Fi service, so the utility improvement decreases.

8.2 Impact of User Utility Weight β

Fig. 8 shows the impact of weight β in the caseof LwLg. When β increases, the FCC emphasizesmore on the user utility compared with the income-centric benchmark. Therefore, the user utility ratioincreases and the income ratio decreases with β. Whenβ becomes very large, S∗

w becomes zero according to(22) and S∗

g = S, hence the two curves in Fig. 8(a)become flat (when β > 3.5). As we have proved, theuser utility ratio is upper-bounded and the incomeratio is lower-bounded as shown in Fig. 8(a). Assummarized in Section 7.3, the consideration of userutility will drive down prices of both services. Thehigher the β, the lower the service prices will be, asshown in Fig. 8(b). No matter how the subscribernumber (Nw and Ng) changes, the total Wi-Fi and4G subscriber number increases using the proposedspectrum allocation, and the higher β is, the higherthe increase will be, as shown in Fig. 8(c).

In the case of HwLg, the influences of β on user util-ity ratio, income ratio, service prices, and subscribernumbers are similar as Fig. 8, but the slopes of thecurves are steeper. This is because the reallocation ofspectrum from Wi-Fi to 4G is more significant as Wi-Fihas a high spectrum level.

In the case of LwHg , the influences of β on userutility ratio, income ratio, service prices are similar asFig. 8(a) and Fig. 8(b), but the change of Wi-Fi and 4Gsubscriber numbers are just opposite to Fig. 8(c) dueto different directions of spectrum reallocation.

In all three cases, the total subscriber number in-creases in β, since the proposed spectrum allocationimproves the aggregate user utility by covering morelow-end users (with a small α) with the Wi-Fi service.

8.3 Impact of Wi-Fi Spectrum Efficiency η

Fig. 9 shows the impact of Wi-Fi spectrum efficiencyη in the case of LwLg. Interestingly, when η increases(hence Wi-Fi can better utilize the spectrum), the userutility ratio decreases, as shown in Fig. 9(a). Thereason is that if η is already high, reallocating spec-trum from Wi-Fi to 4G does not significantly improvethe spectrum utilization, thus does not improve theusers’ aggregate utility much. Therefore, the FCC willreallocate less spectrum from Wi-Fi to 4G, resulting ina smaller utility ratio and a higher income ratio. Asη increases, the Wi-Fi capacity increases, so the Wi-Fi



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0.5 1.5 2.5 3.5 4.5

0.5

1

1.5

2

2.5

3

Weight β

Rat

io

IncomeUser utility

(a) Income and user utility ratio

0.5 1.5 2.5 3.5 4.50.2

0.3

0.4

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0.6

0.7

0.8

0.9

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Weight β

Pric

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(b) Service prices

0.5 1.5 2.5 3.5 4.50

0.1

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0.4

0.5

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Subs

crib

er n

umbe

r

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(c) User distribution

Fig. 8: Impact of weight β.

0.1 0.3 0.5 0.70.8

1

1.2

1.4

1.6

1.8

Wi−Fi spectrum efficiency η

Rat

io

IncomeUser utility

(a) Income and user utility ratio

0.1 0.3 0.5 0.70.4

0.5

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4GWi−Fi

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0.1 0.3 0.5 0.70

0.1

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crib

er n

umbe

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4GWi−FiTotal

(c) User distribution

Fig. 9: Impact of Wi-Fi spectrum efficiency η.

operator decreases pw to attract more users, as shownin Fig. 9(b). This increases the market competition,hence forces 4G operator to decrease pg to try to keep(most of) its subscribers. As shown in Fig. 9(c), the Wi-Fi subscriber number increases due to an increasedWi-Fi capacity; the 4G subscriber number decreasesdue to competition from Wi-Fi; the entire user numberincreases thanks to the improved spectrum efficiencyof Wi-Fi network. In the case of HwLg, the changeof η does not affect anything. The reason is thatthe Wi-Fi spectrum level is high, so the influence ofcapacity constraint (f(ηSw)/δ) does not exist. In thecase of LwHg, the influences of η on user utility ratio,income ratio, service prices, and subscriber numberare similar to those in Fig. 9.

8.4 Impact of Wi-Fi Network Coverage θ

The influence of Wi-Fi network coverage θ is similar tothe influence of Wi-Fi spectrum efficiency η, as theyboth reflect the performance of Wi-Fi network. Themajor difference is the impacts of these two factorson the service prices. Fig. 10 shows the case of LwHg :when the Wi-Fi coverage increases, the two serviceprices change in different directions. The 4G pricepg decreases because Wi-Fi competes with 4G moreintensely, hence the 4G operator has to reduce the4G price to maintain its subscribers. The Wi-Fi serviceprice pw is influenced by two factors: (i) the need todecrease to match the decreased cellular price pg dueto competition, and (ii) the need to increase to reflecta better Wi-Fi coverage. Fig. 10 shows that the secondfactor dominates and pw increases with its coverage.

0.3 0.5 0.7 0.90.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Wi−Fi network coverage θ

Pric

e

4GWi−Fi

Fig. 10: Impact of Wi-Fi network coverage on serviceprices.

9 CONCLUSION

In this paper, we propose a novel spectrum allocationscheme that enables the FCC to take into accountboth the income and the users’ utility. We model thewireless market interactions as a 3-stage game, whichinvolves the FCC, the Wi-Fi and the 4G operators, andall wireless users. We use backward induction to firstcalculate users’ subscription choice in Stage III, thenderive the equilibrium prices of both Wi-Fi and 4Gservices in Stage II, and finally obtain the equilibriumspectrum allocation for the FCC to maximize theweighted sum of the income and the users’ aggregateutility in Stage I. Comparing with the income-centricbenchmark, we show that the consideration of users’aggregate utility will make FCC balance the spectrumallocation between two operators. We further charac-terize the lower-bound of the income loss ratio of theproposed spectrum allocation to the income-centricbenchmark, and provide detailed discussions regard-



14

ing the impacts of weight β, Wi-Fi spectrum efficiencyη, and Wi-Fi network coverage θ on the spectrumallocation, service prices, and user subscription.

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YANJIAO CHEN received her B.E. degree ofelectronic engineering from Tsinghua Univer-sity in 2010. She is currently a Ph.D. can-didate in Hong Kong University of Scienceand Technology. Her research interests in-clude spectrum management for Femtocellnetworks, network economics and quality ofexperience (QoE) of multimedia in wirelessnetworks.

LINGJIE DUAN (S’09-M’12) received thePhD degree from the Chinese Universityof Hong Kong in 2012. He is an assistantprofessor of engineering systems and de-sign pillar, Singapore University of Technol-ogy and Design. During 2011, he was avisiting scholar in the Department of EECS,University of California at Berkeley. His re-search interests include network economicsand game theory, network optimization, andenergy harvesting. He is the TPC Co-Chair

of INFOCOM 2014 Workshop on GCCCN and the Co-Chair of theICCS 2014 Special Issue on “Network Economics and Communica-tion Theory”.

JIANWEI HUANG (S’01-M’06-SM’11) is anAssociate Professor and Director of the Net-work Communications and Economics Lab inthe Department of Information Engineeringat the Chinese University of Hong Kong. Hereceived his Ph.D. degree from Northwest-ern University in 2005. Dr. Huang is the co-recipient of 7 Best Paper Awards. He hasserved as the Editor of IEEE Journal onSelected Areas in Communications - Cog-nitive Radio Series and IEEE Transactions

on Wireless Communications, and Guest Editor of IEEE Journalon Selected Areas in Communications and IEEE CommunicationsMagazine.

QIAN ZHANG is a full Professor in the De-partment of Computer Science and Engi-neering, Hong Kong University of Scienceand Technology. Dr. Zhang has publishedabout 300 refereed papers in internationalleading journals and key conferences in theareas of wireless/Internet multimedia net-working, wireless communications and net-working, and wireless sensor networks. Sheis a Fellow of IEEE for “contribution to themobility and spectrum management of wire-

less networks and mobile communications”. Dr. Zhang has receivedMIT TR100 (MIT Technology Review) worlds top young innovatoraward. She also received the Best Asia Pacific Young ResearcherAward elected by IEEE Communication Society in year 2004.

Documents

Balancing Income and User Utility in Spectrum Allocationncel.ie.cuhk.edu.hk/sites/default/files/07035108.pdfgrant from Huawei-HKUST joint lab, the General Research Funds (Project Number