Economic Analysis of 4G Network UpgradeLingjie Duan, Jianwei Huang, and Jean Walrand

AbstractAs the successor to the 3G standard, 4G providesmuch higher data rates to address cellular users ever-increasingdemands for high-speed multimedia communications. This paperanalyzes the cellular operators timing of network upgrades andmodels that users can switch operators and services. Being thefirst to upgrade 3G to 4G service, an operator increases hismarket share but takes more risk or upgrade cost because 4Gtechnology matures over time. This paper first studies a 4Gmonopoly market with one dominant operator and some smalloperators, where the monopolist decides his upgrade time bytrading off increased market share and upgrade cost. The paperalso considers a 4G competition market and develops a gametheoretic model for studying operators interactions. The analysisshows that operators select different upgrade times to avoidsevere competition. One operator takes the lead to upgrade, usingthe benefit of a larger market share to compensate for the largercost of an early upgrade. This result matches well with manyindustry observations of asymmetric 4G upgrades. The paperfurther shows that the availability of 4G upgrade may decreaseboth operators profits due to increased competition. Perhapssurprisingly, the profits can increase with the upgrade cost.

I. INTRODUCTION

The third generation (3G) of cellular wireless networks waslaunched during the last decade. It has provided users withhigh-quality voice channels and moderate data rates (up to2 Mbps). However, 3G service cannot seamlessly integratethe existing wireless technologies (e.g., GSM, wireless LAN,and Bluetooth) [1], and cannot satisfy users fast growingneeds for high data rates. Thus, most major cellular operatorsworldwide plan to deploy the fourth-generation (4G) networksto provide much higher data rates (up to hundreds of megabitsper second) and integrate heterogeneous wireless technologies.The 4G technology is expected to support new services suchas high-quality video chat and video conferencing [3].

One may expect competitive operators in the same cellularmarket to upgrade to a 4G service at about the same time.However, many industry examples show that symmetric 4Gupgrades do not happen in practice, even when multiple op-

L. Duan is with the Engineering Systems and Design Pillar, SingaporeUniversity of Technology and Design (email: swing.dlj@gmail.com).

J. Huang is with the Network Communications and Economics Lab,Department of Information Engineering, The Chinese University of HongKong (email: jwhuang@ie.cuhk.edu.hk).

J. Walrand is with the Department of Electrical Engineering and ComputerSciences, University of California at Berkeley, California- 94720 (email:wlr@eecs.berkeley.edu).

This work is supported by SUTD-MIT International Design Center Grant(Project no.: IDSF1200106OH) and SUTD Start-up Research Grant (Projectno.: SRG ESD 2012 042). This work is also supported by the GeneralResearch Funds (Project Number CUHK 412710 and CUHK 412511) es-tablished under the University Grant Committee of the Hong Kong SpecialAdministrative Region, China, and the NSF-NetSE grants 1024318 and0910702.

erators have obtained the necessary spectrum and technologypatents for upgrade ( [3], [4]). In South Korea, for example,Korean Telecom took the lead to deploy the worlds first4G network using WiMAX technology in 2006, whereas SKTelecom started to upgrade using more mature LTE technologyin 2011. In US, Sprint deployed the first 4G WiMAX networkin late 2008, Verizon waited until the end of 2010 to deployhis 4G LTE network, and AT&T planed to deploy his 4G LTEnetwork at the end of 2011 [3]. In China, China Mobile andChina Unicom are the two dominant cellular operators, andChina Mobile has decided to first deploy 4G LTE networkduring 2012-2013 [4]. Thus, the key question we want toanswer in this paper is the following: How do the cellularoperators decide the timing to upgrade to 4G networks?

In this paper, we analyze the timing of operators 4Gupgrades in different models, including both a 4G monopolymarket and a 4G competition market. Operators need to paythe cost of 4G upgrade, which decreases over time as 4Gtechnology matures. There are two key factors that affect theoperators upgrade decisions: namely, 4G upgrade cost anduser switching cost. An existing 3G user can switch to the4G service of the same operator or of a different operator,depending on how large the switching cost is. In a monopolymarket where only a dominant operator can choose to upgradeto 4G, this operator can use the 4G service to capture a largermarket share from small operators. In a competition marketwhere multiple operators can choose to upgrade, we analyzethe operators interactions as a non-cooperative game. Westudy how the users inter-network switching cost affects theoperators upgrade decisions, and our findings are consistentwith the asymmetric upgrades observed in the industry.

Our key results and contributions are as follows. A revenue-sharing model between operators: Most ex-

isting works only study a single networks revenue byexploring the network effect (e.g., [7]), and the resultsmay not apply in a competitive market. In Section III, westudy two interconnected networks, where their operatorsshare the revenue of the inter-network traffic.

Monopolists optimal timing of 4G upgrade: By upgrad-ing early, the 4G monopolist in Section IV obtains a largemarket share and a large revenue because of 4Gs Qualityof Service (QoS) improvement, but it cannot benefit fromthe cost depreciation over time. When the upgrade cost isrelatively low, he upgrades at the earliest available time;otherwise he postpones his upgrade.

Competitive operators asymmetric upgrades: In Sec-tions V and VI, users can switch operators. By upgradingearly, an operator captures a large market share and

978-1-4673-5946-7/13/$31.00 2013 IEEE

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the 4Gs QoS improvement which can compensate forthe large upgrade cost. The other operator, however,postpones his upgrade to avoid severe competition andbenefits from cost reduction. The availability of 4Gupgrade may decrease both operators profits because ofthe increased competition, and paradoxically, their profitsmay increase with the upgrade cost.

II. RELATED WORK

A. Network Effect and Network ValueIn telecommunications, the network effect is the added value

that a user derives from the presence of other users [13]. Ina network with N users, each user perceives a value that in-creases with N . A reasonable model was suggested by Briscoeet al. [7], where each user perceives a value of order log(N).In that model, a user ranks the other users in decreasing orderof importance and assigns a value 1/k to the k-th user in thatorder, for a total value 1+1/2+ +1/(N1) log(N). Theresulting total network value isN log(N), which is appropriatefor cellular networks shown by quantitative studies [7].

B. Network UpgradeRecently, there has been a growing interest in studying the

economics of network upgrades [8][10]. Musacchio et al. [8]studied the upgrade timing game between two interconnectedInternet Service Providers (ISPs), where one ISPs architectureupgrade also benefits the other because of the network effect.This free-riding effect may make the second operator postponehis upgrade or even never decide to upgrade. Jiang et al. [9]studied a network security game, where one users investment(upgrade) can reduce the propagation of computer viruses toall users. In our problem, however, one operator may benefitfrom the others upgrade only when he also upgrades, lettinghis 4G users communicate with existing 4G users in othernetworks. Moreover, our model characterizes the dynamicsof users switching between operators and/or services. Thesedynamics imply that an operator can obtain a larger marketshare by upgrading earlier, and this weakens the free-ridingeffect. Sen et al. [10] studied the users adoption and diffusionof a new network technology in the presence of an incumbenttechnology. Our work is different from that study in that weare not focusing on technology competition to attract users,but on the operators competition in upgrade timing to obtaingreater profits. Moreover, the switching cost is not consideredin [10], whereas it is an important parameter of our model.

III. SYSTEM MODEL

A. Value of Cellular NetworksIn this paper, we adopt the N log(N) Law, where the

network value with N users is proportional to N log(N). Theoperator of a cellular network prefers a large network value;this is because the revenue he obtains by charging users canbe proportional to the network value. Notice that the valueof a 4G network is larger than a 3G network even when twonetworks have the same number of users. This is because thecommunication between two 4G users is more efficient and

more frequent than between two 3G users. Because the averagedata rate in the 4G service is 5-10 times faster than the 3G(both downlink and uplink), a 4G network can support manynew applications. We denote the efficiency ratio between 3Gand 4G services as (0, 1). That is, by serving all his usersvia QoS-guaranteed 4G rather than 3G services, an operatorobtains a larger (normalized) revenue N log(N) instead ofN log(N).1 Note that this result holds for a single operatorsnetwork that is not connected to other networks.

Next we discuss the revenues of multiple operators whosenetworks (e.g., two 3G networks) are interconnected. For thepurpose of illustration, we consider two networks that containN1 and N2 users, respectively. The whole market coversN = N1 +N2 users. We assume that two operators 3G (andlater 4G) services are equally good to users, and the efficiencyratio is the same for both operators. The traffic between twousers can be intra-network (when both users belong to thesame operator) or inter-network (when two users belong todifferent operators), and the revenue calculations in the twocases are different. We assume that the user who originatesthe communication session (irrespective of whether the samenetwork or to the other network) pays for the communication.This is motivated by the industry observations in EU and manyAsian countries.2 Before analyzing each operators revenue,we first introduce two practical concepts in cellular market:termination rate and user ignorance.

When two users of the same operator 1 communicate witheach other, the calling user only pays operator 1. But when anoperator 1s user calls an operator 2s user, operator 2 chargesa termination rate for the incoming call [11].3 We denote thetwo operators revenue-sharing portion per inter-network callas , where the value of (0, 1) depends on the agreementbetween the two operators or on governments regulation ontermination rate.

User ignorance is a unique problem in the wireless cellularnetwork, where users are often not able to identify whichspecific network they are calling. Mobile number portabilityfurther exacerbates this problem [14]. Thus a typical usersevaluation of two interconnected 3G networks does not dependon which network he belongs to, and equals log(N) whereN = N1 +N2. We assume a call from any user terminates ata user in network i {1, 2} with a probability of Ni/N asin [14]. The operators revenues when they are both providing3G services are given in Lemma 1.

Lemma 1: When operators 1 and 2 provide 3G services,their revenues are N1 log(N) and N2 log(N), respectively.

The proof of Lemma 1 is given in our online technical report

1We assume that an operators operational cost (proportional to networkvalue) has been deducted already, and thus the revenue in this paper representsa normalized one.

2Our model can also be extended to the case where both involved users in acommunication session pay for their communication. This is what happeningin US cellular market.

3In the US, termination rate follows Bill and Keep and is low. Thenoperator 1 can keep most of the calling users payment. In EU, however,termination rate follows Calling Party Pays and is much higher. Then mostof the calling users payment to operator 1 is used to compensate for thetermination rate charged by operator 2 [11].

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[20]. Both operators revenues are linear in their numbers ofusers (or market share), and are independent of the sharingportion of the inter-network revenue. Intuitively, the inter-network traffic between two networks is bidirectional: whena user originates a call from network 1 to another user innetwork 2, his inter-network traffic generates a fraction ofcorresponding revenue to operator 1; when the other user callsback from network 2 to network 1, he generates a fraction1 of the same amount of revenue to operator 1. Thusan operators total revenue is independent of . Later, inSection IV, we show that such independence on also applieswhen the two operators both provide 4G services or providemixed 3G and 4G services.

B. User Churn during Upgrade from 3G to 4G ServicesWhen 4G service becomes available in the market (offered

by one or both networks), the existing 3G users have anincentive to switch to the new service to experience a betterQoS. Such user churn does not happen simultaneously for allusers, this is because different users have different sensitivitiesto quality improvements and switching costs [12]. We usetwo parameters and to model the user churn within andbetween operators:

Intra-network user churn: If an operator provides 4G inaddition to his existing 3G service, his 3G users need tobuy new mobile phones to use the 4G service. The usersalso spend time to learn how to use the 4G service ontheir new phones. We use to denote the users switchingrate to the 4G service within the same network.

Inter-network user churn: If a 3G user wants to switchto another networks 4G service, he either waits till hiscurrent 3G contract expires, or pays for the penalty ofimmediate contract termination. This means that inter-network user churn incurs an additional cost on top ofthe mobile device update, and thus the switching rate willbe smaller than the intra-network user churn. We use to denote the users inter-network switching rate to 4Gservice, where (0, 1) reflects the transaction cost ofswitching operators.

We illustrate the process of user churn through a continuoustime model. The starting time t = 0 denotes the time whenthe spectrum resource and the 4G technology are available forat least one operator (see Section IV for monopoly marketand Section VI for competition market). We also assume thatthe portion of users switching to the 4G service follows theexponential distribution (at rate for intra-network churn and for inter-network churn).

As an example, assume that operator 1 introduces a 4Gservice at time t = T1 while operator 2 decides not to upgrade.The numbers of operator 1s 4G users and 3G users at any timet 0 are N4G1 (t) and N3G1 (t), respectively. The number ofoperator 2s 3G users at time t 0 is N3G2 (t). As time tincreases (from T1), 3G users in both networks start to churnto 4G service, and t 0N3G1 (t) = N1e

max(tT1,0), N3G2 (t) = N2emax(tT1,0),

(1)

31 ( )GN t

0 1Tt

41 ( )GN t

32 ( )GN t

N

2N

Fig. 1. The numbers of users in the operators different services as functionsof time t. Here, operator 1 upgrades at T1 and operator 2 does not upgrade.

and operator 1s 4G service gains an increasing market share,N4G1 (t) = NN1emax(tT1,0)N2emax(tT1,0). (2)

We illustrate (1) and (2) in Fig. 1. We can see that operator 1searly upgrade attracts users from his competitor and increaseshis market share. Notice that (2) increases with , thusoperator 1 captures a large market share when is large (i.e.,the switching cost is low).

C. Operators Revenues and Upgrade CostsBecause of the time discount, an operator values the current

revenue more than the same amount of revenue in the future.We denote the discount rate over time as S, and the discountfactor is thus eSt at time t according to [16].

We approximate one operators 4G upgrade cost as a one-time investment. This is a practical approximation, as anoperators initial investment of wireless spectrum and infras-tructure can be much higher than the maintenance costs in thefuture. For example, spectrum is a very scarce resource thatis allocated (auctioned) infrequently by government agencies.Thus an operator cannot obtain additional spectrum frequentlyafter his 4G upgrade. To ensure a good initial 4G coverage,an operator also needs to update many base stations to coverat least a whole city all at once. Otherwise, 4G users wouldbe unhappy with the service, and this would damage theoperators reputation. That is why Sprint and Verizon coveredmany markets in their initial launch of their 4G services [18].

More specifically, we denote the 4G upgrade cost at t = 0as K, which discounts over time at a rate U . Thus if anoperator upgrades at time t, he needs to pay an upgradecost KeUt according to [16]. We should point out that theupgrade cost decreases faster than the normal discount rate(i.e., U > S). This happens because the upgrade cost decreasesbecause of both technology improvement and time discount.Very often the advance of technology is the dominant factorin determining U , and this is discussed further in Section VI.

Based on these discussions on revenue and upgrade cost,we define an operators profit as the difference between hisrevenue in the long run and the one-time upgrade cost. Withoutloss of generality, we will normalize an operators revenue rate(at any time t), total revenue, and upgrade cost by N log(N),where N is the total number of users in the market.4

4Our model and analytical results later can be extended to the case whereN increases over time. As new users prefer 4G service to 3G service, andcan easily switch at rate , operators will have more incentives to upgradeearlier.

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IV. 4G MONOPOLY MARKETWe first look at the case where only operator 1 can choose

to upgrade from 3G to 4G, while the other operators (oneor more) always offer the 3G service because of the lack offinancial resources or the necessary technology. This can be areasonable model, for example, for countries such as Mexicoand some Latin American ones, where America Movil is thedominant cellular operator in the 3G market. As the worldsfourth-largest cellular network operator, America Movil hasthe advantage over other small local operators in winningadditional spectrum via auctions and obtaining LTE patents,and he is expected to be the 4G monopolist in that area [15].

The key question in this section is how operator 1 shouldchoose his upgrade time T1 from the 3G service to the 4Gservice. T1 = 0 means that operator 1 upgrades at the earliesttime that the spectrum and technology are available, andT1 > 0 means that operator 1 chooses to upgrade later totake advantage of the reduction in the upgrade cost. Becauseof user churn from the 3G to the 4G service, the operatorsmarket shares and revenue rates change after time T1. For thatreason, we analyze periods t T1 and t > T1 separately.

Before 4G upgrade (t T1): Operator 1s and other op-erators market shares do not change over time. Operator1s revenue rate at time t is

3G3G1 (t) = N1N

,

which is independent of time t. His revenue during thistime period is

3G3G1,tT1 = T10

3G3G1 (t)eStdt =

N1SN

(1 eST1).(3)

After 4G upgrade (t > T1): Operator 1s market shareincreases over time, and the other operators total marketshare (denoted by N3G2 (t)/N ) decreases over time. Wedenote operator 1s numbers of 3G users and 4G users asN3G1 (t) and N

4G1 (t), respectively, and we have N

3G1 (t)+

N4G1 (t) +N3G2 (t) = N . This implies that

N3G2 (t) = (NN1)e(tT1), N3G1 (t) = N1e(tT1),andN4G1 (t) = N N1e(tT1) (N N1)e(tT1).

Note that a 3G users communication with a 3G or a4G user is still based on the 3G standard, and only thecommunication between two 4G users can achieve a high4G standard QoS. Operator 1s revenue rate is

4G3G1 (t) =N3G1 (t)

N+

N4G1 (t)

N

(N4G1 (t) + N

3G1 (t)

N+

N3G2 (t)

N

),

which is independent of the revenue sharing ratio between the calling party and receiving party. Operator1s revenue during this time period is then

4G3G1,t>T1 = T1

4G3G1 (t)eStdt, (4)

where t is an approximation of the long-term 4Gservice provision (e.g., one decade) before the emergenceof the next generation standard. This approximation is

reasonable since the revenue in the distant future becomesless important because of discount.

Figure 1 illustrates how the numbers of users of operatorsdifferent services change over time. Before operator 1s up-grade (e.g., t T1 in Fig. 1), the number of total users in eachnetwork does not change;5 after operator 1s upgrade, operator1s and the other operators 3G users switch to the new 4Gservice at rates and , respectively.

By considering (3), (4), and the decreasing cost KeUT1 ,operator 1s long-term profit when choosing an upgrade timeT1 is1(T1) =

3G3G1,tT1 +

4G3G1,t>T1

KeUT1

= eST1(1

S+ (1 )

(N1N

)22+ S

+ (1 )(NN1

N

)22+ S

)

eST1(2(1 )

N1N

+ S+ (2 )

NN1N

+ S

)KeUT1

+ 2eST1(1 )N1(NN1)

N2

(1 + )+ S+

N1

NS(1 eST1). (5)

We can show that 1(T1) in (5) is strictly concave in T1, thuswe can compute the optimal upgrade time T 1 by solving thefirst-order condition. The optimal upgrade time depends on thefollowing upgrade cost threshold in the monopoly 4G market,

Kmonoth =(1 )(N1N )

2

2+S +(NN1N )

2

2+S 2

N1N

+S +2

N1(NN1)N2

(1+)+S

U/S

+1 N1N (2 )NN1N S+S

U. (6)

Theorem 1: Operator 1s optimal upgrade time in a 4Gmonopoly market is:

Low cost regime (upgrade cost K Kmonoth ): operator 1upgrades at T 1 = 0.

High cost regime (K > Kmonoth ): operator 1 upgrades at

T 1 =1

U S log(

K

Kmonoth

)> 0. (7)

Intuitively, an early upgrade gives operator 1 a larger marketshare and enables him to get a higher revenue via the moreefficient 4G service. Such advantage is especially obvious inthe low cost regime where the upgrade cost K is small.

Next we focus on the high cost regime, and explore howthe network parameters affect operator 1s upgrade time.

Observation 1: Operator 1s optimal upgrade time T 1 in-creases with the upgrade cost K, and decreases with (i.e.,increases with the users inter-network switching cost).

The proofs of Observation 1 and the following observationsare given in our online technical report [20].

Observation 2 (Figure 2): When Kmonoth < K T2): Both operators have upgraded, and3G users only switch to the 4G service of their currentoperator. The numbers of users in operators differentservices are

N3G1 (t) =N

2e(tT1),

N4G1 (t) = N N

2e(tT1) N

2e(T2T1),

N3G2 (t) =N

2e(T2T1)(tT2),

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31 ( ) 2G NN t 32 ( ) 2

G NN t 31 ( )GN t 41 ( )

GN t 32 ( )GN t 31 ( )

GN t 41 ( )GN t 32 ( )

GN t42 ( )GN t

(Phase II)1T0 2T

t

(Phase III)1T0 2T

t

(Phase I)1T0 2T

t

Fig. 4. Users switches over the operators services: Phase I with 3G services only, Phase II with operator 1s 4G service, and Phase III with both operators4G services.

0 1Tt

31 ( )GN t

32 ( )GN t

41 ( )GN t

N

42 ( )GN t

2T

2N

Fig. 5. The numbers of users in the operators different services as functionsof time t. Here, operator 1 chooses his upgrade time at T1 and operator 2chooses T2 with T1 T2.

and

N4G2 (t) =N

2e(T2T1)

(1 e(tT2)

).

Figure 5 summarizes how users churn in the three phases.Similar to Section IV, we can derive operators revenue ratesbased on users churn over time. By integrating each operatorsrevenue rate over all three phases, we obtain that operatorslong-term revenue. Recall that an operators profit is thedifference between his revenue and the one-time upgrade cost.By further considering the symmetric case of T1 T2, wehave the following result.

Lemma 2: Consider two operators i, j {1, 2} (with i = j)upgrading at Ti and Tj . Operator is long-term profit is

i(Ti, Tj) =

{ER(Ti, Tj), if Ti Tj ;LT (Tj , Ti), if Ti Tj ,

(8)

where ER(Ti, Tj) and LT (Tj , Ti) are given in (9) and (10),respectively.

Note that an operators profit i(Ti, Tj) is continuous inhis upgrade time Ti. When operator is upgrade time Ti isless than Tj , he increases his market share at rate duringthe time period from Ti to Tj ; but when Ti > Tj , operatori loses his market share at rate during the period fromTj to Ti. This explains why we need two different functionsER(Ti, Tj) and LT (Tj , Ti) to completely characterize thelong-term profit for each operator.

B. Duopoly Upgrade GameNext we consider the non-cooperative game theoretical

interactions between two operators, where each of them seeksto maximize his long-term profit by choosing the best upgradetime.

Upgrade Game: We model the competition between twooperators as follows:

Players: Operators 1 and 2.

Strategy spaces: Operator i {1, 2} can choose upgradetime Ti from the feasible set Ti = [0,].6

Payoff functions: Operator i {1, 2} wants to maximizehis profit i(Ti, Tj) defined in (8).

Notice that we consider a static game here, where bothoperators decide when to upgrade at the beginning of time.This is motivated by the fact that operators usually makelong-term decisions in practice rather than changing decisionsfrequently, as many upgrade operations (e.g., financial budgetand technological trials) need to be planned and prepared. Aswe consider that each operator has complete information abouthis competitors and users parameters, he will not deviatefrom his initial decision as time goes on. Also, it should bepointed out that in many competition markets operators canobtain available resource for 4G upgrade at a similar time aswe mentioned at the beginning of this section. After makingdecision at t = 0, one operator does not change his decisionlater on. This is reasonable when operators can predict thefuture 4G market adoption.7

In Section VI, we analyze the duopoly upgrade game underdifferent switching costs (i.e., the value of (0, 1)).Nash equilibrium is a commonly used solution concept fora static game. At a Nash equilibrium, no player can increasehis payoff by deviating unilaterally [19]. We are interestedin characterizing the conditions under which an asymmetricupgrade equilibrium emerges between symmetric operators.

VI. 4G COMPETITION MARKET: PRACTICALINTER-NETWORK SWITCHING RATE

In this section, we consider the case of > 0, i.e., 3Gusers may switch to the 4G service of a different operator.The equilibrium analysis in this general case depends on therelationship between U (upgrade cost discount rate) and S(money depreciation rate). We assume that U is much largerthan S, i.e., U > S + . This represents the practical casewhere the advance of technology is the dominant factor indetermining U , and not many 3G users choose to switchoperators when the 4G service is just deployed (i.e., small )[17]. For example, Sprint deployed the first 4G network in USby using WiMAX technology in 2008 when LTE technology

6Note that Ti = means that operator i never upgrades.7Operators can predict the future market adoption by exploring historical

records of the market and some trial of 4G deployment. In the future we willstudy the incomplete information case, where an operator may learn moreinformation as the time goes and revise his upgrade decision (if he has notupgraded yet).

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ER(Ti, Tj) = + (2 )eSTi e(TjTi)STj

2S (2 )(e

STi e(TjTi)STj )2(+ S)

(1 ) eSTi

+ S

+1 2

(eSTi + e(1+)(TjTi)STj

2(2+ S)+

eSTi e(1+)(TjTi)STj(1 + )+ S

+eSTi e2(TjTi)STj

2(2+ S)

)

1 2(+ S)

e(TjTi)STj(1 e

(TjTi) + e(TjTi)

2

)KeUTi . (9)

LT (Tj , Ti) =e(TiTj)STi

2

(1

S 1

+ S 1

2

(e(TiTj) + e(TiTj)

)( 1+ S

12+ S

))+

2S(1 eSTj )

+

2(+ S)(eSTj e(TiTj)STi )KeUTi . (10)

was not mature yet. Only two years later, in 2010, LTE couldalready offer a much lower cost per bit than WiMAX [3]. From2012, LTE is expected to be the leading technology choice for4G networks. This example motivates that U is much largerthan S, and we will study whether the operators symmetric4G upgrades happen in this scenario.

Recall that by upgrading at T1 and T2, the operators receivethe profits given in (8). In game theory, one operators bestresponse function is his upgrade time that achieves the largestlong-term profit, as a function of a fixed upgrade time of theother operator [19]. A fixed point of the two operators bestresponse functions is the Nash equilibrium, and in generalthere can be more than one such fixed point.

We can show that the operators best responses functions(T best1 (T2) and T

best2 (T1)) depend on the upgrade cost K, and

in particular, they depend on two cost thresholds (Kcompth1 S+.

Next we illustrate numerically how the two operators bestresponse functions (T best1 (T2) and T

best2 (T1)) change with the

upgrade cost K.Figure 6 shows that each operators best response function

is discontinuous in the medium cost regime, and the two bestresponse functions with the same value of K intersect at twopoints: 0 = T 1 < T

2 and (symmetrically) 0 = T

2 < T

1 .

To illustrate this situation, consider operator 2s best responseT best2 (T1) in the case K = 0.062. If operator 1 upgrades earlysuch that T1 is less than 0.05, operator 2 does not upgradeat the same time to avoid a severe competition. If operator1 upgrades later such that T1 is larger than 0.05, operator2 chooses to upgrade earlier than operator 1 to increase hismarket share. Thus T best2 (T1) is discontinuous at T1 = 0.05.

Figure 7 shows that each operators best response functionis discontinuous in the high cost regime, and the two functions(with the same value of K) intersect at two points, equilibria0 < T 1 < T

2 and (symmetrically) 0 < T

2 < T

1 . Unlike

Fig. 6, the high cost here prevents any operator from choosingthe upgrade time t = 0.

Figure 8 summarizes how operators upgrade equilibriumchanges as cost K increases: starting with T i = T

j = 0

in low cost regime, then 0 = T i < Tj with increasing T

j in

medium cost regime, and finally 0 < T i < Tj with increasing

T 1 and T2 in high cost regime.

In the following theorem, we prove that the operators donot choose symmetric upgrades as long as the cost is not low.

Theorem 2: The two operators 4G upgrade equilibria sat-

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Operator 1s upgrade time T1

Ope

rato

r 2s

upg

rade

tim

e T 2

K=0.072:T2best(T1)

K=0.072:T1best(T2)

K=0.087:T2best(T1)

K=0.087:T1best(T2)

K=0.072: NE1

K=0.072: NE2

K=0.087: NE2

K=0.087: NE1

Fig. 7. Two operators best upgrade responses to each other according todifferent cost values in the high cost regime. Other parameters are = 0.5,N = 1000, U = 2, S = 1, = 1 and = 0.5 such that U > S + .

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

Operator 1s equilibrium time T1*

Ope

rato

r 2s

equ

ilibriu

m ti

me T 2*

NE 1: T1*T2

*

NE 2: T1*T2

*

Low cost regime:0=T

1*=T

2* as K

Medium cost regime:0=T

1* 0 as follows:

Earlier upgrade gives an operator the advantage to attractmore users (from the other operator), and enables theoperator to collect a higher revenue from the 4G service.

Later upgrade allows an operator to incur a reducedupgrade cost and to take advantage of the network effectin the 4G market (with more existing 4G users) when heupgrades.

In order to fully enjoy the two advantages of earlier or

0.4 0.45 0.5 0.55

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Operator 1s equilibrium profit 1*

Ope

rato

r 2s

equ

ilibriu

m p

rofit

2*

NE 1: T1*T2*

NE 2: T1*T2*

Medium cost regime:

1* S + ).

Next we study how operators equilibrium profits changewith cost K and the economic efficiency ratio between 3Gand 4G services in the three cost regimes. Note that an operatormay or may not be able to charge a significantly higher pricefrom a 4G user though 4G does improve a lot over 3G inQoS. Figures 9 and 10 show operators equilibrium profitsunder large and small values, respectively.

We first study the large scenario in Fig. 9 which means anoperator to charge a slightly higher price in 4G service than3G. Without loss of generality, we focus on the case whereoperator 1 upgrades no later than operator 2 (i.e., T 1 T 2 ).

In the low cost regime, by upgrading at T 1 = T2 = 0,

the two operators profits are the same and decrease withcost K.

In the medium cost regime, Fig. 9 shows that operator1 receives a larger profit than operator 2 by upgradingat T 1 = 0. Perhaps surprisingly, his profit increaseswith K, whereas operator 2s profit decreases with K.Intuitively, the increase of K encourages operator 2 tofurther postpone his upgrade and lose more users tooperator 1. The change of operator 1s profit trades offthe increases of his market share and upgrade cost. Asoperator 1s market share increases, his growing 4G userscommunicate more with his 3G users via the efficient 3Gservice under large . Operator 1s 3G revenue increasesbecause of a more efficient intra-network traffic, whichhelps compensate for the upgrade cost.

In the high cost regime, Fig. 9 shows that both operatorshave to postpone their upgrades and, surprisingly, bothoperators profits increase with K. As K increases,operator 1 further postpones his upgrade and operator

2013 Proceedings IEEE INFOCOM

1077

0.1 0.15 0.2 0.25 0.30.1

0.15

0.2

0.25

0.3

Operator 1s equilibrium profit 1*

Ope

rato

r 2s

equ

ilibriu

m p

rofit

2*

NE 1: T1*T2

*

NE 2: T1*T2

*

Low cost regime:

1*=

2* as K

Medium cost regime:

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