1

Economic Analysis of 4G Upgrade TimingLingjie Duan, Member, IEEE, Jianwei Huang, Senior Member, IEEE, and Jean Walrand, Fellow, IEEE

Abstract—As the successor to the 3G standard, the 4G cellular standard provides much higher data rates to address cellular users’ever-increasing demands for high-speed multimedia communications. This paper analyzes the cellular operators’ timing of networkupgrades, by considering user subscription dynamics induced by switching from 3G to 4G technologies. Being the first to upgrade 3Gto 4G service, an operator increases its market share but takes more risk or upgrade cost as 4G technology matures over time. Thispaper first studies a 4G monopoly market with one dominant operator and some small operators, where the monopolist decides itsupgrade time by trading off increased market share and upgrade cost. The paper also considers a 4G competitive market and developsa game theoretic model for studying operators’ interactions. The analysis shows that operators select different upgrade times to avoidsevere competition. One operator takes the lead to upgrade, using the benefit of a larger market share to compensate for the largercost of an early upgrade. This result matches well with many industry observations of asymmetric 4G upgrades. The paper furthershows that the availability of 4G upgrade may decrease both operators’ profits due to increased competition. Perhaps surprisingly, theprofits can increase with the upgrade cost.

Index Terms—etworks economics, 4Getworks economics, 4GN upgrade timing, user dynamics, game theory, Nash equilibrium.

F

1 INTRODUCTION

THE third generation (3G) of cellular wireless net-works was launched during the last decade. It has

provided users with high-quality voice channels andmoderate data rates (up to 2 Mbps). However, 3G servicecannot seamlessly integrate the existing wireless tech-nologies (e.g., GSM, wireless LAN, and Bluetooth) [2],and cannot satisfy users’ fast growing needs for highdata rates. Thus, most major cellular operators world-wide plan to deploy fourth-generation (4G) networksto provide much higher data rates (up to hundreds ofmegabits per second) and integrate heterogeneous wire-less technologies. 4G technology is expected to supportnew services such as high-quality video chat and videoconferencing [5].

One may expect competitive operators in the samecellular market to upgrade to a 4G service at aboutthe same time. However, many industry examples showthat symmetric 4G upgrades do not happen in practice,even when multiple operators have obtained the neces-sary spectrum and technology patents for upgrade ( [5],[6]). In South Korea, for example, Korean Telecom took

• Lingjie Duan is with the Engineering Systems and Design Pillar, Sin-gapore University of Technology and Design. Jianwei Huang is with theNetwork Communications and Economics Lab, Department of InformationEngineering, The Chinese University of Hong Kong. Jean Walrand iswith the Department of Electrical Engineering and Computer Sciences,University of California at Berkeley, California- 94720.E-mail addresses: lingjie duan@sutd.edu.sg, jwhuang@ie.cuhk.edu.hk, andwlr@eecs.berkeley.edu.

Part of the results will appear in IEEE INFOCOM, Turin, Italy, April 2013[1]. This work is supported by SUTD-MIT International Design CenterGrant (Project no.: IDSF1200106OH) and SUTD Start-up Research Grant(Project no.: SRG ESD 2012 042). This work is also supported by theGeneral Research Funds (Project Number CUHK 412710 and CUHK 412511)established under the University Grant Committee of the Hong Kong SpecialAdministrative Region, China, and the NSF-NetSE grants 1024318 and0910702 in the US.

the lead to deploy the world’s first 4G network usingWiMAX technology in 2006, whereas SK Telecom startedto upgrade using more mature LTE technology in 2011.In US, Sprint deployed the first 4G WiMAX network inlate 2008, Verizon waited until the beginning of 2011to deploy its 4G LTE network, and AT&T formallydeployed its 4G LTE network in 2013 [5]. In China, ChinaMobile and China Unicom are the two dominant cellularoperators, and China Mobile has decided to be the firstto deploy 4G LTE network during 2012-2013 [6]. Thus,the key question we want to answer in this paper is thefollowing: How do the cellular operators decide the timing ofan upgrade to 4G networks?1

In this paper, we analyze the timing of operators’4G upgrades in different settings, including both a 4Gmonopoly market and a 4G competitive market. Operatorsneed to pay the cost of 4G upgrade, which decreases overtime as 4G technology matures. There are two key factorsthat affect the operators’ upgrade decisions: namely, 4Gupgrade cost and user switching cost. An existing 3G usercan switch to the 4G service of the same operator or of adifferent operator, depending on how large the switchingcost is. In a monopoly market where only a dominantoperator can choose to upgrade to 4G, this operator canuse the 4G service to capture a larger market share fromsmall operators. In a competitive market where multi-ple operators can choose to upgrade, we analyze theoperators’ interactions as a non-cooperative game. Westudy how the users’ inter-network switching cost affectsthe operators’ upgrade decisions, and our findings areconsistent with the asymmetric upgrades observed in theindustry.

Our key results and contributions are as follows.

1. We define the upgrade time to be the first time when the 4Gservice is offered by the operator in the market.

2

• A revenue-sharing model between operators: Most ex-isting works only study a single network’s revenueby exploring the network effect (e.g., [4], [9]), andthe results may not apply to a competitive market.In Section 3, we study two interconnected networks,where their operators share the revenue of the inter-network traffic.

• Monopolist’s optimal timing of 4G upgrade: By upgrad-ing early, the 4G monopolist in Section 4 obtains alarge market share and a large revenue because of4G’s Quality of Service (QoS) improvement, but itcannot benefit from the cost depreciation over time.When the upgrade cost is relatively low, it upgradesat the earliest available time; otherwise it postponesits upgrade.

• Competitive operators’ symmetric upgrades under nointer-network switching: In Section 5, we developa game theoretic model for studying competitiveoperators’ interactions. In Section 6, we consideran extreme situation where users do not switchoperators. In this situation, the operators upgrade si-multaneously, irrespectively of whether the upgradecost is high or low.

• Competitive operators’ asymmetric upgrades under prac-tical inter-network switching: In Section 7, users canswitch operators. By upgrading early, an operatorcaptures a large market share and the 4G’s QoSimprovement which can compensate for the largeupgrade cost. The other operator, however, post-pones its 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.

2 RELATED WORK2.1 Network Effect and Network ValueIn telecommunications, the network effect is the addedvalue that a user derives from the presence of other users[25]. In a network with N users, each user perceives avalue that increases with N . If each user attaches thesame value to the possibility of connecting to any oneof the other N − 1 users, it may be considered that heperceives a network value proportional to N−1. Then thetotal value of the network is proportional to N(N − 1),or roughly N2, which is known as the Metcalfe’s Law[4]. A refined model was suggested by Briscoe et al. [9],where each user perceives a value of order log(N). Inthat model, a user ranks the other users in decreasingorder of importance and assigns a value 1/k to the k-th user in that order, for a total value 1 + 1/2 + · · · +1/(N − 1) ≈ log(N). The resulting total network valueis N log(N), which is appropriate for cellular networksshown by quantitative studies [9].

2.2 Network UpgradeRecently, there has been a growing interest in studyingthe economics of network upgrades [10]–[12]. Musacchio

et al. [10] studied the upgrade timing game between twointerconnected Internet Service Providers (ISPs), whereone ISP’s architecture upgrade also benefits the otherbecause of the network effect. This free-riding effect maymake the second operator postpone its upgrade or evennever decide to upgrade. Jiang et al. [11] studied a net-work security game, where one user’s investment (up-grade) can reduce the propagation of computer virusesto all users. In our problem, however, one operatormay benefit from the other’s upgrade only when it alsoupgrades, letting its 4G users communicate with exist-ing 4G users in other networks. Moreover, our modelcharacterizes the dynamics of users switching betweenoperators and/or services. These dynamics imply that anoperator can obtain a larger market share by upgradingearlier, and this weakens the free-riding effect. Sen etal. [12] studied the users’ adoption and diffusion of anew network technology in the presence of an incumbenttechnology. Our work is different from that study in thatwe are not focusing on technology competition to attractusers, but on the operators’ competition in upgradetiming to obtain greater profits. Moreover, the switchingcost is not considered in [12], whereas it is an importantparameter of our model.

Our study is related to the literature on the war ofattrition [34]–[38], which considered how N + K firmscompete for N prizes, and analyzed the exit time forlow-value firms. A key result is that low-value firmswill drop out instantaneously. Our model is differentfrom these models in several aspects. First, our modelconsiders a tighter interplay among operators, where oneoperator can use 4G service to attract subscribers fromanother operator. There is no such interactions amongfirms in the war of attrition games. Second, the userchurn in our model makes the revenue rate time-varyingfor operators, and the maturity of 4G technology overtime provides an incentive for a late upgrade. Thesefactors were not considered in [34]–[38]. Because of thesedifferences, our analysis produces some counter-intuitiveresults, e.g., the availability of the 4G technology mayreduce the operators’ profits.

3 SYSTEM MODEL

3.1 Value of Cellular NetworksIn this paper, we adopt the N log(N) Law [9], where thenetwork value with N users is proportional to N log(N).The operator of a cellular network prefers a large net-work value; this is because the revenue it obtains bycharging users can be proportional to the network value.According to the surveys done by ABIresearch andValidas LLC ( [13], [14]), the 4G data service is usuallypriced significantly higher than the 3G data service, and4G users usually consume much more data comparedwith 3G users. The communication between two 4Gusers is more effiicient and more frequent than betweentwo 3G users. Thus the value of a 4G network is largerthan a 3G network even when two networks have the

3

same number of users. Because the average data rate inthe 4G service is 5-10 times higher than the 3G (bothdownlink and uplink), a 4G network can support manynew applications. We denote the efficiency ratio between3G and 4G services as γ ∈ (0, 1). That is, by serving all itsusers via QoS-guaranteed 4G rather than 3G services, anoperator obtains a larger (normalized) revenue N log(N)instead of γN log(N).2 Note that this result holds for asingle operator’s network that is not connected to othernetworks.

Next we discuss the revenues of multiple operatorswhose networks (e.g., two 3G networks) are intercon-nected. For the purpose of illustration, we consider twonetworks that contain N1 and N2 users, respectively. Thewhole market covers N = N1+N2 users. We assume thattwo operators’ 3G (and later 4G) services are equallygood to users, and the efficiency ratio γ is the samefor both operators. The traffic between two users canbe intra-network (when both users belong to the sameoperator) or inter-network (when two users belong todifferent operators), and the revenue calculations in thetwo cases are different. We assume that the user whooriginates the communication session (irrespective ofwhether the same network or to the other network) paysfor the communication. This is motivated by the industryobservations in EU and many Asian countries.3 Beforeanalyzing each operator’s revenue, we first introducetwo practical concepts in cellular market: “terminationrate” and “user ignorance”.

When two users of the same operator 1 communicatewith each other, the calling user only pays operator 1.But when an operator 1’s user calls an operator 2’s user,operator 2 charges a termination rate for the incomingcall [22].4 We denote the two operators’ revenue-sharingportion per inter-network call as η, where the value ofη ∈ (0, 1) depends on the agreement between the twooperators or on governments’ regulation on terminationrate.

User ignorance is a unique property in the wirelesscellular network, where users are often not able to iden-tify which specific network they are calling [27]. Mobilenumber portability further exacerbates this feature [26].As the data communication becomes dominant in wire-less networks, many recent wireless plans (e.g., providedby AT&T and Verizon) no longer charge intra- and inter-network traffics differently, and this further strenghtensuser ignorance. Thus a typical user’s evaluation of two

2. We assume that an operator’s operational cost (proportional tonetwork value) has been deducted already, and thus the revenue inthis paper represents a normalized one.

3. Our model can also be extended to the case where both involvedusers in a communication session pay for their communication. Thisis what happening in US cellular market.

4. In the US, termination rate follows “Bill and Keep” and is low.Then operator 1 can keep most of the calling user’s payment. In EU,however, termination rate follows “Calling Party Pays” and is muchhigher. Then most of the calling user’s payment to operator 1 is used tocompensate for the termination rate charged by operator 2 [22]. Thoughthese models have different revenue sharing portions η, our followingkey results are independent of η and thus apply to both models.

interconnected 3G networks does not depend on whichnetwork he belongs to, and equals γ log(N) where N =N1 +N2. We assume a call from any user terminates ata user in network i ∈ {1, 2} with a probability of Ni/Nas in [26]. The operators’ revenues when they are bothproviding 3G services are given in Lemma 1.

Lemma 1: When operators 1 and 2 provide 3G services,their revenues are γN1 log(N) and γN2 log(N), respec-tively.

Proof: For a user in operator 1’s network, his com-munication brings N1

N γ log(N) + ηN2N γ log(N) amount of

revenue to operator 1 and (1− η)N2N γ log(N) amount of

revenue to operator 2. Similarly, for a user in operator2’s network, his communication brings (1−η)N1

N γ log(N)amount of revenue to operator 1 and ηN1

N γ log(N) +N2N γ log(N) amount of revenue to operator 2. Thus op-

erator 1’s total revenue isN1(

N1

Nγ log(N) + η

N2

Nγ log(N)) +N2((1− η)

N1

Nγ log(N))

= γN1 log(N),and operator 2’s total revenue is γN2 log(N) due tosymmetry.

Both operators’ revenues are linear in their numbersof users (or market share), and are independent of thesharing portion η of the inter-network revenue. Intu-itively, the inter-network traffic between two networksis bidirectional: when a user originates a call from net-work 1 to another user in network 2, its inter-networktraffic generates a fraction η of corresponding revenue tooperator 1; when the other user calls back from network2 to network 1, he generates a fraction 1− η of the sameamount of revenue to operator 1. Thus an operator’stotal revenue is independent of η. Later, in Section 4, weshow that such independence on η also applies whenthe two operators both provide 4G services or providemixed 3G and 4G services.

3.2 User Churn during Upgrade from 3G to 4G Ser-vicesWhen 4G service becomes available in the market (of-fered by one or both networks), the existing 3G usershave an incentive to switch to the new service to expe-rience a better QoS. Such user churn does not happensimultaneously for all users, as different users have dif-ferent sensitivities to quality improvements and switch-ing costs. We use two parameters λ and α to model theuser churn within and between operators:• Intra-network user churn: If an operator provides 4G

in addition to its existing 3G service, its 3G usersneed to buy new mobile phones to use the 4Gservice. The users also spend time to learn how touse the 4G service on their new phones. We use λto denote the users’ switching rate to the 4G servicewithin the same network as in [20].

• Inter-network user churn: If a 3G user wants toswitch to another network’s 4G service, he eitherwaits till his current 3G contract expires, or paysfor the penalty of immediate contract termination.

4

Fig. 1. Verizon’s new 4G subscribersfrom its own 3G subscribers [21]

Fig. 2. Verizon’s total number of sub-scribers [21]

Fig. 3. Verizon’s 3G user numberbefore the 4G upgrade [21]

TABLE 1Quarterly change (in percentage) of Verizon’s 3G user

number after the 4G upgrade [21]

Q2’11 Q3’11 Q4’11 Q1’12 Q2’12 Q3’12-0.35% -1.72% -1.84% -2.05% -3.88% -5.70%

This means that inter-network user churn incurs anadditional cost on top of the mobile device update,and thus the switching rate will be smaller than theintra-network user churn. We use αλ to denote theusers’ inter-network switching rate to 4G service,where α ∈ (0, 1) reflects the transaction cost ofswitching operators.

We illustrate the process of user churn through acontinuous time model. The starting time t = 0 denotesthe time when the spectrum resource and 4G technologyare available for at least one operator (see Section 4 formonopoly market and Sections 6 and 7 for competitivemarket). Similar to [15], [16], we also assume that theportion of users switching to the 4G service follows theexponential distribution (at a rate λ for intra-networkchurn and a rate of αλ for inter-network churn). Theexponential switching model has been widely used inmarketing research [15]–[17], and can help us deriveclosed-form solutions and engineering insights. We as-sume the user churn model is exogenous given. 5 We donot consider dynamic pricing between strategic opera-tors, and our focus is the upgrade timing in long run.6

Such a model can reasonably match the existing in-dustry data with properly chosen system parameters. Letus take Verizon’s market data from [21] as an example,which captures the changes of Verizon’s subscriberssince its 4G launch in early 2011 (as the second 4Goperator in the US). We analyze this data according tothe methodology in [19]. After Verizon launched its 4Gservice, its 3G users no longer churned to Sprint’s 4G

5. An operator can predict such a model based on historical marketdata and new market survey before committing to the technologyupgrade.

6. In the future work, we plan to study the case where an operatorcan use a low introductory price to promote the 4G service, in order toattract users from the other operators. It can also reduce its 3G serviceprice, to reduce the user churn of its 3G users to other operators’4G services. In this case, we can use a utility-based modeling tounderstand how users churn under different prices, similar as [12].

service. Furthermore, Verizon’s 4G service attracted itsown 3G users and other operators’ 3G users at differentrates.

Figure 1 illustrates the number of users churning fromVerizon’s 3G services to 4G services in each quarter. Wecan accurately match the intra-network churning datausing an exponential function with a rate of λ = 0.045million per quarter. Similarly, Figure 2 shows how theVerizon’s total subscriber number increases due to userchurn from other operators. Note that Verizon’s increaseof total user number in Fig. 2 is mainly due to theintroduction of 4G LTE service, as its 3G market hasalready saturated ([21], [39]). Fig. 3 shows that between2005 and 2010, Verizon’s 3G retail user number increasesmore and more slowly over time, and such a changebecomes negligible before its 4G upgrade (i.e., less than0.6% increase from 4Q’09 to 1Q’10). After its 4G upgrade,Table 3.2 shows the change (in percentage) of Verizon’s3G user number. We can see that the 3G user numberdecreases faster over time, as more and more Verizon’s3G users churn to its 4G service (which is consistentwith Fig. 1). Finally, as Fig. 2 approximates well thenumber of subscribers switching to Verizon’s 4G servicefrom other operators, we can estimate the parameter ofthe exponential growth to be αλ = 0.011 million perquarter for inter-network churning. Thus, these industryobservations validate our model of exponential userintra- and inter-network churns. The α value (equalto 0.24 in this example) reflects the transaction cost ofswitching operators (e.g., breaking existing contract).7

Let us consider an example to illustrate how usersswitch between two operators and two services. Weassume that operator 1 introduces a 4G service at timet = T1 while operator 2 decides not to upgrade. Thenumbers of operator 1’s 4G users and 3G users at anytime t ≥ 0 are N4G

1 (t) and N3G1 (t), respectively. The

number of operator 2’s 3G users at time t ≥ 0 is N3G2 (t).

As time t increases (from T1), 3G users in both networksstart to switch to 4G service, and ∀t ≥ 0

N3G1 (t) = N1e

−λ·max(t−T1,0), N3G2 (t) = N2e

−αλ·max(t−T1,0),(1)

7. This paper does not focus on the analysis of the contract’s impacton user churning behaviors across networks. Interested readers canrefer to [23], [24] for more discussions.

5

3

1 ( )GN t

Operator 2’s 3G users:

Operator 1’s

3G users:

01T

t

4

1 ( )GN t

3

2 ( )GN t

N

2

NOperator 1’s 4G users:

Fig. 4. The numbers of users in the operators’ differentservices as functions of time t. Here, operator 1 upgradesat T1 and operator 2 does not upgrade.

and operator 1’s 4G service gains an increasing marketshare,N4G

1 (t) = N−N1e−λ·max(t−T1,0)−N2e

−αλ·max(t−T1,0). (2)We illustrate (1) and (2) in Fig. 4. We can see that operator1’s early upgrade attracts users from its competitor andincreases its market share. Notice that (2) increases withα, thus operator 1 captures a large market share whenα is large (i.e., the switching cost is low).

3.3 Operators’ Revenues and Upgrade CostsOwing to depreciation over time, an operator values thecurrent revenue more than the same amount of revenuein the future. We denote the discount rate over time as S,and the discount factor is thus e−St at time t accordingto [29].

We approximate one operator’s 4G upgrade cost as aone-time investment. This is a practical approximation,as an operator’s initial investment of wireless spectrumand infrastructure can be much higher than the mainte-nance costs in the future. For example, spectrum is a veryscarce resource that is allocated (auctioned) infrequentlyby government agencies. It is time-consuming and noteasy for an operator to obtain sufficient spectrum fromanother operator for its 4G upgrade.8 To ensure a goodinitial 4G coverage, an operator also needs to updatemany base stations to cover at least a whole city allat once. Otherwise, 4G users would be unhappy withthe service, and this would damage the operator’s rep-utation. That is why Sprint and Verizon covered manymarkets in their initial launch of their 4G services [31].

More specifically, we denote the 4G upgrade cost att = 0 as K, which reduces over time at a rate U .Thus if an operator upgrades at time t, it needs to payan upgrade cost Ke−Ut according to [29]. We shouldpoint out that the upgrade cost decreases faster than thenormal discount rate (i.e., U > S).9 This happens becausethe upgrade cost decreases because of both technologyimprovement and depreciation. Very often the advance

8. It is possible for one operator to purchase from another operatorwho has extra spectrum resources. For example, the FCC approvedT-Mobile’s purchase of some 4G spectrum from Verizon in 2012.However, such a purchase does not happen often for the increasinglycompetitive 4G spectrum market, and is a length process that requiresFCC’s approval.

9. The parameter S can also be viewed as the inflation rate, whichis usually very small according to [32].

of technology is the dominant factor in determining U ,and this is discussed further in Section 7.

Based on these discussions on revenue and upgradecost, we define an operator’s profit as the differencebetween its revenue in the long run and the one-time upgrade cost. Without loss of generality, we willnormalize an operator’s revenue rate (at any time t),total revenue, and upgrade cost by N log(N), where Nis the total number of users in the market.10 Becauseof this normalization, our main results derived in thefollowing sections are robust to the choice of 4G networkvaluation function. For example, if the majority of 4Gtraffic is related to mobile Internet instead of directcommunications among users, the network effect willbe weaker and network value function f(N) satisfiesN < f(N) < N log(N).11 In this case, by normalizing theoperator’s revenue rate, total revenue, and upgrade costby f(N), we can still derive the same optimal upgradetiming for the monopoly case and prove asymmetrictiming equilibrium for the duopoly case by followingthe same analysis as follows.

4 4G MONOPOLY MARKET

We first look at the case where only operator 1 canchoose to upgrade from 3G to 4G, while the otheroperators (one or more) always offer 3G service becauseof the lack of financial resources or the necessary tech-nology. This can be a reasonable model, for example,for countries such as Mexico and some other LatinAmerican ones, where America Movil is the dominantcellular operator in the 3G market. As the world’s fourth-largest cellular network operator, America Movil has theadvantage over other small local operators in winningadditional spectrum via auctions and obtaining LTEpatents, and it is expected to be the 4G monopolist inthat area [28].

The key question in this section is how operator 1should choose its upgrade time T1 from the 3G service tothe 4G service. T1 = 0 means that operator 1 upgradesat the earliest time that the spectrum and technologyare available, and T1 > 0 means that operator 1 choosesto upgrade later to take advantage of the reduction inthe upgrade cost. Because of user churn from the 3G tothe 4G service, the operators’ market shares and revenuerates change after time T1. For that reason, we analyzeperiods t ≤ T1 and t > T1 separately.• Before 4G upgrade (t ≤ T1): Operator 1’s and other

operators’ market shares do not change over time.Operator 1’s revenue rate at time t is

π3G−3G1 (t) = γ

N1

N,

10. Our model and all analytical results in the following can beextended to the case where N increases over time. As new users prefer4G service to 3G service, and can easily switch at rate λ, operators willhave more incentives to upgrade earlier.

11. The linear network value function N assumes no communica-tions among users.

6

which is independent of time t. Its revenue duringthis time period is

π3G−3G1,t≤T1

=∫ T1

0

π3G−3G1 (t)e−Stdt =

γN1

SN(1− e−ST1).

(3)• After 4G upgrade (t > T1): Operator 1’s market share

increases over time, and the other operators’ totalmarket share (denoted by N3G

2 (t)/N ) decreases overtime. We denote operator 1’s numbers of 3G usersand 4G users as N3G

1 (t) and N4G1 (t), respectively,

and we have N3G1 (t) + N4G

1 (t) + N3G2 (t) = N . This

implies thatN3G

2 (t) = (N−N1)e−αλ(t−T1), N3G1 (t) = N1e

−λ(t−T1),

andN4G

1 (t) = N −N1e−λ(t−T1) − (N −N1)e−αλ(t−T1).

Note that a 3G user’s communication with a 3Gor a 4G user is still based on the 3G standard,and only the communication between two 4G userscan achieve a high 4G standard QoS. Operator 1’srevenue rate is

π4G−3G1 (t) =

γN3G1 (t)N

+N4G

1 (t)N

·(N4G

1 (t) + γN3G1 (t)

N+γN3G

2 (t)N

),

which is independent of the revenue sharing ratioη between the calling party and receiving party.Operator 1’s revenue during this time period is then

π4G−3G1,t>T1

=∫ ∞T1

π4G−3G1 (t)e−Stdt, (4)

where t → ∞ is an approximation of the long-term 4G service provision (e.g., one decade) beforethe emergence of the next generation standard. Thisapproximation is reasonable since the revenue inthe distant future becomes less important becauseof discounting.

Figure 4 illustrates how the numbers of users of opera-tors’ different services change over time. Before operator1’s upgrade (e.g., t ≤ T1 in Fig. 4), the number of totalusers in each network does not change;12 after operator1’s upgrade, operator 1’s and the other operators’ 3Gusers switch to the new 4G service at rates λ and αλ,respectively.

By considering (3), (4), and the decreasing costKe−UT1 , operator 1’s long-term profit when choosing anupgrade time T1 isπ1(T1) = π3G−3G

1,t≤T1+ π4G−3G

1,t>T1−Ke−UT1

= e−ST1

(1S

+ (1− γ)(N1N

)22λ+ S

+ (1− γ)(N−N1N

)22αλ+ S

)

− e−ST1

(2(1− γ)

N1N

λ+ S+ (2− γ)

N−N1N

αλ+ S

)−Ke−UT1

+ 2e−ST1(1− γ)N1(N−N1)

N2

(1 + α)λ+ S+N1γ

NS(1− e−ST1). (5)

12. It is possible for users to switch between networks in the same3G service class, but such minor switching does not change the averagemarket shares.

We can show that π1(T1) in (5) is strictly concave in T1,thus we can compute the optimal upgrade time T ∗1 bysolving the first-order condition. The optimal upgradetime depends on the following upgrade cost thresholdin the monopoly 4G market,

Kmonoth =(1− γ)

(N1N )2

2λ+S + (N−N1N )2

2αλ+S −2

N1N

λ+S +2

N1(N−N1)N2

(1+α)λ+S

U/S

+1− γN1

N − (2− γ)N−N1N

Sαλ+S

U. (6)

Theorem 1: Operator 1’s optimal upgrade time in a 4Gmonopoly market is:• Low cost regime (upgrade cost K ≤ Kmono

th ): operator1 upgrades at T ∗1 = 0.

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

atT ∗1 =

1U − S

log(

K

Kmonoth

)> 0. (7)

Intuitively, an early upgrade gives operator 1 a largermarket share and enables him to get a higher revenuevia the more efficient 4G service. Such advantage isespecially obvious in the low cost regime where theupgrade cost K is small.

Next we focus on the high cost regime, and explorehow the network parameters affect operator 1’s upgradetime.

Observation 1: Operator 1’s optimal upgrade time T ∗1increases with the upgrade cost K, and decreases withα (i.e., increases with the users’ inter-network switchingcost).

Proof: Equation (7) shows that T ∗1 is increasing inK. Next we show the relation between T ∗1 and α. Inorder to show that T ∗1 is decreasing in α, we need tofirst show that T ∗1 is decreasing in Kmono

th , and then showthat Kmono

th is increasing in α. Notice that (7) shows thatT ∗1 is decreasing in Kmono

th . Consequently, we only needto examine the relation between Kmono

th and α. Noticethat the partial derivative ∂Kmono

th /∂α is linear in γ, andits values at the two boundaries of γ ∈ (0, 1) are bothpositive. Indeed,∂Kmono

th

∂α|γ=0 =

2λ(N −N1)UN

(1

(αλ+ S)2− 1

(2αλ+ S)2

)+

2λN1(N −N1)UN2

(1

(2αλ+ S)2− 1

((1 + α)λ+ S)2

)> 0,

and∂Kmono

th

∂α|γ=1 =

N −N1

N

λ

U(αλ+ S)2> 0.

Thus ∂Kmonoth /∂α is positive for any γ, and Kmono

th isincreasing in α. This completes the proof.

Observation 2 (Figure 5): When Kmonoth < K <

USK

monoth , T ∗1 first increases and then decreases in U .

When K ≥ USK

monoth , T ∗1 monotonically decreases in U .

Figure 5 shows T ∗1 as a function of U and K. When Kis large or U is large, operator 1 wants to postpone itsupgrade until the upgrade cost decreases significantly. Alarger U (thus a faster cost-decreasing rate) strengthensits willingness to postpone. When K is small and U issmall, the upgrade cost is small and does not decrease

7

1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Discounting rate of upgrade cost U

Opt

imal

upg

rade

dec

isio

n T

1 *

K=0.14K=0.18K=0.22

Fig. 5. Operator 1’s optimal upgrade time T ∗1 as afunction of cost K and the discount rate U of cost. Otherparameters are N = 1000, N1 = 500, S = 1, λ = 1, andα = 0.5.

0.2 0.4 0.6 0.80.8

0.9

1

1.1

1.2

1.3

1.4

1.5

Efficiency ratio between 3G and 4G services (γ)

Opt

imal

upg

rade

tim

e T

1 *

N1/N=30%

N1/N=50%

N1/N=70%

Fig. 6. Operator 1’s optimal upgrade time T ∗1 as a functionof the efficiency ratio between 3G and 4G services (γ)and its original market share N1

N . Other parameters areN = 1000, K = 2000, S = 1, λ = 1, and U = 2.

fast enough. In this case, operator 1 chooses to upgradeearly if the revenue increase can compensate for thesmall upgrade cost.

By checking the first order derivative of T ∗1 in (7) overγ under different N1/N values, we having the following.

Observation 3 (Figure 6): When operator 1’s originalmarket share N1

N is large, T ∗1 increases in the efficiencyratio γ (between 3G and 4G). When N1

N is small, T ∗1decreases in γ.

Figure 6 shows T ∗1 as a function of γ and N1N . When

N1N is large, operator 1 cannot attract many users from

other operators and has a smaller incentive to upgrade.As γ increases (and thus the QoS gap between 3G and4G shrinks), it is less interested in 4G service and thuspostpones its upgrade. When N1

N is small, operator 1 haslimited market share and can attract many more usersby upgrading early. As γ increases, operator 1 obtainsa higher revenue between its existing 3G users and theincreasing number of 4G users. The other small operatorswithout market power, however, do not benefit fromthis, as they lose their market shares due to operator

1’s 4G service. Thus T ∗1 decreases in γ.

5 4G COMPETITIVE MARKET: DUOPOLYMODEL AND GAME FORMULATIONIn this section, we focus on the competition betweenmultiple operators who can choose to upgrade to 4Gservices. To make the analysis tractable and to deriveclear engineering insights, we focus on the case of twooperators (duopoly) in this paper. This analysis servesas the first step in understanding the more generaloligopoly case. This duopoly model is reasonable ina country like China, where China Mobile and ChinaUnicom are the two dominant cellular operators in the3G market. A similar situation exists in several otherAsian and European countries as well.

The focus of this and the following section is tounderstand why in so many existing industry examples(e.g., [5]–[7]) operators choose to upgrade to 4G servicesat different times even though they have the resourcesto upgrade simultaneously. In particular, we examinewhether such asymmetric upgrades emerge even whenthe two operators are similar (e.g., having similar marketshares before upgrades (e.g., Verizon has 106.3 millionusers and AT&T has 98.6 million users in the US). InSection 8, we also examine the case where networks areheterogeneous in nature, and we show that asymmetricoperators (e.g., with different market shares) have moreincentives to upgrade at different times. Thus in thefollowing analysis we can consider two operators thathave the same market shares before the 4G upgrades(N1 = N2), the same upgrade cost K, and the samecost discount rate U . We will first derive the operators’profits under any upgrade decisions, and then analyzethe duopoly game where each operator chooses the bestupgrade time to maximize its profit.

5.1 Operators’ Long-term ProfitsLet us denote two operators’ upgrade times as T1 and T2,respectively. Because the two operators are symmetric,without loss of generality, we assume in the followingexample (before Lemma 2) that operator 1 upgrades nolater than operator 2 (i.e., T1 ≤ T2). To calculate theoperators’ profits, we first need to understand how userschurn from 3G to 4G services, and how this affects theoperators’ revenue rates over time. Figure 7 shows thatuser churn is different in three phases, depending onhow many operators have upgraded.• Phase I (0 ≤ t ≤ T1): No operator has upgraded and

both operators’ market shares do not change. Thetwo operators’ revenue rates are

π3G−3G1 (t) = π3G−3G

2 (t) =γ

2.

• Phase II (T1 < t ≤ T2): Operator 1 has upgraded to4G service but operator 2 has not. The 3G users oftwo operators switch to operator 1’s 4G service atdifferent rates. The numbers of users in the opera-tors’ different services are

N3G1 (t) =

N

2e−λ(t−T1), N3G

2 (t) =N

2e−αλ(t−T1),

8

3

1 ( )2

G NN t 3

2 ( )2

G NN t

4G-4G

3

1 ( )GN t 4

1 ( )GN t3

2 ( )GN t

3

1 ( )GN t 4

1 ( )GN t 3

2 ( )GN t

4

2 ( )GN t

4G-4G 4G-4G

(Phase II)

1T

3G-3G

0

4G-3G

2Tt

(Phase III)

1T

3G-3G

0

4G-3G

2Tt

(Phase I)

1T

3G-3G

0

4G-3G

2Tt

Fig. 7. Users’ switches over the operators’ services: Phase I with 3G services only, Phase II with operator 1’s 4Gservice, and Phase III with both operators’ 4G services.

and

N4G1 (t) = N − N

2e−λ(t−T1) − N

2e−αλ(t−T1).

The two operators’ revenue rates are

π4G−3G1 (t) =

(1− 1− γ

2(e−λ(t−T1) + e−αλ(t−T1))

)·(

1− e−λ(t−T1)e−αλ(t−T1)

2

)+γ

2e−λ(t−T1),

andπ4G−3G

2 (t) =γ

2e−αλ(t−T1).

• Phase III (t > T2): Both operators have upgraded,and 3G users only switch to the 4G service of theircurrent operator. The numbers of users in operators’different services are

N3G1 (t) =

N

2e−λ(t−T1),

N4G1 (t) = N − N

2e−λ(t−T1) − N

2e−αλ(T2−T1),

N3G2 (t) =

N

2eαλ(T2−T1)−λ(t−T2),

and

N4G2 (t) =

N

2eαλ(T2−T1)

(1− e−λ(t−T2)

).

The two operators’ revenue rates are

π4G−4G1 (t) =

γ

2e−λ(t−T1) +

(1− e−λ(t−T1) + e−λ(t−T2)

2

)·(

1− 1− γ2

e−λ(t−T1) − 1− γ2

e−αλ(T2−T1)−λ(t−T2)

),

and

π4G−4G2 (t) =

γe−αλ(T2−T1)−λ(t−T2)

2+

1− e−λ(t−T2)

2

·1− 1−γ

2 e−λ(t−T1) − 1−λ2 e−αλ(T2−T1)−λ(t−T2)

2.

Figure 8 summarizes how users churn in the threephases. Similar to Section 4, we can derive operators’revenue rates based on users’ churn over time. By in-tegrating each operator’s revenue rate over all threephases, we obtain that operator’s long-term revenue.Recall that an operator’s profit is the difference betweenits revenue and the one-time upgrade cost. By furtherconsidering the symmetric case of T1 ≤ T2, we have thefollowing result.

01T

t

3

1 ( )GN t

3

2 ( )GN t

4

1 ( )GN t

N

4

2 ( )GN t

2T

2

N

Operator 2’s 3G users:

Operator 1’s

3G users:

Operator 2’s 4G

users:

Operator 1’s 4G users:

Fig. 8. The numbers of users in the operators’ differentservices as functions of time t. Here, operator 1 choosesits upgrade time at T1 and operator 2 chooses T2 withT1 ≤ T2.

Lemma 2: Consider two operators i, j ∈ {1, 2} (withi 6= j) upgrading at Ti and Tj . Operator i’s long-termprofit is

πi(Ti, Tj) =

{πER(Ti, Tj), if Ti ≤ Tj ;πLT (Tj , Ti), if Ti ≥ Tj ,

(8)

where πER(Ti, Tj) and πLT (Tj , Ti) are given in (9) and(10), respectively.

Note that an operator’s profit πi(Ti, Tj) is continuousin its upgrade time Ti. When operator i’s upgrade timeTi is less than Tj , it increases its market share at rateαλ during the time period from Ti to Tj ; but whenTi > Tj , operator i loses its market share at rate αλduring the period from Tj to Ti. This explains why weneed two different functions πER(Ti, Tj) and πLT (Tj , Ti)to completely characterize the long-term profit for eachoperator.

5.2 Duopoly Upgrade GameNext we consider the non-cooperative game that modelsthe interactions between two operators, where each ofthem seeks to maximize its long-term profit by choosingthe best upgrade time.

Upgrade Game: We model the competition between twooperators as follows:• Players: Operators 1 and 2.• Strategy spaces: Operator i ∈ {1, 2} can choose

upgrade time Ti from the feasible set Ti = [0,∞].13

• Payoff functions: Operator i ∈ {1, 2} wants to max-imize its profit πi(Ti, Tj) defined in (8).

13. Note that Ti =∞ means that operator i never upgrades.

9

πER(Ti, Tj) =γ + (2− γ)e−STi − e−αλ(Tj−Ti)−STj

2S−

(2− γ)(e−STi − e−αλ(Tj−Ti)−STj )

2(αλ+ S)− (1− γ)

e−STi

λ+ S

+1− γ

2

e−STi + e−(1+α)λ(Tj−Ti)−STj

2(2λ+ S)+e−STi − e−(1+α)λ(Tj−Ti)−STj

(1 + α)λ+ S+e−STi − e−2αλ(Tj−Ti)−STj

2(2αλ+ S)

!

−1− γ

2(λ+ S)e−αλ(Tj−Ti)−STj

1−

e−λ(Tj−Ti) + e−αλ(Tj−Ti)

2

!−Ke−UTi . (9)

πLT (Tj , Ti) =e−αλ(Ti−Tj)−STi

2

1

S−

1− γλ+ S

−1− γ

2

“e−λ(Ti−Tj) + e−αλ(Ti−Tj)

”„ 1

λ+ S−

1

2λ+ S

«!+

γ

2S(1− e−STj )

+γ

2(αλ+ S)(e−STj − e−αλ(Ti−Tj)−STi )−Ke−UTi . (10)

Notice that we consider a static game here, whereboth operators decide when to upgrade at the beginningof time. This is motivated by the fact that operatorsusually make long-term decisions in practice rather thanchanging decisions frequently, as many upgrade opera-tions (e.g., financial budget and technological trials) needto be planned and prepared. As we consider that eachoperator has complete information about its competitor’sand users’ parameters, it will not deviate from its initialdecision as time goes on. Also, it should be pointed outthat in many competitive markets operators can obtainavailable resources for 4G upgrade at a similar time aswe mentioned at the beginning of this section. Aftermaking decision at t = 0, one operator does not changeits decision later on. This is reasonable when operatorscan predict the future 4G market adoption.14

For a static game, Nash equilibrium is a commonlyused solution concept. At a Nash equilibrium, no playercan increase its payoff by deviating unilaterally [33].We are interested in characterizing the conditions un-der which an asymmetric upgrade equilibrium emergesbetween symmetric operators. As users’ subscriptiondynamics and the corresponding analysis depend on theswitching costs (i.e., the value of α ∈ [0, 1)), we divideour analysis into Sections 6 and 7.

6 4G COMPETITIVE MARKET: NO INTER-NETWORK SWITCHING

In this section, we consider the case of α = 0, i.e., nouser switches operators because of a very high switchingcost. This corresponds to the case, for example, wherethe penalty for terminating a 3G contract to the otheroperator’s 4G service is very high or where a contractlasts for a very long time.15 The analysis here helps usbetter understand the more general case of α > 0 inSection 7.

14. Operators can predict the future market adoption by exploringhistorical records of the market and some trial of 4G deployment. Inthe future we will study the incomplete information case, where anoperator may learn more information as the time goes and revise itsupgrade decision (if it has not upgraded yet).

15. The penalty can also involve the inconvenience resulted from thechange of cell phone number if one cannot keep the existing numberafter switching to the other operator. Note that the penalty of 3Gcontract termination is not high if the involved user is switching tothe 4G service provided by the same operator. Actually, an operatorprefers its 3G users switching to its 4G service.

Let us denote the cost threshold in the 4G competitivemarket under α = 0 as

Kcompth1,α=0 =

3(1− γ)λ2

4U(λ+ S)(2λ+ S). (11)

Theorem 2: With symmetric operators and α = 0, thereexists a unique symmetric 4G upgrade equilibrium withthe following characteristics:• Low cost regime (upgrade cost K ≤ Kcomp

th1,α=0): Bothoperators upgrade at T ∗1 = T ∗2 = 0.

• High cost regime (upgrade K > Kcompth1,α=0): Both

operators upgrade at

T ∗1 = T ∗2 =1

U − Slog

(K

Kcompth1,α=0

)> 0, (12)

which increases with K.The proof of Theorem 2 is given in Appendix A.Intuitively, in the low cost regime, both operators are

willing to upgrade at t = 0 to maximize the revenuefrom providing high-quality 4G services. In the high costregime, on the other hand, both operators postpone theirupgrades until the upgrade cost is small enough. If onlyone operator upgrades, its total market share does notchange (since α = 0) and only its own 3G users switchto the 4G service. Thus the revenue from the 4G servicedoes not compensate for the high upgrade cost. Thismakes the operator reluctant to upgrade until the 4Gtechnology is mature. As one operator cannot attract theother operator’s users, the two operators independentlydecide their upgrade timings.

It should be noted that the duopoly case here is not aspecial case of the monopoly case of α = 0 in Section 4.Unlike Section 4, both operators here can upgrade to 4Gservices, and the positive network effect of technologyupgrade exists between two operators even when α = 0.More precisely, when one operator upgrades to 4G,though its market share (total number of 3G and 4Gusers) remains the same, its new 4G users will enjoythe addition of intercommunication benefits with the4G users of the other operator (once that operator haschosen to upgrade to 4G). Such positive network effectis mutual between the operators.

To mathematically demonstrate the the differencesbetween the two sections, we compare the cost thresholdKcompth1,α=0 in (11) and Kmono

th in (6) for initial upgrade asfollows. For a fair comparison between the two sections,we assume that the monopolist in Section 4 covers half ofthe market (N1 = N/2). In this section, we have assumed

10

that each competitive operator covers half of the market.By substituting N1 = N/2 and α = 0 into (6), we have

Kmonoth |α=0,N1=N/2=

(1− γ)λ2

2U(λ+ S)(2λ+ S),

which is smaller than Kcompth1,α=0 in (11). This shows

that each of the two competitive operators here has ahigher incentive to upgrade immediately (even withoutincreasing its market share). Intuitively, a competitiveoperator can benefit from the 4G upgrade by the otheroperator due to the positive network effect, and thisurges both operators to upgrade earlier. Unlike thissection, in Section 4 the monopolist’s 4G users have no4G users in the other networks to communicate with,which reduces the monopolist’s incentive to upgrade.

7 4G COMPETITION MARKET: PRACTICALINTER-NETWORK SWITCHING RATE

In this section, we consider the case of α > 0, i.e., 3Gusers may switch to the 4G service of a different operator.The equilibrium analysis in this general case depends onthe relationship between U (upgrade cost discount rate)and S (money depreciation rate). We assume that U ismuch larger than S, i.e., U > S + αλ. This representsthe practical case where the advance of technology isthe dominant factor in determining U , and not many 3Gusers choose to switch operators when the 4G serviceis just deployed (i.e., small α) [30]. For example, Sprintdeployed the first 4G network in US by using WiMAXtechnology in 2008 when LTE technology was not matureyet. Only two years later, in 2010, LTE could already offera much lower cost per bit than WiMAX [5]. From 2012,LTE is expected to be the leading technology choice for4G networks. This example motivates that U is muchlarger than S, and we will study whether the operators’symmetric 4G upgrades happen in this scenario.

Recall that by upgrading at T1 and T2, the operatorsreceive the profits given in (8). In our game theoreticmodel, one operator’s best response function is its upgradetime that achieves the largest long-term profit, as afunction of a fixed upgrade time of the other operator[33]. A fixed point of the two operators’ best responsefunctions is the Nash equilibrium, and in general therecan be more than one such fixed point.

We can show that the operators’ best response func-tions (T best1 (T2) and T best2 (T1)) depend on the upgradecost K, and in particular, they depend on two costthresholds (Kcomp

th1 < Kcompth2 ) that lead to three cost

regimes: low, medium, and high.When the upgrade cost K is less than the first thresh-

old Kcompth1 (i.e., low cost regime), both operators will

upgrade at t = 0 to maximize the revenue from the 4Gservice. By solving

∂πLT (0, Ti)∂Ti

|Ti=0 = 0,∀i ∈ {1, 2},

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

Operator 1’s upgrade time T1

Ope

rato

r 2’

s up

grad

e tim

e T

2

K=0.058: T2best(T

1)

K=0.058: T1best(T

2)

K=0.062: T2best(T

1)

K=0.062: T1best(T

2)

K=0.058: NE1

K=0.058: NE2K=0.062: NE2

K=0.062: NE1

Fig. 9. Two operators’ best upgrade responses to eachother according to different cost values in the medium costregime. Other parameters are α = 0.5, N = 1000, U = 2,S = 1, λ = 1 and γ = 0.5 (hence U > S + αλ).

we have

Kcompth1 =

((1− γ) (1− α)λ

λ+ S+αλ

S− 1− γ

2(λ+ 3αλ+ 2S)

·(

1λ+ S

− 12λ+ S

))1

2U. (13)

When the upgrade cost K is larger than Kcompth1 (i.e.,

medium or high cost regimes), at least one operatorpostpones its upgrade until the upgrade cost decreasessufficiently. In particular, when K is larger than thesecond threshold Kcomp

th2 (i.e., high cost regime), bothoperators postpone their upgrades. When operator i ∈{1, 2} upgrades at t = 0, operator j 6= i postpones itsupgrade to T bestj (0), which is the unique solution to

∂πLT (0, Tj)∂Tj

|Tj=T bestj (0) = 0. (14)

The threshold Kcompth2 can be obtained by solving

∂πER(Ti, T bestj (0))∂Ti

|Ti=0 = 0. (15)

Next we illustrate numerically how the two operators’best response functions (T best1 (T2) and T best2 (T1)) changewith the upgrade cost K.

Figure 9 shows that each operator’s best responsefunction is discontinuous in the medium cost regime,and the two best response functions with the samevalue of K intersect at two points: 0 = T ∗1 < T ∗2 and(symmetrically) 0 = T ∗2 < T ∗1 . To illustrate this situation,consider operator 2’s best response T best2 (T1) in the caseK = 0.062. If operator 1 upgrades early such that T1

is less than 0.05, operator 2 does not upgrade at thesame time to avoid a severe competition. If operator 1upgrades later such that T1 is larger than 0.05, operator2 chooses to upgrade earlier than operator 1 to increaseits market share. Thus T best2 (T1) is discontinuous atT1 = 0.05.

Figure 10 shows that each operator’s best responsefunction is discontinuous in the high cost regime, andthe two functions (with the same value of K) intersect attwo points, equilibria 0 < T ∗1 < T ∗2 and (symmetrically)

11

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 1’s upgrade time T1

Ope

rato

r 2’

s up

grad

e tim

e T

2

K=0.072:T

2best(T

1)

K=0.072:T1best(T

2)

K=0.087:T2best(T

1)

K=0.087:T1best(T

2)

K=0.072: NE1

K=0.072: NE2

K=0.087: NE2

K=0.087: NE1

Fig. 10. Two operators’ best upgrade responses to eachother according to different cost values in the high costregime. Other parameters are α = 0.5, N = 1000, U = 2,S = 1, λ = 1 and γ = 0.5 (hence U > S + αλ).

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 1’s equilibrium time T1*

Ope

rato

r 2’

s eq

uilib

rium

tim

e T

2*

NE 1: T1* ≤T

2*

NE 2: T1* ≥T

2*

Low cost regime:0=T

1*=T

2* as K↑

Medium cost regime:0=T

1*<T

2*↑ as K↑

High cost regime:0<T

1*↑<T

2*↑: as K↑

Fig. 11. How equilibrium (T ∗1 ,T ∗2 ) changes as cost Kincreases. Other parameters are α = 0.5, N = 1000,U = 2, S = 1, λ = 1, and γ = 0.5 (hence U > S + αλ).

0 < T ∗2 < T ∗1 . Unlike Fig. 9, the high cost here preventsany operator from choosing the upgrade time t = 0.

Figure 11 summarizes how operators’ upgrade equi-librium changes as cost K increases: starting with T ∗i =T ∗j = 0 in low cost regime, then 0 = T ∗i < T ∗jwith increasing T ∗j in medium cost regime, and finally0 < T ∗i < T ∗j with increasing T ∗1 and T ∗2 in high costregime.

In the following theorem, we prove that the operatorsdo not choose symmetric upgrades as long as the cost isnot low.

Theorem 3: The two operators’ 4G upgrade equilibriasatisfy the following properties:• Low cost regime (K ≤ Kcomp

th1 ): Both operators up-grade at T ∗1 = T ∗2 = 0.

• Medium cost regime (Kcompth1 < K ≤ Kcomp

th2 ): Opera-tors do not upgrade at the same time, and only oneoperator may upgrade at t = 0. The possible equi-libria can only be 0 ≤ T ∗1 < T ∗2 and (symmetrically)0 ≤ T ∗2 < T ∗1 .

• High cost regime (K > Kcompth2 ): Operators do not

upgrade at the same time, and none of them up-grade at t = 0. The possible equilibria can only be0 < T ∗1 < T ∗2 and (symmetrically) 0 < T ∗2 < T ∗1 .Proof: We divide our proof into three parts. First, we

prove the equilibrium in the low cost regime. Then weprove the asymmetric upgrade structure in the other tworegimes. Finally, we prove that both operators postponetheir upgrades in the high cost regime.

Proof of equilibrium in the low cost regime: We want toprove that T ∗1 = T ∗2 = 0 is the unique equilibrium underK ≤ Kcomp

th1 .• We first show T ∗1 = T ∗2 = 0 is an equilibrium by

proving that no operator has an incentive to deviate.Given T ∗1 = 0, operator 2’s profit is πLT (0, T2) whenupgrading at T2. We can show that πLT (0, T2) has aunique maximum T best2 (0) = 0. Thus operator 2 willnot deviate from T ∗2 = 0. Similarly, we can provethat given T ∗2 , operator 1 will not deviate from T ∗1 =0.

• We then prove the uniqueness of the equilibrium bycontradiction. Without the loss of generality, we sup-pose there is another equilibrium with 0 < T ∗1 ≤ T ∗2 .Note that T ∗1 cannot equal zero, otherwise operator2 will also upgrade at t = 0 and we reach the sameequilibrium. Next we show that 0 < T ∗1 < T ∗2 cannotbe an equilibrium by proving operator 2 will chooseits upgrade at the same time. This can be provedby showing operator 2’s profit πLT (T1, T2) with anyT1 < T2 decreases with T2. Thus the possible otherequilibrium can only be T ∗1 = T ∗2 > 0. However, thisis not possible since operator 1’s profit πER(T1, T2)with any T1 = T2 decreases with T1, i.e.,

∂πER(T1, T2)∂T1

|T1=T2

≤ −1− γ4

e−ST13λ2

(2λ+ S)(λ+ S)+Kcomp

th1 Ue−UT1

< −e−ST13(1− γ)λ2

4(2λ+ S)(λ+ S)+Kcomp

th1 Ue−ST1 = 0.

Proof of asymmetric upgrades in medium and high costregimes: Next we focus on K > Kcomp

th1 , and prove bycontradiction that symmetric upgrades T ∗1 = T ∗2 cannotbe an equilibrium. Suppose T ∗1 = T ∗2 > 0 is an equi-librium, where no operator upgrades at t = 0 becauseK > Kcomp

th1 . Next we check if an operator (e.g., operator1) will deviate from this equilibrium.• The supposed equilibrium T ∗1 = T ∗2 should guaran-

tee that operator 1 will not choose to upgrade earlierthan T ∗2 , i.e.,

∂πER(T1, T∗2 )

∂T1|T1=T∗2

≥ 0. (16)

• The supposed equilibrium T ∗1 = T ∗2 should guaran-tee that operator 1 will not choose to upgrade laterthan T ∗2 , i.e.,

∂πLT (T ∗2 , T1)∂T1

|T1=T∗2≤ 0. (17)

12

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 1’s equilibrium profit π1*

Ope

rato

r 2’

s eq

uilib

rium

pro

fit π

2*

NE 1: T1*≤T

2*

NE 2: T1*≥T

2*

Medium cost regime:π

1*↑<π

2*↓ as K↑

High cost regime:π

1*↑<π

2*↑ as K↑

Low cost regime:π

1*↓=π

2*↓ as K↑

Fig. 12. The equilibrium profits (π∗1 ,π∗2) change as cost Kincreases under a large γ = 0.5. Other parameters areα = 0.5, N = 1000, U = 2, S = 1, and λ = 1 (henceU > S + αλ).

However, we can show that∂πER(T1, T

∗2 )

∂T1|T1=T∗2

− ∂πLT (T ∗2 , T1)∂T1

|T1=T∗2

= αλe−ST∗2

(1

2λ+ S− 1λ+ S

)< 0, (18)

which means that (16) and (17) cannot be satisfied at thesame time. Thus T ∗1 = T ∗2 cannot be an equilibrium.

Proof of postponed upgrades in the high cost regime: Fi-nally, we prove that no operator upgrades at t = 0by using a contradiction argument. Since K > Kcomp

th1 ,at least one operator postpones its upgrade. Withoutloss of generality, we suppose that 0 = T ∗1 < T ∗2 isan equilibrium, where we have proved that only asym-metric upgrades exist at any possible equilibrium. Weneed to explore whether any operator has an incentiveto deviate.• Given T ∗1 = 0, operator 2’s profit is πLT (0, T2). It

will not deviate only when T ∗2 already maximizesπLT (T1, T2) (i.e., T ∗2 = T best2 (0)).

• Given T ∗2 = T best2 (0), operator 1’s profit isπER(T1, T

best2 (0)) by upgrading at T1 ≤ T best2 (0). We

find that operator 1 has an incentive to deviate fromT ∗1 = 0. Indeed, since K > Kcomp

th2 ,∂πER(T1, T

best2 (0))

∂T1|T1=0 > 0. (19)

Thus 0 = T ∗1 < T ∗2 cannot be an equilibrium, and bothoperators postpone their upgrades.

To understand the intuition behind the asymmetricstructure, we summarize the advantages of earlier andlater upgrades with α > 0 as follows:• Earlier upgrade gives an operator the advantage to

attract more users (from the other operator), andenables the operator to collect a higher revenue fromthe 4G service.

• Later upgrade allows an operator to incur a reducedupgrade cost when it upgrades.

In order to fully enjoy the two advantages of earlier orlater upgrades, operators will avoid symmetric upgrade.

If one operator upgrades much earlier to capture alarger market share that can compensate for a largeupgrade cost, the second operator will not upgrade at thesame time to avoid severe competition in market share;instead, the second operator will wait until its loss ofusers and revenue is compensated by the reduction ofupgrade cost (with U > S + αλ).

Next we study how operators’ equilibrium profitschange with cost K and the economic efficiency ratioγ between 3G and 4G services in the three cost regimes.Note that an operator may or may not be able to charge asignificantly higher price from a 4G user though 4G doesimprove a lot over 3G in QoS. Figures 12 and 13 showoperators’ equilibrium profits under large and small γvalues, respectively.

We first study the large γ scenario in Fig. 12 whichmeans that an operator only charges a slightly higherprice in 4G service than 3G. Without loss of generality,we focus on the case where operator 1 upgrades no laterthan operator 2 (i.e., T ∗1 ≤ T ∗2 ).• In the low cost regime, by upgrading at T ∗1 =T ∗2 = 0, the two operators’ profits are the same anddecrease with cost K.

• In the medium cost regime, Fig. 12 shows thatoperator 1 receives a larger profit than operator 2 byupgrading at T ∗1 = 0. Perhaps surprisingly, its profitincreases with K, whereas operator 2’s profit decreaseswith K. Intuitively, the increase of K encouragesoperator 2 to further postpone its upgrade and losemore users to operator 1. The change of operator1’s profit trades off the increase of its market shareand upgrade cost. As operator 1’s market shareincreases, its growing 4G users communicate morewith its 3G users via the efficient 3G service underlarge γ. Operator 1’s 3G revenue increases becauseof a more efficient intra-network traffic, which helpscompensate for the upgrade cost.

• In the high cost regime, Fig. 12 shows that bothoperators have to postpone their upgrades and,surprisingly, both operators’ profits increase with K.As K increases, operator 1 further postpones itsupgrade and operator 2’s market share decreasesmore slowly. Thus operators’ competition in themarket share is postponed, and under large γ op-erator 2 can obtain more 3G revenue before oper-ator 1’s upgrade. Operator 1, on the other hand,also benefits from its postponement to decrease itsupgrade cost. Since operator 2 also postpones itsupgrade, operator 1 can still capture a large marketshare even though it upgrades later. As K →∞, nooperator upgrades and operators’ profits approachthe symmetric 3G profits. Under large γ, the 4Gservice is not much better than 3G and the avail-ability of 4G upgrade only intensifies operators’competition. Compared to traditional 3G scenario,both operators’ profits decrease when the upgradecost is high. In other words, both operators will be betteroff if 4G technology is not available in this case.

13

0.1 0.15 0.2 0.25 0.30.1

0.15

0.2

0.25

0.3

Operator 1’s equilibrium profit π1*

Ope

rato

r 2’

s eq

uilib

rium

pro

fit π

2*

NE 1: T1* ≤T

2*

NE 2: T1* ≥T

2*

Low cost regime:π

1*↓=π

2*↓ as K↑

Medium cost regime:π

1*↓<π

2*↓ as K↑

High cost regime:π

1*↓<π

2*↓ as K↑

Fig. 13. Two operators’ equilibrium profits (π∗1 ,π∗2) changeas cost K increases under small γ = 0.1. Other parame-ters are α = 0.5, N = 1000, U = 2, S = 1, and λ = 1 suchthat U > S + αλ.

Figure 13 shows how operators’ profits change withK under small γ. The results are more intuitive; thisis because the operators’ profits decrease with K in allthree cost regimes. Under small γ, the availability of the4G upgrade significantly improves the revenue in eachnetwork. A larger K reduces the benefit of upgrades.However, the operators’ profits will not be smaller afterthe 4G upgrade under any value of K.

8 EXTENSION TO THE HETEROGENEOUS OP-ERATOR SCENARIO

Here we present some preliminary results by consideringoperators’ network heterogeneity, and show its impacton our previous results. More specifically, we assumethat the numbers of 3G users in the two networks aredifferent (N1 6= N2 with N1 + N2 = N ), and users haveasymmetric switching costs between the two operatorsto 4G service. We denote the switching cost factor fromoperator 1’s 3G service to operator 2’s 4G service asα1, and the switching rate is thus α1λ. Similarly, theswitching cost factor from operator 2’s 3G service tooperator 1’s 4G service is α2, which may be differentfrom α1. We still focus on the most practical setting withU > max(α1, α2)λ+ S as in Section 7.

We first fix operator 1’s upgrade decision at T1 = 0 andstudy operator 2’s best response. Similar to Section 5.1,we can derive operator 2’s real time revenue before andafter T1 as

π4G−3G2,t≤T2

(t) = γN2

Ne−α2λt,

and

π4G−4G2,t>T2

(t) =N2

Ne−α2λT2

(γe−λ(t−T2) + (1− e−λ(t−T2))

·

(1− (1− γ)N1

Ne−λt − (1− γ)N2

Ne(1−α2)λT2−λt

)).

Operator 2’s profit is

π2(0, T2)

=

Z T2

0

π4G−3G2,t≤T2

(t)e−Stdt+

Z ∞T2

π4G−4G2,t>T2

(t)e−Stdt−Ke−UT2

=

γ

1− e−(α2λ+S)T2

S + α2λ+e−(α2λ+S)T2

S− (1− γ)e−(α2λ+S)T2

λ+ S

− λ(1− γ)(λ+ S)(2λ+ S)

N1

Ne((1+α2)λ+S)T2+

N2

Ne(2α2λ+S)T2

!!N2

N,

which is not concave. However, we notice that π2(0, T2)is first monotonically increasing and then monotonicallydecreasing in T2, we can show that π2(0, T2) is has aunique extremum on T2, which maximizes operator 2’sprofit. If cost K is small enough such that

dπ2(0, T2)/dT2|T2=0 ≤ 0,

then operator 2’s best response is T2(0) = 0. Otherwise,operator 2 will postpone its upgrade (i.e., T2(0) > 0).Thus we can derive the first cost threshold for operator2 as

K̃compth1 (N2, α2) =

1U

((1− γ) (1− α2)λ

λ+ S−

1λ+S −

12λ+S

N

· (1− γ) (N −N2((1 + α2)λ+ S) +N2(2α2λ+ S)) +α2λ

S

),

(20)

which depends on N2 where N1 = N−N2. Similarly, wehave K̃comp

th1 (N1) for operator 1. We can further check

∂K̃compth1 (N2, α2)∂N2

=1− γU

(1− α2)λN

(1

λ+ S− 1

2λ+ S

)> 0,

where all three terms in the multiplication are positive.Thus K̃comp

th1 (N2, α2) is increasing in N2. As a linear func-tion of α2, we can check that K̃comp

th1 (N2, α2) is linearlyincreasing in α2 as long as S is small. Similarly, we candevelop the cost threshold K̃comp

th1 (N1, α1) for operator1. Both operators will upgrade immediately only whenthe upgrade cost K is less than both cost thresholds, orequivalently, the minimum of the two thresholds. Thenwe have the following result.

Theorem 4: An operator having a larger subscribernumber (than its competitor) or having a smaller powerof keeping its subscribers has more incentive to upgradeimmediately. This characterizes its fear to lose many ofits 3G users to its competitor. Two operators will onlyreach the symmetric equilibrium T ∗1 = T ∗2 = 0 when

K ≤ K̃compth1 := min(K̃comp

th1 (N1, α1), K̃compth1 (N2, α2)).

The fact that K̃compth1 < Kcomp

th1 (with Kcompth1 in (13))

under N1 6= N2 shows that the competition betweenoperators become less severe if they have asymmetricmarket power. When K is larger than K̃comp

th1 , we cansimilarly show that operators will only reach an asym-metric equilibrium as in Theorem 3. This means that twooperators with heterogeneous parameters are more likelyto reach an asymmetric equilibrium in 4G upgrades.

14

9 CONCLUSIONS AND FUTURE WORK

This paper presents the first analytical study of opera-tors’ 4G upgrade decisions. In a 4G monopoly market,we show that the monopolist’s optimal upgrade timetrades off an increased market share and the decreasingupgrade cost. In a non-cooperative upgrading game withtwo operators, we show that operators select differentupgrade times to avoid the severe competition. Wefurther show that the availability of 4G upgrade maydecrease both operators’ profits due to competition, andtheir profits may increase with the upgrade cost.

There are several possible ways to extend the resultsin this paper. First, we could consider an oligopolymarket with more than two competitive operators. Weexpect the asymmetric upgrading equilibrium will stilloccur. Second, our model can also be extended to includedifferent pricing plans (e.g., student plan, corporate plan,and residential plan), which contribute differently to anoperator’s revenue. By following a similar revenue anal-ysis for more than one network as in Section 3, we canderive the total revenue that accounts for the differentprices and weights of different plans. Finally, there maybe some correlation in users’ usage patterns (e.g., fromthose belonging to the same family account). Accordingto [21], one Verizon account has 2.72 connections on aver-age, and users belong to the same account communicatewith each other more often. We may approximate usersof the same account as a super user, and model trafficand revenue collection from many super users.

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15

Lingjie Duan (S’09-M’12) is an Assistant Pro-fessor in Engineering Systems and Design Pillar,Singapore University of Technology and Design.He received Ph.D. degree from The ChineseUniversity of Hong Kong in 2012. During 2011,he was a visiting scholar in the Dept. of EECSat University of California at Berkeley. His re-search interests include network economics andgame theory. He is the TPC Co-Chair of INFO-COM’2014 Workshop on GCCCN and a TPCmember for multiple top-tier conferences.

Jianwei Huang (S’01-M’06-SM’11) is an Asso-ciate Professor in the Department of InformationEngineering at the Chinese University of HongKong. He received Ph.D. from Northwestern Uni-versity in 2005. He received the IEEE MarconiPrize Paper Award in Wireless Communications2011, and another six best paper awards in lead-ing international conferences. He is currently theAssociate Editor-in-Chief of IEEE ComSoc Tech-nology News, and an Editor of IEEE Journal onSelected Areas in Communications - Cognitive

Radio Series and IEEE Transactions on Wireless Communications.

Jean Walrand received his Ph.D. in EECS fromUC Berkeley and has been on the faculty ofthat department since 1982. He is the author ofAn Introduction to Queueing Networks (PrenticeHall, 1988) and of Communication Networks: AFirst Course (2nd ed. McGraw-Hill,1998) andco-author of High-Performance CommunicationNetworks (2nd ed, Morgan Kaufman, 2000), ofCommunication Networks: A Concise Introduc-tion (Morgan & Claypool, 2010), and of Schedul-ing and Congestion Control for Communication

and Processing networks (Morgan & Claypool, 2010). His researchinterests include stochastic processes, queuing theory, communicationnetworks, game theory and the economics of the Internet. Prof. Walrandis a Fellow of the Belgian American Education Foundation and of theIEEE and a recipient of the Lanchester Prize, the Stephen O. RicePrize, the IEEE Kobayashi Award and the ACM Sigmetrics AchievementAward.

APPENDIX APROOF OF THEOREM 2

Proof: We establish the properties of the operators’equilibria in the two cost regimes, respectively.

Proof of equilibrium in the low cost regime: We want toprove that T ∗1 = T ∗2 = 0 is the unique equilibrium underK ≤ Kcomp

th1,α=0.• We first show T ∗1 = T ∗2 = 0 is an equilibrium by

proving that no operator has an incentive to deviate.Given T ∗1 = 0, operator 2’s profit is πLT (0, T2) whenupgrading at T2. We can show that πLT (0, T2) has aunique maximum T best2 (0) = 0. Thus operator 2 willnot deviate from T ∗2 = 0. Similarly, we can provethat given T ∗2 , operator 1 will not deviate from T ∗1 =0.

• We then prove the uniqueness of the equilibrium bycontradiction. Without the loss of generality, we sup-pose there is another equilibrium with 0 < T ∗1 ≤ T ∗2 .Note that T ∗1 cannot equal zero, otherwise operator2 will also upgrade at t = 0 and we reach the sameequilibrium. Next we show that 0 < T ∗1 < T ∗2 cannotbe an equilibrium by proving operator 2 will chooseits upgrade at the same time. This can be provedby showing operator 2’s profit πLT (T1, T2) with anyT1 < T2 decreases with T2. Thus the possible otherequilibrium can only be T ∗1 = T ∗2 > 0. However, thisis not possible since operator 1’s profit πER(T1, T2)with any T1 = T2 decreases with T1, i.e.,∂πER(T1, T2)

∂T1|T1=T2

≤ −1− γ4

e−ST13λ2

(2λ+ S)(λ+ S)+Kcomp

th1,α=0Ue−UT1

< −e−ST13(1− γ)λ2

4(2λ+ S)(λ+ S)+Kcomp

th1,α=0Ue−ST1 = 0.

Proof of equilibrium in high cost regime: By following asimilar proof as in the low cost regime, we can provethat T ∗1 = T ∗2 in (12) is the unique equilibrium underK > Kcomp

th1,α=0. The difference is that given T ∗i , weneed to consider the possibilities of the other operatorj upgrading earlier or later than T ∗i .