Economics of Mobile Data OffloadingLin Gao∗, George Iosifidis†, Jianwei Huang∗, and Leandros Tassiulas†

∗Dept. of Information Engineering, The Chinese University of Hong Kong, Hong Kong†Dept. of Computer and Communication Engineering, University of Thessaly, and CERTH, Greece

Abstract— Mobile data offloading is a promising approach toalleviate network congestion and enhance quality of service (QoS)in mobile cellular networks. In this paper, we investigate theeconomics of mobile data offloading through third-party WiFi orfemtocell access points (APs). Specifically, we consider a market-based data offloading solution, where macrocellular base stations(BSs) pay APs for offloading traffic. The key questions arising insuch a marketplace are following: (i) how much traffic should eachAP offload for each BS? and (ii) what is the corresponding paymentof each BS to each AP? We answer these questions by using thenon-cooperative game theory. In particular, we define a multi-leader multi-follower data offloading game (DOFF), where BSs(leaders) propose market prices, and accordingly APs (followers)determine the traffic volumes they are willing to offload. We char-acterize the subgame perfect equilibrium (SPE) of this game, andfurther compare the SPE with two other classic market outcomes:(i) the market balance (MB) in a perfect competition market(i.e., without price participation), and (ii) the monopoly outcome(MO) in a monopoly market (i.e., without price competition). Ourresults analytically show that (i) the price participation (of BSs)will drive market prices down, compared to those under the MBoutcome, and (ii) the price competition (among BSs) will drivemarket prices up, compared to those under the MO outcome.

I. INTRODUCTION

We are witnessing an unprecedented worldwide growth ofmobile data traffic, which is expected to reach 10.8 exabytesper month by 2016, an 18-fold increase over 2011 [1]. How-ever, traditional network expansion methods by acquiring morespectrum licenses, deploying new macrocells of small size, andupgrading technologies (e.g., from WCDMA to LTE/LTE-A)are costly and time-consuming [2]. Clearly, network operatorsmust find novel methods to resolve the mismatch between de-mand and supply growth, and mobile data offloading appearsas one of the most attractive solutions.

Simply speaking, mobile data offloading is the use of com-plementary network technologies, such as WiFi and femtocell,for delivering data traffic originally targeted for cellular net-works. A growing number of studies have been devoted tothe potential performance benefits of mobile data offloadingand the technologies to support it [3]–[12]. Specifically, thebenefit of macrocellular data offloading through WiFi networks(called WiFi offloading) was studied and quantified using realdata traces in [3]–[6]. It is shown that in a typical urbanenvironment, WiFi can offload about 65% cellular trafficand save 55% battery energy for mobile users (MUs) [3].

This work is supported by the General Research Funds (Project NumberCUHK 412710 and CUHK 412511) established under the University GrantCommittee of the Hong Kong Special Administrative Region, China. Thiswork is also supported by the EINS, the Network of Excellence in InternetScience, through FP7-ICT grant 288021 from the European Commission. Thiswork is also supported by the “ARISTEIA” Action of the “OPERATIONALPROGRAMME EDUCATION AND LIFELONG LEARNING”, and is co-funded by the European Social Fund (ESF) and National Resources.

BS3

BS1

BS2

AP4 MU13

AP6

AP2

AP5

AP1

AP3MU23MU22

MU33

MU31

MU12

MU11

MU21MU32

Fig. 1. An instance of the data offloading scenario with M = 3 macrocellularBSs and N = 6 APs. Each AP can offload the macrocellular traffic generatedby those MUs within its coverage area. In this example, BS 1 offloads thetraffic of MU11 to AP 1, and the traffic of MU13 to AP 4; and BS 2 and BS3 offload the traffic of MU23 and MU31 to AP 6, respectively.

This performance gain can be further enlarged with the useof delaying transmission [3]–[5] and the prediction of WiFiavailability [6]. In addition to WiFi offloading, femtocellsare another primary option for macrocellular data offloading.The potential benefits and costs of deploying femtocells weresurveyed in [7]–[9]. The authors in [10] and [11] investigatedthe network operators’ profit gains from offering dual servicesthrough both macrocells and femtocells. In [12], the authorsstudied the network cost savings by femtocell networks. All ofthese studies have reached consensus that offloading throughWiFi and femtocell networks is a cost-effective and energy-prudent solution for network congestion.1

However, the above results focused on the technical aspectof data offloading, without considering the economic incen-tive for WiFi or femtocells access points (APs) to admitmacrocellular traffic (i.e., to operate in the so-called openaccess mode [13]). This incentive issue is particularly im-portant for the scenario where APs are privately owned bythird-party entities, who are expected to be reluctant to admitnon-registered users’ traffic without proper incentives. In [14]and [15], the authors considered the incentive framework forthe so-called user-initiated data offloading, where MUs initiatethe offloading process (i.e., when and where to offload theirtraffic), and hence MUs offer necessary incentives in order toaccess to APs. In this paper, we consider the network-initiateddata offloading, where cellular networks initiate the offloadingprocess (of each MUs), and hence the network operators areresponsible of incentivizing APs. In our model, we assumethat MUs are either (i) compliant or properly incentivized,2

such that they will offload their traffic exactly as the networksintended, or (ii) unaware of the offloading process at all, that

1As evidence, many network operators have already started to deploy theirown WiFi or femtocell networks for data offloading (e.g., AT&T [17]), orinitiate collaborations with other WiFi networks (as O2 did with BT [18]).

2For example, network operators can offer MUs certain rebate for compen-sating their QoS degradations (e.g., delay) induced by the data offloading.

2

is, data offloading is totally transparent to MUs.3

Specifically, we consider the network-initiated mobile dataoffloading through third-party WiFi or femtocell APs, andfocus on the economic interactions between network operatorsand APs. In particular, we focus on the incentives that networkoperators need to provide to APs in order to encourage coop-erative data offloading. Figure 1 illustrates an instance of dataoffloading between 3 macrocellular base stations (BSs) and 6APs. Each BS can offload traffic (of its MUs) to multiple APs,and each AP can admit traffic from multiple BSs.

We study the incentive design problem by adopting amarket-based data offloading solution, where network opera-tors share the data offloading benefits with APs through marketmechanisms. Namely, network operators pay APs with certainmonetary compensation for offloading traffic. Therefore, thekey problems for such a market are to determine: (a) trafficoffloading volumes: how much traffic should each AP offloadfor each BS? and (b) pricing rules: what is the correspondingpayment of each BS to each AP?4 We answer these questionsby using the non-cooperative game theory.

More specifically, we introduce a two-stage multi-leadermulti-follower game, called data offloading game (DOFF),where BSs act as leaders and APs as followers. In the firststage, each BS proposes the price that it is willing to pay toeach AP for offloading its traffic. In the second stage, eachAP decides how much traffic to offload for each BS. Wecharacterize the Nash equilibrium, NE, (or more precisely, thesubgame perfect equilibrium, SPE) of this game. We showthat under the SPE, (i) every AP rejects all BSs except thoseproposing the highest price, and (ii) the highest (BS) price toevery AP equalizes the maximum marginal cost reduction (ofall BSs) and the marginal payment (to the AP).5

We further compare the SPE with the outcomes of two otherclassic market settings: (i) the perfect competition market,and (ii) the monopoly market. The first case corresponds tothe scenario where the market prices are determined by anexternal controller (e.g., a price regulator), and BSs and APsact as price-takers, i.e. do not estimate the impact of theirstrategies on the prices. In this sense, the market players donot participate directly in the determination of prices (no priceparticipation). The second case arises when all BSs act asone single player, either because they are managed by thesame operator or because they collude and create an effectivemonopoly. Thus, there is no price competition. We analyticallyshow that the equilibrium price (arises in the SPE of our DOFFgame) is upper-bounded by the market clearing price (arises in

3This is natural for the femtocell offloading. For the WiFi offloading,this approach can be achieved by recent technological advances such as theHybrid Calling technology used by Republic Wireless Inc. [19], which addressseamless roaming between WiFi networks and cellular networks.

4Our data offloading market model differs from the spectrum marketmodels in wireless networks (e.g., [20]–[22]) in the following aspects: (i)the offloading benefit for each BS is AP-specific, and the offloading cost foreach AP is BS-specific; (ii) the offloading solution is constrained by the BSs’traffic profiles and the APs’ resource limits; and (iii) the offloading solutionsbetween different BSs and APs are coupled.

5The detailed definitions of the marginal cost reduction and the marginalpayment are given in Definitions 1 and 5, respectively.

the equilibrium of the perfect competition market), and lower-bounded by the monopoly price (arises in the equilibrium ofthe monopoly market).

The main contributions of this work are the following:• To the best of our knowledge, this is the first work that

considers a mobile data offloading market with multiplemacrocellular BSs and multiple APs, and applies gametheoretic modeling and analysis of the equilibrium.

• We systematically study the economic interactions of BSsand APs. Specifically, we define a multi-leader multi-follower data offloading game–DOFF, with BSs as leadersand APs as followers; and analytically characterize thesubgame perfect equilibrium–SPE.

• We compare the SPE of the DOFF game with theoutcomes of perfect competition market (no price partici-pation) and monopoly market (no price competition). Ourresults analytically show the impact of price participationabd price competition on market prices.

The rest of this paper is organized as follows. In Section II,we present the system model. In Sections III, we provide thesocial optimality and market balance. In Section IV, we studythe DOFF game in details. Finally, we conclude in Section V.

II. SYSTEM MODEL

A. System Description

We consider a system that includes a set M , {1, ...,M}of macrocellular base stations (BSs) and a set N , {1, ..., N}of third-party WiFi or femtocell access points (APs). BSs mayhave overlapping coverage areas, while APs are assumed tobe non-overlapping with each other. Each BS serves a set ofmobile users (MUs) randomly distributed within its coveragearea. The traffic of an MU can be offloaded to an AP, only ifit is covered by that AP (e.g., MU11 and AP 1 in Figure 1).Figure 1 illustrates such a two-tier network.

Let Smn denote the total (maximum) BS m’s traffic thatcan be offloaded to AP n, i.e., those generated by all MUsof BS m in the coverage area of AP n. Let Sm0 denote thetotal BS m’s traffic that cannot be offloaded to any AP, i.e.,those generated by all MUs of BS m not covered by any AP.Obviously, Smn ≡ 0 if there is no overlap between a BS mand an AP n (e.g., BS 1 and AP 2 in Figure 1). The trafficprofile of BS m is denoted by

Sm , (Sm0, Sm1, ..., SmN ).

Due to the uncertainty of MUs’ data usage and mobility,Sm changes randomly over time. We consider a quasi-staticnetwork scenario, where Sm,∀m ∈ M, remains unchangedwithin every data offloading period (e.g., one minute in oursimulation), while changes across periods.

We define the transmission efficiency of a communicationlink (between a BS and an MU, or an AP and an MU) as theaverage volume of traffic (Mbits) that can be supported by oneunit of spectrum resource (MHz), denoted by θ (measured inMbits/MHz). For convenience, we normalize the transmissionefficiency within θ ∈ [0, 1]. Since each AP’s coverage areais very small, we assume that: (i) the transmission efficiency

3

parameters between each AP and its covered MUs are iden-tical, and attain the maximum value θ = 1; and (ii) thetransmission efficiency parameters between each BS m andits served MUs covered by the same AP (say n) are alsoidentical, denoted by θmn ∈ [0, 1]. We further denote θm0

as the average transmission efficiency between BS m andall of its served MUs whose traffic cannot be offloaded. Thetransmission efficiency profile of BS m is denoted by

θm , (θm0, θm1, ..., θmN ).

We similarly assume that θm,∀m remains unchanged withinevery offloading period, while changes across periods.

Let xmn ∈ [0, Smn] denote the traffic offloaded from BSm to AP n, and zmn ∈ [0,+∞) denote the correspondingpayment from BS m to AP n. Then, the traffic offloadingmatrix can be written as X , (xmn)m∈M,n∈N , and thepayment matrix can be written as Z , (zmn)m∈M,n∈N .

B. Macrocellular BS Modeling

Let Cm(b) denote the serving cost of BS m for b units ofspectrum resource consumption. Such a serving cost may in-clude the energy cost, operating cost, coordinating cost, etc.We generally assume that Cm(b) is strictly increasing andconvex, that is, C′m(b) > 0 and C′′m(b) ≥ 0, ∀m ∈M.

Denote xm , (xm1, ..., xmN ) and zm , (zm1, ..., zmN ) asthe traffic offloading profile and payment profile of BS m, cor-responding to the m-th rows of X and Z, respectively. Givenany xm, the BS m’s total spectrum resource consumption fordelivering all remaining un-offloaded traffic is

bm(xm,Sm,θm) = Sm0

θm0+∑Nn=1

Smn−xmn

θmn. (1)

For clarity, we will write bm(xm,Sm,θm) as bm hereafter.Therefore, the total cost of BS m (including both the servingcost and the payment to APs) under xm and zm is

CTOTm (xm; zm) = Cm(bm) +

∑Nn=1 zmn. (2)

The payoff of every BS m is defined as the cost reductionachieved from data offloading. Formally,

UBSm(xm; zm) = CTOT

m (0;0)− CTOTm (xm; zm)

= Rm(xm)−∑Nn=1 zmn,

(3)

where Rm(xm) = Cm(b0m)− Cm(bm) is the BS m’s servingcost reduction, and b0m = bm(0,Sm,θm) =

∑Nn=0

Smn

θmnis the

BS m’s total resource consumption without data offloading.

C. AP Modeling

Each AP is owned by a private owner, and has its owntraffic demand. Let ξn denote the AP n’s own traffic demand,which changes randomly over time. Denote fn(ξ) and Fn(ξ)as the PDF and CDF of ξn, respectively.6 Let Bn denote thetotal spectrum resource available for AP n. Then, the expectedprofit of AP n (from its own traffic) is7

WAPn (Bn) = (wn − cn) · Eξn

[min{Bn, ξn}

], (4)

6Note that the deterministic case where ξn keeps unchanged within a singledata offloading period is a special case of our model here.

7Here we consider the linear serving cost for APs (as used in [16]).Extending to a general serving cost function is straightforward.

where wn is the average unit revenue from its own traffic,cn is the unit cost for its resource consumption, and Eξn [·]denotes the expectation with respect to ξn.

Denote xn , (x1n, ..., xMn) and zn , (z1n, ..., zMn) asthe traffic offloading profile and payment profile to AP n,corresponding to the n-th columns of X and Z, respectively.When admitting xn, the resource for BSs’ traffic is xn ,∑Mm=1 xmn, and thus the resource left for its own traffic is

Bn − xn. Obviously, a feasible xn satisfies: xn ≤ Bn. Underany feasible xn and zn, the total profit of AP n is

WTOTn (xn; zn) = WAP

n (Bn−xn)− cn ·xn +∑Mm=1 zmn. (5)

The payoff of every AP n is defined as the profit improve-ment achieved from data offloading, i.e.,

UAPn (xn; zn) = WTOT

n (xn; zn)−WTOTn (0;0)

= Qn(xn) +∑Mm=1 zmn,

(6)

where Qn(xn) = WAPn (Bn − xn) − WAP

n (Bn) − cnxn is theAP n’s own profit loss by offloading BSs’ traffic.

D. Social Welfare

The social welfare is defined as the aggregate payoff of allBSs and APs, and denoted byΨ(X;Z) =

∑m∈M UBS

m(xm; zm) +∑n∈N UAP

n (xn; zn)

=∑m∈M Rm(xm) +

∑n∈N Qn(xn).

(7)

We will also write Ψ(X;Z) as Ψ(X), since it is independentof the payment transfers between BSs and APs.

III. SOCIAL OPTIMALITY AND MARKET BALANCE

In this section, we first derive the centralized optimal so-lution (social optimality) which maximizes the social welfare.Then we study how to achieve the social optimality in a perfectcompetition market with price-taking BSs and APs.

A. Social Optimality

The social welfare optimization problem (SWO) is(SWO) max

XΨ(X)

s.t. (i) 0 ≤ xmn ≤ Smn, ∀m ∈M, n ∈ N ;

(ii)∑m∈M xmn ≤ Bn, ∀n ∈ N .

(8)

For convenience, we first introduce the following concepts.

Definition 1 (Marginal Cost Reduction of BS m).MCBS

m(xm) , C′m(bm). (9)

Definition 2 (Marginal Profit Loss of AP n).MLAP

n (xn) , wn − (wn − cn) · Fn(Bn − xn). (10)

Intuitively, the marginal cost reduction MCBSm represents the

decrease of BS m’s serving cost induced by reducing oneadditional unit of resource consumption. The marginal profitloss MLAP

n represents the decrease of AP n’s profit induced byoffloading one additional units of traffic for BSs.

Then, the first-order derivative of Ψ(X) can be written as:∂Ψ(X)∂xmn

= θ−1mn ·MCBS

m(xm)−MLAPn (xn). (11)

We can further show that the SWO (8) is a convex optimization(see Appendix-A in [24]), and thus can be solved by theKarush-Kuhn-Tucker (KKT) conditions [23]. Formally,

4

Lemma 1 (Social Optimality). The socially optimal solutionX◦ is given by the following optimality conditions:(A.1) θ−1

mn ·MCBSm(x◦m)−MLAP

n (x◦n)−µ◦mn + λ◦mn− η◦n = 0,(A.2) µ◦mn · (Smn − x◦mn) = 0, λ◦mn · x◦mn = 0,

η◦n · (Bn −∑Mm=1 x

◦mn) = 0,

for all m ∈ M and n ∈ N , where µ◦mn, λ◦mn and η◦n are the

corresponding optimal Lagrange multipliers.

B. Market Balance in Perfect Competition Market

Now we consider a perfect competition market, also calledprice-taking market, where market prices (pricing rules) areset by a central controller (e.g., price regulator) in order tomatch (balance) the market supply and demand. BSs and APsare price takers, responding to the announced prices withoutconsidering the impact of their strategies on the market prices.The individual optimization problems for price-taking BSs(BSO) and APs (APO) are, respectively,8

(BSO) maxxm

UBSm(xm; zm(xm)), s.t. xmn ∈ [0, Smn],∀n ∈ N .

(APO) maxxn

UAPn (xn; zn(xn)), s.t.

∑m∈M xmn ≤ Bn.

Let x†m , (x†m1, ..., x†mN ) denote the optimal individual

decision of BS m (i.e., the solution of BSO), and x‡n ,(x‡1n, ..., x

‡Mn) denote the optimal individual decision of AP n

(i.e., the solution of APO). Intuitively, we can also view x†mnas the demand of BS m for AP n’s resource, and x‡mn as thesupply of AP n’s resource intended for BS m.

Definition 3 (Market Balance - MB). The offloading market isconsidered balanced when x†mn = x‡mn, ∀m ∈M, n ∈ N .

By interpreting the Lagrange multipliers as shadow prices,we define the following market clearing prices:

p◦n , MLAPn (x◦n) + η◦n, ∀n ∈ N , (12)

each associated with one AP.9 It is easy to show that under theabove market clearing prices, we have: x†mn = x‡mn = x◦mnfor all m ∈M and n ∈ N . Therefore:

Observation 1 (Market Balance). The market clearing prices(12) lead to the market balance. In addition, under the marketbalance, the social welfare is maximized.

IV. GAME THEORETIC ANALYSIS IN THE FREE MARKET

In this section, we consider a free market, where there isno central controller, and both traffic offloading volumes andpricing rules are determined freely by BSs and APs.

In this case, some players (e.g., BSs in our analysis10) aregiven the authority to set market prices, and thus become price-setters. This is the so-called price participation. Naturally,price competition may occur between the price-setters. Weare interested in addressing the following problems: what isthe impact of price participation and price competition on themarket outcome and on the social welfare achieved.

8Here zm(xm) , (zmn(·))n∈N and zn(xn) , (zmn(·))m∈M.9This implies the following pricing rules: zmn(xmn) = p◦n ·xmn, ∀m,n.10The analysis is quite similar when APs are given this authority (see [24]).

A. Data Offloading Game Formulation

We formulate the interactions between BSs and APs in thefree market as a two-stage non-cooperative game, called dataoffloading game (DOFF). In the first stage (Stage-I), every BSproposes a price to every AP within its coverage area. In thesecond stage (Stage-II), every AP indicates the traffic volumeit is willing to offload for every BS who has proposed to it. Inthe context of multi-stage game, we refer to BSs as leaders,and APs as followers. Obviously, under such a game situation,BSs are price-setters, while APs are still price-takers.11

Let Nm denote the set of APs within the coverage area ofBS m, and Mn denote the set of BSs whose coverage areascontains AP n. In the example of Figure 1, N1 = {AP1, AP4,AP5, AP6}, andM6 = {BS1, BS2, BS3}. The strategy of BSm is the price profile pm , (pmn)n∈Nm , where pmn is theprice BS m proposes to AP n; and the strategy of AP n isthe traffic profile xn , (xmn)m∈Mn

, where xmn is the trafficoffloading volume AP n responds to BS m.

Definition 4 (Subgame Perfect Equilibrium - SPE). A strategyprofile

({p∗m}m∈M, {x∗n}n∈N

)is a subgame perfect equilib-

rium if it represents a Nash equilibrium at every stage, i.e.,I: p∗m = arg max

pm

UBSm

(x∗m(P∗−m,pm);pm � x∗m(P∗−m,pm)

),

II: x∗n = x∗n(p∗n) = arg maxxn

UAPn

(xn;p∗n � xn

),

for all BSs m ∈M at Stage-I and APs n ∈ N at Stage-II.12

B. An Illustrative Model with One BS and One AP

We first look at an illustrative model with one BS and oneAP. In this case, the DOFF game degenerates to a conventionalStackelberg game. We use this simple model to illustrate (i) thebest individual decisions of APs and BSs, and (ii) the impactof price participation on the market outcome.

For notational consistency, we denote the BS as m and theAP as n, i.e., M = {m} and N = {n}. We solve the SPE ofthis game by the backward induction method.

1) Stage-II: Given the BS’s price pmn, the AP’s optimiza-tion problem (APO) and the corresponding solution are

(APO) maxxmn

UAPn (xmn; pmn · xmn), s.t. xmn ∈ [0, Bn].

Lemma 2 (AP’s Optimal Decision).

x∗mn =

0 if pmn < cn

Bn − F (-1)n

(wn−pmn

wn−cn

)if pmn ∈ [cn, wn]

Bn if pmn > wn

(13)

where cn , cn + (wn − cn) · [1− Fn(Bn)].

11This is due to the two-stage modeling assumption (i.e., BSs and APs arenot making decisions simultaneously). Note that for a data offloading market,the two-stage assumption is more reasonable than assuming that BSs and APsmake decisions at the same time. The key reason is that BSs usually havemore market power, and thus have the priority to determine market prices.

12Here P∗−m , (p∗1, ...,p∗m−1,p

∗m+1, ...,p

∗M ), and p�x represents the

inner product of two vectors p and x. Recall that (i) pm = (pm1, ..., pmN ),and xm = (xm1, ..., xmN ) are all prices and traffic offloading volumesrelated to BS m, and (ii) pn = (p1n, ..., pMn), and xn = (x1n, ..., xMn)are those related to AP n. By (II), every x∗mn in x∗n is a function of p∗n.Thus, x∗m is a function of P∗ = (p∗n)n∈N , which can also be written asP∗ = (p∗m)m∈M = (P∗−m,p

∗m).

5

Traffic volume

Price

0 x∗mn x◦mn dmn

p∗mn

p◦mn

cnMCBS

m · θ−1mn

MLAPn

MPAPn

(B)

MB

(E)

SPE

Fig. 2. Illustration of SPE and MB for the single-BS single-AP model. Thehollow circles marked with (E) and (B) denote the conditions for SPE and MB,respectively. Specifically, (i) under the SPE, the BS’s marginal cost reductionMCBS

m · θ−1mn equals to the marginal payment MPAP

n to the AP; (ii) under theMB, the BS’s marginal cost reduction MCBS

m ·θ−1mn equals to the AP’s marginal

profit loss MLAPn (and the BS demand equals to AP supply).

Intuitively, if the BS proposes a large enough price pmn(i.e., pmn > wn), AP n will sell all of its resource to the BS.On the other hand, if the price pmn is smaller than a thresholdcn, AP n will sell none of its resource to the BS. In this sense,we refer to cn as the critical price of AP n.

2) Stage-I: Knowing the AP’s response, the BS’s optimiza-tion problem (BSO$) in Stage-I is

(BSO$) maxpmn

UBSm

(x∗mn(pmn); pmn · x∗mn(pmn)

)s.t. 0 ≤ x∗mn(pmn) ≤ Smn.

(14)

For convenience, we introduce the following concept.

Definition 5 (Marginal Payment of BS m to AP n).MPAP

n (pmn) , x∗mn ·∂pmn

∂x∗mn+ pmn (15)

Intuitively, the marginal payment MPAPn represents the increase

of BS’s total payment (not unit price) in order to have AP noffload one additional unit of traffic.13

Then, the first-order derivative of UBSm(·; ·) can be written as

∂UBSm

∂pmn=(θ−1mn ·MCBS

m(x∗mn)−MPAPn (pmn)

)· ∂x

∗mn

∂pmn. (16)

We can further show that problem (14) is a convex optimiza-tion, and thus can be solved by KKT analysis. Formally,

Lemma 3 (BS’s Optimal Decision). The BS m’s optimal pricep∗mn satisfies the following conditions:(D.1) θ−1

mn ·MCBSm(x∗mn)−MPAP

n (p∗mn)− µ∗mn + λ∗mn = 0,(D.2) µ∗mn · (Smn − x∗mn) = 0, λ∗mn · x∗mn = 0,where µ∗mn and λ∗mn are the optimal Lagrange multipliers.

3) Observation: Figure 2 illustrates the SPE and MB, fromwhich we can see: x∗mn ≤ x◦mn and p∗mn ≤ p◦mn. Therefore,

Observation 2. The price participation (of BSs) drives themarket price down from the market clearing price (as shownby the SPE and MB in Figure 2).

C. A General Model with Multiple BSs and Multiple APs

We now return to the general model with multiple BSs andmultiple APs. Similarly, we apply the backward induction.

13To achieve this, the BS needs to increase the price by ∆pmn = ∂pmn∂x∗mn

.This leads to an additional payment that consists of two parts: (i) pmn +∆pmn by offloading an additional unit of traffic at the new price pmn +∆pmn, and (ii) x∗mn ·∆pmn for the existing x∗mn units of offloading traffic.

1) Stage-II: First, we study every AP n’s optimal decisionx∗n when receiving multiple price proposals from BSs. Letpn , (pmn)m∈Mn

denote the prices AP n receives. Theoptimization problem (APO) for AP n is

(APO) maxxn

UAPn (xn; pn � xn)

s.t. (i) xmn ≥ 0, ∀m ∈Mn;

(ii)∑m∈Mn

xmn ≤ Bn.(17)

Note that (17) is a direct extension of the APO in SectionIII-B. Next we provide some additional properties for thesolution of (17). Denote p+

n , maxm∈Mnpmn as the highest

price of all BSs,M+n , {m ∈Mn | pmn = p+

n } as the set ofBSs proposing the highest price, and Xn ,

∑m∈M+

nxmn as

the total resource for those BSs proposing the highest price.

Lemma 4. The optimal solution x∗n of problem (17) satisfies:(a) x∗mn = 0, ∀m /∈M+

n ;(b) X∗n is given by (13) by replacing pmn as p+

n .

Lemma 4 shows that only the highest price is accepted bythe AP. We further notice that any feasible division of X∗namong BSs M+

n leads to the same payoff for the AP. Thatis, the allocation of X∗n is irrelevant to the AP’s payoff. Thisimplies that the conditions in Lemma 4 are also sufficient.

Although the AP n’s benefit does not depend on the detailedallocation of X∗n (among M+

n ), an explicit allocation mecha-nism is still necessary for analyzing the BSs’ best decisions.We adopt the following proportional allocation:

x∗mn , ψmn ·X∗n, ∀m ∈M+n , (18)

where ψmn , dmn∑k∈M+

ndkn

, and dmn is the BS m’s traffic of-

floading demand (to AP n) under the price p+n (which is given

by the BSO in Section III-B). Intuitively, we can view thatAP n offloads X∗n units of traffic picked randomly and uni-formly from the total demand of BSsM+

n . Note that both ψmnand X∗n depends on the price p+

n , and so does x∗mn. Figure 2illustrates the BS’s demand dmn under the price p∗mn.

2) Stage-I: We now study the price competition among BSsin the first stage, and the resultant SPE.

We first focus on the price competition of BSs for a partic-ular AP n, assuming that their prices to other APs are fixed. Aprice competition equilibrium in AP n is a price profile p∗n ,(p∗mn)m∈Mn

such that no BSs m ∈Mn can improve its pay-off by changing its price pmn. Note that if every AP reachesits equilibrium, we obtain the SPE of the whole game.

By the previous analysis, we can find that for any priceprofile pn, only the highest price is accepted by AP n. Thatis, only those BSs proposing the highest price survive in theprice competition. This implies that for any price profile, thoseprice proposals smaller than the highest price are irrelevant.Therefore, to characterize the equilibrium p∗n, we only needto find (i) the highest price p+

n = maxm∈Mnp∗mn, and (ii) the

BSs M+n proposing this price (and thus survive).

We have the following necessary conditions for p∗n.

Lemma 5. The equilibrium p∗n in AP n satisfies:θ−1mn ·MCBS

m(0) ≤ p+n , ∀m /∈M+

n . (19)

6

Traffic volume

Price

0 XMn X◦nX∗n

pMn

p+n

p◦n

cn(c)

(a)MCBS1

θ1n

(b)MCBS

2

θ2n

MLAPn

MPAPn

(B)

MB

(O)(E)

MO SPE

x∗1n = ψ1n ·X∗n

x∗2n

Fig. 3. Illustration of SPE, MB, and MO for the two-BS single-APmodel. The curve (c) is the horizontal stack of curves (a) and (b), denotingthe integrated marginal cost reduction of both BSs. The hollow circlesmarked with (E), (B), and (O) denote the conditions for SPE, MB, andMO, respectively. Specifically, (i) under the MO, the marginal cost reductionθ−1mn ·MCBS

m of every BS m ∈ {1, 2} equals to the marginal payment MPAPn

to the AP; (ii) under the MB, the marginal cost reduction θ−1mn · MCBS

m ofevery BS m ∈ {1, 2} equals to the AP’s marginal profit loss MLAP

n (and theAP supply X◦n equals to the total BS demand); and (iii) under the SPE, thelargest marginal cost reduction of both BSs (i.e., θ−1

1n ·MCBS1 ) equals to the

marginal payment MPAPn to the AP. It is easy to show that the market price

under the SPE (p+n ) must be between those under the MO (pMn) and MB (p◦n).

Here θ−1mn·MCBS

m(0) represents the highest price under whichBS m is willing to demand a positive amount of resource fromAP n (as shown by the initial point of curve MCBS

m · θ−1mn in

Figure 2). Thus, Lemma 5 implies that any BS proposing aprice lower than the highest price p+

n (i.e., those BSs m /∈M+

n ) must have a zero demand under p+n . Accordingly, any BS

m with a positive demand under p+n (i.e., θ−1

mn ·MCBSm(0) > p+

n )must propose this highest price (and thus survive). Next westudy what will be the highest price p+

n that they can propose.

Lemma 6. The highest price p+n under the equilibrium p∗n is

given by the following condition:MPAP

n (p+n ) = max

m∈M+n

θ−1mn ·MCBS

m

(x∗mn(p+

n )), (20)

where x∗mn(p+n ) = ψmn ·X∗n defined by (18).

Intuitively, if there exists one BS m ∈ M+n that has an

incentive to increase its price (i.e., θ−1mn · MCBS

m(x∗mn) >MPAP

n (p+n )), then by Lemma 5, all other BSs with positive

demand will naturally following the newly increased price.The highest price p+

n will continuous to increase, until thelargest θ−1

mn ·MCBSm(x∗mn) meets the MPAP

n (p+n ).

3) Observation: We compare the equilibrium p∗n with twoclassic market outcomes: (i) the market balance (MB) de-fined in Section III-B, and (ii) the monopoly outcome (MO),wherein BSs act as an integrated individual (monopolist)and seek to maximize their aggregate payoff (see [24] fordetails). Figure 3 illustrates these solution concepts for thescenario of two BSs and one AP, from which we can find thatpMn ≤ p+

n ≤ p◦n. Therefore, we have the following observation.Observation 3. The price competition (of BSs) drives the mar-ket price up from the monopoly price (as shown by points MOand SPE in Figure 3). Moreover, similar to Observation 2, theprice participation (of BSs) drives the market price down fromthe market clearing price (as shown by MB and SPE).

V. CONCLUSION

The increasing traffic demand and the network operators’growing needs for reducing their capital and operational ex-penditures (CAPEX and OPEX) are expected to boost third-party offloading solutions. Studying the economic interactionsof the involved entities is an important step towards therealization of this promising solution. In this paper, we studythe economics of mobile data offloading through third-partyWiFi or femtocell APs. We propose a market-based offloadingscenario, and study the market outcome by using the non-cooperative game theory. We further study the impact of priceparticipation and competition on the market outcome. Ouranalysis sheds light to many different aspects of this interestingproblem. There are many directions for future work in thisarea. For example, it would be useful to study the price ofanarchy of the resulting SPE. Another interesting direction isto study the offloading market under incomplete information.

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