CELLULAR data traffic has been growing at an unprecedented

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1 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 A Icetive Framework for Mobile Data Offloadig Market uder Price Competitio Hamed Shah-Maouri, Member, IEEE, Vicet W.S. Wog, Fellow, IEEE, ad Jiawei Huag, Fellow, IEEE Abtract Mobile data offloadig ca help the mobile etwork operator MNO) cope with the exploive growth of cellular traffic, by deliverig mobile traffic through third-party acce poit. However, the acce poit ower APO) would eed proper icetive to participate i data offloadig. I thi paper, we coider a data offloadig market that iclude both price-takig ad price-ettig APO. We formulate the iteractio amog the MNO ad thee two type of APO a a three-tage Stackelberg game, ad tudy the MNO profit maximizatio problem. Due to a o-covex trategy pace, it i i geeral a o-covex game. Neverthele, we traform the trategy pace ito a covex et ad prove that a uique ubgame perfect equilibrium exit. We further propoe iterative algorithm for the MNO ad price-ettig APO to obtai the equilibrium. Employig the propoed algorithm, the APO do ot eed to obtai full iformatio about the MNO ad other APO. Through umerical tudie, we how that the MNO profit ca icreae up to three time comparig with the o-offloadig cae. Furthermore, our propoed icetive mechaim outperform a exitig algorithm by 8% i term of the MNO profit. Reult further how that price competitio amog price-ettig APO drive the equilibrium market price dow. Idex Term Mobile data offloadig, etwork ecoomic, o-covex game, Stackelberg game, ubgame perfect equilibrium. INTRODUCTION. Backgroud CELLULAR data traffic ha bee growig at a uprecedeted rate over the pat few year. The icreaig data traffic force the mobile etwork operator MNO) to employ differet method to fill the gap betwee the fat growig demad ad the low growig capacity of their deployed etwork. Acquirig more pectrum licee, itallig ew macro/micro/femto-cell bae tatio BS), ad deployig ew techologie ca alleviate etwork cogetio ad icreae the MNO capacity. Neverthele, thee method are both cotly ad time coumig to implemet. Mobile data offloadig, which refer to deliverig data traffic of the MNO to third-party etwork, i a promiig alterative to addre thi gap. The MNO ca deliver traffic of it ow ubcriber through WiFi, femtocell, or microcell etwork to upport the growig traffic demad. Mobile traffic offloaded oto WiFi ad femtocell etwork exceeded the cellular traffic for the firt time i 25 []. The performace beefit of mobile data offloadig [2], [3], [4], [5], [6] ha motivated the MNO to deploy their ow WiFi etwork [7]. However, a ubiquitou deploymet of WiFi acce poit AP) with a good coverage ca be very expeive due to the dyamic behavior of mobile traffic, the mall coverage area of each AP a well a the cot of AP H. Shah-Maouri ad Vicet W.S. Wog are with the Departmet of Electrical ad Computer Egieerig, the Uiverity of Britih Columbia, Vacouver, Caada. {hhahmaour, vicetw}@ece.ubc.ca. J. Huag i with the Departmet of Iformatio Egieerig, the Chiee Uiverity of Hog Kog, Hog Kog. jwhuag@ie.cuhk.edu.hk. Maucript received 6 Mar. 26; revied 23 Dec. 26; accepted 9 Mar. 27. Date of publicatio xx xxx. 27; date of curret verio xx xxx. 27. For iformatio o obtaiig reprit of thi article, pleae ed to: reprit@ieee.org, ad referece the Digital Object Idetifier below. Digital Object Idetifier o..9/tmc.27.xxxxxx deploymet ad ite acquiitio. To fully exploit the beefit of data offloadig, the MNO ca take advatage of the third-party AP. I retur, the MNO hould provide proper ecoomic icetive to the acce poit ower APO), i order to compeate their cot i term of eergy coumptio, backhaul cot, ad the potetial impact o their ow cutomer..2 Motivatio ad Cotributio Several exitig tudie o mobile data offloadig market e.g., [8], [9], []) coidered the leader-follower model aumig that the MNO alway ha a larger market power tha the APO. I thee work, data offloadig market i modeled a a two-tage Stackelberg game, where the MNO act a the leader ad et the price, while the APO are price-takig follower. However, i practice, ome APO ca have igificat market power ad hece are price-ettig itead of pricig-takig player. The aforemetioed exitig model caot be ued to maage uch market. It i thu importat to udertad how the MNO hould iteract with both price-takig ad price-ettig APO i a mobile data offloadig market. I thi paper, we propoe a icetive framework which allow the MNO to iteract with both price-ettig ad price-takig APO. Reidetial uer or mall compaie which ow WiFi/femtocell AP are coidered a pricetakig player. The MNO ha the priority to determie market price for thi type of APO. O the other had, price-ettig player are thoe APO who have more market power tha MNO. A example, large etwork provider uch a BT WiFi [] with over 5 millio hotpot i Uited Kigdom ad chai tore uch a Starbuck Corp. may have more market power tha a igle MNO. The followig

2 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 Stage I: Price-ettig APO move firt ad et the price. Stage II: The MNO determie the offloadig trategy ad et price for the price-takig APO. Stage III: Price-takig APO determie the amout of traffic they ca admit. Fig. : Three-tage Stackelberg game model. Price-takig ad priceettig APO are repreeted by differet ico for preetatio clarity. coideratio harply ditict our work from the exitig literature: a) The MNO egotiate with both price-ettig ad pricetakig APO. b) The MNO aim to maximize it ow profit, which i obtaied by icreaig the aggregate etwork capacity ad imultaeouly reducig the cot. To facilitate the aalyi, we formulate the iteractio betwee the MNO ad APO a a three-tage Stackelberg game a how i Fig.. Through thi game, we tudy the price competitio amog price-ettig APO a well a the competitio betwee the price-takig APO for the amout of traffic they ca deliver. I the firt tage, the price-ettig APO determie their price with the goal of maximizig their ow payoff. I the ecod tage, the MNO decide whether to utilize the price-ettig APO by either acceptig or decliig their offered price, ad how much traffic to offload i cae of acceptig the price. It the et the price for the price-takig APO with the goal of maximizig it ow profit. I the third tage, price-takig APO follow the MNO price to determie how much traffic they ca admit to maximize their ow payoff. We coider two verio of thi game formulatio with differet MNO objective fuctio, amely: i) cot reductio, ad ii) etwork expaio. I the cot reductio problem, give a fixed admitted traffic demad, the MNO offload mobile traffic to the APO to reduce the cot of data delivery. The etwork expaio problem further allow the MNO to optimize the amout of traffic to be admitted ito the ytem, which will be delivered through it ow macrocell BS ad third-party AP. The etwork expaio problem ca be viewed a a geeralizatio of the cot reductio problem. For the ake of preetatio clarity, we firt tudy the cot reductio problem ad the exted the aalyi to tudy the etwork expaio problem. I ummary, the key cotributio of thi paper are a follow: Uique icetive framework: We model a data offloadig market that icorporate both price-takig ad priceettig APO. We model the iteractio amog the two type of APO ad MNO a a three-tage Stackelberg game, ad derive the correpodig ubgame perfect equilibrium which determie the market price. Equilibrium Aalyi: We how that the game i i geeral a o-covex game due to a o-covex trategy pace. Neverthele, we traform the trategy pace. It hould be oted that i the exitig work e.g., [8], [9], []), it wa aumed that the APO oly compete for the amout of traffic they ca offload. However, utilizig our propoed framework, we are able to model the price competitio amog the APO a well. ito a compact covex et via bijectio, baed o which we prove the exitece of a ubgame perfect equilibrium. We further prove that a uique equilibrium trategy ca be obtaied i the cot reductio problem. Algorithm Deig: To obtai the ubgame perfect equilibrium, we propoe iterative algorithm, oe for priceettig APO to determie their bet repoe trategie ad oe for the MNO to facilitate iformatio exchage amog the APO. Utilizig the propoed algorithm, the price-ettig APO do ot eed to obtai full iformatio about the MNO ad other APO, at the expee of a mall commuicatio overhead. We prove that the propoed algorithm coverge to the uique ubgame perfect equilibrium trategy. Equilibrium Efficiecy: We evaluate the efficiecy of the equilibrium through exteive umerical tudie, by comparig the ocial welfare obtaied by the equilibrium with the ocial welfare of a market without price competitio. Reult how that the equilibrium efficiecy icreae a the umber of APO icreae due to a higher competitio i the market. Moreover, our propoed icetive framework i able to achieve a cloe-to-optimal ocial welfare whe a large umber of APO participate i the data offloadig market. Performace Evaluatio: Simulatio reult how that the propoed market model ca igificatly improve the MNO profit. Our propoed framework icreae the profit by up to three time comparig to the ooffloadig cae i the cot reductio problem. Moreover, our propoed framework outperform the cheme propoed i [8] by 8% i term of the MNO profit. For the etwork expaio problem, our propoed framework ca icreae the MNO profit by up to four time through deliverig 3% more traffic. We alo how that a higher data delivery cot of the MNO lead to higher payoff for the APO. Furthermore, we how that the propoed iterative algorithm coverge quickly to the uique ubgame perfect equilibrium trategy. Thi paper i orgaized a follow. I Sectio 2, we review the related literature. I Sectio 3, we itroduce the ytem model. We formulate the three-tage Stackelberg game i Sectio 4. I Sectio 5, we tudy the exitece ad uiquee of equilibrium ad develop the iterative algorithm. We evaluate the performace of our framework through exteive imulatio i Sectio 6. Fially, we coclude i Sectio 7. 2 RELATED LITERATURE The ecoomic apect of mobile data offloadig have recetly bee tudied o two differet approache, amely uer-iitiated ad operator-iitiated offloadig. The former approach coider the ceario where mobile ubcriber egotiate with the MNO ad APO to offload their traffic. The MNO ha to leae the AP badwidth ad provide icetive for mobile uer to iitialize the offloadig [9], [], [2], [3], [4], [5]. I particular, the mechaim propoed i [9], [] are baed o leader-follower game. Lee et al. i [9] coidered cellular traffic offloadig via freely available WiFi etwork. Zhag et al. i [] modeled the data offloadig market a a leader-follower game,

3 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION where macrocell, mall cell, ad WiFi etwork ower are leader, ad mobile ubcriber are price-takig follower. I additio to the aforemetioed tudie i mobile data offloadig, leader-follower game have bee widely ued to model pricig ad ecoomic apect i differet wirele etwork [6], [7], [8], [9], [2]. Thee work modeled the market a two-tage Stackelberg game. The operator-iitiated offloadig approach focue o the offloadig deciio made by the MNO ad APO o behalf of the uer. Such offloadig deciio i traparet to the mobile uer. Although everal icetive mechaim have bee propoed for operator-iitiated offloadig, oe of them coidered price-ettig ad price-takig player imultaeouly. Gao et al. i [8] propoed a market-baed data offloadig olutio coiderig oly price-takig APO. Gao et al. i [2] further coidered a bargaiig-baed mobile data offloadig approach, where the MNO i give the authority to iitiate the market iteractio. Wag et al. i [22] propoed a ditributed icetive mechaim to model the iteractio betwee offloadig ervice provider i.e., APO) ad offloadig ervice coumer i.e., data flow). Kag et al. i [23] propoed a icetive mechaim to motivate WiFi APO to deliver the MNO traffic. I thi work, WiFi APO are rewarded ot oly baed o the amout of traffic they deliver but alo baed o the quality of their offloadig ervice. The work i [24], [25], [26] propoed everal auctio mechaim for data offloadig, where the APO are aumed to be price-takig. I thee tudie, the objective of the MNO i to miimize the cot of data delivery. I our work, however, we tudy a more geeral problem of improvig the MNO profit by admittig more traffic ad reducig the cot imultaeouly. Such a ditictio make our aalyi of the etwork expaio problem much more challegig tha thoe tudied i the literature. I additio to the ecoomic apect of data offloadig, Che et al. [27] tudied eergy-efficiecy orieted traffic offloadig mechaim. They propoed a olie reiforcemet learig framework for the problem of traffic offloadig i a tochatic heterogeeou cellular etwork. Their objective wa to miimize the total eergy coumptio of the heterogeeou cellular etwork while maitaiig the quality-of-ervice experieced by mobile uer. However, they did ot coider the trategic behavior of APO. 3 SYSTEM MODEL We focu o the iteractio betwee a macrocell BS of a MNO ad a et of third-party APO 2. The AP have overlappig coverage area with the BS, hece the MNO may offload it traffic to them. There are N t price-takig ad N price-ettig APO, deoted by et N t = {,..., N t } ad N = {N t +,..., N t +N }, repectively. Thu, we have the et N = N t N of third-party APO, ad N = N t +N. Sice each APO coverage area i relatively mall, we aume the APO are patially o-overlappig ad they 2. Our aalyi ca be exteded to the cae of multiple macrocell BS. To do o, the cot to be itroduced later) impoed to all BS by deliverig traffic to mobile uer a well a the total traffic delivered through all BS hould be coidered. Price-takig AP Price-ettig AP MNO BS Fig. 2: A ytem with oe macrocell BS, two price-ettig APO, ad eve price-takig APO. do ot iterfere with each other 3, imilar a the model i [8], [2], [28]. We further aume that APO operate i pectrum differet from the BS, ad hece do ot geerate iterferece to the BS. I particular, WiFi AP operate i the uliceed bad, ad femtocell AP are allocated with differet frequecy bad from the BS pectrum bad [29], [3]. Fig. 2 illutrate a example of the etwork with oe macrocell BS, two pricig-ettig APO, ad eve pricetakig APO. We divide the coverage area of the BS ito N + regio repreeted by the et N {}. Amog them, N regio are aociated with APO, where traffic of mobile ubcriber ca either be erved by the BS or be offloaded to the correpodig APO. Set {} repreet the regio which i oly covered by the BS but ot by ay APO, where offloadig i ot poible. The total dowlik traffic demad i a regio m N {} i deoted by S m, which i iitially teered toward the MNO 4. Hece, the traffic demad vector of the MNO i S = S,..., S N ). The traffic demad S m varie over time due to the tochatic ature of mobile ubcriber traffic. We coider a quaitatic etwork ceario, ad aalyze the market mechaim i a data offloadig period e.g., two ecod), durig which S m remai uchaged for all m. For differet approache to meaure ad etimate the traffic demad, ee [3], [32]. We defie the amout of data traffic that ca be delivered by oe uit of pectrum reource i Hz) per uit time a the tramiio efficiecy. The tramiio efficiecy of each lik deped o variou factor uch a path lo, hadowig, ad fadig. However, we ue the tramiio efficiecy to determie the aggregate badwidth required to deliver the traffic demad to all uer withi each regio. We do ot pecify how the tramiio efficiecy i computed, a our framework i applicable for geeral tramiio model. The tramiio efficiecy ca be obtaied through meauremet ad variou predictio method [33], [34], [35]. Let θ deote the tramiio efficiecy i bit/ec/hz) of commuicatio lik betwee 3. Exteio of the curret model to a geeral model with overlappig APO require coiderig the iterferece amog the APO. Moreover, the umber of ditict overlappig regio partially covered by differet APO grow expoetially with N, which further make the aalyi complicated. We leave thi exteio a future work. 4. Similar to other operator-iitiated offloadig mechaim e.g., [8], [5], [2], [22], [23], [24]), we coider the total traffic demad of each regio, a the mobile ubcriber are ot directly ivolved i market egotiatio.

4 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 the BS ad mobile ubcriber located i regio N, which i covered by APO. Sice the coverage area of each APO i relatively mall, we aume that differet mobile ubcriber i the ame regio have the ame tramiio efficiecy. Furthermore, θ repreet the average tramiio efficiecy betwee the BS ad thoe mobile ubcriber i regio {} hece, ot covered by ay APO) 5. The BS badwidth reource coumed for deliverig oe bit of data i regio m N {} withi oe uit of time i θ m i Hz). The tramiio efficiecy profile of the BS i differet regio i θ ) θ,..., θ N. We alo deote the average tramiio efficiecy betwee APO N ad mobile ubcriber located i the correpodig regio by φ. The tramiio efficiecy profile of APO i φ φ,..., φ N t, φ N t +,..., φ N t +N ). The tramiio efficiecy profile may vary over time due to mobility of the ubcriber. We coider a quai-tatic etwork ceario, where the tramiio efficiecy profile remai uchaged withi a igle time period ad may chage i differet period. We alo aume that they ca be meaured by the BS ad the correpodig APO 6. The above model ad aumptio about tramiio efficiecy have bee widely ued i mobile data offloadig tudie for operator-iitiated mechaim e.g., [8], [5], [2], [24]). 3. MNO Modelig The MNO leae the badwidth from third-party APO to offload the traffic of mobile ubcriber. The MNO chooe it offloadig trategy to maximize it ow profit, which i the total reveue obtaied from providig etwork badwidth to the mobile ubcriber) miu the cot of deliverig mobile traffic). Let x t deote the amout of traffic i term of total delivered bit i offloadig period) offloaded to a price-takig APO i regio N t. Similarly, x deote the amout of traffic offloaded to a price-ettig APO i regio N. The, we ue y to deote the traffic i regio N, that i ot offloaded to ay APO but erved by the MNO directly. Notice that x t + y x + y, repectively), which repreet the total amout of traffic i regio N t N, repectively), i equal to S ad i a cotat i the later formulated cot reductio problem. However, i the etwork expaio problem, x t +y x +y ) i the total admitted traffic by the etire ytem i regio, ad i a deciio variable itead of a cotat. Sice each regio N i aociated with a APO, we ue the term regio ad APO iterchageably for ay =,..., N. We deote the amout of traffic of ubcriber located i regio {} a y, which ca oly be erved by the BS. The MNO offloadig trategy i captured 5. We aume regio {} i relatively large ad may mobile ubcriber exit withi thi regio. I thi cae, the average tramiio efficiecy θ accurately reflect the coumed reource. 6. A imperfect meauremet of tramiio efficiecy profile may affect the trategy of the player ad degrade the performace of the framework. We may coider a robut optimizatio framework to addre uch imperfect iformatio, ad we will leave thi exteio a future work a the curret model i already rich eough. by vector y = y m ) m N {} ad x = [x t, x ], where x t = x ) N t ad x = x ) N. The cot of MNO coit of the reource coumptio cot ad the paymet provided to the APO for offloadig the MNO traffic. The reource coumptio cot i due to deliverig u-offloaded traffic, which coume the followig amout of reource i the BS. m N {} y m θ m. ) Notice that for each m, y m /θ m i the amout of badwidth reource required for tramittig y m bit i oe uit of time. The reource coumptio cot of the MNO, deoted by cy), i cy) = c b y m. 2) m N θ m {} We aume that fuctio c b ) i trictly icreaig ad covex i.e., c b ) >, c b ) > ) [36]. Thi covex cot fuctio idicate that the margial cot for deliverig oe more uit of data to the mobile ubcriber i icreaig. We further aume that the margial cot fuctio ad it firt derivative are weakly covex 7. For the APO, they eed to receive proper paymet from the MNO to be compeated for their cot icurred for offloadig the traffic. A price-ettig APO ca determie uch a paymet, while a price-takig APO eed to decide whether to accept or reject the paymet determied by the MNO. Although the paymet fuctio are uually aumed to be liear i the demad, there exit everal tudie that coider oliear paymet fuctio of the demad e.g., [37], [38], [39], [4], [4], [42]). I thi paper, we ue the followig differet paymet fuctio to reflect the market power of differet player. Price-ettig APO: MNO paymet fuctio for APO N i qx ) = p x ) 2, where p i the price et by the APO. Price-takig APO: MNO paymet fuctio for APO N t i qx t t ) = p t x t, where pt i the price et by the MNO. The covex paymet fuctio related to price-ettig APO reflect their market power ad the deire to get compeated more a they act a leader for the MNO. However, for price-takig APO, the liear paymet fuctio expree the MNO iteret of payig le, ice the MNO act a the leader for thee APO ad ha more market power to et the price. We deote the price vector of price-takig ad priceettig APO a p t = p t ) N t ad p = p ) N, repectively. We further deote the reveue of MNO obtaied from deliverig z bit to the mobile ubcriber withi the offloadig period a rz), which i a icreaig weakly cocave fuctio. We defie zx, y) m N {} y m + N t xt + N x, which repreet the amout of 7. The cot fuctio ued i the exitig work [6], [8], [9], [] alo atify thee coditio.

5 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION total mobile traffic delivered through the MNO ad differet APO. The MNO profit i V x, y, p) = r zx, y)) cy) qx ) qx t t ), N N t where vector p = p t, p ). 3.2 APO Modelig The AP are maaged by elfih ower. They hare their badwidth with the MNO to maximize their ow profit. We aume that each APO N ha a maximum capacity of B, which ca be aiged to erve the MNO traffic a well a it ow ubcriber traffic. We deote APO profit from ervig it ow ubcriber a r ), which i a fuctio of the APO available capacity ad may vary acro differet offloadig period. The APO determie the profit fuctio r ) coiderig it ow ubcriber traffic. Similar to [43], we aume that r ) i a o-decreaig ad weakly cocave fuctio i the APO available capacity. APO N will icur a profit lo from local ubcriber whe it admit x bit of traffic from the MNO. We deote the APO profit lo i thi cae by J x) a follow: J x) r B x ) ) x c, 4) φ φ where c ) i the APO cot of deliverig the traffic of MNO. I each APO N, the amout of badwidth reource required for tramittig x bit i a uit of time i x/φ, ad B x/φ repreet the available capacity which ca be allocated to the APO ubcriber. Similar to [36], we aume that c ) i differetiable, icreaig, ad covex, which reflect the fact that the margial cot for admittig oe more uit of data i o-decreaig. Therefore, the fuctio J ) i a decreaig cocave fuctio, where J ) > ad J φ B ) <. We further aume that J ), which repreet the margial profit lo, ad it firt ad ecod derivative are weakly cocave a well 8. A APO payoff obtaied from both offloadig MNO traffic ad deliverig it ow ubcriber traffic i a follow: V t x t, p t ) = J x t ) + p t x t, N t 5) V x, p ) = J x ) + p x ) 2, N. 6) A feaible offloadig trategy eed to atify the iequalitie x t φ B ad x φ B. 3.3 Data Offloadig Game We model the iteractio betwee the MNO ad APO a a three-tage Stackelberg game a how i Fig.. I each tage, each player determie it trategy with the goal of maximizig it payoff. We formally defie the followig game: 8. Weakly cocave r ) ad r ) ad weakly covex c ) ad c ) ca atify thee coditio although they are ot ecearily required. The reveue ad cot fuctio ued i the exitig work [6], [8], [9], [] atify thee coditio. However, we aume that the margial profit lo ad it firt ad ecod derivative are weakly cocave, which i le retrictive tha the aumptio ued i the aforemetioed exitig work. 3) Stage I: Player: price-ettig APO N ; Strategy: price vector p ; Payoff : V x, p ) give i 6). Stage II: Player: MNO; Strategy: offloadig vector x, y, ad price vector p t ; Payoff : V x, y, p) give i 3). Stage III: Player: price-takig APO N t ; Strategy: offloadig vector x t ; Payoff : V t x t, p t ) give i 5). Our goal i to determie the ubgame perfect equilibrium SPE) of the multi-tage game, where either the MNO or the APO have icetive to deviate uilaterally. Defiitio Subgame Perfect Equilibrium [44]). A trategy profile x NE, y NE, p NE) icludig the offloadig trategie x NE ad y NE ad price vector p NE i a ubgame perfect equilibrium if it repreet a Nah equilibrium NE) i every ubgame of the origial game. I the ext ectio, we will formulate the three-tage Stackelberg game for both cot reductio ad etwork expaio problem. We will ue backward iductio [44] to obtai the correpodig SPE. 4 THREE-STAGE GAME FORMULATION I thi ectio, we tart with Stage III ad aalyze the behavior of price-takig APO. We the tur to Stage II to obtai the MNO trategy. Fially, we tudy the priceettig APO trategie i Stage I. 4. Stage III Price-takig APO) I Stage III, give the price-vector p t et by the MNO, the price-takig APO determie how much traffic they ca admit i order to maximize their ow payoff. I particular, each APO N t elect it trategy x t withi the trategy pace E x t = [, φ B ] to maximize it payoff V t x t, p t ). Thi lead to the followig optimizatio problem. AP t : maximize x t V t x t, p t ) ubject to x t φ B. 7a) 7b) We ca how that problem 7) i a cocave maximizatio problem. Let x t p t ) deote the uique optimal olutio of problem 7), which i give i Theorem. Theorem. The optimal offloadig traffic deciio of price-takig APO i ) x t p t =, if p t < p t,mi L p t ), if p t,mi p t p t,max φ B, if p t > p t,max, where p t,mi J ) ad p t,max J φ B ) with J x) = dj dx. Furthermore, L p t ) i a icreaig fuctio i p t ad we kow L ) x) = J x). The proof of Theorem i give i Appedix A. We ow characterize the propertie of the optimal trategy x t p t ) a a corollary of Theorem. Thee propertie clarify the behavior of the price-takig APO ad will be ued later i Stage II. Corollary. x t p t ) i a icreaig ad cocave fuctio i p t over iterval [pt,mi, p t,max ]. The proof of Corollary ca be foud i Appedix B. 8)

6 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 x t p t ) APO Available Capacity.2 p t,mi p t,max p t Fig. 3: The optimal trategy x t pt ). APO will ot admit traffic of the MNO for ay price le tha p t,mi. Moreover, the MNO will ot aouce ay price greater tha p t,max, ice APO i ot able to deliver traffic more tha it capacity. Obervatio: Fig. 3 how the optimal trategy of a price-takig APO. A prove i Corollary ad illutrated i Fig. 3, x t p t ) i cocave ad icreaig for ay p t [p t,mi, p t,max ]. Notice that the MNO will ot et ay price greater tha p t,max, ice the APO i ot able to admit more traffic. Therefore, the trategy pace of the MNO i limited to p t [p t,mi, p t,max ]. Moreover, we will how that the aalyi of Stage II become complicated due to the cocavity of optimal trategy of price-takig APO. 4.2 Stage II MNO Problem) We ow aalyze the MNO behavior i Stage II. We firt formulate the cot reductio problem. We the exted the reult ad formulate the etwork expaio problem Cot Reductio Problem I the cot reductio problem, we have x t + y = S for all N t ad x + y = S for all N. Sice S i a cotat for each APO, the total delivered traffic S = m N {} S m i alo a cotat. Thu, the MNO profit i V crp x, y, p) = rs) cy) p x ) 2 p t x t, N N t 9) where rs) i the cotat reveue obtaied i thi cae. The MNO trategy pace i E = E x E y E p t, where E x = {[, S ]} N, E y = {[, S m ]} m N {}, while accordig to Corollary, we have E p t = {[p t,mi, p t,max ]} N t. Give the trategie of the priceettig APO i.e., p ) ad with the kowledge of how the MNO trategy would affect the price-takig APO trategie i.e., x t p t ), N t ), the MNO maximize it ow profit i Stage II. Thi lead to the followig optimizatio problem. MNO: maximize V crp x t p t ), x ), y, p t, p )) a) x,y,p t ) E ) ubject to x t p t + y = S, N t b) x + y = S, N. c) The optimal olutio of problem ) exit ad i uique if the problem i a trictly cocave maximizatio problem. Cocavity of thi problem deped o the cocavity of the objective fuctio V crp give by a) ad covexity of it cotrait. Accordig to Corollary, we kow that cotrait b) i ot a affie equality i.e., covex) cotrait, a x t p t ) i cocave i p t. Thu, problem ) i ot a cocave maximizatio problem. To traform problem ) ito a cocave maximizatio problem, we replace cotrait b) by a affie equality cotrait. To do o, we itroduce ew variable x t = x t ) N t, which belog to the pace et E x t {[, φ B ]} N t. We the replace x t p t ) by x t for ay N t i cotrait b), ad obtai the followig equality which i a affie cotrait. x t + y = S, N t. ) We further replace p t, N t by x t ) x t ) i the objective fuctio of problem ), where x t ) i the ivere of fuctio x t p t ). Ivere of x t : [p t,mi, p t,max ] [, φ B ] exit due to Corollary. We ow traform problem ) ito the followig equivalet problem. maximize rs) cy) p x ) 2 x t x t ) x t ) x,y, x t ) Ẽ N N t 2a) ubject to c), ), 2b) where Ẽ = E x E y E x t. Sice problem ) ad 2) are equivalet, we ca obtai the MNO optimal trategy by olvig problem 2). Through the followig theorem, we prove that problem 2) i a trictly cocave maximizatio problem. Theorem 2. Problem 2) i a trictly cocave maximizatio problem uder ay fixed p ad ha a uique optimal olutio. The proof of Theorem 2 i give i Appedix C. We deote the optimal) olutio of problem 2) a x p ), y p ), x t p ). Thi olutio deped o the trategy of price-ettig APO, i.e., p, which will be obtaied i Stage I Network Expaio Problem The etwork expaio problem coider a more geeral ceario tha the cot reductio problem. Through etwork expaio, the MNO maximize it ow profit by reducig the cot ad expadig the etwork, where the total delivered mobile traffic ca be higher tha the MNO ow etwork capacity. To obtai the trategy of the MNO, we firt determie it profit. Note that the total delivered traffic i o loger a cotat i thi cae. Similar to the cot reductio problem, we itroduce ew variable x t = x t ) N t ad replace the optimal trategy of price-takig APO i.e., x t p t )) by x t. The MNO profit i V ep x t, x ), y, p ) = r z x t, x )) cy) p x ) 2 N x t x t ) x t ), 3) N t where the firt term i 3) i the reveue obtaied from deliverig data to mobile ubcriber. The MNO optimal

7 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION trategy ca be determied from the followig problem, where we take the limited capacity of the BS ito accout. MNO: maximize V ep x t, x ), y, p ) x,y, x t ) Ẽ 4a) ubject to x t + y S, N t 4b) x + y S, N 4c) y S, 4d) y m B, 4e) θ m m N {} where B deote the capacity of the BS, ad cotrait 4e) eure that the total reource coumed i the BS i le tha it available capacity. Sice the etwork expaio problem predict how much traffic the MNO ca ultimately deliver, cotrait 4b) 4d) eure that the total traffic i greater tha or equal to the admitted traffic. The etwork capacity obtaied through etwork expaio ca either be allocated to the bet effort traffic or be ued to admit more mobile ubcriber. The feaible regio of problem 4) i differet from 2). However, we ca how that it i till compact ad covex. Notice that all cotrait are affie. Therefore, imilar to Theorem 2, problem 4) i a trictly cocave maximizatio problem ad ha a uique optimal olutio. Similarly, we deote the optimal ) olutio of problem 4) a x p ), y p ), x t p ). Notice that if the iequality cotrait of problem 4) are replaced by equality cotrait ad cotrait 4e) i removed, the the problem will be traformed ito the cot reductio problem. The MNO profit maximizatio problem i both cot reductio ad etwork expaio problem are how to be cocave maximizatio problem. Therefore, their optimal olutio ca be obtaied uig tadard optimizatio techique uch a the iterior-poit method [45]. Moreover, the payoff maximizatio problem of price-takig APO formulated i Sectio 4. ad the oe of price-ettig APO which will be itroduced i Sectio 4.3 have oly oe variable i.e., the APO trategy). Thee together imply the calability of the propoed icetive framework. 4.3 Stage I Price-ettig APO) We ow aalyze the price competitio amog price-ettig APO, which determie their trategie with the kowledge of how their trategie would affect the MNO trategy. For price-ettig APO N, the optimal repoe of the MNO obtaied i Stage II i.e., x p )) deped ot oly o price p, but alo o the price ubmitted by other player. Thi reflect the iterdepedece of APO pricig deciio. Let p deote the trategy of the priceettig APO excludig. Thu, we have p = p, p ). Sice the trategie of price-ettig player are coupled, we form the price-ettig o-cooperative game PS-NCG) G N, E, V ), i which E repreet the trategy pace. Moreover, the APO payoff fuctio i V x ) p, p ), p = J x p, p ) ) + p x p, p ) ) 2, 5) where the optimal trategy of the MNO i.e., x p, p ) ) obtaied i Stage II i ubtituted ito the payoff fuctio. To obtai the optimal trategie of price-ettig APO, we firt itroduce the cocept of bet repoe trategy ad NE. Defiitio 2 Bet Repoe Strategy [46]). Give p, player bet repoe trategy i: p = argmax V ) p x p, p ), p 6a) V x ubject to x p, p ) φ B, 6b) which i the choice of p that maximize ) p, p ), p. Defiitio 3 Nah Equilibrium [46]). A trategy profile p NE = p NE,..., p NE N ) i a NE if it i a fixed poit of bet repoe, i.e., for all p, N ) ) ) ) x p NE, p NE, p NE V x p, p NE, p. V To obtai the bet repoe trategie of price-ettig APO, we firt eed to kow x p, p ), which i the MNO optimal trategy i repoe to the price vector p, p ). We determie the MNO optimal trategy ad it propertie through Lemma to 3 to aalyze the exitece ad uiquee of the NE i Sectio 5. Lemma. I the cot reductio problem, give p, the MNO optimal trategy i x { p, p ) S, if p < p,mi = L p,p ) κ p +, otherwie,, 7) where L p, p ) i a fuctio of p ad p, ad p,mi ad κ > are uer depedet cotat. The proof of Lemma i give i Appedix D, where we alo obtai L p, p ). We ow derive the followig propertie of MNO optimal trategy give i 7). Lemma 2. Give p, the MNO optimal trategy x p, p ) obtaied i 7) atifie the followig propertie. ) x p, p ) i decreaig i p for p p,mi ; 2) x p, p ) i covex i p for p p,mi ; 3) L p, p ) i decreaig ad covex i p for p p,mi. The proof of Lemma 2 i give i Appedix E. Lemma 3. I the etwork expaio problem, the MNO optimal trategy x p, p ) atifie the followig propertie. ) lim p x p, p ) = for ay p. 2) x p, p ) i trictly decreaig i p for ay p. The proof of Lemma 3 ca eaily be obtaied imilar to the proof of Lemma 2 give i Appedix E, hece i omitted due to page limit. Obervatio: Fig. 4a) illutrate the optimal olutio x p, p ) i the cot reductio problem, which cofirm the propertie i Lemma 2. Accordig to Fig. 4a), price-ettig APO will ever et it price p < p,mi, ice the MNO offload the ame amout of traffic eve whe ettig p = p,mi. I) other word, the payoff fuctio V x p, p ), p i le tha

8 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 p,p ) x Traffic Demad.2 p,mi p a) x p,p ) p Fig. 4: MNO optimal trategy, which determie the amout of data offloaded to APO a) x p, p ) i the cot reductio problem. APO doe ot ubmit ay price le tha p,mi, ice the MNO doe ot offload more traffic tha it demad. b) x p, p ) i the etwork expaio problem. The amout of offloaded traffic approache ifiity whe p. ) V x p,mi, p ), p,mi for ay p < p,mi, ice x p, p ) = x p,mi, p ). Therefore, we ca limit the trategy pace of APO to p [p,mi, ), where the MNO optimal trategy x p, p ) i decreaig ad covex. Fig. 4b) how the optimal trategy x p, p ) i the etwork expaio problem. Accordig to Fig. 4, the MNO optimal trategy x p, p ) i Stage II ca be greater tha the capacity of the APO i.e., φ B ) for mall value of the price p. Thi would ot be a problem however, a price-ettig APO will chooe it price p i Stage I uch that the amout of traffic that MNO chooe to offload i Stage II will be o larger tha the APO capacity. 5 EQUILIBRIUM ANALYSIS I thi ectio, we tudy the exitece ad uiquee of the equilibrium i PS-NCG. We the develop iterative algorithm for the price-ettig APO ad MNO i the cot reductio problem ad prove that they coverge to the uique NE. 5. Exitece of Nah Equilibria We tudy PS-NCG formulated i Stage I for price-ettig APO, to how whether there exit a NE trategy. To derive the NE of the game, we ca compute the APO bet repoe trategie i thi game by olvig problem 6). However, PS-NCG i geerally a o-covex game due to the o-covexity of the payoff fuctio. Furthermore, cotrait 6b) may alo be o-covex i the etwork expaio problem. To tackle thi iue, we ue the trategy pace mappig ad traform the o-covex trategy pace ito a compact covex oe via bijectio. To traform cotrait 6b) ito a covex cotrait, we itroduce a ew variable x for each price-ettig APO N, ad replace x p, p ) by x. We further replace p by ) x x ) i problem 6), where x = x ) N. The ivere fuctio x ) x ) alway exit due to Lemma 2 ad 3. Therefore, the APO trategy pace i mapped oe-to-oe from p [, + ) ito x [, + ). The trategy pace i further limited to x [, φ B ], to take the limited capacity of APO ito accout. Thu, problem 6) ca be coverted ito the followig equivalet b) problem, from which the bet repoe trategy of APO will be obtaied. AP : x = argmax x V x, x ) x )) 8a) ubject to x φ B. 8b) Sice there i a oe-to-oe mappig betwee x ad price p, the bet pricig trategy of player ca be obtaied from p = x ) x ). Although the cotrait of problem 8) i affie, the objective fuctio may ot be cocave. We prove the exitece of a NE uig the Brouwer fixed poit theorem [46, Ch. 3]. Theorem 3. There exit a NE i Game PS-NCG. Proof. To prove thi theorem, we firt how that the fixed poit olutio of the bet repoe trategie for all price-ettig APO exit. To determie the fixed poit olutio, we firt repreet the APO bet repoe trategy obtaied from problem 8) a x = f x ), where x i the amout of traffic offloaded to the priceettig APO excludig. We further defie the fuctio x = F x ), where F = f ) N : {[, φ B ]} N {, φ B ]} N. We ow apply the Brouwer fixed poit theorem [46, Ch. 3], which tate that for ay cotiuou fuctio F that map a cloed covex et ito itelf, there i a poit x uch that Fx ) = x. It i clear that the et {[, φ B ]} N i cloed ad covex ad F i cotiuou. Therefore, there exit a fixed poit olutio for fuctio F = f ) N : {[, φ B ]} N {, φ B ]} N, which correpod to the bet repoe trategie of price-ettig APO a it i obtaied from problem 8). Accordig to Defiitio 3, we prove the exitece of a NE. 5.2 Uiquee of the Nah Equilibrium I thi ubectio, we focu o the cot reductio problem ad how that a uique NE exit for thi problem. Notice that i practice, the MNO determie the offloadig trategy baed o the cot reductio problem, while the etwork expaio problem i oly formulated to predict how much traffic the MNO ca ultimately deliver. To obtai the NE, we determie the bet repoe trategie of price-ettig APO through the followig problem, whe we ubtitute x p, p ) give by 7) ito problem 6). = argmax J L p, p ) ) p κ p + p L p, p ) 2 ) + κ p + 9a) p ubject to L p, p ) κ p + φ B. 9b) Due to Lemma 2, cotrait 9b) i a covex cotrait. However, the objective fuctio may ot be cocave a x p, p ) = Lp,p ) κ p + i covex i p. Neverthele, through the followig lemma, we prove that problem 9) i a trictly cocave maximizatio problem ad admit a uique optimal olutio. Lemma 4. There exit p,max p,mi, uch that problem 9) i a trictly cocave maximizatio problem over the cloed iterval [p,mi, p,max ], ad the objective fuctio i decreaig for p >

9 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION p,max. Thu, problem 9) ha a uique optimal olutio over the ame iterval. The proof of Lemma 4 i give i Appedix F. Utilizig Lemma 4, we ca replace the feaible regio of problem 9) by [p,mi, p,max ], which i itroduced i Appedix F. Through the followig theorem, we ow prove the uiquee of the NE i PS-NCG. Theorem 4. There exit a uique NE i Game PS-NCG. Proof. The proof i baed o the followig lemma [47]. Lemma 5 [47]). A uique NE exit i Game PS-NCG if for all N The trategy pace i a oempty, covex, ad compact ubet of ome Euclidea pace. Player payoff V i cotiuou ad trictly cocave i it ow trategy p. Accordig to Lemma 4, PS-NCG atifie the above propertie, ad we ca coclude the uiquee of the NE. 5.3 Algorithm Deig I thi ubectio, we develop iterative algorithm for priceettig APO ad the MNO to obtai the NE of PS-NCG uder the cot reductio problem. Utilizig the propoed algorithm, price-ettig APO do ot eed to obtai full iformatio about the MNO e.g., ier operatio of the MNO, traffic demad, ad cot). The propoed algorithm for the MNO alo facilitate iformatio exchage amog the price-ettig APO i a ditributed maer. We prove that the propoed algorithm coverge to the uique NE determied i Sectio 4. We firt develop a iterative algorithm for the priceettig APO to update their bet repoe trategie. We coider a price-ettig APO N, whoe bet repoe trategy ca be obtaied by olvig problem 9). Let p t) ad x t) deote the APO ad MNO trategie at the t-th iteratio. We update the APO trategy uig the followig rule, which i obtaied baed o the gradiet method: p t+) = P p t) + ϕ dj dx t) x p x t) p t) + 2p t) t) x ) + x t) ) 2 ) where ϕ t) i the tep ize at iteratio t ad P deote the projectio oto the feaible regio of problem 9). We deote thi update rule a p t+) = F p t) ). 2) The propoed gradiet-baed algorithm i illutrated i Algorithm. Price-ettig APO firt radomly iitialize p ) Step ). I each iteratio, the APO ubmit it price to the MNO via the commuicatio lik etablihed betwee the APO ad MNO Step 3). Meawhile, the MNO compute x t) ad x ad aouce them to the p p t) APO. Upo receptio of the MNO repoe Step 4), the, Algorithm : BR p ): Iterative Bet Repoe Adaptatio for a Price-ettig APO. iitializatio: t, p ), ad ɛ 2 do 3 The APO ubmit the price trategy p t) 4 The APO collect the MNO repoe x t) p t), p ) ad x. p 5 The APO update it ) price trategy. p t+) F p t) a i 2) 6 t t + 7 while p t) 8 output: p t) p t ) > ɛ to the MNO. Algorithm 2: Ditributed Iterative Algorithm for Iformatio Exchage amog the MNO ad APO. iitializatio: k, cov flag, ad ɛ 2 while cov flag = 3 The MNO collect p k) from all APO N. 4 Each APO N update it bet repoe trategy via Algorithm. p k+) BR p k) ), N 5 The MNO check the termiatio criterio. if p k+) cov flag ed 6 k k + 7 ed 8 output: p k) p k) < ɛ, N the APO update the price baed o rule 2) Step 5). Thi procedure cotiue util covergece. The APO check the termiatio criterio Step 7) ad top the algorithm whe the relative chage of the price durig coecutive iteratio are ufficietly mall a determied by the poitive cotat ɛ). Utilizig thi iterative algorithm, price-ettig APO oly eed to kow the MNO repoe to their trategie. The propoed iterative algorithm ha a relatively mall commuicatio overhead, ice a few meage eed to be exchaged i the market. Utilizig Algorithm 2, the MNO facilitate iformatio exchage amog the price-ettig APO. The MNO repoe to each price-ettig APO reflect the trategie of other APO. A a reult, the price-ettig APO do ot eed to kow or meaure) the trategie of other APO. I each iteratio, the MNO hare the iformatio eeded by the price-ettig APO ad receive their updated bet repoe trategie. Thi procedure cotiue util covergece. We ow prove the covergece of our algorithm. A tated i Lemma 4, problem 9) i a triclty cocave maximizatio problem. Therefore, the gradiet-baed algorithm how i Algorithm coverge to the optimal olutio of thi problem [48]. To tudy the covergece of Algorithm 2, we follow the ratioale of the proof i [49], which ha bee widely ued i the literature e.g., [5]). Theorem 5. The propoed iterative algorithm how i Algorithm 2 coverge to the uique NE of PS-NCG. Proof. Algorithm 2 coverge whe the followig coditio are atified [49]: Firt, a fixed poit olutio

10 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 that i NE) mut exit. Secod, the payoff fuctio V ) x p, p ), p i cocave i p, p ). Notice that accordig to [49, Def. ], cocavity i ufficiet for the covergece, while it i ot a eceary coditio. Similar to Lemma 4, we ow how that V i cocave i the feaible regio of the payoff maximizatio problem aroud the NE. Lemma 6. For each price-ettig player N, the payoff fuctio V ) x p, p ), p i cocave i p, p ). The proof of Lemma 6 i give i Appedix G. A tated i Theorem 4, a uique NE exit for PS-NCG. Moreover, Lemma 6 tate that V ) x p, p ), p i cocave. Thee together complete the proof of Theorem 5. 6 PERFORMANCE EVALUATION I thi ectio, we evaluate the MNO profit whe offloadig it ubcriber traffic to the third-party APO, ad APO payoff by admittig the MNO traffic. Through exteive umerical tudie, we compare the SPE with the outcome of data offloadig game DOFF) propoed i [8], which oly coider motivatig price-takig APO to admit the MNO traffic. To validate the SPE efficiecy, we compare our propoed framework with the ocially optimal olutio. The efficiecy of SPE i defied a the ratio betwee the ocial welfare obtaied by the SPE ad the maximum ocial welfare. Thi value i alway le tha or equal to, ad illutrate how the market outcome degrade due to player elfih behavior i problem coidered i thi paper. The ocial welfare i defied a the ummatio of all player payoff a follow. Ψx, y) = r zx, y)) cy) + J x t ) + J x ). N t N The maximum ocial welfare, deoted by Ψ, ca be obtaied from the followig problem. Ψ = maximize x,y) R Ψx, y), where R i the feaible regio of x, y). We deote the equilibrium trategie derived i our model a x NE ad y NE. Defiitio 4 SPE Efficiecy). The SPE efficiecy i the ratio betwee the ocial welfare obtaied by the equilibrium ad the maximum ocial welfare, ad equal Ψx NE, y NE ) Ψ. 2) The price-of-aarchy PoA) i aother cocept i game theory that meaure how the efficiecy of a ytem degrade due to elfih behavior of it aget. For the cot reductio problem that admit a uique SPE, the PoA i the ivere of SPE efficiecy. 6. Simulatio Setup We coider a MNO, repreeted by a macrocell BS. Ule tated otherwie, there are N t = price-takig WiFi APO ad N = 2 price-ettig WiFi APO. We ow decribe the meauremet-baed model imilar a i [9], [24]) ued to evaluate the performace of our propoed framework. We coider a model of WiFi coectio probability baed o the meauremet i [2], ad a model of traffic demad of mobile ubcriber baed o the iformatio from [], [5], [52]. There are a total of 4 mobile ubcriber. The APO are located i thoe area with high mobile ubcriber deity, ad hece high traffic demad. Accordig to [2], the average WiFi cotact probability i.7. Similar to [8], [2], we coider the ormalized tramiio efficiecy profile of MNO ad APO, which follow a uiform ditributio i the iterval [.3, ] ad [.4, ], repectively. We further aume that the badwidth of macrocell BS i 2 MHz, while the ormalized badwidth of each WiFi AP i radomly choe from {, 2, 5.5, } MHz imilar to [2]. The achievable data rate i the product of badwidth ad tramiio efficiecy. To geerate realitic traffic demad, we refer to the meauremet of mobile data reported i [5], [52] ad the projectio of future mobile data predicted by Cico []. The average martphoe will geerate 4.4 GB of traffic per moth by 22 []. Accordig to [5], [52], the uer traffic volume follow a upper-trucated power-law ditributio. We chooe the parameter of uch a ditributio to match the per-moth average data traffic of 4.4 GB 9. We aume that the reveue ad cot fuctio for the MNO are rz) = rz ad c b y) = cy 2, repectively, where r = $/Mb ad the cotat c i the MNO cot factor. For each APO N, we aume r x) = a log + x) ad c x) = c x 2, where a ad c are choe radomly ad uiformly from the iterval [,.5] ad [.2,.5], repectively. We tudy the market for a offloadig period that correpod to two ecod. 6.2 Cot Reductio Problem We firt evaluate the MNO profit ad the total traffic delivered to mobile ubcriber. Fig. 5 how the profit for differet value of the MNO cot factor i.e., c), while Fig. 6 illutrate the correpodig amout of traffic delivered by the macrocell BS ad offloaded to the APO. From thee figure, we ca oberve that data offloadig igificatly icreae the MNO profit. The profit i icreaed up to three time whe c =.4, wherea 25% of traffic i offloaded to the APO. Whe the cot of data delivery i the MNO icreae, the MNO leae more reource from APO to offload more traffic. Moreover, the propoed threetage game outperform the DOFF game by up to 8% whe c =.4. Such improvemet i achieved by the participatio of price-ettig APO, which provide more reource ad drive the price for price-takig APO dow. We the evaluate the efficiecy of SPE. Fig. 7 how the ratio betwee the ocial welfare obtaied at a SPE ad the maximum ocial welfare. A the umber of APO icreae, the efficiecy ratio icreae due to a higher competitio i the market. Furthermore, the SPE efficiecy degrade whe the cot of data delivery become higher. Thi i becaue more traffic will be delivered to the APO, which 9. The daily traffic of each mobile ubcriber follow the power-law ditributio f S ) = σ) σ /S max σ for S max, where σ =.57 a oberved i [52] ad S max i determied uch that the permoth average data traffic i 4.4 GB aumig mobile uer are active at daytime i.e., 8: a.m. 8: p.m.)

11 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION MNO Profit tage Game DOFF [8] No-offloadig APO Average Payoff Price-ettig APO Price-takig APO MNO Cot Factor MNO Cot Factor Fig. 5: The MNO profit for differet value of MNO cot factor. The propoed framework igificatly improve the MNO profit i compario with DOFF [8] ad o-offloadig cae. Fig. 8: Average payoff per APO for price-ettig ad price-takig APO veru the MNO cot factor. Total Delivered Traffic Mb) Macrocell Traffic Offloaded Traffic MNO Cot Factor Fig. 6: The correpodig total delivered traffic for differet value of MNO cot factor. More traffic will be offloaded to the APO a the cot of data delivery i the MNO icreae. Equilibrium Efficiecy N,N t ) = 5,) 5,5) 5,2) 2,) MNO Cot Factor Fig. 7: Efficiecy of equilibrium. Three-tage game efficietly geerate a cloe-to-optimal ocial welfare. ca maipulate the market ad make the SPE le efficiet. Whe the cot of data delivery i zero, the ocial welfare at the SPE i the ame a the maximum ocial welfare. I thi cae, we have x NE, y NE ) =, S), which reult i the efficiecy of. We ow ivetigate the payoff achieved by the APO whe deliverig the traffic. To evaluate the payoff of differet APO i a fair maer, we chooe the ame et of parameter for all APO. Fig. 8 illutrate the average Average Payoff of Price-takig APO Payoff Price N = N = 2 N = 5 Fig. 9: Average payoff that a price-takig APO ca obtai for differet value of N ad the average of optimal price p t et by the MNO for price-takig APO. N t =, c =.4) payoff per APO obtaied by price-ettig ad price-takig APO. The larger cot of data delivery of the MNO, the higher payoff that the APO ca obtai. Furthermore, the price-ettig APO ca achieve a higher average payoff tha the price-takig APO due to more market power. We further evaluate how the price competitio amog price-ettig APO affect the market price et by the MNO for price-takig APO i.e., p t ). Fig. 9 how the average payoff of price-takig APO ad the correpodig price et by the MNO, whe differet umber of price-ettig APO participate i the market. From thi figure, we ca oberve that the participatio of price-ettig APO will drive the market price dow a they have more market power. Coequetly, the price-takig APO obtai le payoff. We fially ivetigate the covergece rate of Algorithm 2. I each iteratio, the price-ettig APO update ad ubmit their bet repoe trategie to the MNO. Fig. illutrate the price dyamic choe by the price-ettig APO, ad how that Algorithm 2 coverge fairly quickly. A Theorem 5 tate that thi algorithm coverge to the uique NE of Game PS-NCG, we coclude that Algorithm 2 coverge quickly to the SPE. 6.3 Network Expaio Problem The etwork expaio problem tudie how much traffic the ytem ca deliver, whe the MNO i able to expad the Average Optimal Price

12 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 Price p APO # APO #2 Total Delivered Traffic Mb) Macrocell Traffic Offloaded Traffic Iteratio k MNO Cot Factor Fig. : The price choe by price-ettig APO i differet iteratio a obtaied by Algorithm 2. ɛ = ɛ =., ϕ t) = /t 2 ) Fig. 2: The amout of total traffic that ca be ultimately delivered to the mobile ubcriber. MNO Profit tage Game No-offloadig Equilibrium Efficiecy N,N t ) = 5,) 5,5) 5,2) 2,) MNO Cot Factor Fig. : The MNO profit for differet value of c. The propoed framework igificatly improve the profit i compario with ooffloadig cae. etwork by offloadig more mobile traffic to the APO. Fig. how the MNO profit obtaied by data offloadig, ad Fig. 2 how the total traffic delivered by the macrocell BS ad offloaded to the APO for differet value of c. Fig. 2 how that 3% more traffic ca be delivered to the mobile ubcriber by leaig the APO reource whe c =.4, uder the etwork expaio equilibrium reult. Fig. how the MNO ca obtai up to 4 time more profit i thi cae. By comparig Fig. 2 ad Fig. 6, we ee that the coideratio of etwork expaio allow u to deliver 2% more traffic whe c =.4, comparig with the cae of ot coiderig etwork expaio. Fig. 3 how the efficiecy of the equilibrium of the etwork expaio problem. Sice the amout of traffic offloaded to the APO i the etwork expaio problem i higher tha i the cot reductio problem, the equilibrium i le efficiet i the etwork expaio problem. Ulike Fig. 7, the efficiecy ratio i le tha eve whe c =. Thi i becaue the MNO i till itereted i leaig the APO reource to expad it etwork, eve if it cot of data delivery i zero. 7 CONCLUSION I thi paper, we tudied the ecoomic of mobile data offloadig i cellular etwork, where a MNO iteract with both price-ettig ad price-takig APO to offload the MNO Cot Factor Fig. 3: Efficiecy of equilibrium veru c. The equilibrium efficiecy degrade whe the cot of data delivery become higher. mobile traffic. We propoed a icetive framework for the data offloadig market, ad modeled the iteractio i the market a a three-tage Stackelberg game. We aalyzed two game with differet MNO objective fuctio via the cot reductio ad etwork expaio problem. I the cot reductio problem, give fixed total admitted traffic demad, the MNO aim to offload it traffic to reduce the cot of data delivery. Through the etwork expaio problem, the MNO i able to icreae the aggregate etwork capacity ad imultaeouly reduce the cot of data delivery. Numerical reult howed our propoed icetive framework outperform a exitig market model by 8% i term of the MNO profit i the cot reductio problem. Moreover, the MNO ca obtai 4 time more profit through etwork expaio. Reult further howed that price competitio amog price-ettig APO will drive the market price dow, comparig to the cae where all APO are pricetakig. Although we oly coidered a igle repreetative BS i thi model, the reult ca be directly exteded to the cae where a MNO ha multiple BS. For future work, we will coider a market whe multiple MNO compete to offload their traffic to differet type of APO. Moreover, we will coider iterferece iduced by overlappig AP, whe they are deely deployed. We will further take the latecy ad reliability ito accout to eable data offloadig i future fifth geeratio 5G) wirele ytem.

13 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION ACKNOWLEDGMENTS Thi work i upported by the Natural Sciece ad Egieerig Reearch Coucil of Caada NSERC) ad the Geeral Reearch Fud Project Number CUHK ad 4296) etablihed uder the Uiverity Grat Committee of the Hog Kog Special Admiitrative Regio, Chia. APPENDIX A PROOF OF THEOREM The optimal olutio of problem 7) ca be obtaied by Karuh-Kuh-Tucker KKT) aalyi, which atifie the followig coditio. J x t ) + p t λ =, 22a) φ ) x t φ B =, 22b) λ x t φ B, 22c) λ, 22d) where J x t ) dj dx x t ad λ i the optimal Lagrage multiplier. Accordig to the KKT coditio, we coider two differet cae: Cae. λ = : I thi cae, the optimal olutio atifie J x t ) + p t =. 23) We deote the value of x t p t ) obtaied from 23) a L p t ), where we kow L ) x) = J x). Although it i ot i cloed form, we till ca extract the required propertie. If p t < p t,mi J ), 23) doe ot have ay olutio ad we obtai x t =. Cae 2. λ > : I thi cae, accordig to the complemetary lacke coditio give by 22b), the optimal olutio i x t = φ B. It mea that the total capacity of APO i allocated to the MNO traffic. Thi coditio i atified whe p t > p t,max J φ B ). Thi complete the proof of Theorem. APPENDIX B PROOF OF COROLLARY The optimal trategy x t i icreaig for p t [p t,mi, p t,max ] ice J, ad hece L p t ), are icreaig. To how that x t i cocave, we take the derivative twice from both ide of 23). We have d 2 x t dp t2 dj dx xt ) + ) dx t 2 d 2 J dp t dx 2 xt ) =. 24) Accordig to the aumptio metioed i Sectio 3, we kow that J ) <, dj dx <, ad d2 J dx. Therefore, we 2 ca coclude that d2 x t, which complete the proof. dp t2 APPENDIX C PROOF OF THEOREM 2 The trategy pace of the MNO i a compact ad covex et. I additio, all cotrait of problem 2) icludig ) are affie cotrait. Therefore, to prove that problem 2) i a trictly cocave maximizatio problem, we eed to how that it objective fuctio, give a follow, i cocave. V crp x t, x ), y, p ) = rs) cy) N p x ) 2 x t x t ) x t ). 25) N t Let h x t ) N t xt x t ) x t ). To ivetigate the cocavity of 25), we firt repreet the variable et of problem 2) a η = y, x, x t ) ad form the Heia a follow. [ 2 V crp ] H y H = = H x, 26) η i η j η i,η j η H xt where [ H y 2 ] cy) = y m y k θ = c... y) θ N m,k N {},..., θ H x = 2 diag p,..., p N ), ) 2 h 2 h H xt = diag,...,. x t2 x t2 N t θ b, N ) = c y)θ T θ, It i clear that H i egative emidefiite, if H y, H x, ad H xt are all egative emidefiite. We firt focu o H y. Give ay o-zero vector w = w,..., w N ), we calculate Sice c y) = c N wh y w T = c y) θ ) b w T T ) θ b w T. 27) b N ) y = θ > ad θ b w T ) T θb w T ) = ) 2 = θ w >, we have wh y w T <. Thu, matrix H y i egative emidefiite. Furthermore, we ca eaily how that H x i egative emidefiite. A metioed i Sectio 4 ad illutrated i Fig. 4, we alway kow p > for all N. To prove that diagoal matrix H xt i egative emidefiite, it i ufficiet to how that for ay <. We firt prove the followig lemma., 2 h x) x t2 Lemma 7. For ay ivertible cocave fuctio f, f ) x) i covex if f i a icreaig fuctio, where f ) i the ivere of fuctio f. Proof. To prove Lemma 7, we derive the ecod derivative of f ) x). The firt derivative of f ) x) i f f ) x)), hece the ecod derivative i f f ) x)), which i f f ) x))) 3 poitive whe f i a icreaig ad cocave fuctio. Thi complete the proof of Lemma 7.

14 IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 We ow proceed the proof of Theorem 2. Each diagoal elemet of H xt i 2 h x t ) x t2 = 2 dxt ) d x t x t d 2 x t ). 28) d x t2 A tated i Corollary, x t p t ) i a icreaig cocave fuctio. Accordig to Lemma 7, x t ) x t ) i a icreaig covex fuctio. Thu, we ca coclude that <, which reult i the egative emidefiitee 2 h x t ) x t2 of H xt. Therefore, the objective fuctio 25) i cocave, which complete the proof. APPENDIX D PROOF OF LEMMA We firt prove that there exit a p,mi > uch that x p, p ) = S whe p [, p,mi ). We the obtai x p, p ) for p p,mi. Accordig to cotrait c), we kow that x p, p ) S. To prove that p,mi exit, by cotradictio, we aume that x < S for p =. Therefore, we have y > to atify x + y = S. We kow that x, y ) = S, ) i alo a feaible olutio of problem 2). By ubtitutig x, y ) = S, ) ito the objective fuctio i.e., MNO profit) of problem 2), we obtai a higher MNO profit tha x, y). Note that the paymet p x 2 i zero whe p =. Therefore, x < S caot be the optimal olutio, ad we have x = S whe p =. To obtai the MNO optimal trategy x p, p ), we form the Lagragia of problem 2). L crp x t, x ), y, p, λ, α, β ) = rs) cy) p x ) 2 x t x t ) x t ) N N t λ x ) t + y S ) λ x + y S N N t + α y + β x + β x t, N N N t where λ = λ ) N, α = α ) N, ad β = β ) N are the Lagrage multiplier. Let h x t ) x t ) x t ). The optimal olutio atifie the followig KKT coditio. A.) Lcrp = c y ) λ + α =, N, y θ A.2) Lcrp x A.3) Lcrp = 2p x λ + β =, N, x t = h x t ) λ + β =, N t, A.4) αy =, α, N, A.5) βx =, β, N, A.6) β x t =, β, N t, A.7) x t + y = S, N t, A.8) x + y = S, N, where h dh d x. We focu o the MNO optimal trategy t x p, p ) for N. We kow x p, p ) = S whe p [, p,mi ). I thi cae, we have β = accordig to A.5). Whe we further icreae p, x p, p ) decreae, wherea accordig to A.2), λ. Sice x + y = S, we have y >, which force α to be zero. I thi cae, by combiig A.) ad A.2), we have ) x p, p ) = c y ) 2θ p = c b m N y m {} θm 2θ p. 29) To determie 29), we categorize the APO excludig ito the followig et: Q = {m N ym = }, Q 2 = {m N {} \ {} ym = S m }, Q 3 = {m N \ {} < ym < S m }. We ow rewrite m N {} a follow: m Q 2 S m + S x + θ m θ + y m θm m N Q 3 m N t Q 3 S m x m θ m S m x t m θ m. 3) Notice that x m + ym = S m for all m N, x t m + ym = S m for all m N t, ad y = S. Moreover, we kow x m > for all m N Q 3 ad xt m > for all m N t Q 3. Accordig to A.4) A.6), we have αm = ad βm = for all m Q 3. By combiig A.) A.3), we have θ p x = θ m p mx m, m N Q 3, m, 3) θ p x = h m x t m), m N t Q 3. 32) We deote the ivere of fuctio h m ), which exit due to Corollary, a h ) m ). By ubtitutig the above equatio ito 3), we obtai m N {} ym = A x θ m θ m N t Q 3 θ p x h ) m m N Q 3 p mθ 2 m θ p x, 33) θ m where A m Q 2 S m/θ m + S /θ + m N Q 3 S m/θ m i cotat. By ubtitutig 33) ito 29) ad ome mathematical maipulatio, we have x θ + θ p x m N Q 3 p mθ 2 m = A c ) b 2θ p x ) which ca be rewritte a: κ p + )x where κ m N Q 3 m N t Q 3 ) h ) m θ p x ), θ m = g θ p x ), 34) θ 2 p ad m θ2 m g θ p x ) θ A θ c ) 2θ p x θ b m N t Q 3 h ) m ) θ p x ). 35) θ m Thu, by olvig equatio 34), the MNO optimal trategy x p, p ) ca be obtaied a follow: x { p, p ) S, if p < p,mi = L p,p ) κ p +, otherwie,,

15 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION which complete the proof. APPENDIX E PROOF OF LEMMA 2 To prove that x p, p ) i decreaig i p, by cotradictio, we aume that x p, p ) i o-decreaig. Thu, p x y m θm p, p ) i icreaig i p, while m N {} give i 3) i decreaig i p. Notice that accordig to Corollary, h ) m i a icreaig fuctio. Therefore, the right had ide of 29) i decreaig i p, which cotradict the aumptio that x p, p ) i o-decreaig. To how that the ecod ad third propertie tated i Lemma 2 alo hold, we take the derivative twice from both ide of 34). dx 2κ dp Thu, 2 dx dp + p d 2 x dp 2 + κ p + ) d2 x dp 2 dx dp = θ 2 x + p + θ 2 dx dp + p ) κ θ dg = θ 2 x dp + p ) 2 d 2 g d 2 x dp 2 dp 2 ) dg ) + d2 x dx dp To proceed, we eed the followig lemma. dp 2 dp. 36) ) 2 d 2 g dp 2. 37) Lemma 8. The fuctio g x) = θ A θ c ) b 2x) θ m N t h ) Q 3 m x) θ m i decreaig ad covex. Proof. Sice c b ) i a icreaig covex fuctio, imilar to Lemma 7, we ca coclude that the ivere fuctio ) i icreaig but cocave. We ow how that h ) m i alo a icreaig cocave fuctio. Recall that h m x) = x t ) m x) = J mx). Accordig to the aumptio metioed i Sectio 3, h m ) i a icreaig covex fuctio, while h ) m i a icreaig cocave fuctio due to Lemma 7. Thu, g ) i decreaig ad covex. c ) b Accordig to Lemma 8, dg dp ad d2 g dp are both poitive. 2 To how that x p, p ) i covex i p, by cotradictio, we aume that there i at leat oe poit, where d2 x dp <. 2 I thi cae, the left had ide of 37) become egative ice x p, p ) i o-icreaig, while we kow d2 g dp. 2 Therefore, we ca coclude that x i covex i p. Similarly, if there exit a poit at which the ecod derivative of L p, p ) = κ p + )x p, p ), a how i 36), i egative, the ecod derivative of p x p, p ) will be egative a well, ice x i covex. However, thi cotradict with 36). Thu, L p, p ) i covex i p. We fially how that L p, p ) = κ p + )x p, p ) i decreaig. Whe p icreae, p x p, p ) icreae a well, while g θ p x ) decreae accordig to Lemma 8. Thu, κ p + )x p, p ) = g θ p x ), a give by 34), i decreaig i p. APPENDIX F PROOF OF LEMMA 4 Accordig to Lemma 2, the feaible regio of problem 9) i compact ad covex. With the help of propertie tated i Lemma 2, we how that there exit p,max p,mi uch that for ay p [p,mi, p,max ], the objective fuctio i cocave ad the optimal price p i located withi thi regio. For the ake of preetatio clarity, we abue the otatio ad ue L p ) ice p i cotat i problem 9). To obtai the optimal olutio of problem 9), we et the firt derivative of the objective fuctio 9a) to zero. dv dp = dx dp ) J x ) + 2p x + ) 2 x =. 38) We ow how that the objective fuctio V i cocave at ay tatioary poit atifyig the above equatio. The ecod derivative of V i d 2 V dp 2 = d2 x dp 2 J x + dx dp By ubtitutig x ) ) + 2p x ) dx dp J x ) + 2 dx dp = Lp ) κ p + ito 39) ad ome mathemat- ca be rewritte a follow: ical maipulatio, d2 V dp 2 + 2x dx dp. 39) d 2 V dp 2 = L p ) κ p J L p ) ) + κ p + 2p L p ) ) + κ p + ) 2 dx ) ) 2 J κ p + dp x ) + 2x + x L + p ) κ p + L p ) ) κ p + ) 2 J L p ) ) L p ) κ p + κ p + L p ) ) κ p + ) 2 + 2L p ) κ p + + 2p L p ) κ p + 2pL ) p ) κ p + ) 2 + 2L p )L p ) κ p + ) 2. 4) Due to 38), the ecod term of d2 V dp 2 rewrite d2 V dp a follow: 2 d 2 V dp 2 = L p ) κ p + J L p ) ) κ p + L + p ) κ p + L p ) κ p + ) 2 + 2L p ) κ p + ) 2 p L p ) + L p )) L + p ) κ p + L p ) ) κ p + ) 2 i zero. We further ) 2 J L p ) ) κ p +

16 2L p ) κ p + + 2p L p ) κ p + 2pL p ) ) κ p + ) 2. d 2 V dp 2 4) The firt ad ecod term of are both egative ice J ) i a decreaig cocave fuctio while L ) i covex. Similar to Lemma 2, we ca prove that L i alo covex i p, while the proof i omitted due to page limit. Therefore, the third term of 4) i alo egative ice p L p ) + L p ) L 2p ) due to covexity of L. Fially, the fourth term of 4) i egative a well, ice it d dp dp p x ) <. Therefore, V ca be rewritte a dx i cocave at ay tatioary poit implyig that there exit at mot oe uch poit. We ow coider two poible cae: Cae. There i oe tatioary poit a deoted by p. Thu, we ca coclude that there exit a iterval where the objective fuctio V i locally cocave ad p belog to thi iterval. Furthermore, V i decreaig for p > p,max due to the uiquee of the tatioary poit. Cae 2. There i o tatioary poit, which implie that the objective fuctio i either icreaig or decreaig. If the objective fuctio i trictly decreaig, the optimal olutio of problem 9) i p,mi ad we have p,max = p,mi. If the objective fuctio i icreaig, the ecod term of 4) i alway egative. Thu, V i cocave i [p,mi, + ) ad = +. p,max I cocluio, we ca limit the feaible regio of problem 9) to the iterval [p,mi, p,max ], which complete the proof. APPENDIX G PROOF OF LEMMA 6 ) x p, p ), p i To how that the payoff fuctio V cocave i p = p, p ) over the feaible regio of APO payoff maximizatio problem, we ue the firt order coditio for the cocavity. The fuctio fp ) i cocave if for ay p ad ˆp, we have f ˆp ) < f p ) + ˆp p ) T f p ). 42) Therefore, for each price-ettig APO N, we eed to how that J x ˆp, ˆp ) ) + ˆp x ˆp, ˆp ) ) 2 < J x + p, p ) ) + p x ˆp m p m) V p. m N m p, p ) ) 2 Due to Lemma 4, V i cocave i p. Therefore, to prove that V i cocave i p, it i ufficiet to how that m N \{} ˆp m p m) V p. Without lo of geerality, m we aume ˆp p. If V p for all m N \ {}, the m above coditio hold. We ow derive V p a follow: m V p = x m p J x p, p ) ) + 2p x p, p ) ). m A tated i Lemma 2, x m i o-icreaig i p m, while p mx m i icreaig. Thu, baed o 3) give i Appedix IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27 D, x i o-decreaig i p m x ad we have p m x p, p ) ) +. Furthermore, accordig to 38), J 2p x p, p ) i poitive aroud the NE, which complete the proof. REFERENCES [] Cico, Cico viual etworkig idex: Global mobile data traffic forecat update 25-22, Feb. 26. [2] K. Lee, J. Lee, Y. Yi, I. Rhee, ad S. Chog, Mobile data offloadig: How much ca WiFi deliver? IEEE/ACM Tra. o Networkig, vol. 2, o. 2, pp , Aug. 23. [3] M. Bei, M. Simek, A. Czylwik, W. Saad, S. Valeti, ad M. Debbah, Whe cellular meet WiFi i wirele mall cell etwork, IEEE Commu. Magazie, vol. 5, o. 6, pp. 44 5, Ju. 23. [4] E. Bulut ad B. K. Szymaki, WiFi acce poit deploymet for efficiet mobile data offloadig, ACM SIGMOBILE Mobile Computig ad Commuicatio Review, vol. 7, o., pp. 7 78, Jul. 23. [5] J. Adrew, H. Claue, M. Dohler, S. Raga, ad M. Reed, Femtocell: Pat, preet, ad future, IEEE J. o Selected Area i Commu., vol. 3, o. 3, pp , Apr. 22. [6] L. Dua, J. Huag, ad B. Shou, Ecoomic of femtocell ervice proviio, IEEE Tra. o Mobile Computig, vol. 2, o., pp , Nov. 23. [7] AT&T Pre Releae, New AT&T Wi-Fi olutio coect buiee ad their cutomer, Mar. 25. [8] L. Gao, G. Ioifidi, J. Huag, ad L. Taiula, Ecoomic of mobile data offloadig, i Proc. of IEEE INFOCOM, Turi, Italy, Apr. 23. [9] J. Lee, Y. Yi, S. Chog, ad Y. Ji, Ecoomic of WiFi offloadig: Tradig delay for cellular capacity, IEEE Tra. o Wirele Commu., vol. 3, o. 3, pp , Mar. 24. [] H. Zhag, M. Bei, L. A. DaSilva, ad Z. Ha, Multi-leader multi-follower Stackelberg game amog Wi-Fi, mall cell ad macrocell etwork, i Proc. of IEEE GLOBECOM, Auti, TX, Dec. 24. [] BT WiFi, [2] M. H. Cheug ad J. Huag, DAWN: Delay-aware Wi-Fi offloadig ad etwork electio, IEEE J. o Selected Area i Commu., vol. 33, o. 6, pp , Ju. 25. [3] X. Zhuo, W. Gao, G. Cao, ad S. Hua, A icetive framework for cellular traffic offloadig, IEEE Tra. o Mobile Computig, vol. 3, o. 3, pp , Mar. 24. [4] C. Joe-Wog, S. Se, ad S. Ha, Offerig upplemetary etwork techologie: Adoptio behavior ad offloadig beefit, IEEE/ACM Tra. o Networkig, vol. 23, o. 2, pp , Apr. 25. [5] W. Wag, X. Wu, L. Xie, ad S. Lu, Femto-Matchig: Efficiet traffic offloadig i heterogeeou cellular etwork, i Proc. of IEEE INFOCOM, Hog Kog, Chia, Apr. 25. [6] L. Roe, E. V. Belmega, W. Saad, ad M. Debbah, Pricig i heterogeeou wirele etwork: Hierarchical game ad dyamic, IEEE Tra. o Wirele Commu., vol. 3, o. 9, pp , Sep. 24. [7] X. Kag ad Y. Wu, Icetive mechaim deig for heterogeeou peer-to-peer etwork: A Stackelberg game approach, IEEE Tra. o Mobile Computig, vol. 4, o. 5, pp. 8 3, May 25. [8] M. Haddad, P. Wicek, H. B. A. Sidi, ad E. Altma, A automated dyamic offet for etwork electio i heterogeeou etwork, IEEE Tra. o Mobile Computig, vol. 5, o. 9, pp , Sep. 26. [9] Z. Che, Y. Liu, B. Zhou, ad M. Tao, Cachig icetive deig i wirele D2D etwork: A Stackelberg game approach, i Proc. of IEEE ICC, Kuala Lumpur, Malayia, May 26. [2] J. Li, H. Che, Y. Che, Z. Li, B. Vucetic, ad L. Hazo, Pricig ad reource allocatio via game theory for a mall-cell video cachig ytem, IEEE J. o Selected Area i Commu., vol. 34, o. 8, pp , Aug. 26. [2] L. Gao, G. Ioifidi, J. Huag, L. Taiula, ad D. Li, Bargaiigbaed mobile data offloadig, IEEE J. o Selected Area i Commu., vol. 32, o. 6, pp. 4 25, Ju. 24.

17 SHAH-MANSOURI et al.: AN INCENTIVE FRAMEWORK FOR MOBILE DATA OFFLOADING MARKET UNDER PRICE COMPETITION [22] K. Wag, F. C. M. Lau, L. Che, ad R. Schober, Pricig mobile data offloadig: A ditributed market framework, IEEE Tra. o Wirele Commu., vol. 5, o. 2, pp , Feb. 26. [23] X. Kag ad S. Su, Icetive mechaim deig for mobile data offloadig i heterogeeou etwork, i Proc. of IEEE ICC, Lodo, UK, Ju. 25. [24] W. Dog, S. Rallapalli, R. Jaa, L. Qiu, K. Ramakriha, L. Razoumov, Y. Zhag, ad T. W. Cho, ideal: Icetivized dyamic cellular offloadig via auctio, IEEE/ACM Tra. o Networkig, vol. 22, o. 4, pp , Aug. 24. [25] S. Pari, F. Martigo, I. Filippii, ad L. Che, A efficiet auctio-baed mechaim for mobile data offloadig, IEEE Tra. o Mobile Computig, vol. 4, o. 8, pp , Aug. 25. [26] G. Ioifidi, L. Gao, J. Huag, ad L. Taiula, A double-auctio mechaim for mobile data-offloadig market, IEEE/ACM Tra. o Networkig, vol. 23, o. 5, pp , Oct. 25. [27] X. Che, J. Wu, Y. Cai, H. Zhag, ad T. Che, Eergy-efficiecy orieted traffic offloadig i wirele etwork: A brief urvey ad a learig approach for heterogeeou cellular etwork, IEEE J. o Selected Area i Commu., vol. 33, o. 4, pp , Apr. 25. [28] L. Dua, J. Huag, ad B. Shou, Ecoomic of femtocell ervice proviio, IEEE Tra. o Mobile Computig, vol. 2, o., pp , Nov. 23. [29] J. D. Hobby ad H. Claue, Deploymet optio for femtocell ad their impact o exitig macrocellular etwork, Bell Lab Techical Joural, vol. 3, o. 4, pp. 45 6, Feb. 29. [3] N. Shetty, S. Parekh, ad J. Walrad, Ecoomic of femtocell, i Proc. of IEEE GLOBECOM, Hoolulu, HI, Nov. 29. [3] U. Paul, A. Subramaia, M. Buddhikot, ad S. Da, Udertadig traffic dyamic i cellular data etwork, i Proc. of IEEE INFOCOM, Shaghai, Chia, Apr. 2. [32] U. Paul, L. Ortiz, S. Da, G. Fuco, ad M. Buddhikot, Learig probabilitic model of cellular etwork traffic with applicatio to reource maagemet, i Proc. of IEEE DYSPAN, McLea, VA, Apr. 24. [33] Y. J. Hog, J. Kim, ad D. K. Sug, Two-dimeioal chael etimatio ad predictio for chedulig i cellular etwork, IEEE Tra. o Vehicular Techology, vol. 58, o. 6, pp , Jul. 29. [34] W. Xi, X. Yu, S. Nagata, ad L. Che, Novel CQI update method i CoMP tramiio, i Proc. of IEEE PIMRC, Lodo, UK, Sep. 23. [35] R. Margolie, A. Sridhara, V. Aggarwal, R. Jaa, N. K. Shakaraarayaa, V. A. Vaihampaya, ad G. Zuma, Exploitig mobility i proportioal fair cellular chedulig: Meauremet ad algorithm, IEEE/ACM Tra. o Networkig, vol. 24, o., pp , Feb. 26. [36] K. So, H. Kim, Y. Yi, ad B. Krihamachari, Toward eergyefficiet operatio of bae tatio i cellular wirele etwork, i Gree Commuicatio: Theoretical Fudametal, Algorithm ad Applicatio, J. Wu, S. Raga, ad H. Zhag, Ed. CRC Pre, Taylor - Fraci, pp , Sep. 22. [37] H. She ad T. Baar, Optimal oliear pricig for a moopolitic etwork ervice provider with complete ad icomplete iformatio, IEEE J. o Selected Area i Commu., vol. 25, o. 6, pp , Aug. 27. [38] S. Shakkottai, R. Srikat, A. Ozdaglar, ad D. Acemoglu, The price of implicity, IEEE J. o Selected Area i Commu., vol. 26, o. 7, pp , Sep. 28. [39] P. Hade, M. Chiag, R. Calderbak, ad J. Zhag, Pricig uder cotrait i acce etwork: Reveue maximizatio ad cogetio maagemet, i Porc. of IEEE INFOCOM, Sa Diego, CA, Mar. 2. [4] R. B. Wilo, Noliear Pricig. Oxford Uiverity Pre, 993. [4] N. El Karoui ad M. Queez, No-liear pricig theory ad backward tochatic differetial equatio, Lecture Note i Mathematic, Fiacial Mathematic, pp , Oct. 26. [42] C. Boet ad P. Duboi, Iferece o vertical cotract betwee maufacturer ad retailer allowig for oliear pricig ad reale price maiteace, The RAND Joural of Ecoomic, vol. 4, o., pp , Ja. 2. [43] C. Courcoubeti ad R. Weber, Pricig Commuicatio Network: Ecoomic, Techology ad Modellig. Wiley Pre, 23. [44] A. Ma-Colell, M. D. Whito, ad J. R. Gree, Microecoomic Theory. Oxford Uiverity Pre, New York, NY, 995. [45] A. Be-Tal ad A. S. Nemirovki, Lecture o Moder Covex Optimizatio: Aalyi, Algorithm, ad Egieerig Applicatio. Society for Idutrial ad Applied Mathematic SIAM), 2. [46] Y. Shoham ad K. Leyto-Brow, Multiaget Sytem: Algorithmic, Game-Theoretic, ad Logical Foudatio. Cambridge Uiverity Pre, 28. [47] J. B. Roe, Exitece ad uiquee of equilibrium poit for cocave N-pero game, Ecoometrica, vol. 33, o. 3, pp , Ju [48] S. Boyd, Covex Optimizatio I, Lecture Note. [Olie]. Available: [49] R. D. Yate, A framework for uplik power cotrol i cellular radio ytem, IEEE J. o Selected Area i Commu., vol. 3, o. 7, pp , Sep [5] D. Niyato ad E. Hoai, A ocooperative game-theoretic framework for radio reource maagemet i 4G heterogeeou wirele acce etwork, IEEE Tra. o Mobile Computig, vol. 7, o. 3, pp , Mar. 28. [5] U. Paul, A. P. Subramaia, M. M. Buddhikot, ad S. R. Da, Udertadig traffic dyamic i cellular data etwork, i Proc. of IEEE INFOCOM, Shaghai, Chia, Apr. 2. [52] M. Z. Shafiq, L. Ji, A. X. Liu, ad J. Wag, Characterizig ad modelig Iteret traffic dyamic of cellular device, i Proc. of ACM SIGMETRICS, Sa Joe, CA, Ju. 2. Hamed Shah-Maouri S 6, M 4) received the B.Sc., M.Sc., ad Ph.D. degree from Sharif Uiverity of Techology, Tehra, Ira, i 25, 27, ad 22, repectively all i electrical egieerig. He raked firt amog the graduate tudet. From 22 to 23, he wa with Parma Co., Tehra, Ira. Curretly, Dr. Shah-Maouri i a Pot-doctoral Reearch ad Teachig Fellow at the Uiverity of Britih Columbia, Vacouver, Caada. Hi reearch iteret are i the area of tochatic aalyi, optimizatio ad game theory ad their applicatio i ecoomic of cellular etwork ad mobile cloud computig ytem. He ha erved a the publicatio co-chair for the IEEE Caadia Coferece o Electrical ad Computer Egieerig 26 ad a the techical program committee TPC) member for everal coferece icludig the IEEE Global Commuicatio Coferece GLOBECOM) 25 ad the IEEE Vehicular Techology Coferece VTC Fall) 26 ad 27. Vicet W.S. Wog S 94, M, SM 7, F 6) received the B.Sc. degree from the Uiverity of Maitoba, Wiipeg, MB, Caada, i 994, the M.A.Sc. degree from the Uiverity of Waterloo, Waterloo, ON, Caada, i 996, ad the Ph.D. degree from the Uiverity of Britih Columbia UBC), Vacouver, BC, Caada, i 2. From 2 to 2, he worked a a ytem egieer at PMC-Sierra Ic. ow Microemi). He joied the Departmet of Electrical ad Computer Egieerig at UBC i 22 ad i curretly a Profeor. Hi reearch area iclude protocol deig, optimizatio, ad reource maagemet of commuicatio etwork, with applicatio to wirele etwork, mart grid, mobile cloud computig, ad Iteret of Thig. Dr. Wog co-edited the book Key Techologie for 5G Wirele Sytem, Cambridge Uiverity Pre, 27. He i a Editor of the IEEE Traactio o Commuicatio. He ha erved a a Guet Editor of IEEE Joural o Selected Area i Commuicatio ad IEEE Wirele Commuicatio. He ha erved o the editorial board of IEEE Traactio o Vehicular Techology ad Joural of Commuicatio ad Network. He ha erved a a Techical Program Co-chair of IEEE SmartGridComm 4, a well a a Sympoium Co-chair of IEEE SmartGridComm 3 ad IEEE Globecom 3. He i the Chair of the IEEE Commuicatio Society Emergig Techical Sub-Committee o Smart Grid Commuicatio ad the IEEE Vacouver Joit Commuicatio Chapter. He received the 24 UBC Killam Faculty Reearch Fellowhip.

18 Jiawei Huag S, M 6, SM, F 6) i a IEEE Fellow, a Ditiguihed Lecturer of IEEE Commuicatio Society, ad a Thomo Reuter Highly Cited Reearcher i Computer Sciece. He i a Aociate Profeor ad Director of the Network Commuicatio ad Ecoomic Lab cel.ie.cuhk.edu.hk), i the Departmet of Iformatio Egieerig at the Chiee Uiverity of Hog Kog. He received the Ph.D. degree from Northweter Uiverity i 25, ad worked a a Potdoc Reearch Aociate at Priceto Uiverity durig He i the co-recipiet of 8 Bet Paper Award, icludig IEEE Marcoi Prize Paper Award i Wirele Commuicatio i 2. He ha co-authored ix book, icludig the textbook o Wirele Network Pricig. He received the CUHK Youg Reearcher Award i 24 ad IEEE ComSoc Aia-Pacific Outtadig Youg Reearcher Award i 29. He ha erved a a Aociate Editor of IEEE/ACM Traactio o Networkig, IEEE Traactio o Wirele Commuicatio, IEEE Joural o Selected Area i Commuicatio - Cogitive Radio Serie, ad IEEE Traactio o Cogitive Commuicatio ad Networkig. He ha erved a a Co-Editor-i-Chief of Wiley Iformatio ad Commuicatio Techology Serie, a Area Editor of Spriger Ecyclopedia of Wirele Network, ad a Sectio Editor for Spriger Hadbook of Cogitive Radio. He ha erved a the Chair of IEEE ComSoc Cogitive Network Techical Committee ad Multimedia Commuicatio Techical Committee. He i the recipiet of IEEE ComSoc Multimedia Commuicatio Techical Committee Ditiguihed Service Award i 25 ad IEEE GLOBECOM Outtadig Service Award i 2. IEEE TRANSACTIONS ON MOBILE COMPUTING, VOL. PP, NO. 99, MONTH 27

u t u 0 ( 7) Intuitively, the maximum principles can be explained by the following observation. Recall

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