Analysis of duopoly price competition between WLAN providers. Citation Ieee International Conference On Communications, 2009

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1 Tle Aalyss of duopoly prce compeo bewee WLAN provders Auhor(s Kog, Z; Tuff, B; Kwok, YK; Wag, J Cao Ieee Ieraoal Coferece O Commucaos, 9 Issued Dae 9 URL hp://hdl.hadle.e/7/635 Rghs Ths work s lcesed uder a Creave Commos Arbuo- NoCommercal-NoDervaves 4. Ieraoal Lcese.; IEEE Ieraoal Coferece o Commucaos Proceedgs. Copyrgh IEEE.; 9 IEEE. Persoal use of hs maeral s permed. However, permsso o repr/republsh hs maeral for adversg or promooal purposes or for creag ew collecve works for resale or redsrbuo o servers or lss, or o reuse ay copyrghed compoe of hs work oher works mus be obaed from he IEEE.

2 Ths full ex paper was peer revewed a he dreco of IEEE Commucaos ocey subjec maer expers for publcao he IEEE ICC 9 proceedgs Aalyss of Duopoly Prce Compeo bewee WLAN Provders Zhe Kog Deparme of EEE Uv. of Hog Kog Hog Kog, CHINA zkog@eee.hku.hk Bruo Tuff INRIA Rees Campus Uv. de Beauleu Rees, FRANCE buff@rsa.fr Yu-Kwog Kwok Deparme of ECE Colorado ae Uv. For Colls, CO, UA rcky.kwok@colosae.edu Jagzhou Wag Deparme of Elecrocs Uversy of Ke Caerbury, Ke, UK j.z.wag@ke.ac.uk Absrac Wh he rapd developme of wreless Iere servces, several WLAN servce provders may coexs oe publc hospo o compee for he same group of cusomers, leadg o a evable prce compeo. The charged prce ad he provsoed packe loss a each provder are major facors deermg users demads ad behavors, whch ur wll affec provders reveue ad socal welfare. I hs paper, we se up a ovel game model o aalyze a duopoly prce compeo. We frs show he users demads are dsrbued bewee provders accordg o a Wardrop Equlbrum ad he prove he exsece of a Nash equlbrum o provders charged prces. Through aalyss, we furher fd ha Nash equlbrum sae he socal welfare s very close o s maxmal value cooperave suao. Furhermore, he provders aggregae reveues also do o decrease whe he users have hgh sesvy abou he charged prces. Thus he compeve duopoly WLAN marke ca sll ru a effce way eve he absece of complex regulao schemes. Keywords: Prce compeo; Prce of aarchy; Wardrop equlbrum; Nash equlbrum; WLAN; hospo. I. INTRODUCTION gfca vesmes broadbad wreless eworks are prolferag aroud he world rece years, ad have led o a hghly compeve evrome for wreless LAN (WLAN servce provders. They have o compee wh each oher ad arac users order o geerae profs. I a ypcal compeve hospo evrome (e.g., a publc cyber-café, several WLAN provders may coexs o provde wreless access servces for he same group of users. I aempg o arac users ad opmze her reveues, hey eed o prce her servces by akg o accou a wde rage of facors, cludg prefereces of ed-users, he qualy-of-servce (Qo lmaos of used echology, ad he poeal compeo from oher provders. Furhermore, whe he Qo ad prces are observed, wreless users wll allocae her demads across he provders. Ths process wll ur affec he reveue of provders. Thus, wh he capably o affec users demad ad provders reveue, prcg wreless eworks has gaed much aeo recely []. udyg he mpac of prce compeo s recevg creasg aeo he eworkg commuy. For geeral deas o prcg, he reader s referred o []. The prcg game amog wreless provders wh fxed capacy s aalyzed [], ad s show ha he prce of aarchy (PoA [3] s. Acemoglu e al. [4] sudy a prcg problem whch he users are sesve o he prce ad a covex cogeso delay, ad gve gh bouds o PoA. I [5] Hayrapeya e al. gves a mproved boud for he specal case whch packe delay s a pure egave effec, whch s lear or covex. Nyao e al. [6] vesgae wo levels of compeos cogve wreless mesh ework, ad propose game heorec soluos o choose prce ad rasmsso rae for prmary users ad secodary users, bu he demad dsrbuo of secodary users s o aalyzed. I [7] a cos-prce mechasm based o users chael occupacy s proposed WLAN o maxmze he sysem hroughpu, bu he compeo bewee dffere WLAN provders s o aalyzed. Furhermore, because WLAN sadard employs a coeo based radom mulple access mehod based o CMA/CA,.e. DCF, as he fudameal MAC echque, here exss packe collso, whch wll resul packe loss. O he oher had, wreless rasmsso error may lead o packe loss oo. Thus, hs paper, we cosder packe loss rae (PLR sead of delay or rae as egave exeraly [], ad fd ha s deed cocave oher ha covex WLAN evrome. Thus he prevous fdg cao be used WLAN suaos drecly. To he bes of our kowledge, he problem of prcg compeo for WLAN provders wh he cosderao of prce ad PLR s relavely uexplored. I hs paper, we focus o a duopoly compeo, whch wo WLAN provders compee for a same group of wreless users by adjusg he prce charged. The provder s am s o maxmze s ow prof, whle he users, drve by self-eress, raoally choose he servce provder offerg he bes combao of Qo ad prce. We assume ha users are sesve o he expereced PLR as well as charged prces. The PLR a each access po (AP occurs because of packes collso or packe rasmsso error. Geerally, he users are more lkely o coec o he provder wh good chael codo ad/or low prce, bu whe more users coec o a provder, he probably of collso wll crease oo. Ths feaure s also kow as egave exeraly [], sce he decso made by a user wll have a egave effec o he payoff of ohers coeced o he same provder. Thus, ca affec user s wllgess o accep a provder s servce. Cosequely, each user has o calculae he expereced cos o choose a provder accordg o he charged prce ad esmaed PLR. We assume each user lkes o choose he provder wh he mmum expereced cos ad he umber of users (.e., demad decreases wh he cos. Ths leads us o research some mpora quesos: how he users demad s spl amog provders whe her prce sraeges are gve ad he PLRs are esmaed? Does here exs Nash equlbrum ( [9] prce vecor, uder whch o provder ca ulaerally mprove s ow reveue by chagg s prce? Furher, wheher here exss he effcecy loss,.e., PoA, as wha s show by may pror works, e.g., [4], [5]? To aswer hese quesos, we frs se up a game model o aalyze egave exerales duced by PLR, gve s approxmao ad fd s cocave deed. The we characerze he Wardrop equlbrum ( [8] for he dsrbuo of users demads. Afer ha, we prove he exsece of o provders charged prces. The hrough umercal aalyss we fd ha PoA ca be close o., whch dcaes ha he socal welfare does o suffer oo much hs compeve evrome. Furhermore, we fd /9/$5. 9 IEEE Auhorzed lcesed use lmed o: The Uversy of Hog Kog Lbrares. Dowloaded o May 7, a 8:5:58 UTC from IEEE Xplore. Resrcos apply.

3 Ths full ex paper was peer revewed a he dreco of IEEE Commucaos ocey subjec maer expers for publcao he IEEE ICC 9 proceedgs ha he compeo does o lead o reveue loss whe he users are very sesve o charged prces. The res of hs paper s orgazed as follows. We prese he sysem model eco II. eco III aalyses he demad dsrbuo erms of. The prcg game s researched eco IV. eco V gves umercal aalyss of effcecy ad we coclude eco VI. II. MODEL A. Nework model We cosder a IEEE 8. hospo area covered by APs, ad le Ω= {, } be he se of APs. Each AP Ω s corolled by a dsc provder, whch charges a prce p per user for usg s servce. Thus here he erms of AP ad provder are erchageable. We assume APs share he same frequecy badwdh he ework, whereas dffere APs are beg operaed o dffere frequecy chaels ad usg dffere PHY mode. Thus here s o erferece amog dffere users. DCF s used as MAC level muluser access mehod. We also gore hdde ode problem, so ha every user ca sese all oher s rasmsso as []. Furhermore we operae saurao codos,.e., he rasmsso queue of each user s assumed o be always oempy. Ad here s oly oe user ha ca rasm s packes successfully o he beloged AP a oe me slo. Le x be he umber of users coeced o AP. Noe ha users are assumed o be fesmal, each of hem havg a eglgble mpac o ohers. We assume ha whe he h x user coecs o he AP, wll ge a uly of U ( x, whch s s wllgess o pay o ge servce. mlar o wha s doe [4], we use he assumpo ha whe a user decdes o receve he WLAN access servce, wll ge a reservao preferece of R,.e. U( x = R. Bu a cos c( x wll also be expereced whe here are x users, whch s drecly relaed wh charged prce p ad packe loss rae PLR( x. Furhermore, hs user wll o coec o AP f he expereced cos exceeds s reservao preferece. The he aggregae user s surplus AP ca be = ( R c( x x ( where he cos c( x ca be expressed as c( x = p + f( PLR( x = p + f( x ( where f ( s used o express he expereced cos resulg from PLR. For he sake of smplcy, we assume users are homogeeous ad le f ( x = PLR ( x, where. The expereced cos s a mpora parameer sce users demad ad provders reveue are hghly relaed o. I hs work, we defe he reveue of provder as Π ( p = p x (3 As defed [4], he socal welfare (W ca be expressed by addg he ules of all users ad provders,.e. W = ( R f( x x (4 Ω B. Packe loss rae I 8. sysems, PLR ca be expressed as l c c c PLR ( x = P = ( P ( P = P + P P P (5 where l P ad c P are packe loss rae ad packe collso rae expereced a AP wh x users, respecvely, ad P s he mmal packe rasmsso error rae suppored by AP. I fac, due o locao-depede characerscs, he rasmsso error rae wll be dffere for dffere users. To focus o he prce seg for dffere provders ad ease he aalyss, he exogeous formao used hs game s he mmal rasmsso error rae P ha AP ca suppor uder s geography suao ad provsoed rasmsso echques, e.g. codg ad modulao. Furhermore, a user may also auomacally move o a sasfacory place for a accepable rasmsso codo. The because here s o erferece bewee provders hs model, we ca assume ha P a a parcular provder s maaed as a cosa value ad dffere provders wll suppor dffere P. I erms of packe collso rae P c, here we refer o as he packe collso observed a each dvdual user [],.e. s he probably ha oe user ecouers collsos whe rasms packes. Defe P as he rasmsso probably for each user. The f x users are coeced o c x he provder, we have P ( P =. ce he provders wa o maxmze her profs, he assumpo of maxmum saurao hroughpu s reasoable hs research. I [], a approxmao for P uder maxmum achevable saurao hroughpu s gve by * [ x + ( x ( Tc ]/ x P = = (6 * ( x ( T * x T / xk c c where K = T * c /, ad T * c s he average me whe he chael s sesed busy by each sao durg a collso, whch s deermed as a cosa by gve PHY ad MAC mechasm. Thus we have l c x PLR( x = P = ( P ( P = ( ( P xk x + xk ( xk K x + K K ( l( ( x xk = ( P e ( P e e ( P e We ca furher yeld l x = PLR ( P = l (8 l ( P + ω where ω = l ( P, = +. Remark ha K K K PLR( x s srcly creasg wh regard o x ad P. Also, sce he secod dervave PLR '' ( x <, s cocave. III. DITRIBUTION OF UER DEMAND Whe a user was o access o a WLAN, wll face several compeve provders wh dffere charged prces ad provsoed Qo, ad herefore feels dffere expereced coss c( x. Naurally prefers o choose he provder wh he mmum cos C = m c ( x. ( (7 Auhorzed lcesed use lmed o: The Uversy of Hog Kog Lbrares. Dowloaded o May 7, a 8:5:58 UTC from IEEE Xplore. Resrcos apply.

4 Ths full ex paper was peer revewed a he dreco of IEEE Commucaos ocey subjec maer expers for publcao he IEEE ICC 9 proceedgs Furhermore, f a provder s cos s oo hgh, here wll o addoal user wllg o choose, ad eve already coeced oes wll be expeced o swch. The a equlbrum sae, he expereced cos wll be decal a all provders havg a posve demad, or he umber of users coeced o oe provder wll be zero because of a oo hgh access prce. Ths s also kow as Wardrop equlbrum (, whch ca be mahemacally defed as follows. Defo : The users demad X = ( x, x s sad o acheve f x ( c( x C = Ω,ad C = m ( C( x, x > Ω Also he oal level of demad verfes x = X( C. Ω Ths equao meas ha he provders wh posve demad wll have he same expereced cos C ; oherwse he provder wll have zero demad due o he hgh cos. ce c ( x C ad x = X( C, he exsece of Ω our model ca be verfed as ha [4], [8]. Nex we wll derve he uqueess of hs research. We assume he oal umber of users he sysem decreases wh he mmum expereced cos. The we ca X C as express he aggregae demad fuco ( X ( C = X d C, for d (9 where d s a demad parameer o express user s sesvy o he expereced cos. The we ca characerze he vecor X = ( x, x as follows. Proposo : Wh demad fuco X ( C ad prce vecor P = ( p, p, vecor X = ( x, x ca be characerzed as he smalles soluo of he equao XC ( x( C wh C [, X ( X ], ad Ω p X ( C PLR ( X p p x = PLR PLR( X PLR( > > ( p PLR ( Proof: ce sae, C = p + f ( x for >, ad p he oal demad lm s X, we have x = PLR, p whe PLR( X > > PLR (. Oherwse whe p p s oo hgh, ad PLR (, we ca ge x = ; ad whe p s oo small, ad all users wa o coec wh, we have x X ( C =. The we ca ge x ( C as he expresso (. Wha s more, afer oe user choose a provder, he mmum expereced cos C = m( c ( x creases ad XC ( s decreased. If XC ( > x( C, ew user ca eer he sysem o choose a provder; oherwse, he demad lm s reached, ad he correspodg s acheved oo. x IV. PRICE COMPETITION ANALYI I hs duopoly evrome, each provder seeks o se s prce order o maxmze s ow reveue. Thus we se up a prce game ad he se of provders Ω represes he players. For provder Ω= {, }, s aco s defed as he prce choce p.the he aco profle s deoed as he prce vecor p = ( p, p. Is uly s expressed by he reveue Π ( p, p, where p deoes he prces chose by provders else oher ha. The we defe as Defo : A prce vecor p = ( p, p s sad o be a Nash equlbrum f for all Ω, Π ( p, p Π ( p, p p > We frs prove a lemma ha esablshes some resuls wh a prce cofgurao p. Lemma : Assume ha boh provders have posve demad x = ( x, x for prce p = ( p, p. If provder decreases s prce o p = p ε < p, ad p j for provder j s uchaged, where superscrp s used o refer he values correspodg o a ew suao; he l, l ( he PLR creases o P > P ( he leas expereced cos decreases o C = φ < C, adφ < ε, whereφ >, ε >. Proof: ce p < pad p j = p j, more users wll swch from provder j o. Thus x > x ad x < x.ce PLR s j j l, P = m( ( j( j = j + j j Γ creasg relaed o user umber, we have Le C c x c x p f x ( < p + f x = c ( x = C j j j j j l > P. The ( = ad = φ = + ( = ε + ( ω x ω x ω x p ε e e e e e e ω x ω x p p ε C = φ, he we ge C C p f x p f x = + ( + ( = ε + ( e e e e > ε Thus φ < ε. Wh hs observao, we ca show he exsece of. Proposo : Whe packe loss rae s much smaller ha, here exss a prce vecor P ( p, p = o acheve Nash equlbrum hs prce compeo. Proof: Whou loss of geeraly, cosder a possble chage of provder from p o p = p ε < p. l, l, The P > P, we ca have ce p x p x Π Π = = ( p p x l, ε ω + l l, ( P P, wh Taylor expaso, we have Auhorzed lcesed use lmed o: The Uversy of Hog Kog Lbrares. Dowloaded o May 7, a 8:5:58 UTC from IEEE Xplore. Resrcos apply.

5 Ths full ex paper was peer revewed a he dreco of IEEE Commucaos ocey subjec maer expers for publcao he IEEE ICC 9 proceedgs Π Π ( p ε p l, x ω P < ( p p x ε l, ω P ε ( p x p x < Wh hs prce seg, o provder ca mprove s reveue by ulaerally chagg s prce, hus resulg a. V. EFFICIENCY ANALYI We vesgae he effcecy of hs game erms of socal welfare ad provders reveue. Here we use PoA as a measure of he wors case dfferece bewee he socal welfare of a cooperavely opmal soluo (socal opmum ad ha a o-cooperave sae. mlar o ha [5], we defe PoA as follows. P = p, p s he prce vecor ha Defo 4: If ( maxmzes socal welfare, P ( p, p = s he prce W ( P vecor sae; he PoA s defed as PoA =. W ( P ce W ( P W ( P, we ge PoA. Whe PoA s close o, meas he socal welfare arrved a compeve evrome s early as good as ha reached hrough cooperave opmzao. Whereas a large PoA meas he compeo s less effce. I o-cooperave suao, we ca frs ge prce vecor P by fdg he erseco of boh users bes respose fucos as roduced [9], ad he fd he correspodg dsrbuo X wh Proposo. Afer ha we ca calculae W ( P hrough P ad X drecly. Whle for W ( P cooperave suao, ca be foud by solvg he followg opmzao problem. W ( P = max ( R f( x x ( subjec o: Γ x ( p + f( x C = (I x X ( C (II Γ where he cosra I meas he soluo should sasfy ; ad he cosra II s used o cosra he user demad. ce he above problem has a couous objecve fuco ad a compac cosra se, he exsece of a socal opmum s guaraeed. Based o above aalyss, we ca vesgae he PoA hrough umercal mehods wh wo APs ha eher compee or cooperae wh each oher, whch he oal umber of users X s ad R s. We also se K=9.334 as ha []. I Fgure, we show he bes respose uder dffere se of prces. Boh of he packe rasmsso error raes for hese wo provders are se o be., ad demad parameer s. We fd ha here exs a uque a prces (.4,.4. A smlar resul wh a prces (.3,. also exs Fgure, where P =.ad P =.3, ad d = 3. These verfy he exsece of uder dffere suaos. Fg.3 ad Fg.4 show he socal welfare uder dffere rasmsso error rae ad demad parameer combaos. We ca see ha W creases wh demad parameer, whle sae s close o ha cooperave suaos. From Fg.5 ad Fg.6, he PoA s ear.. The leg he provders compee wh each oher wll yeld almos he same socal welfare as a global marke regulaor would have gve. Thus hs compeo s o as effce as wha we usually look a o-cooperave suao. Prce charged by AP Bes respose fuco (P=. P=. d= B(P B(P Prce charged by AP Fg. : Bes respose fuco ( P = P =., d =. Prce charged by AP Bes respose fuco (P=. P=.3 d=3 B(P B(P Prce charged by AP Fg. : Bes respose fuco ( P =., P =.3, d = 3. ocal welfare ocal welfare (P=. P=. ocal Opmum ocal welfare Demad parameer Fg. 3: W vs. Demad parameer ( P = P =.. ocal welfare ocal welfare (P=. P=.3 ocal Opmum ocal welfare Demad parameer Fg. 4: W vs. Demad parameer ( P =., P =.3. Auhorzed lcesed use lmed o: The Uversy of Hog Kog Lbrares. Dowloaded o May 7, a 8:5:58 UTC from IEEE Xplore. Resrcos apply.

6 Ths full ex paper was peer revewed a he dreco of IEEE Commucaos ocey subjec maer expers for publcao he IEEE ICC 9 proceedgs.8 Prce of aarchy (P=. P=. Prce of aarchy.8 Prce of aarchy (P=. P=.3 Prce of aarchy PoA PoA Demad parameer Fg. 5: PoA vs. Demad parameer ( P = P =.. 5 Aggregae reveue (P=. P=. Cooperave opmal aggragee reveue Aggregae Reveue Demad parameer Fg. 6: PoA vs. Demad parameer ( P =., P =.3. 5 Aggregae reveue (P=. P=.3 Cooperave opmal aggragee reveue Aggregae Reveue Aggregae reveue 5 Aggregae reveue Demad parameer Fg. 7: Aggregae reveue vs. Demad parameer ( P = P =.. Though he effcecy of W s o decreased much uder compeo, he provders reveue sae s much smaller ha ha cooperave suaos whe he demad parameer s small as show Fg.7 ad Fg, 8. Bu whe he demad parameer creases, he aggregae reveues sae wll eveually approach o he cooperave maxmal reveue. Ths s because he larger he demad parameer, he more sesvely ha he user demad respods o he chage of prces. If he user s o sesve abou he prce, he provders would lke o cooperae or eve collude wh each oher o ga hgh reveue by seg hgh prce. Bu whe he user has hgher prce sesvy, wll o receve ay provders servce a all f he charged prces are oo hgh. The he provders wll be more raoal ad prefer o se relavely lower prces so as o arac users. Though he reveue s decreased wh demad parameer, wll cosequely be same as cooperave maxmal reveue. Furhermore, as show Fg.3 ad Fg.4, he socal welfare also mproves wh he creasg of user s prce sesvy. Usually he users are very sesve o he prce of wreless servces, hus all ees cludg provders ad users wll o suffer from hs compeo. VI. CONCLUION I hs paper, he mpac of charged prce ad he provsoed PLR by dffere WLAN provders o users demad, provders reveue ad socal welfare a duopoly evrome s vesgaed. Based o a game-heorec model, we frs aalyzed he egave exerales assocaed wh PLR. The we foud he exsece ad uqueess of a for users demad dsrbuo bewee provders, ad deermed he exsece of he of prce compeo. Furhermore, hrough umercal aalyss, we showed ha he socal welfare wll o suffer oo much Demad parameer Fg. 8: Aggregae reveue vs. Demad parameer ( P =., P =.3. uder prce compeo. Thus eve whou admsrave eforceme, he compeve marke self ca sll deerme he rgh prce whou degradg socal welfare or eve provders reveues, especally whe he users are very sesve o he prce (I fac hs s jus he ormal case for user s aude o prce for wreless servces. I he ex sep, we wa o mprove he sysem model ad sudy hs compeo a more realsc seg. Aoher eresg ssue cocers he compeo a more geeral olgopoly evrome, where here exs more ha wo compeve provders. REFERENCE [] C. Courcoubes ad R. Weber, Prcg Commucao Neworks: Ecoomcs, Techology ad Modellg, Joh Wley & os, 3 [] P. Malle ad B. Tuff, Aalyss of Prce Compeo a loed Resource Allocao Game, I Proc. IEEE INFOCOM 8, Phoex, UA, Apr. 8. [3] C. Papadmrou, Algorhms, Games, ad he Iere, I Proc. ACM TOC, pp , Hersossos, Greece, July. [4] D. Acemoglu ad A. Ozdaglar, Compeo ad Effcecy Cogesed Markes, Mahemacs of Operaos Research, vol. 3, o., pp. -3, Feb. 7. [5] A. Hayrapeya, E. Tardos ad T. Wexler, A Nework Prcg Game for elfsh Traffc, I Proc. ACM PODC 5, pp. 84-9, Las Vegas, NV, UA, July 5. [6] D. Nyao, E. Hossa, ad L. Le, Compeve pecrum harg ad Prcg Cogve Wreless Mesh Neworks, Proc. IEEE WCNC 8, pp , Las Vegas, NV, UA, Mar.-Apr. 8. [7]. hakkoa, E. Alma, ad A. Kumar, Mulhomg of Users o Access Pos WLANs: A Populao Game Perspecve, IEEE J. elec. Areas Commu., vol. 5, o. 6, pp.7-5, Aug. 7. [8] J. G. Wardrop, ome Theorecal Aspecs of Road Traffc Research, Proceedgs of he Isue of Cvl Egeers, Par II, vol., pp , Lodo, UK, 95. [9] M. J. Osbore ad A. Rubse, A Course Game Theory, MIT Press, 994. [] G. Bach, Performace Aalyss of he IEEE 8. Dsrbued Coordao Fuco, IEEE J. elec. Areas Commu., vol. 8, o.3, pp , Mar.. Auhorzed lcesed use lmed o: The Uversy of Hog Kog Lbrares. Dowloaded o May 7, a 8:5:58 UTC from IEEE Xplore. Resrcos apply.