Competition in Parallel-Serial Networks

Transcription

1 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX Competton n Parallel-Seral Networs Daron Acemoglu and Asuman Ozdaglar, Member, IEEE Abstract We study the effcency mplcatons of competton among proft-maxmzng servce provders n communcaton networs Servce provders set prces for transmsson of flows through ther (subnetwor The central queston s whether the presence of prces wll help or hnder networ performance We nvestgate ths queston by consderng the dfference between users wllngness to pay and delay costs as the effcency metrc Prevous wor has demonstrated that n networs consstng of parallel lns, effcency losses from competton are bounded Nevertheless, parallel-ln networs are specal, and n most networs, traffc has to smultaneously traverse lns (or subnetwors operated by ndependent servce provders The smplest networ topology allowng for ths feature s the parallel-seral structure, whch we study n ths paper In contrast to exstng results, we show that n the presence of seral lns, the effcency loss relatve to the socal optmum can be arbtrarly large The reason for ths degradaton of performance s the double margnalzaton problem, whereby each seral provder charges hgh prces not tang nto account the effect of ths strategy on the profts of other provders along the same path Nevertheless, when there are no delay costs wthout transmsson (e, latences at zero are equal to zero, rrespectve of the number of seral and parallel provders, the effcency of strong olgopoly equlbra can be bounded by /, where strong olgopoly equlbra are equlbra n whch each provder plays a strct best response and all of the traffc s transmtted However, even wth strong olgopoly equlbra, neffcency can be arbtrarly large when the assumpton of no delay costs wthout transmsson s relaxed Index Terms Prcng, competton, congeston externaltes, Wardrop equlbrum, socal optmum, olgopoly equlbrum, effcency, prce of anarchy I INTRODUCTION There has been growng nterest n prcng as a method of allocatng scarce networ resources The standard approach assumes that prces may used as control parameters to acheve a system optmum n a decentralzed manner (see, eg, [, [, [3 Nevertheless, n practce many prces are controlled by for-proft servce provders, whose objectve s, at least n part, to ncrease ther revenues and profts A central queston s whether prces chosen to maxmze servce provder revenues wll also play a useful role n the allocaton of networ resources across users Research to date suggests that proft-maxmzng prcng may mprove the allocaton of resources n communcaton networs wth self-nterested users Let the metrc of effcency be the dfference between users wllngness to pay and delay Manuscrpt receved May, 006; revsed January 5, 007 Ths research was funded n part by the Natonal Scence Foundaton CAREER award DMI A prelmnary verson of ths wor was presented at INFOCOM, 006 D Acemoglu s wth the Department of Economcs, Massachusetts Insttute of Technology, Cambrdge, MA, 039 (e-mal: daron@mtedu A Ozdaglar s wth the Department of Electrcal Engneerng and Computer Scence, Massachusetts Insttute of Technology, Cambrdge, MA, 039 (emal: asuman@mtedu costs n the decentralzed equlbrum relatve to that n the socal optmum (whch would be chosen by a networ planner wth full nformaton and full control over users In the absence of prces, the decentralzed equlbrum can be hghly neffcent (see [4 In contrast, Acemoglu and Ozdaglar [5 show that wth nelastc and homogeneous users, prcng by a monopolst controllng all lns n a parallel-ln networ always acheves effcency (e, the effcency metrc s equal to Huang, Ozdaglar and Acemoglu [6 extend ths result to a general networ topology More recently, Acemoglu and Ozdaglar [7 show that n a parallel-ln networ wth nelastc and homogeneous users, the effcency metrc wth an arbtrary number of competng networ provders s always greater than or equal to 5/6 Most communcaton networs cannot be represented by parallel-ln topologes, however A gven source-destnaton par wll typcally transmt through multple nterconnected subnetwors (or lns, potentally operated by dfferent servce provders Exstng results on the parallel-ln topology do not address how the cooperaton and competton between servce provders wll mpact on effcency n such general networs In ths paper, we tae a step n ths drecton by consderng the smplest networ topology that allows for seral nterconnecton of multple lns/subnetwors, whch s the parallelseral topology We focus on a sngle source-destnaton par, wth flows choosng one of multple parallel paths We allow each path to consst of multple lns/subnetwors operated by ndependent servce provders Our man results show that that the effcency losses resultng from competton are consderably hgher wth ths topology The source of addtonal neffcency s the presence of seral servce provders and wll thus be present n more general networ topologes Ths suggests that unregulated competton n general communcaton networs may have consderable costs n terms of the effcency of resource allocaton and certan types of regulaton may be necessary to mae sure that servce provder competton does not lead to sgnfcant degradaton of networ performance In our model, an orgn-destnaton par s lned by multple parallel paths, each potentally consstng of an arbtrary number of seral lns Congeston costs are captured by lnspecfc non-decreasng convex latency functons, denoted by l ( for ln Each ln s owned by a dfferent servce provder All users are nelastc and homogeneous Ths envronment nduces the followng two-stage game: each servce provder smultaneously sets the prce for transmsson of bandwdth on ts ln, denoted by p Observng all the prces, n the second stage users route ther nformaton through the path wth the lowest effectve cost, where effectve cost conssts of the sum of prces and latences of the lns

2 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX along a path [e, sum of p +l ( s over the lns comprsng a path Our objectve s to study the effcency propertes of the subgame perfect equlbra of ths game The man novel aspect of ths model compared to the parallel-ln topology s the prcng decsons of dfferent (seral servce provders along a sngle path When a partcular provder charges a hgher prce, t creates a negatve externalty on other provders along the same path, because ths hgher prce reduces the transmsson that all the provders along ths path receve Ths s the equvalent of the double margnalzaton problem n economc models wth multple monopoles and s the source of the sgnfcant degradaton n the effcency performance of the networ In ts most extreme form, the double margnalzaton problem leads to a type of coordnaton falure, whereby all provders, expectng others to charge hgh prces, also charge prohbtvely hgh prces, effectvely llng all data transmsson on a gven path Such coordnaton falures can lead to arbtrarly low effcency Ths type of pathologcal behavor can happen n subgame perfect equlbra (what we refer to as olgopoly equlbra, OE, but we show that t cannot happen n strct subgame perfect equlbra, strct OE, whch follows the noton of strct equlbrum ntroduced n Harsany [8 In strct OE, each servce provder must play a strct best response to the prcng strateges of other servce provders We show that ths requrement s suffcent to rule out the pathologcal coordnaton falures mentoned above Nevertheless, we show that strct OE can also have arbtrarly large effcency losses agan owng to the double margnalzaton problem Even n a strct OE, seral provders gnore the negatve externalty they create on other provders along the same path and charge too hgh prces, whch can once agan prevent any transmsson on a partcular path, even when such transmsson s socally optmal Interestngly, however, these extreme neffcent outcomes occur when hgh prces on a partcular path prevent the entre avalable traffc from beng transmtted To nvestgate mplcatons of prce competton when all traffc s transmtted, we defne an even stronger noton of equlbrum, strong OE, as a strct OE n whch all traffc s transmtted We show that when latency wthout any traffc s equal to zero [e, l (0 = 0, there s a tght bound of / on the effcency of strong OE rrespectve of the number of paths and servce provders n the networ Ths bound s reached by smple examples In strong OE, the double margnalzaton problem s stll present, and ths s the reason why the bound of / s lower than the 5/6 bound n our prevous wor, [7 However, the assumpton that l (0 = 0 s mportant for ths result We show that when ths assumpton s relaxed, the effcency loss of strong OE relatve to the socal optmum can be agan arbtrarly large These results shed doubt on the conjecture that unregulated competton among servce provders mght lead to prces approxmatng those that would be set as control parameters Models of selfsh routng wthout prces, eg, [4 or [9, assume that all traffc s always transmtted Our model ncorporates a reservaton utlty for users, so that ths s not necessarly the case All traffc wll be transmtted n equlbrum when ths reservaton utlty s suffcently large by a centralzed networ operator Instead, they show that competton among servce provders wth general networ topologes can lead to sgnfcant degradaton of networ performance (Example 3 below can be part of any networ topology and cause arbtrary effcency losses Nevertheless, t has to be borne n mnd that the examples that have very poor performance relatve to the socal optmum are somewhat pathologcal, and ths begs the queston of whether better performance bounds could be obtaned n more realstc topologes, whch s an area left for future wor Wor related to our paper ncludes studes quantfyng effcency losses of selfsh routng wthout prces (eg, Koutsoupas and Papadmtrou [0, Roughgarden and Tardos [4, Correa, Schulz, and Ster-Moses [9, and Fredman [; of resource allocaton by dfferent maret mechansms (eg, Johar and Tstsls [, Sanghav and Haje [3; and of networ desgn (eg, Anshelevch et al [4 Basar and Srant [5 analyze monopoly prcng n a networ context under specfc assumptons on the utlty and latency functons, whle He and Walrand [6 study competton and cooperaton among Internet servce provders under specfc demand models Most closely related to the current paper are our prevous wor [7, where we study the exstence and effcency of olgopoly equlbra n parallel-ln networs, as well as Hayrapetyan, Tardos, and Wexler [7 and Ozdaglar [8, who study prcng n a parallel-ln networ wth elastc demand No other paper has nvestgated prce competton n the presence of seral provders or more general topologes The rest of the paper s organzed as follows Secton II outlnes the basc envronment It defnes the concept of Wardrop equlbrum for the routng of flows gven prces set by servce provders Secton III defnes the concept of equlbrum n the game among the servce provders and establshes the exstence of a pure strategy equlbrum wth lnear latency functons, and the exstence of a mxed strategy equlbrum more generally Secton IV focuses on the effcency analyss of olgopoly equlbrum and contans the man results of the paper Ths secton frst shows that an olgopoly equlbrum can be arbtrarly neffcent because of the coordnaton falures resultng from double margnalzaton It then ntroduces the concepts of strct and strong olgopoly equlbra, and provdes a characterzaton of equlbrum prces n strct olgopoly equlbra Ths secton also establshes the exstence of a unque strong olgopoly equlbrum wth lnear latences and a suffcently hgh reservaton utlty, and presents bounds on the neffcency of strong olgopoly equlbra It concludes by showng how even strong olgopoly equlbra can be arbtrarly neffcent when latences at zero congeston are postve Secton V concludes, whle the Appendces contan some of the proofs not provded n the text II MODEL We consder a networ wth I parallel paths that connect a sngle source-destnaton par Each path conssts of n lns Let I = {,, I} denote the set of paths and N denote the set of lns on path Let x denote the flow on path, and x = [x,, x I denote the vector of path flows Each ln

3 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 3 l j (x, p j u (x d unts Reservaton utlty : R x x 3 x Rd Fg A networ wth seral and parallel lns d x n the networ has a flow-dependent latency functon l (x, whch measures the delay as a functon of the total flow on ln (see Fgure We denote the prce per unt flow (bandwdth of ln j by p j Let p = [p j j N, I denote the vector of prces We are nterested n the problem of routng d unts of flow across the I paths We assume that ths s the aggregate flow of many small users and thus adopt the Wardrop s prncple (see [9 n characterzng the flow dstrbuton n the networ; e, the flows are routed along paths wth mnmum effectve cost, defned as the sum of the latences and prces of the lns along that path (see the defnton below Wardrop s prncple s used extensvely n modellng traffc behavor n transportaton networs ( [0, [, [ and communcaton networs ( [4, [9 We also assume that users have a reservaton utlty R and decde not to send ther flow f the effectve cost exceeds the reservaton utlty Ths mples that user preferences can be represented by the pecewse lnear aggregate utlty functon u ( depcted n Fgure Ths assumpton also mples that all users are homogeneous snce they all have the same reservaton utlty, R Defnton For a gven prce vector p 0, a vector x W E R I + s a Wardrop equlbrum (WE f I xw E d and for all wth x W E > 0, ( l j (x W E + p j j N j N ( lj (x W E + p j { ( = mn l j (x W E + p j },( I j N R, wth { I xw E } = d f mn I j N l j (x W E + p j < R We denote the set of WE at a gven p by W (p We adopt the followng assumpton on the latency functons throughout the paper except n Secton IV-F Assumpton For each I, the latency functon l : [0, [0, s convex, contnuously dfferentable, nondecreasng, and satsfes l (0 = 0 Proposton (Exstence and Contnuty Let Assumpton hold For any prce vector p 0, the set of WE, W (p, s nonempty Moreover, the correspondence W : R I + R I + s upper semcontnuous We dscuss potental ssues n extendng ths wor to users wth elastc demands and heterogeneous qualty-of-servce requrements n the concludng secton Fg Aggregate utlty functon Proof setch: Gven any p 0, the proof s based on usng Assumpton (n partcular the nondecreasng assumpton on the latency functons to show that the set of WE s gven by the set of optmal solutons of the followng optmzaton problem QED max x 0 st ((R p j x I j N x d I x 0 j N l j (zdz ( For a gven prce vector p, the WE need not be unque n general Under further restrctons on the l, we obtan: Proposton (Unqueness Let Assumpton hold Assume further that for all I, there exsts some j N, such that l j s strctly ncreasng Then, for any prce vector p 0, the set of WE, W (p, s a sngleton Moreover, the functon W : R I + R I + s contnuous Proof: Under the gven assumptons, for any p 0, the objectve functon of problem ( s strctly convex, and therefore has a unque optmal soluton Ths shows the unqueness of the WE at a gven p Snce the correspondence W s upper semcontnuous from Proposton and sngle-valued, t s contnuous QED We next defne the socal problem and the socal optmum, whch s the routng (flow allocaton that would be chosen by a central networ planner that has full control and nformaton about the networ Defnton A flow vector x S s a socal optmum f t s an optmal soluton of the socal problem maxmze x 0 (R l j (x x (3 I j N subject to x d I By Assumpton, the socal problem has a contnuous objectve functon and a compact constrant set, guaranteeng the exstence of a socal optmum, x S Moreover, usng the

4 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 4 optmalty condtons for a convex program, we see that a vector x S R I + s a socal optmum f and only f I xs d and there exsts a λ S 0 such that λ S( I = xs d = 0 and for each I, R l j (x S x S l j(x S λ S f x S = 0, j N j N = λ S f x S > 0 (4 For a gven vector x R I +, we defne the value of the objectve functon n the socal problem, S(x = (R l j (x x, (5 I j N as the socal surplus, e, the dfference between the users wllngness to pay and the total latency III OLIGOPOLY PRICING AND EQUILIBRIUM We assume that there are multple servce provders, each ownng one of the lns on the paths n the networ Each ln may represent a more general subnetwor operated by ndependent servce provders 3 Servce provder j charges a prce p j per unt bandwdth on ln j N Gven the vector of prces of lns owned by other servce provders, p j = [p, the proft of servce provder j wth j N s Π j (p j, p j, x = p j x, where x W (p j, p j The objectve of each servce provder s to maxmze profts Because ther profts depend on the prces set by other servce provders, each servce provder forms conjectures about the actons of other servce provders, as well as the behavor of users, whch they do accordng to the noton of subgame perfect Nash equlbrum We refer to the game among servce provders as the prce competton game Defnton 3 A vector (, x OE 0 s a (pure strategy Olgopoly Equlbrum (OE f x OE W ( j, j and for all I, j N, and p j 0, x W (p j, j, Π j ( j, j, x OE Π j (p j, j, x (6 We refer to as the OE prce The next proposton shows that for lnear latency functons, there exsts a pure strategy OE The proof reles on the explct characterzaton of the OE prces (see Proposton 4 below, and therefore s provded n Appendx B Proposton 3 Let Assumpton hold and assume that the latency functons are lnear Then the prce competton game has a pure strategy OE The exstence result cannot be generalzed to general convex latency functons as shown n the followng example 3 Ths s because users care about the overall latency of a subnetwor, not about patterns of routng flows wthn the subnetwor operated by a servce provder The possblty of servce provders operatng multple paths s consdered n [7 and s ruled out here to smplfy the notaton Example Consder a two path networ wth one ln on each path Let the total flow be d = Assume that the latency functons are gven by { 0 f 0 x δ l (x = 0, l (x = x δ ɛ x δ, for some ɛ > 0 and δ > /, wth the conventon that when ɛ = 0, l (x = for x > δ It can be easly verfed that there exsts no pure strategy OE for small ɛ (see [7 for detals Nevertheless, t can be shown that the prce competton game always has a mxed strategy OE (see the analyss for a parallel ln networ n [7 A Ineffcency of OE IV EFFICIENCY ANALYSIS In ths secton, we study the effcency propertes of OE, and strct and strong OE (defned below We consder prce competton games that have pure strategy OE or strct OE (ths set ncludes, but s larger than, networs wth lnear latency functons covered by Proposton 3 Gven a parallelln networ wth I paths, n lns on path, and latency functons {l j } (j N, I, let OE({lj } denote the set of flow allocatons x OE = [x OE I at an OE (or strct OE dependng on the context We defne the effcency metrc at some x OE OE({lj } as r I ({l j }, x OE = S(xOE S(x S, (7 where x S s a socal optmum gven the latency functons {l j } [cf Eq (5 We adopt the conventon that for the effcency metrc 0/0 = Followng the lterature on the prce of anarchy, (see [0, we are nterested n the worst performance of an olgopoly equlbrum, so we loo for a lower bound on nf nf {l j } x OE OE({lj } r I ({l j }, x OE We frst show that the performance of an OE can be arbtrarly bad Example Consder a two path networ, whch has 3 lns on path wth dentcally 0 latency functons and one ln on path wth latency functon l(x = x, where / Let the total flow be d = and the reservaton utlty be R = The unque socal optmum for ths example s x S = (, 0 Now consder the followng strategy combnaton Each of the three servce provders on path, denoted by =,, and 3, charge prce p =, whle the servce provder on path charges p = / It can be verfed that there s no devaton that s proftable for any of the servce provders Frst, consder the seral provders on path ; gven the prces of two of the seral servce provders, there wll always be zero traffc on path, so the remanng servce provder s playng a best response (snce any prce for ths provder would lead to zero profts Moreover, t can be verfed that these strateges are not wealy domnated, snce f =, were to play p = 0 and the provder on path were to set a hgh enough p, = 3

5 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 5 would choose to play p 3 = (Ths also establshes that the OE wll be tremblng hand perfect, see [3, pp for a defnton Fnally, let us consder the provder on path Gven the strateges of the seral provders on path and a fxed > 0, t can be verfed that the optmal strategy of ths provder s to set p = / The resultng equlbrum flow allocaton s x OE = [ 0,, whch nvolves routng all the admtted traffc on path (though not all of the traffc s necessarly admtted Therefore, the effcency metrc for ths example s r ({l j }, x OE = = xoe whch goes to 0 as l (x OE x OE = /4, Example establshes that pure strategy OE wth the parallel-seral ln topology can be arbtrarly neffcent Ths result s at some level pathologcal, however The reason for the arbtrary neffcency of the pure strategy OE s the unreasonably hgh prces charged by the servce provders along path It s a best response (even wealy undomnated strategy for them to do so, because other provders also charge unreasonably hgh prces, so there s no transmsson on ths path and they suffer no adverse consequences from chargng unreasonable prces We may expect ths pathologcal stuaton not to arse n practce for a number of reasons Frst, frms may not coordnate on such an equlbrum (especally when other equlbra exst In ths case, for example, we may expect them to realze that f they all reduced ther prces, they would all mae hgher profts and would stll be playng equlbrum actons Second and relatedly, we may expect provders on a path to form a coalton and coordnate ther prcng decsons (at least wthn some bounds The coalton-proof Nash equlbrum concept of Bernhem, Peleg and Whnston [4 naturally leads to such an outcome and can be shown to rule out the stuaton n Example The applcaton of ths concept to the current setup s further dscussed n Secton V Fnally, an applcaton of the stronger concept of equlbrum ntroduced by Harsany, the strct equlbrum (see [8, or [3, pp -, wll be shown to rule out the pattern n Example In the current paper, we pursue ths last approach and strengthen the concept of equlbrum to strct olgopoly equlbrum B Strct OE and Prce Characterzaton Harsany s concept of strct equlbrum requres each player s best response to be unque Recall that the standard Nash equlbrum and our OE concept only requre each player, n partcular each servce provder, to play a wea best response We now strengthen ths condton Defnton 4 A vector (, x OE 0 s a strct OE (Olgopoly Equlbrum f x OE W ( j, j and for all I, j N, and for all p j 0, p j j, x W (p j, j, Π j ( j, j, x OE > Π j (p j, j, x (8 We refer to as the strct OE prce In the remander of ths paper, we focus on strct OE and we use the notaton OE({lj } to denote the set of flow allocatons x OE = [x OE I at a strct OE for a networ wth latency functons {l j } (j N, I The dfference between Defntons 3 and 4 s obvous The latter requres servce provders to play a strct best response, whle the former does not Notce that n both equlbra, we have not changed the behavor of the users gven by the WE (as n Defnton Notce also that we have removed the qualfer pure strategy, snce as s well nown, strct equlbra always have to be pure strategy (because mxed strategy equlbra, by defnton, nvolve players beng ndfferent among the strateges over whch they are mxng Therefore, there are stuatons n whch a mxed strategy OE exsts, but strct OE does not Moreover, t can be verfed that there are also stuatons n whch a pure strategy OE exsts, but a strct OE does not We do not vew ths as a serous shortcomng, however, snce, as Example above showed, even pure strategy OE do not always exst Moreover, Proposton 5 below establshes that when latency functons are lnear, a unque strong OE (whch s a stronger verson of strct OE exsts We next provde an explct characterzaton of the strct OE prces, whch wll be essental for the subsequent effcency analyss The followng lemma establshes that all path flows are postve at a strct OE Lemma Let (, x OE be a strct OE Let Assumpton hold Then j x OE > 0 for all I and j N Proof: Assume to arrve at a contradcton that j x OE = 0 for some I and j N Then, at any prce p j wth p j > j, we have Π j ( p j, j, x = Π j ( j, j, x OE, contradctng the defnton of the strct OE (cf Defnton 8 QED As shown n Example, the result of the precedng lemma does not extend to non-strct OE prces, e, there may be OE n whch some of the provders mae zero proft whle others are mang postve profts We have shown n [7 that for parallel-ln topology, f at any OE one of the provders maes postve proft, all of the provders mae postve profts (see [7, Lemma 4 Example shows that ths result no longer holds for non-strct OE for the parallel-seral topology Lemma, on the other hand, ensures that t holds for strct OE and allows us to wrte the optmzaton problems for each provder n terms of equalty and nequalty constrants We can then use the frst order optmalty condtons to obtan an explct characterzaton of the strct OE prces (see Appendx A for the proof

6 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 6 Proposton 4 Let (, x OE be a strct OE Let Assumpton hold Then, for all I, j N, we have (a (b j = x OE mn j x OE l (x OE l (xoe, f l (xoe s = 0, { for some s, for all N s, [ n R l (x OE, [ x OE N l (xoe + otherwse s Ns l (xoe s }, In partcular, for two lns, when the mnmum effectve cost s less than R, for =,, j N, the strct OE prces are gven by j = x OE [ N l (x OE + N l (x OE The prce characterzaton n Proposton 4 s a generalzaton of the prce characterzaton n [7, and as n that paper, t wll be useful n provdng bounds on the neffcency of prce competton However, the next example shows that even wth strct OE, effcency losses can be arbtrarly large C Ineffcency of Strct OE Example 3 Consder a one path networ, whch has n lns wth dentcal latency functons l(x = x/n Let the total flow be d = and the reservaton utlty be R = For any n, the unque socal optmum for ths example s x S = /, wth a correspondng socal surplus S(x S = /4 Usng the prce characterzaton gven n Proposton 4 and the defnton of a WE, t follows that there exsts a unque strct OE, n whch all provders charge the prce = /(n +, and the equlbrum flow s x OE = /(n + The effcency metrc for ths example s therefore ( r ({l j }, x OE = whch goes to 0 as n n+ ( n+ = 4n (n +, Ths example establshes that even wth strct OE, whch rules out the pathologcal coordnaton falures dscussed above, effcency losses can be arbtrarly large The reason for ths s agan the double margnalzaton problem, whch ncreases the cost of transmsson so much that there s no transmsson n equlbrum along certan paths (eg, along the sngle path n the example as n It s also evdence that the structure depcted n Example 3 can be part of any (9 general networ topology, and thus establshes that strct OE can be arbtrarly neffcent n general networs Despte ths smplcty and potental generalty, the behavor n Example 3 s stll somewhat pathologcal, snce t reles on the double margnalzaton problem reducng the transmsson Ths may be thought to be unlely partcularly n networs where the reservaton utlty, R, of users s hgh enough Ths leads us to defne an even stronger noton of equlbrum, strong OE 4 Defnton 5 A vector (, x OE 0 s a strong OE (Olgopoly Equlbrum f t s a strct OE, and I xoe = d In ths case, we refer to as the strong OE prce and denote the set of strong OE flow allocatons n a networ wth latency functons by {l j } by OE d ({l j } The only dfference between Defnton 4 and Defnton 5 s that n the latter we requre all of the potental flow, d, to be transmtted Ths wll be the case when the reservaton utlty, R, of users s large enough The followng proposton establshes that ths s ndeed the case and n fact when R s large enough there exsts a unque strong OE The proof of the proposton s provded n Appendx C Proposton 5 Let Assumpton hold Assume further that the latency functons are lnear, e, l j (x = a j x for all j N, and all I Defne the set { } I 0 = I a j = 0 j N Let I 0 denote the cardnalty of set I 0 and assume that I 0 Then, there exsts some R, such that for all reservaton utltes R R, the prce competton game has a unque strong OE Note that the assumpton I 0 n the precedng proposton cannot be dspensed wth, e, wthout ths assumpton, we can have stuatons n whch there exsts a pure strategy OE, but no strct OE To see ths, consder a two-ln parallel networ where both latency functons are dentcally equal to zero [e, l (x = 0, for =, and for all x In ths case, there exsts a unque pure strategy OE, dentcal to the standard Bertrand equlbrum, where both servce provders charge p = p = 0 It can be verfed, however, that ths s not a strct best response for ether of them, thus a strct OE does not exst (there are no other pure strategy OE and mxed strategy OE cannot be strct D Effcency of Strong OE wth Two Paths We now characterze the effcency propertes of strong OE We start wth a two path networ, wth n lns on path =,, where each ln s owned by a dfferent provder Frst, consder the followng example, whch llustrates that even 4 Note that ths noton s unrelated to Aumann s noton of strong equlbrum, whch requres a Nash equlbrum to have the property that no coalton of players should be able to jontly devate, tang the actons of all other players as gven, and ncrease the payoffs to all the members of the coalton (see [5, [4 The noton of coalton-proof Nash equlbrum dscussed n Secton V s a weaer verson of Aumann s strong equlbrum

7 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 7 wth strong OE the effcency loss can be worse than that n parallel ln networs (whch was shown to be bounded below by 5/6 n [7 Example 4 Consder a two path networ, whch has n lns on path wth dentcally 0 latency functons and one ln on path wth latency functon l(x = x / Let the total flow be d = and the reservaton utlty be R = The unque socal optmum for ths example s x S = (, 0 Usng Proposton 4 and the defnton of a WE, OE flows x OE must satsfy [ l j (x OE + x OE l j(x OE + l j(x OE j N j N j N = [ l j (x OE + x OE l j(x OE + l j(x OE j N j N j N Substtutng for the latency functons and solvng the above together wth x OE + x OE = shows that unque strong OE nvolves ( x OE = n +, n, n + whch goes to (0, as n The socal surplus at the socal optmum s, whle the socal surplus at the strong OE goes to / as n We next present two lemmas, whch wll be useful n provdng a bound on the effcency metrc for strong OE Note that these lemmas are vald for all OE The frst lemma s straghtforward and allows us to assume wthout loss of generalty that R I = xs I = l (x S xs > 0 n the subsequent analyss Lemma Gven a set of latency functons {l j } j N, I, assume that ( l j (x S x S = R x S, I j N I for some socal optmum x S Then every x OE OE({lj } s a socal optmum, mplyng that r I ({l j }, x OE = The followng lemma provdes a relaton between the total flow admtted at an OE and at a socal optmum Lemma 3 For a set of latency functons {l j } (j N, I, let Assumpton hold Let (, x OE be an OE and x S be a socal optmum Then x OE x S I I Proof: Assume to arrve at a contradcton that I I xoe xs Ths mples that xoe > x S for some Hence, l j (x OE l j (x S, j N > We also have l j (x OE > l j (x S for some j N [Otherwse, we would have l j (x S = l j (xs = 0 for all j N, whch yelds a contradcton by the optmalty condtons (4 and the fact that I xs < d Usng the defnton of the WE and the optmalty condtons (4, we obtan R ( l j (x OE j R ( l j (x S x S l j(x S j N j N Combnng the precedng wth l j (x OE l j (x S N, wth strct nequalty for some j, and j x OE l j(x OE x S l j(x S, for all j [usng Proposton 4(a and the fact that xl (x s nondecreasng, cf Assumpton, we obtan a contradcton QED The next theorem provdes a tght lower bound on r ({l j }, x OE [cf (7 for a strong OE In the followng, we assume wthout loss of generalty that d = Theorem Consder a two path networ, wth n lns on path =,, where each ln s owned by a dfferent provder, and ln j N has a latency functon l j Suppose that Assumpton holds and the prce competton game has a strong OE Then r ({l j }, x OE, xoe OE d ({l j } (0 Moreover, the bound s tght, e, there exsts {l j } and x OE OE d ({l j } that attans the lower bound n (0 Proof: The proof follows a number of steps: Step : We are nterested n fndng a lower bound for the problem nf nf r ({l j }, x OE ( {l j } x OE OE d ({l j} Gven {l j }, let x OE OE({lj } and let x S be a socal optmum By Lemma 3 and the fact that x OE OE d ({l j } (e, t s a strong OE, we have x OE = x S = = Ths mples that there exsts some such that x OE < x S Snce the problem s symmetrc, we can restrct ourselves to {l j } for whch x OE < x S We clam that nf nf {l } L x OE OE d ({l } where problem (E s gven by r OE = mn l S,j,(l S,j 0 l,j,l,j 0 subject to ( y S,yOE 0 j N l S,j = ( R y OE l j N,j R y S ( j N ls,j r ({l }, x OE r OE, ( ( y OE l j N,j y S ( j N ls,j l,j S y S (l,j S, =,, j N, (3 ( ( ( +y S (l,j S = l,j S +y S (l,j S, j N j N j N (4

8 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 8 ( j N l S,j ( + y S (l,j S R, j N y S =, = l,j + l,j(y S y OE l S,j, j N, (5 l,j y OE l,j, =,, j N, (6 y OE =, = + Strct OE Constrants Problem (E can be vewed as a fnte dmensonal problem that captures the equlbrum and socal optmum characterstcs of the nfnte dmensonal problem gven n ( Ths mples that nstead of optmzng over the entre functon l j for some j N, I, we optmze over the possble values of l j ( and l j ( at the equlbrum and the socal optmum, whch we denote by l,j, l,j, ls,j, (ls,j The constrants of the problem guarantee that these values satsfy the necessary optmalty condtons for a socal optmum and a strct OE (whch are the same as the condtons for a strong OE In partcular, condtons (3 and (6 capture the convexty assumpton on l j ( by relatng the values l,j, l,j and ls,j, (ls,j [note that the assumpton l j (0 = 0 s essental here Condton (4 s the optmalty condton for the socal optmum Condton (5 uses the nondecreasng and the convexty assumpton on the latency functons; snce we are focusng on {l j ( } such that x OE x S, we must have l,j + l,j(y S y OE l S,j, for all j N Fnally, the last set of constrants are the necessary condtons for a pure strategy OE In partcular, for a two path networ, usng Proposton 4, the Strct OE Constrants are gven by n y OE = n y OE [ l,j + j N [ l,j + j N j N l,j j N l,j + j N l,j + j N l,j, [and therefore n and n are also decson varables n problem (E Note that gven any feasble soluton of problem (, we have a feasble soluton for problem (E wth the same objectve functon value Therefore, the optmum value of problem (E s ndeed a lower bound on the optmum value of problem ( Step : Consder the followng change of varables for problem (E l S = l,j, S l S = l,j S j N j N l = l,j, l = l,j, j N j N (l S = (l,j S, (l S = (l,j S, j N j N l = and rewrte problem (E as j N l,j, l = r OE = mnmze l S,(l S 0 l,l 0 subject to y S,yOE 0 j N l,j, R l y OE l y OE R l SyS ls ys l S y S (l S, =,, l S + y S (l S = l S + y S (l S, l S + y S (l S R, y S =, = l + l (y S y OE l S, l y OE l, =,, = y OE =, + Strct OE Constrants (E Note that ths problem has a very smlar structure to the fnte-dmensonal problem consdered n the proof of Theorem of [7 for parallel-ln networs Let ( l S, ( l S, l, l, ȳs, ȳoe denote the optmal soluton of problem (E We have shown n [7 that l S = 0 for =, Step 3: Usng l S = 0 for =,, and l = 0, l = 0, we see that r OE = mn l,l y OE, y OE 0 n, n subject to l y OE R l y OE l, l + n y OE l = n y OE l, n y OE l R = Next, usng the transformaton m n y OE to wrte: r OE = mn l,l y OE, y OE 0 m, m 0 subject to y OE l y OE R (7 = n y OE and m = l y OE l, l + m l = m l, m l R y OE =, = (8 though we also have to ensure that the soluton to ths program ensures that n and n are ntegers

9 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 9 Now t can verfed that ( l, l, ȳ OE, ȳ OE, m, m = ( R, R, 0,,, s an optmal soluton to the program (8, and moreover, t satsfes n, n, thus t s also a soluton to (7 The correspondng optmum value s r OE = / By (, ths mples that nf nf {l j } x OE OE({lj } r ({l j }, x OE Fnally, Example 4 shows that ths bound s tght, e, QED mn {l j} mn x OE OE({lj } r ({l j }, x OE = Ths theorem shows that there exsts a tght bound of / for strong OE under the assumpton of zero latency wthout any congeston [l (0 = 0 In contrast to the case n Example 3, strong OE ensures that all of the traffc s transmtted n equlbrum, whch s the ey to the exstence of a bound on the neffcency of equlbrum The bound wth strong OE s nonetheless worse than the effcency bound n the parallel-ln topology consdered n [7 Ths s agan because of the double margnalzaton problem: each provder along path has a greater ncentve to ncrease ts prce (relatve to the benchmar where all these lns are owned by the same provder, because t does not nternalze the reducton n the profts of the other ln owners along the same path Consequently, n Example 4, there are hgher prces along path, and ths nduces greater fracton of users to choose path, ncreasng neffcency To see the role of seral lns more clearly, consder a modfed verson of Example 4, where all n lns along path are owned by the same servce provder Ths would mae the example equvalent to a parallel-ln topology In ths case the unque strct OE flows are gven by x OE = /3 and x OE = /3, and ths example reaches the 5/6 bound of [7 rather than / bound of Example 4 E Effcency of Strong OE wth Multple Paths We next consder a more general networ consstng of I paths, wth n lns on path, where each ln s owned by a dfferent provder The followng example llustrates the effcency propertes of a strong OE n an I path networ Example 5 Consder an I path networ, whch has n lns on path wth dentcally 0 latency functons and one ln on each of the paths,, I wth the same latency functon l(x = x(i / Let the total flow be d = and the reservaton utlty be R = Clearly, the unque socal optmum for ths example s x S = [, 0,, 0 Usng Proposton 4 and the defnton of a WE, t can be seen that the flow allocaton at the unque strct (strong OE s [ x OE = /n + /n, (I ( + /n,, (I ( + /n Hence the effcency metrc for ths example s r I ({l j }, x OE = (, + /n whch goes to / as n The next theorem generalzes Theorem The proof s smlar to that of Theorem and s omtted Theorem Consder a general I path networ, wth n lns on path I, where each ln s owned by a dfferent provder, and ln j, j N, has a latency functon l j Suppose that Assumpton holds and the prce competton game has a strong OE Then r I ({l j }, x OE, xoe OE d ({l j } (9 Moreover, the bound s tght, e, there exsts {l j } and x OE OE d ({l j } that attans the lower bound n (9 An mportant mplcaton of ths theorem and of Example 5 s that the bound of / s tght even wth an arbtrarly large number of paths Naturally, such a tght bound could be obtaned trvally when all paths except two have arbtrarly hgh latences Nevertheless, Example 5 shows that ths bound s reached wth postve flows on all paths for arbtrarly large networs Ths result s nterestng n part because models where a large number of olgopolsts compete often converge to a compettve and effcent equlbrum, and yet ths example shows ths not to be the case n our model It s mportant to note, however, that such a convergence result would apply as n when a gven networ s replcated n tmes Instead, n examples where the metrc of effcency remans at /, the networ n queston s not a n-replcaton of another networ F Postve Latency at 0 Congeston Interestngly, the bound on the effcency loss of strong OE does not generalze once we relax the assumpton that l (0 = 0 Example 6 Consder a two path networ, whch has n lns on path wth dentcally 0 latency functons and one ln on path wth latency functon l(x = ɛx + b for some scalars ɛ > 0 and b > 0 Agan the unque socal optmum s x S = (, 0 The flows at the unque strct (strong OE are gven by ( ɛ + b x OE = ɛ(n +, ɛn b ɛ(n + Let ɛ = b/ n Then, as b and n, we have that x OE (0,, and the effcency metrc r ({l j }, x OE 0 Ths example shows that the effcency loss could be arbtrarly hgh even at a strong OE for a networ that nvolves parallel and seral lns f the assumpton l (0 = 0 s relaxed Ths establshes: Proposton 6 In the presence of postve latency at zero congeston, strong OE wth the parallel-seral topology can be arbtrarly neffcent

10 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX 0 It s useful to note that n the same example wth the parallel-ln topology (e, all n lns along path owned by the same provder, we would have { ( b+ɛ x OE = 3ɛ, ɛ b 3ɛ, f ɛ b, (, 0, otherwse Consequently, b and ɛ 0, we have that x OE (, 0, and r ({l j }, x OE Therefore, the hghly neffcent equlbrum s a result of the parallel-seral topology, not of the assumpton that there s postve latency at 0 congeston In fact, [7 shows that wth parallel topology, but postve latency at 0 congeston, there s agan a tght bound of on effcency, whch s qute close to, but slghtly lower than 5/6 V CONCLUSIONS In ths paper, we presented an analyss of prce competton n communcaton networs wth congeston The focus has been the effcency mplcatons of prce competton n networs wth the seral-parallel topology Our major result s that contrary to the case of parallelln topology studed n [7, the parallel-seral topology leads to sgnfcant effcency losses relatve to the socal optmal In partcular, OE can now be arbtrarly neffcent Ths s partly due to an extreme (pathologcal form of double margnalzaton, whereby all seral provders on a partcular path charge prohbtvely hgh prces expectng others on that path to do so as well We showed that the concept of strct OE, whch requres all servce provders to play strct best responses, removes ths pathologcal behavor, but the effcency loss of strct OE s also unbounded because of the related double margnalzaton problem In partcular, the total cost of transmsson on a path consstng of many seral provders can be prohbtvely hgh that most of the users do not transmt n equlbrum, even though transmsson of all the traffc s socally optmal Yet, when users value transmsson suffcently, we may expect them to transmt even wth hgh costs Motvated by ths, we defned a stronger noton of equlbrum, strong OE, whch s a strct OE wth all of the traffc transmtted n equlbrum For strong OE, we showed that as long as there s zero latency at zero congeston, there s a tght bound of / on the neffcency resultng from prce competton Once the zero latency at zero congeston assumpton s removed, however, there s no such tght bound even wth strong OE, and the equlbrum can once agan be arbtrarly neffcent The results n ths paper add to the growng lterature on the mpact of game-theoretc nteractons between servce provders and users n communcaton networs A number of questons n ths area requre further analyss: All the examples of arbtrarly hgh neffcency presented n ths paper are under extreme confguratons Therefore, we suspect that these worst-case results may not be nformatve about the extent of degradaton of performance n real-world networ structures An open area for further study s the quantfcaton of neffcency arsng from prce competton n average or typcal networs The methods used by Fredman n hs analyss of genercty of neffcency of selfsh routng are lely to be useful n ths context as well (see [ Our results suggest that competton between servce provders can have sgnfcant costs n more general topologes, as long as these nclude seral servce provders, causng the double margnalzaton problem In fact, Example 3, whch shows the possblty of arbtrarly large neffcences, can be part of any general networ topology Nevertheless, a characterzaton of the structure of networs that would lead to worst-case scenaros s an area for future research 3 The most mportant smplfyng feature of our analyss s the assumpton that users are homogeneous n the sense that the same reservaton utlty, R, apples to all users It s possble to conduct a smlar analyss wth elastc and heterogeneous users (or traffc as n [7, [8, assumng that servce provders are restrcted to charge unform prces to all users Perhaps a more attractve alternatve s to allow non-lnear prcng and prce dfferentaton, whereby servce provders may charge dfferent prces for dfferent qualtes of servce and dfferent delay guarantees (and let users sort between dfferent types of servces or contracts Ths s an mportant research area for understandng equlbra n communcaton networs, where users often have heterogeneous qualty of servce requrements 4 All of our effcency bounds concern pure strategy equlbrum Possble bounds for mxed strategy olgopoly equlbra s another open research queston 5 As dscussed above, another nterestng equlbrum noton to consder n models of competton n parallelseral or more general topologes would be the coaltonproof Nash equlbrum concept of Bernhem, Peleg and Whnston, [4 It can be shown that the pure strategy olgopoly equlbrum n Example where all seral servce provders charge prohbtvely hgh prces s not coalton proof In partcular, n that game, the coalton consstng of all the seral provders along path would have a self-enforcng devaton that would ncrease the payoff to each of them (ths would be smply to reduce ther prces smultaneously to allow postve transmsson through path The concept of coalton-proof Nash equlbrum may be attractve n the context of competton n general communcaton networs, snce we may expect self-enforcng agreements between provders that are both cooperatng and competng to emerge The problem wth coalton-proof Nash equlbra, however, s that such equlbra often fal to exst Despte ths potental shortcomng, an nvestgaton of the structure and effcency of coalton-proof Nash equlbra n communcaton networs wth general topologes would be an nterestng area for further study VI APPENDIX A: PROOF OF PROPOSITION 4 Snce (, x OE s a strct OE, t follows by Lemma that > 0 for all I and j N Usng the defnton j x OE

11 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX of a Wardrop equlbrum (cf Defnton, we have that for all I and j N, ( j, x OE s an optmal soluton of the problem p j x (0 subject to p j + N + max (p j,x 0 l (x = p j + N x s d s I + N s + l (x s, s, l (x R, ( Usng the frst order optmalty condtons of the precedng problem, we obtan j = x OE l (x OE θ, ( where θ = 0, f l (xoe s = 0 for some s, for all N s, s x OE ξ Ns l (xoe s, otherwse, and ξ 0 s the Lagrange multpler assocated wth constrant ( Snce θ{ 0, Eq ( yelds part (a of the proposton } If mn s I N s +l (x OE s = R, then, n vew of the symmetry of the optmzaton problems of each provder on a seral ln, t follows that = [ R l (x OE, N, I (3 n { Assume next that mn s I N s + l (x OE s < R Ths mples that the constrant n Eq ( s slac, and therefore ξ = 0 Combnng Eq ( wth the relaton n (3 yelds the desred result QED VII APPENDIX B: PROOF OF PROPOSITION 3 Let l j (x = a j x for some a j 0 Defne the set { } I 0 = I a j = 0, j N (or equvalently, I 0 s the set of I such that a j = 0 for all j N Let I 0 denote the cardnalty of set I 0 There are two cases to consder: Case : I 0 In ths case, t s evdent that a vector (, x OE wth j = 0 for all I 0, j N and x OE W ( s an OE Case : I 0 In ths case, for some j N, let B j ( be the set of j such that } j ( j, x OE arg max p j 0 x W (p j, j p j x (4 Let B( = [B j ( j By Berge s Theorem of the Maxmum (see [6, t follows that B( s an upper semcontnuous correspondence We next show that t s convexvalued Lemma 4 For all I, j N, and j B j ( j s a convex set 0, the set Proof: Let p j B j ( j and p j B j ( j Consder x W (p j, j and x W ( p j, j such that (p j, x and ( p j, x are optmal solutons of problem (4 If p j x = p j x = 0, then γp j + ( γ p j B j ( j for all γ [0,, establshng convexty Assume next that p j x = p j x > 0 Assume to arrve at a contradcton that p j > p j, (5 whch mples that x < x There are two cases to consder: p j + a x : Snce p j + + a x < p j a x < R, t follows by the defnton of a Wardrop equlbrum that s I x s = d Moreover, n vew of the relaton between the effectve costs, t can be seen that x s x s, for all s, whch combned wth x < x, mples that s I x s < s I x s, yeldng a contradcton p j + + N a x p j + + a x : If both effectve costs are equal to R, e, p j + + a x = p j + N + a x = R, then the optmzaton problem of provder j [cf Problem 0 can be shown to have a strctly concave objectve functon over polyhedral constrants, thus mplyng that p j = p j Assume next that p j + and p j + sets L = and L = { s I p j + N { s I p j + N + a x R, + a x < R Defne the + a x < + a x < N s N s Followng the lne of proof of Proposton 4 (see Appendx A, we can show that [ p j x a +, p j = x [ a + s s / L s s / L Ns a Ns a }, }

12 JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL XX, NO XX, MONTH 0XX Moreover, n vew of the relaton between the effectve costs, t can be seen that L L Snce x < x, the precedng mples that p j < p j, yeldng a contradcton [cf Eq (5 QED Next, n vew of the fact that B( s upper semcontnuous, convex-valued and maps a compact set nto a compact set, we can use Kautan s fxed pont theorem (eg, [6to assert the exstence of a such that B( = To complete the proof, t remans to show that there exsts x OE W ( such that (6 holds If I 0 =, we have by Proposton that W ( s a sngleton, and therefore (6 holds and (, W ( s an OE Assume fnally that I 0 =, and that wthout loss of generalty I 0 We show that for all x, x W (, we have x = x, for all Let EC(x, = mn I { l j (x + j j N If at least one of the followng nequaltes EC( x, < R, or EC( x, < R holds, then one can show that I = x = I = x = d Substtutng x = d I, x n problem (, we see that the objectve functon of problem ( s strctly convex n x = [x, thus showng that x = x If both EC( x, = R and EC( x, = R, then l j ( x = l j ( x, j N j N whch, by the assumpton that l j s strctly ncreasng for some j N, mples that x = x for all, establshng our clam For some x W (, consder the vector x OE = ( d x, x Snce x s unquely defned and x s chosen such that the provders on ln have no ncentve to devate, t follows that (, x OE s an OE QED VIII APPENDIX C: PROOF OF PROPOSITION 5 For brevty, we provde a setch of ths proof The proof reles on the prce characterzaton provded n Appendx A Consder the followng system of lnear equatons: x [ and a + = x h [ s N h a + Ns a s h Ns a } (6, h I, h, x = d (7 I It s straghtforward to see that, under the assumpton I 0, the precedng set of equatons has a unque soluton, whch we denote by x, that satsfes x > 0 for all Consder some j N, I For all l N h, h I wth l j, defne [ p l = x h a + N h s h Consder the optmzaton problem subject to p j + N p + Ns a max p jx (8 (p j,x 0 l (x = p + l (x s, s, N s x s d s I An argument analogous to that n the proof of Proposton 4 n Appendx A mmedately establshes that the vector ( p j, x such that [ p j = x a + s Ns a s the unque optmal soluton of problem (8 It also follows from Proposton 4 that for R suffcently large, ( p, x satsfes the necessary condtons for a strct OE and x = d It therefore follows that there exsts some R <, such that for all R R, ( p, x s a strong OE To see that ths s the unque strong OE for R R, note that any strong OE s also a strct OE, and Lemma establshes that (p, x wth p j x = 0 for some j and cannot be a strct OE Moreover, for R R problem (8 has a unque soluton, gven by the unque soluton to Equatons (6 and (7, whch s the unque canddate for a strong OE, thus completng the proof QED ACKNOWLEDGMENTS The authors would le to than Attla Ambrus, Muhamet Yldz and varous semnar partcpants for very useful comments and suggestons REFERENCES [ F Kelly, A Maulloo, and D Tan, Rate control for communcaton networs: shadow prces, proportonal farness, and stablty, Journal of the Operatonal Research Socety, vol 49, pp 37 5, 998 [ S Low and D Lapsley, Optmzaton flow control, I: Basc algorthm and convergence, IEEE/ACM Transactons on Networng, vol 7, no 6, pp , 999 [3 H Yache, R Mazumdar, and C Rosenberg, A game theoretc framewor for bandwdth allocaton and prcng n broadband networs, IEEE/ACM Transactons on Networng, vol 9, no 5, 000 [4 T Roughgarden and É Tardos, How bad s selfsh routng? Journal of the ACM, vol 49, no, pp 36 59, 00 [5 D Acemoglu and A Ozdaglar, Flow control, routng, and performance from servce provder vewpont, LIDS, Massachusetts Insttute of Technology, Cambrdge, MA, Tech Rep WP-696, 004 [6 X Huang, A Ozdaglar, and D Acemoglu, Effcency and Braess paradox under prcng n general networs, IEEE Journal on Selected Areas n Communcaton, specal ssue on Prce-Based Access Control and Economcs, vol 4, no 5, pp , 006 [7 D Acemoglu and A Ozdaglar, Competton and effcency n congested marets, Mathematcs of Operatons Research, 007