Competition yields efficiency in load balancing games

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1 Competton yelds effcency n load balancng games Jonatha Anselm 1, Urtz Ayesta 1,2, Adam Werman 3 1 BCAM Basque Center for Appled Mathematcs, Dero, 48170, Basque Country, Span 2 IKERBASQUE, Basque Foundaton for Scence, Blbao, 48170, Basque Country, Span 3 Computer and Mathematcal Scences, Calforna Insttute of Technology, Pasadena, CA 91125, USA ABSTRACT We study a nonatomc congeston game wth N parallel lnks, wth each lnk under the control of a proft maxmzng provder. Wthn ths load balancng game, each provder has the freedom to set a prce, or toll, for access to the lnk and seeks to maxmze ts own proft. Gven prces, a Wardrop equlbra among users s assumed, under whch users all choose paths of mnmal cost, whch s defned as latency plus prce. Wthn ths model, we have olgopolstc prce competton whch n equlbrum gves rse to stuatons where nether provders nor users have ncentves to adjust ther prces or routes, respectvely. In ths context, we provde new results about exstence and effcency of olgopolstc equlbra. Our man theorem shows that, when the number of provders s small, olgopolstc equlbra can be extremely neffcent; however as the number of provders N grows, the olgopolstc equlbra become ncreasngly effcent (at a rate of 1/N) and, as N, the olgopolstc equlbrum matches the socally optmal allocaton. 1. INTRODUCTION As computer scentsts have become ncreasngly nterested n economcs and game theory, one of the topcs that has spurred sgnfcant research s that of congeston games (a.k.a. routng games). Applcatons of congeston games are numerous, ncludng settngs such as communcaton networks and transportaton networks, and resultantly a large lterature has grown studyng the exstence and effcency of equlbra n a varety of congeston games. See [22] for a recent survey. In the classc formulaton of a congeston game there are noncooperatve agents sendng traffc through a network and the agent strateges consst of possble routes through the network. In the nonatomc verson of these games, there are a contnuous number of players, each of whch has a neglgble effect on the others. The utlty of routes to agents corresponds to the latency or congeston experenced on the path. Most typcally, the Wardrop equlbra, under whch every agent sends traffc along ts smallest latency path(s), s consdered as the soluton concept. There s a large lterature studyng nonatomc congeston games and, at ths pont propertes related to the exstence and neffcency (a.k.a. prce of anarchy) of the Wardrop equlbra are well understood. Interestngly, n many cases, there s lttle effcency loss between the socal optmal, whch mnmzes the total latency across all flows, and the Wardrop equlbrum. For example, for the case of affne latency functons the prce of anarchy s bounded by 4/3 [25],.e., the equlbra routes have total latency bounded by 4/3 that of the optmal latency. However, under more realstc latency functons there can be sgnfcant effcency loss. For example, f one consders latency functons defned by queueng models (as s approprate for the Internet or transportaton applcatons) the prce of anarchy becomes unbounded: In the case of a load balancng game wth N parallel M/M/1 queues the prce of anarchy s N [13] and when more general queueng models are consdered the prce of anarchy can be unbounded even n the case of two parallel queues [4]. Snce the prce of anarchy can be large n many practcal settngs, sgnfcant research has gone nto understandng how to reduce the prce of anarchy. Classc results n ths drecton nclude () usng a Pgouvan tax (toll) [22], () ncreasng the capacty of each lnk n the network [25], and () controllng centrally a small fracton of the traffc [20, 24]. However, each of these approaches has sgnfcant practcal lmtatons for most settngs. More specfcally, optons () and() requre centralzed control that smply s not possble n nternet or transportaton settngs, whle opton () requres sgnfcant expense for overprovsonng capacty. In ths paper, our goal s to llustrate that a dfferent approach can elmnate neffcency n routng games wthout mposng centralzed authorty. Broadly, our results hghlght that competton among proft-maxmzng provders (correspondng to the lnks n the network) can nduce effcent routng. More specfcally, the model we study n ths paper conssts of a nonatomc congeston game wth N parallel lnks, wth each lnk under the control of a proft maxmzng provder. Wthn ths load balancng game, each provder has the freedom to set a prce, or toll, for access to the lnk and seeks to maxmze ts own proft (.e., the product of prce and traffc per tme). Gven prces, a Wardrop equlbra among users s assumed, under whch users all choose paths of mnmal cost, whch s defned as latency plus prce. See Secton 2 for a formal descrpton of the model. Wthn ths model, we have olgopolstc prce competton among the provders and we seek to understand the exstence and effcency of olgopolstc equlbra, whch are

2 stuatons where nether provders nor users have ncentves to adjust ther prces or routes, respectvely. Notce that the model we consder s perhaps more representatve of many settngs than classc congeston games. In partcular, t provdes a smple framework n whch to study the mpact of prce and competton among provders, and as such has applcaton to () competton among cloud servce provders[22] (Chapter 22), () transportaton networks where toll roads are not centrally controlled [12], () competton among ISPs n communcaton networks [29], and (v) freght transportaton [10]. More generally, the model captures the mpact of competton and prce n congested markets, see [2] for a dscusson of such markets. In fact, the model we consder (or slght varants of t) has been studed n a number of recent papers [3, 14, 12]. Interestngly, the model has proven to be hghly non-trval analytcally, and lttle s known about the exstence and unqueness of olgopolstc equlbra [12, 2]. Ths s because the provder optmzaton problems are not convex and, thus, classcal exstence proofs do not apply. Despte lmted understandng of exstence, some progress has been made n the analyss of ths model. In partcular, [2] has studed the model from the perspectve of the socal surplus (or dfference between the users wllngness to pay and delay costs), whch s dfferent from the prce of anarchy, and shown that the rato of the optmal attaned socal surplus to the one attaned at the olgopolstc equlbrum (f t exsts) s bounded from below by 5/6. In parallel-seral networks, ths rato s lower bounded by 1/2 f the latency functons are zero when evaluated at zero, otherwse t can be arbtrarly large [3]. If provders can compete over both prces and capactes, ths rato can be agan arbtrarly large [1]. Inthspaperwe maketwoman contrbutonstothestudy of olgopolstc equlbra n congeston games. Frst, we provde new results about exstence and nonexstence of olgopolstc equlbra (Secton 3). It turns out that understandng the exstence of olgopolstc equlbra s non-trval because of the non-convex nature of the game, and pror to the current work the exstence of olgopolstc equlbra has only been shown n the case of lnear latency functons [2], whch allows the use of Kakutan s theorem. In ths work, we also showexstence n the case of homogeneous latency functons and, more nterestngly, we also show nonexstence n the case where the system s under-provsoned n the sense that the capacty of the N 1 slowest servers s not enough to handle the full traffc (see Proposton 3). Second, we provde a new understandng of the effcency of olgopolstc equlbra (Secton 4). Specfcally, we show that f there are only two provders the olgopolstc equlbra can be arbtrarly worse than the optmal routng n terms of total latency (.e., the prce of anarchy can be unbounded). However, as the number of provders N grows, the prce of anarchy drops at a rate of 1/N and, as N, the prce of anarchy converges to 1. Note that as we scale N we allow the ncomng traffc to scale proportonally wth N, thus mantanng the same load for the system whle N grows. Ths hghlghts that the mprovement n the prce of anarchys notdue toashrnkngof thecongeston n the system (as would happen f the traffc grow sublnearly wth N), but s n fact due to the ncreasng competton among the provders. So, broadly speakng, ths result can be seen as provdng an example of the maxm that competton yelds effcency. For both our exstence and effcency results, we use queuengbased latency functons as llustratve examples. These examples are hghly relevant for communcaton and transportaton applcatons. Further, they hghlght the power of competton among provders because the neffcency that s evdent for these latency functons n classc congeston games dsappears wth even a small amount of competton among provders. However, the queueng examples also hghlght the complexty of understandng exstence of olgopolstc equlbra. The paper s organzed as follows. In Secton 2, we descrbe the prce competton game under nvestgaton ntroducng the socal optmum, Wardrop and olgopolstc equlbra. Secton 3 presents our results on the exstence of olgopolstc equlbra and Secton 4 analyzes ther effcency from a performance standpont. Fnally, Secton 5 draws the conclusons of our work and outlnes future research. 2. MODEL AND NOTATION As mentoned n the ntroducton, the model consdered n ths paper falls nto the category of olgopolstc prcng games [22, 2]. More specfcally, we consder a network wth N > 1 provders workng n parallel. Each provder s assumed to own a sngle network resource and there s an nfnte stream of nfntely-many arrvng users that follows some stochastc process wth ntensty λ. Upon jonng the network each user selects exactly one provder from whch to receve servce. Each user, thus, carres an nfntesmally small amount of traffc. Let x denote the mean amount of traffc per unt of tme to provder n some statonary regme, whch we defne n the followng, and denote the traffc vector by x = (x 1,...,x N), wth N =1x = λ. In the remander of the paper, ndces and j mplctly range from 1 to N f not otherwse specfed. Wheneverauserselectsprovder, tmustpayanamountp 0 to. Let p = (p 1,...,p N) denote the vector of prces and defne p j = (p ) j. We measure the proft per tme unt of provder by p x. Our focus throughout s on the mean tme t takes for a user to be served at. We denote latency of (a.k.a., the response tme, sojourn tme, or flow tme of ) by l (x ). The form of the latency functons can dffer greatly dependng on the applcaton consdered. However, n transportaton and communcaton networks, latency functons comng from queueng models are most typcally used for modelng purposes. The form of the resultng functon stll depends on several factors, such as the schedulng dscplne mplemented by each provder and the detals of the arrval process. One common model, whch we use for llustratve purposes throughout the paper, s the case of () a Posson arrval process, () Processor-Sharng schedulng 1 [19], () 1 Under Processor-Sharng schedulng a resource s shared evenly among all users present,.e., f there are m users present they each receve 1/mth of the resource.

3 servce tmes are..d., (v) there s no lmt to the number of users that each provder can handle smultaneously, and (v) the selecton of a provder follows an..d. probablty law (as n, e.g., [13]). In ths settng, for all, { (µ x ) 1 f 0 x < µ l (x ) = (1) + otherwse Notethattheparameterµ snterpretedasthemeanservce rate ofprovder. Thelatencyfunctonn(1)s verypopular n communcaton network performance modelng and corresponds to the mean response tme of an M/GI/1/Processor- Sharng queue and an M/M/1/Frst-come-frst-served queue [8]. Though we use the latency functon n (1) for llustratve purposes, the man results of the paper hold more generally. Specfcally, n the remander of the paper the followng assumptons wll be always used for the latency functons. Assumpton 1. The functons l (x ), = 1,...,N, are contnuously dfferentable over [0,µ ), µ, ncreasng, and such that x l (x ) s strctly convex. Assumpton 2. If there exsts a vector of postve real numbers [µ 1,...,µ N] such that lm x µ l (x ) = + for all, then λ < µ. Clearly the latency functon n (1) satsfes these assumptons, as do latency functons based on many other queueng models. Further, note that Assumpton 1 does not necessarly requre the convexty of l, whch s typcally assumed (e.g., n [2, 12]). Addtonally, Assumpton 2 guarantees the stablty of the system n x Opt. Gven the latency functons for each, we defne the latency of the network under the traffc allocaton x as l(x) = x λ l(x). Note that the network latency can be nterpreted as a drect measure of the overall qualty of servce at allocaton x. Gven the setup descrbed to ths pont, we can now defne the optmal traffc allocaton, whch serves as a benchmark for the performance of the olgopolstc equlbra we defne later. Defnton 1. A traffc allocaton x Opt s sad socally optmal, or a socal optmum, f t mnmzes the latency. That s, f x Opt = argmn x: x =λ,x 0, l(x). (2) In the remander of ths secton we ntroduce the Wardrop and olgopolstc equlbra n the context of the model descrbed above. 2.1 Wardrop Equlbrum Snce we assume that the ncomng traffc s the aggregate flow of nfntely many selfsh users that carry an nfntesmal amount of traffc, for a gven prce vector p the statonary regme s a Wardrop equlbrum (WE). Ths s characterzed by Wardrop s frst and second prncples [27] (see also [11]): The dstrbuton of traffc among the provders s such that the sum of the response tme and the prce of each provder,.e., the effectve cost ncurred by each user, s mnmum and equal at each provder. Wardrop s prncples are used extensvely n modelng the traffc dstrbuton of communcaton networks and transportaton networks and has receved sgnfcant attenton n the algorthmc game theory communty, e.g., see the recent survey n [22]. In our context, a Wardrop equlbrum s defned as follows. Defnton 2. For a gven prce vector p, a vector x WE R N s a Wardrop equlbrum f { l (x WE )+p = mn j lj(x WE j xwe = λ 0,. x WE )+p j }, : x WE > 0 Throughout, we denote by W(p) the Wardrop equlbrum acheved wth prce vector p. Note that our settng s essentally equvalent to nonatomc load balancng games, whch have been studed n a number of recent papers [22]. In our context, t s mmedate to conclude the exstence and unqueness of a Wardrop equlbrum, see [6]. Proposton 1. For a fxed prce vector p, there exsts exactly one Wardrop equlbrum. Though the proposton s well-known wthn our assumptons, t s useful to consder the proof. In partcular, the set of Wardrop equlbra under p s gven by the set of mnmzers of x l (z)dz +p x (4) 0 subject to x = λ, x 0,. Snce (4) s a strctly convex functon (l s ncreasng by Assumpton 1) defned over lnear constrants, a unque Wardrop equlbra exsts. 2.2 Olgopolstc Equlbrum The key nteracton that we wsh to model s that of the competton among proft maxmzng provders. Gven a prce vector p, a provder may wsh change ts prce to ncrease ts ndvdual proft n the Wardrop equlbrum that wll result. Thus, proft maxmzng provders compete wth each other playng the so-called prce competton game, and an olgopolstc equlbrum among the provders can arse. An olgopolstc equlbrum represents the statonary stuaton where each provder has no ncentves n unlaterally changng ts prce because otherwse t would decrease ts proft. Specfcally, we defne an olgopolstc equlbrum n our settng as follows. Defnton 3. A vector s an olgopolstc equlbrum f for all (3) [W( )] = max p 0 p[w(p,poe )] (5) Provded that t exsts, an olgopolstc equlbrum provdes a predcton of the pont where the strategc actons of both users and provders should converge. Throughout, we use to be an olgopolstc equlbrum and = W( ) to be the correspondng traffc allocaton.

4 The queston of exstence of an olgopolstc equlbrum s not as smple as that of exstence of a Wardrop equlbrum. However, the followng proposton provdes necessary condtons for a prce vector to be an olgopolstc equlbrum. Further ssues related to exstence are our focus n Secton 3. Proposton 2. Let be an olgopolstc equlbrum and = W( ). The followng condtons must hold = l (x ) x x = + : A = 0, = 0, : A > 0 l ( )+ A x = 0,A 0, xoe = B +A, j : A j =0 ( l j (x j ) x j xj = j ) 1 = λ (6) for some B, A,, f at least two provders are used. Otherwse, = (mn k l (0) l k (λ))e k, where k = argmn jl j(λ) and e k s the unt vector n drecton k. Ths proposton wll play a key role n the remander of the paper and the proof s provded n the appendx. Note that t follows from (6) that the varable A s zero f and only f provder s used n an olgopolstc equlbrum. Further, B s nterpreted as the effectve cost (delay plus prce) ncurred by users. 3. EQUILIBRIA EXISTENCE = λ/n and the equlbrum prces of all provders must be equal,.e., =. Ths allows us to use the system (6) to obtan The prmary questons to address about olgopolstc equlbra are those of exstence and unqueness. As mentoned above, understandng whether olgopolstc equlbra exst n our settng s non-trval because of the non-convex structure of the equlbrum pont (5). Further, t s easy to see that there are many natural cases where olgopolstc equlbra do not exst, whch means that one cannot hope for as strong an exstence result as holds for Wardrop equlbra. The pror lterature has begun to study the queston of exstence; however exstence has only been proven formally n the smple case of lnear functons of the type l (x ) = a x ([2], Proposton 7), whch allows the use of Kakutan s theorem. In ths secton we provde two other small, but mportant cases where somethng can be proven about the exstence/nonexstence of olgopolstc equlbra. Frst, we focus on exstence. It s straghtforward to see that f all provders are homogeneous,.e., l ( ) = l( ),, then a unque olgopolstc equlbrum always exsts. Further, we can provde an exact expresson for the equlbra prces and profts n ths case. To see ths, note that by symmetryx OE = λ l(x ) x =λ/n (7) N 1 x From the above, the equlbrum proft follows easly, and can be seen to be ncreasng wth λ. Profts Iteratons (n) Fgure 1: Provders profts startng from ntal prce vectors p(0) = (1,1) (the two lnes on the bottom) and p(0) = (100,100) (the two lnes at the top). The contnuous (dashed) lnes refer to the proft of provder 1 (2). Second, let us consder nonexstence. One common property of latency functons used n communcaton and transportaton networks s that provders have some capacty on the traffc that can be handled above whch the latency becomes nfnte. For example, n the case of the queuengbased latency functons n (1), when x µ. It turns out that ths capacty constrant on the provders can lead to settngs where olgopolstc equlbra do not exst. Proposton 3. Assume that lm x µ l (x ) = +, for all > 1, and that µ 1 > µ, for all > 1. If λ >1 µ, then there exsts no olgopolstc equlbrum. Proof. Wthn the Wardrop equlbrum condtons (3), provder argmn j j µ can put any arbtrary (but fnte) prce because t wll get at least λ >1 µ + ǫ, ǫ > 0, amount of the traffc. Therefore, for any choce of a fnte prce by that provder, t has the ncentve to devate ts strategy, whch mples that there can be no olgopolstc equlbrum. An nterpretaton of Proposton 3 s that, f the system s under-provsoned, then t s possble for the provder wth the hghest capacty to explot ths fact and obtan arbtrarly large profts. As mentoned, ths fact s partcularly relevant n the case of queueng-based latency functons. On the other hand, let us consder the smple case of N = 2 and wth latency functon (1) and λ < µ 2. Thus, Proposton 3 does not apply. In ths scenaro, for any prce set by provder, provder j cannot choose a prce unboundedly large because provder could handle all the traffc wth an effectve cost less than the prce of j, whch would cause the proft of j to become zero. Therefore, the prces must be bounded n ths case. Note, however, that ths argument s not suffcent to establsh the exstence of an olgopolstc equlbrum, whch s challengng even n ths smple scenaro. Case study: Processor sharng queues Snce the known results about the exstence of olgopolstc equlbra are lmted, n ths secton we consder the specfc

5 λ p 1 ( )[W(p( ))] 1 p 2 ( )[W(p( ))] Table 1: Provders profts by varyngλ wth µ 1 = 6,µ 2 = 4. example of latency functon(1) and use numerc experments to llustrate the exstence of olgopolstc equlbrum n nonhomogeneous scenaros. In order to fnd olgopolstc equlbra, we apply best response(br) dynamcs, whch are known to converge to Nash equlbra n many settngs, see [23] for example. Under best response dynamcs the system starts wth some (nonequlbrum) prce vector p(0). At tme n 0, provder 1 observes the resultng Wardrop equlbrum W(p(n)) and chooses the prce p 1(n+1) that maxmzes ts proft condtoned on the prces of the others provders,.e., p 1(n+1) = argmax p 0 p[w(p,p 1(n))] 1 (8) In turn, provder 2 then observes the Wardrop equlbrum resultng from [p 1(n+1),p 1(n)] and chooses ts best prce p 2(n + 1). In accordance wth (5), the sequence of prces p (n) represent the best response of. In Fgure 1, we llustrate the applcaton of best response dynamcs n the case of N = 2 and latency functons (1). Specfcally, Fgure 1 shows the sequence of profts [p (n) [W(p(n))] ] when µ 1 = 6, µ 2 = 4, λ = 3 and dfferent ntal vectors p(0). Notce that the ntal prce vectors seem to not affect the asymptotc profts. In fact, ths s no accdent as we can prove that n ths settng an olgopolstc equlbrum s unque f t exsts. Specfcally, the (necessary) condtons (6) allows us to mmedately derve condtons for the number of possble olgopolstc equlbra, and we have the followng proposton. Proposton 4. Assume N = 2, latency functons (1) wth µ 1 > µ 2, and λ < µ 2. The number of olgopolstc equlbra s less than or equal to the number of roots n the nterval (λ µ 2,µ 1) of the cubc (µ 1 λ+x 1)(µ 2 λ+x 1) 2 = (µ 2 x 1)(µ 1 x 1) 2. (9) From ths proposton, we can see that f µ 1 s suffcently large, t s possble to show that the dscrmnant of the above cubc s negatve (the domnant term s O(µ 6 1)). Therefore, only one real root exsts, meanng that at most one olgopolstc equlbrum can arse n the prce competton game. The results of other experments wth best response dynamcs are summarzed n Table 1, whch shows the equlbrum prces of all servers as λ s vared. Agan, these prces do not seem to be dependent of the ntal condtons. As one mght expect, the profts ncreases wth the load. Further, when λ 4, ths system becomes under-provsoned and Proposton 3 apples. 4. EQUILIBRIA EFFICIENCY We now move from the exstence of olgopolstc equlbra to the effcency of olgopolstc equlbra. Before movng to the results t s worth placng them n the context of results about the effcency of Wardrop equlbra for nonatomc load balancng games. In partcular, there are some settngs where Wardrop equlbra are qute effcent, forexamplenthecase ofaffnel wheretheprceof anarchy s 4/3 ndependent of N. However, more commonly the prce of anarchy can be large. For example, when polynomal l are consdered the prce of anarchy approxmately grows wth the degree of the polynomal [25]. Further, when queueng latency functons are consdered the prce of anarchy can be unbounded. In the case of M/GI/1/Processor- Sharng queues the prce of anarchy s N [13] and f other schedulng polces such as Shortest-Remanng-Processng- Tme frst are consdered the prce of anarchy can be unbounded already n the case of N = 2 [9]. One motvaton for studyng olgopolstc prcng s to understand how competton among proft maxmzng provders affects ths neffcency. Thus, we focus agan on the prce of anarchy (PoA), defned formally as follows n our settng: OE ) PoA(N;l 1,...,l N) def l(x = sup l(x Opt ) 1, (10) where s an olgopolstc equlbrum. Ths s a natural measure of effcency to consder for applcatons n communcaton and transportaton networks where latency provdes a measure of the user experence (or qualty of servce). For example, n the context of a cloud-computng nfrastructure-as-a-servce provder, user experence s of key mportance and any ncreased latency compared to the optmal allocaton ndcates suboptmal resource allocaton. Though we focus on the prce of anarchy as defned above, other complementary effcency measures have been looked at prevously n the lterature. For example, the olgopolstc prcng game under nvestgaton has been studed n[2] wth a focus on an effcency measure that s defned as the rato between the socal surplus at the worst olgopolstc equlbrum and at the optmal allocaton. Under ths measure, t s shown that the effcency s remarkably hgh, bounded from below by 5/6. 2 Thus, the olgopolstc equlbrum s qute effcent under that measure. However, those results do not mply any bound on the prce of anarchy defned above. The man concluson of the results n ths secton s that, though the prce of anarchy can be arbtrarly large f N s small (and so there s lttle competton), t converges to 1 wth rate 1/N as the number of provders N ncreases to nfnty. Ths holds even under queueng latency functons, where the contrast s stark. If N s large the Wardrop equlbrum can be extremely neffcent, whle the olgopolstc equlbrum s extremely effcent. Ths hghlghts the beneft of ncreasng competton among provders to the underlyng archtecture that hosts the provders. In the context 2 We observe that our noton of socally optmal allocaton slghtly dffers from the one n [2], where the authors use the addtonal parameter R called reservaton utlty. However, f R s suffcently large, then both notons of socal optmum are equvalent.

6 of cloud-computng, an nfrastructure provder such as [15, 16, 17] should therefore encourage competton n order to optmally explot ts avalable resources. 4.1 Ineffcency under lmted competton The fact that Wardrop equlbra can be extremely neffcent already provdes ntuton for the fact that the olgopolstc equlbra can be neffcent for a small number of provders. However, the stuaton s even worse than one mght expect. We show n the followng that the prce of anarchy of olgopolstc equlbra s unbounded even n the case when affne latency functons are consdered, whch we noted earler have a 4/3 prce of anarchy n the case of the Wardrop equlbrum. Thus, lmted competton can nduce neffcency even n stuatons that are, n some sense, nherently effcent. Theorem 1. sup N;l1,...,l N PoA(N;l 1,...,l N) =. Proof. To prove the result t suffces to gve a famly of examples for whch the prce of anarchy are unbounded. For ths consder N = 2 and l 1(x 1) = a 1x 1, l 2(x 2) = x 2. We frst calculate the optmal allocaton. Solvng we obtan x Opt = 1 λ argmna1x2 1 +x 2 2 s.t. : x 1 +x 2 = λ,x 1,x 2 0, x Opt 1 = λ a (11) We next calculate the olgopolstc equlbrum. For the assumed latency functons, n an olgopolstc equlbrum we have (see [2], equaton (20)) ( l1(x 1) l (x )+x + l2(x2) ) = B, x 1 x 2 (12) x = λ, x 0. whch gves.e., { 2a1x 1 +x 1 = 2x 2 +b 2 +a 1x 2 x = λ, x 0. (13) 1 = λ a a (14) To show that the prce of anarchy can be unbounded, we need to bound the followng quantty l( ) l(x Opt ) = a1 (λ(a 1 +2)) 2 +(λ λ(a 1+2) 9(a 1 +1) 2 3(a 1 +1) )2 +(λ λ(a 1+2) ) 3(a 1 +1) λ a 2 1 (a 1 +(λ λ +1) 2 a 1 +1 )2 +(λ λ ). a 1 +1 Notce that f a 1 s large n the above, then (15) l( ) l(x Opt ) a1 λ 2 +(λ λ 9 3 )2 a 1/9, (16) λ 2 whch mples that the prce of anarchy can be made arbtrarly large as desred. 4.2 Effcency under ncreased competton Havng shown that neffcency may be present when the number of provders s small, we now turn to the case where there s sgnfcant competton,.e., the case of large N. Our goal n ths secton s to hghlght that as N the ncreased competton among provders yelds effcent traffc allocatons. However, to show such a result we need to frst defne a scalng of our system that allows us to take N. Thescalngwe defnenordertoconsder large systemss the followng. We consder there are C 1 provder types or classes, where all provders belongng to a gven class have the same latency functon. In other words, provders of the same class are statstcally equvalent. We then defne β c as theproporton ofclass-c provders, and consder almt where N grows but β = (β 1,...,β C) remans fxed. Ths does not completely defne the scalng however, t s addtonally mportant that the arrval rate grows as the sze of the system N grows, otherwse the system becomes trval. In partcular, f the arrval rate s not scaled, then as N grows the congeston of the system dsappears. Thus, we seek to scale the arrval rate so that the congeston of the system remans constant through the scalng. To acheve ths we choose the arrval rate when there are N servers as λn. Ths scalng s natural because t keeps the load,.e., rato of the arrval rate to the total servce capacty, of the system the same as N s scaled. Addtonally, ths scalng models the fact that the ncreasng number of provders s lkely a response to a growng market,.e., arrval rate. In the queueng theory lterature, ths scalng of the arrval rate wth the system capacty s qute popular, e.g., [28, 18]. Other scalngs can be also consdered [21] and generalzng the results n ths secton to such scalngs s an nterestng topc for future work. However, wth respect to heavy-traffc scalng where the arrval rate approaches the overall network capacty, Proposton 3 already says that an olgopolstc equlbrum cannot exst. In ths context of the scalng descrbed above, we can now state our man result. Theorem 2. As N, there exsts at most one olgopolstc equlbrum and, for any C and β, lm PoA(N;l1,...,lN) = 1 (17) N,β Proof. Let c range from 1 to C, the number of classes, and let c be the olgopolstc-equlbrum traffc that a provder of class c gets,.e., c = for each provder of class c. Let also S N be the set of all olgopolstcequlbrum traffc allocatons assumng that the total number of provders s N. It s clear that S N T N = {x : def x s a soluton of (6)}. In the followng, we prove the theorem by showng that T N 1 as N (keepng fxed β) and that the lmtng allocaton s x Opt. We have that, as N : () wthn the constrants n (6), l (x ) x x = s strctly postve by Assumpton 1 and t can be bounded from above by means of Assumpton 2 (n the condtons of Assumpton 2, for any fxed arrval and servce rates, one has l (x ) x enough), x = < l (x ) x x =µ ǫ, for ǫ small () provders of the same class must have the same equlbrum prce and traffc allocaton by symmetry, and

7 () at least one class of provders s used. Thus, for all c we have ( c ) 1 l = (x ) j:a =0 x x = ( c C l (N 1 {c c =c} )β c (x c ) c x =1: c xc = c A c =0 ) 1 0, (18) where c s the traffc to one provder of class c. Therefore, n the lmt, c = c l (x c) x c c = 0, : A c > 0 xc=, c : A c = 0 c l c( c )+ c = B OE +A c, c A cx c = 0,A c 0, c c βcxoe c = λ, (19) In the same lmtng regme, the optmalty condtons of (2) become l l c(x c)+x c(x c) c x c = λ(w +Y c), c Y cx c = 0,Y c 0, c (20) c βcxc = λ,xc 0, c, where W and Y c are Lagrange multplers (Y c = 0 f and only f c s used), and x c s the traffc to one provder of class c. Gven that the optmzaton problem (2) s strctly convex (by Assumpton 1), the KKT condtons (20) are both necessary and suffcent for the exstence of a unque soluton. Ths means that the condtons (19) must also have a unque soluton concdng wth the soluton of (20). In fact, a soluton for (19) exsts f and only f A c = λy c and B OE = λw because both (20) and(19) have the same structure. Provded that an olgopolstc equlbrum exsts, ths mmedately mples lm PoA(N;l1,...,lN) = 1. N,β It s mportant to notce that the proof of Theorem 2 provdes more nformaton than s summarzed n the theorem statement. In partcular, the proof hghlghts that, as N grows, the allocaton at any olgopolstc equlbrum (provded that t exsts) converges to ts asymptotc value wth rate 1/N. Ths s evdent gven (18). Ths hghlghts that, not only are olgopolstc equlbra effcent when N s large, but they become effcent quckly as N grows. Another observaton that follows from the proof of Theorem 2 s that the prces charged by the provders n the olgopolstc equlbrum converge to the Pgouvan taxes [22], whch are known to nduce optmal behavor. Further, the convergence to these prces happens at a rate of 1/N, so even for small N the prces are close to the socal optmal ones. We explore both of these observatons further n the context of queueng-based latency functons n Secton 4.3 To conclude ths secton, let us brefly remark about one smple, but mportant, extenson of Theorem 2. Specfcally, note that the optmal allocaton we have used as a benchmark to ths pont, though t s commonly used as a benchmark for load balancng and congeston games, s not truly the optmal allocaton. That s, t assumes that requests Boxplot of our bound on the PoA N Fgure 2: Boxplot of our bound on the PoA (23) under a varyng number of provders. The servce rates are chosen unformly at random from [1, 10000]. are allocated to servers probablstcally n a..d. manner,.e., that the probablty a job s forwarded to provder s x Opt /λ. In practce, for example n cloud computng systems, upon the arrval of a user request t s possble to explot nformaton about where recent requests were sent n order to mprove the overall latency. For example, f l = l j for all,j, then a central broker mplementng a Round- Robn allocaton 3 outperforms any probablstc routng polcy, see [26]. Thus, to understand the true neffcency among all possble routng polces (rather than among just probablstc routng polces), the task s more complex. To llustrate ths, note that the arrval process of users to provders changes consderably and ths affects the latency functons n a non-trval manner. However, recent results have provded a bound on the dfference between latences of the probablstc and non-probablstc optmal routng schemes, see [5]. In partcular, f we let x Opt represent the non-probablstc optmal traffc allocaton and l represent the nduced latency functons, then [5] proves that l( x Opt ) l(x Opt )/2. Applyng ths result to the settng of the current paper then smply doubles the prce of anarchy proven n Theorem 2. Thus, as N, the olgopolstc equlbrum provdes latency wthn a factor of 2 of that provded by the optmal, non-probablstc allocaton. Ths s mportant when contrastng the dstrbuted settng modeled by an olgopolstc equlbrum (whch cannot nduce non-probablstc routng) wth a centralzed approach where non-probablstc routng s standard. 4.3 Case study: Processor sharng queues To ths pont we have characterzed the effcency of olgopolstc equlbra under a wde class of latency functons, and consequently, the results provde nsght nto only the extreme cases (very large and very small systems). To obtan more nsght nto the effcency of olgopolstc equlbra we now focus on one specfc class of latency functons where explct results can be obtaned the M/GI/1/Processor- 3 Under Round-Robn, request s sent to provder mod N.

8 Sharng queue (1). Further, ths class represents a partcularly mportant model for communcaton systems. The socal optmal allocaton n ths settng s well understood; see, for example, [7, 13]. Assumng for smplcty that > µ 1 > µ 2 > > µ N > 0, we have x Opt j = µ j µ j ī =1 µ λ ī =1 j=1 µ, j ī, (21) where µ N+1 = 0, { } ī = mn 1 : µ +1 ( µ j λ)/ µj. j=1 and x Opt j = 0, for all j > ī. The above allocaton yelds a socally-optmal cost of l(x Opt ) = 1 ( ī ī λ =1 =1 µ) 2 µ λ ī λ. (22) The Wardrop equlbrum s also well understood for ths settng, [7, 13]. Addtonally, the prce of anarchy for the Wardrop equlbrum has been bounded by N, whch has been shown to be tght [13]. We now move to characterzng the olgopolstc equlbrum. Wthn ths settng, we have already seen n the numercal experments of Secton 3 that olgopolstc equlbra exst f the system s not over-provsoned. Further, we have seen that there s often a unque olgopolstc equlbrum. Wenowderveexplctboundsontheprceof anarchy, traffc allocaton, and prces that emerge n an olgopolstc equlbrum. Theorem 3. For each, let l (x ) as n (1). Assume also that (at least) one olgopolstc equlbrum exsts. If both the socally-optmal and olgopolstc equlbrum traffc allocatons use only the frst k > 1 provders, then PoA(N;l 1,...,l N) ( max k 1 µ j k:j µj ) 1/2. (23) Further, f denotes the prce vector at some OE that uses k provders, then the followng nequaltes hold true for each j k: x Opt µ x Opt xopt + µ x Opt µ (µ x Opt ) B µ 2 (µ x Opt µ µ B ) 2 ( µ j k:j µj B (24) max k 1 (25) µ j k:j µj (26) Proof. Wthout loss of generalty, assume that the number of provders used s N (f only k provders are used, the followng sums and ndces should be up to k). Usng Proposton 2, n an olgopolstc equlbrum the followng ) 1 condtons must hold (µ ) 2 poe (µ xoe = λ. ) 1 + (µ = B, OE ) + x 2 (µ j j ) 2, j Usng the frst nequalty, we have the condtons (µ ) 1 + xoe = λ. (µ B, ) 2 (27) (28) whch concde wth the KKT condtons of (2) provded that s replaced by=. Ths means thatx Opt satsfes (28) when B B Opt = (µ x Opt ) 1 x Opt + (29) (µ x Opt ) 2. Usng the left-hand nequalty and the second equalty of (27), we fnd µ (B ) 2 B 0. (30) Snce B > 0, we get B B = (µ ) 1. (31) µ We use (31) to boundthe second term n the second nequalty of (27) obtanng Substtutng (32) n (27), we fnd (µ ) 1 + xoe = λ. (µ ) + µ B (32) 2 j µj (µ amaxb, ) 2 (33) where a max = max 1 µ. Condtons (33) are agan j µ j the KKT condtons of (2) provded that s replaced by =. Ths means that x Opt satsfes (33) when a maxb B Opt. (34) Inequaltes (24) and (26) follow by the frst nequalty n (27) and (29), (32), (34). Now, from (31) and (34), we fnd µ µ B µ µ BOptamax. (35) Snce (µ x ) 1 s ncreasng n x, substtutng (35) n l( ) l( ) 1 µ B Opt N λ a max λ, (36) and usng (22) we fnally get 1 µ B Opt l( ) l(x Opt ) λ a max = µb Opt 1 λ 1 amax. (37)

9 Inequaltes (24) and (26) provde bounds on the equlbrum prce, traffc allocaton, and proft of each provder, and these bounds are asymptotcally exact snce, as N grows, µ / j µj approaches zero. Ths s n agreement wth Theorem 2. To provde some more nsght nto Theorem 3, we provde the results of some numercal experments n Fgure 2. Specfcally, Fgure 2 vares the network sze N and plots our bound on the prce of anarchy (23) for a random nstance of a load balancng game where the servce rates have been randomly generated from set [1, 10000] accordng to a unform dstrbuton. The fgure shows a boxplot for each N, whch refers to 10,000 experments. The pont of ths fgure s to llustrate the speed wth whch the prce of anarchy converges to 1. In the fgure, we observe that the average performance loss at the worst olgopolstc equlbrum s remarkably small even when N s very small. 5. CONCLUDING REMARKS In ths paper we have studed the exstence and effcency of olgopolstc equlbra n load balancng games. Ths model s meant to provde nsght nto the mpact of competton and prce n congested markets, e.g., transportaton networks and communcaton networks. One partcularly nterestng settng that the model studed here provdes nsght nto s that of the cloud computng nfrastructure-asa-servce model, where a nfrastructure provder such as [15, 16, 17] rents out resources to content provders, who then serve traffc requests ndependently, and are often n competton wth each other. The man nsghts that stem from our results on exstence and effcency of olgopolstc equlbra are that () competton can be dsastrous f the system s under-provsoned,.e., an olgopolstc equlbrum wll not exst and the provder prces wll grow unboundedly, and () f the system s properly provsoned and an olgopolstc equlbrum exsts, then competton yelds effcency,.e., as the N, the prce of anarchy converges to 1 at a rate of 1/N. One mportant techncal contrbuton of ths work s that we defne and analyze a scalng of the load balancng game whch allows a non-trval lmt to emerge as N, and thus allows the proof of result () above. Ths scalng s qute natural and allows the traffc to grow lnearly wth N. In the queueng theory lterature, ths s a popular scalng for studyng mult-server systems[28, 18, 12]. An nterestng drecton for extendng ths work s to study other scalngs of the system n order to understand how robust the concluson that competton yelds effcency s for load balancng games. 6. REFERENCES [1] D. Acemoglu, K. Bmpks, and A. Ozdaglar. Prce and capacty competton. Games and Economc Behavor, 66(1):1 26, May [2] D. Acemoglu and A. Ozdaglar. Competton and effcency n congested markets. Math. Oper. Res., 32(1):1 31, [3] D. Acemoglu and A. Ozdaglar. Competton n parallel-seral networks. Games and Economc Behavor, 25: , August [4] E. Altman, U. Ayesta, and B. Prabhu. Load balancng n processor sharng systems. To appear n Telecommuncaton Systems, (DOI: /s ), [5] J. Anselm and B. Gaujal. Optmal routng n parallel, non-observable queues and the prce of anarchy revsted. In 22nd Internatonal Teletraffc Congress, ITC-22. IEEE, [6] M. J. Beckmann, C. B. McGure, and C. B. Wnsten. Studes n the Economcs of Transportaton. Yale Unversty Press, New Haven, CT, [7] C. H. Bell and S. Stdham. Indvdual versus socal optmzaton n the allocaton of customers to alternatve servers. Management Scence, pages 29:83 839, [8] U. N. Bhat. An Introducton to Queueng Theory: Modelng and Analyss n Applcatons. Brkhauser Verlag, [9] H. Chen, J. Marden, and A. Werman. The effect of local schedulng n load balancng desgns. In Proceedngs of IEEE Infocom, [10] R. Comnett, J. R. Correa, and N. E. Ster-Moses. The mpact of olgopolstc competton n networks. Oper. Res., 57: , November [11] J. R. Correa, A. S. Schulz, and N. E. Ster-Moses. Selfsh routng n capactated networks. Math. Oper. Res., 29(4): , [12] E. Engel, R. Fscher, and A. Galetovc. Toll competton among congested roads. NBER Techncal Workng Papers 0239, Natonal Bureau of Economc Research, Inc, May [13] M. Havv and T. Roughgarden. The prce of anarchy n an exponental mult-server. Op. Res. Lett., 35(4): , [14] A. Hayrapetyan, E. Tardos, and T. Wexler. A network prcng game for selfsh traffc. In Proceedngs of the twenty-fourth annual ACM symposum on Prncples of dstrbuted computng, PODC 05, pages , New York, NY, USA, ACM. [15] [16] [17] [18] F. Kelly. Loss networks. Ann. Appl. Prob., 1: , [19] L. Klenrock. Queueng Systems, vol. 2. John Wley and Sons, [20] Y. Korls, A. Lazar, and A. Orda. Achevng network optma usng stackelberg routng strateges. IEEE Transactons on Networkng, 5(1): , [21] H. Kushner. Heavy Traffc Analyss of Controlled Queueng and Communcaton Networks. Sprnger-Verlag, New York, [22] N. Nsan, T. Roughgarden, E. Tardos, and V. V. Vazran. Algorthmc Game Theory. Cambrdge Unversty Press, New York, NY, USA, [23] A. Orda, R. Rom, and N. Shmkn. Compettve routng n multuser communcaton networks.

10 IEEE/ACM Trans. Netw., 1: , October [24] T. Roughgarden. Stackelberg schedulng strateges. SIAM J. Comput., 33(2): , [25] T. Roughgarden and E. Tardos. How bad s selfsh routng? J. ACM, 49: , March [26] J. Walrand. An Introducton to Queueng Networks. Prentce-Hall, [27] J. G. Wardrop. Some theoretcal aspects of road traffc research. Proceedngs of the Insttute of Cvl Engneers, 1: , [28] W. Whtt. Stochastc Process Lmts. Sprnger, New York, [29] T. Wu and D. Starobnsk. A comparatve analyss of server selecton n content replcaton networks. IEEE/ACM Trans. Netw., 16(6): , Appendx Proof of Proposton 2 As done for the proof of Proposton 9 n [2], the proof derves the KKT condtons of optmzaton problem(5). Pont W( ) s unquely gven by the mnmzers of (4) whose KKT condtons are l (x )+ A = B, x = λ, x 0, A x = 0, A 0, (38) where A and B are Lagrange multplers. Assume that an olgopolstc equlbrum exsts and denote t by. Usng the optmalty condtons (38), for provder j j j = max p jx j s.t.: l j(x j)+p j A j = B, l (x )+ A = B, j x = λ A x = 0, A 0 (39) and after some algebra we obtan l p j = x j (x j ) j x j + Ñ =1: j x j ( ) 1 l (x ) x l j(x j)+p j = B l (x )+ = B, j : Ñ x = λ. (42) If s such that = 0 for all k, then the bestresponse of k s to set the prce (mn k l (0) l k (λ))e k (over ths threshold at least one other provder would have the ncentve to devate ts prce). Provder k must be such that k = argmn j l j(λ). In fact, f provder k s such that l k (λ) l k (λ), thentwouldgetanon-nullproportonofthe traffc n the correspondng Wardrop equlbrum, aganst the assumpton. where each provder j has fxed equlbra prce (a constant here). The constrant x 0,, does not appear because t s redundant (for a fxed prce vector there exsts a unque Wardrop equlbra). Wthout loss of generalty, assume that the frst Ñ provders are used. Snce poe > 0 f and only f Ñ f and only f A = 0 f and only f x > 0, (39) can be rewrtten as (for j Ñ) j j = max p jx j s.t.: l j(x j)+p j = B, l (x )+ = B, j : Ñ x = λ (40) Clearly, f j > Ñ, then poe j j = 0. Wth respect to multplers V and W, the KKT condtons of (40) become x j +V j = 0 l p j +V j (x j ) j x j +W = 0 l V (x ) x +W = 0, j : Ñ Ñ =1 V = 0 (41) l j(x j)+p j = B l (x )+ = B, j : Ñ x = λ,