Competition and Efficiency in Congested Markets

Transcription

1 Competton and Effcency n Congested Markets Daron Acemoglu Department of Economcs Massachusetts Insttute of Technology Asuman E. Ozdaglar Department of Electrcal Engneerng and Computer Scence Massachusetts Insttute of Technology February 7, 2005 Abstract We study the effcency of olgopoly equlbra n congested markets. The motvatng examples are the allocaton of network flows n a communcaton network or of traffc n a transportaton network. We show that ncreasng competton among olgopolsts can reduce effcency, measured as the dfference between users wllngness to pay and delay costs. We characterze a tght bound of 5/6 on effcency n pure strategy equlbra. Ths bound s tght even when the number of routes and olgopolsts s arbtrarly large. We also study the effcency propertes of mxed strategy equlbra. We thank Xn Huang, Ramesh Johar, Erc Maskn, Ncolas Ster Moses, Jean Trole, John Tstskls, Ivan Wernng, Muhamet Yldz and partcpants at the INFORMS conference, Denver 2004 for useful comments.

2 Introducton We analyze prce competton n the presence of congeston costs. Consder the followng envronment: one unt of traffc can use one of I alternatve routes. More traffc on a partcular route causes delays, exertng a negatve (congeston) externalty on exstng traffc. Congeston costs are captured by a route-specfc non-decreasng convex latency functon, l ( ). Proft-maxmzng olgopolsts set prces (tolls) for travel on each route denoted by p. We analyze subgame perfect Nash equlbra of ths envronment, where for each prce vector, p, all traffc chooses the path that has mnmum (toll plus delay) cost, l + p, and olgopolsts choose prces to maxmze profts. The envronment we analyze s of practcal mportance for a number of settngs. These nclude transportaton and communcaton networks, where addtonal use of a route (path) generates greater congeston for all users, and markets n whch there are snob effects, so that goods consumed by fewer other consumers are more valuable (see for example, [38]). The key feature of these envronments s the negatve congeston externalty that users exert on others. Ths externalty has been well-recognzed snce the work by Pgou [27] n economcs, by Samuelson [33], Wardrop [4], Beckmann et al. [4] n transportaton networks, and by Orda et. al. [23], Korls et. al. [8], Kelly [7], Low [20] n communcaton networks. More recently, there has been a growng lterature that focuses on quantfcaton of effcency loss (referred to as the prce of anarchy) that results from externaltes and strategc behavor n dfferent classes of problems; selfsh routng (Koutsoupas and Papadmtrou [9], Roughgarden and Tardos [3], Correa, Schulz, and Ster-Moses [8], Peraks [26], and Fredman [2]), resource allocaton by market mechansms (Johar and Tstskls [6], Sanghav and Haek [32]), and network desgn (Anshelevch et.al. [2]). Nevertheless, the game-theoretc nteractons between (multple) servce provders and users, or the effects of competton among the provders on the effcency loss has not been consdered. Ths s an mportant area for analyss snce n most applcatons, (competng) proft-maxmzng enttes charge prces for use. Moreover, we wll show that the nature of the analyss changes sgnfcantly n the presence of prce competton. We provde a general framework for the analyss of prce competton among provders n a congested (and potentally capactated) network, study exstence of pure strategy and mxed strategy equlbra, and characterze and quantfy the effcency propertes of equlbra. There are four sets of maor results from our analyss. Frst, though the equlbrum of traffc assgnment wthout prces can be hghly neffcent (e.g., [27], [3], [8]), prce-settng by a monopolst nternalzes the negatve externalty and acheves effcency. Second, ncreasng competton can ncrease neffcency. In fact, changng the market structure from monopoly to duopoly almost always ncreases neffcency. Ths result contrasts wth most exstng results n the economcs lterature where greater competton tends to mprove the allocaton of resources (e.g. see Trole [36]). The ntuton for ths result s drven by the presence of congeston and s llustrated by the example we dscuss below. 2 An externalty arses when the actons of the player n a game affects the payoff of other players. 2 Because users are homogeneous and have a constant reservaton utlty n our model, n the absence

3 Thrd, we provde a tght bound on the extent of neffcency n the presence of prcng, whch apples rrespectve of the number of routes, I. We show that socal surplus (defned as the dfference between users wllngness to pay and the delay cost) n any pure strategy olgopoly equlbrum s always greater than 5/6 of the maxmum socal surplus. Smple examples reach ths 5/6 bound. Interestngly, ths bound s obtaned even when the number of routes, I, s arbtrarly large. Fourth, pure strategy equlbra may fal to exst. Ths s not surprsng n vew of the fact that what we have here s a verson of a Bertrand-Edgeworth game where pure strategy equlbra do not exst n the presence of convex costs of producton or capacty constrants (e.g., Edgeworth [], Shubk [35], Benassy [6], Vves [40]). However, n our olgopoly envronment we show that when latency functons are lnear, a pure strategy equlbrum always exsts, essentally because some amount of congeston externaltes remove the payoff dscontnutes nherent n the Bertrand-Edgeworth game. Non-exstence becomes an ssue n our envronment when latency functons are hghly convex. In ths case, we prove that mxed strategy equlbra always exst. We also show that mxed strategy equlbra can lead to arbtrarly neffcent worst-case realzatons; n partcular, socal surplus can become arbtrarly small relatve to the maxmum socal surplus. The followng example llustrates some of these results. Example Fgure shows a stuaton smlar to the one frst analyzed by Pgou [27] to hghlght the neffcency due to congeston externaltes. One unt of traffc wll travel from orgn A to destnaton B, usng ether route or route 2. The latency functons are gven by l (x) = x2 3, l 2 (x) = 2 3 x. It s straghtforward to see that the effcent allocaton [.e., one that mnmzes the total delay cost l (x )x ] s x S = 2/3 and x S 2 = /3, whle the (Wardrop) equlbrum allocaton that equates delay on the two paths s x W E.73 > x S and x W 2 E.27 < x S 2. The source of the neffcency s that each unt of traffc does not nternalze the greater ncrease n delay from travel on route, so there s too much use of ths route relatve to the effcent allocaton. Now consder a monopolst controllng both routes and settng prces for travel to maxmze ts profts. We wll show n greater detal below that n ths case, the monopolst wll set a prce ncludng a markup, x l (when l s dfferentable), whch exactly nternalzes the congeston externalty. In other words, ths markup s equvalent to the Pgovan tax that a socal planner would set n order to nduce decentralzed traffc to choose the effcent allocaton. Consequently, n ths smple example, monopoly prces wll be p ME = (2/3) 3 +k and p ME 2 = (2/3 2 )+k, for some constant k. The resultng traffc n the Wardrop equlbrum wll be dentcal to the effcent allocaton,.e., x ME = 2/3 and x ME 2 = /3. Next consder a duopoly stuaton, where each route s controlled by a dfferent proftmaxmzng provder. In ths case, t can be shown that equlbrum prces wll take the of congeston externaltes, all market structures would acheve effcency, and a change from monopoly to duopoly, for example, would have no effcency consequence. 2

4 2 l (x)=x /3 unt of traffc l (x)=(2/3)x Fgure : A two lnk network wth congeston-dependant latency functons. 2 form p OE = x (l + l 2) [see Eq. (20) n Secton 4], or more specfcally, p OE 0.6 and p OE The resultng equlbrum traffc s x OE.58 < x S and x OE 2.42 > x S 2, whch also dffers from the effcent allocaton. We wll show that ths s generally the case n the olgopoly equlbrum. Interestngly, whle n the Wardrop equlbrum wthout prces, there was too much traffc on route, now there s too lttle traffc because of ts greater markup. It s also noteworthy that although the duopoly equlbrum s neffcent relatve to the monopoly equlbrum, n the monopoly equlbrum k s chosen such that all of the consumer surplus s captured by the monopolst, whle n the olgopoly equlbrum users may have postve consumer surplus. 3 The ntuton for the neffcency of duopoly relatve to monopoly can be obtaned as follows. There s now a new source of (dfferental) monopoly power for each duopolst, whch they explot by dstortng the pattern of traffc: when provder, controllng route, charges a hgher prce, t realzes that ths wll push some traffc from route to route 2, rasng congeston on route 2. But ths makes the traffc usng route become more locked-n, because ther outsde opton, travel on the route 2, has become worse. 4 As a result, the optmal prce that each duopolst charges wll nclude an addtonal markup over the Pgovan markup. These are x l 2 for route and x 2 l for route 2. Snce these two markups are generally dfferent, they wll dstort the pattern of traffc away from the effcent allocaton. Naturally, however, prces are typcally lower wth duopoly, so even though socal surplus declnes, users wll be better off than n monopoly (.e., they wll command a postve consumer surplus). Although there s a large lterature on models of congeston both n transportaton and communcaton networks, very few studes have nvestgated the mplcatons of havng the property rghts over routes assgned to proft-maxmzng provders. In [3], Basar and Srkant analyze monopoly prcng under specfc assumptons on the utlty and latency functons. He and Walrand [5] study competton and cooperaton among nternet servce provders under specfc demand models. Issues of effcent allocaton of 3 Consumer surplus s defned as the dfference between users wllngness to pay and effectve costs, p + l (x ), and s thus dfferent from socal surplus (whch s the dfference between users wllngness to pay and latency cost, l (x ), thus also takes nto account producer surplus/profts). See Mas-Colell, Wnston, and Green [2]. 4 Usng economcs termnology, we could also say that the demand for route becomes more nelastc. Snce ths term has a dfferent meanng n the communcaton networks lterature (see [34]), we do not use t here. 3

5 flows or traffc across routes do not arse n these papers. Our prevous work [] studes the monopoly problem and contans the effcency of the monopoly result (wth nelastc traffc and more restrctve assumptons on latences), but none of the other results here. To mnmze overlap, we dscuss the monopoly problem only brefly n ths paper. In the rest of the paper, we use the termnology of a (communcaton) network, though all of the analyss apples to resource allocaton n transportaton networks, electrcty markets, and other economc applcatons. Secton 2 descrbes the basc envronment. Secton 3 brefly characterzes the monopoly equlbrum and establshes ts effcency. Secton 4 defnes and characterzes the olgopoly equlbra wth competng proft-maxmzng provders. Secton 5 contans the man results and characterzes the effcency propertes of the olgopoly equlbrum and provde bounds on effcency. Secton 6 contans concludng comments. Regardng notaton, all vectors are vewed as column vectors, and nequaltes are to be nterpreted componentwse. We denote by R I + the set of nonnegatve I-dmensonal vectors. Let C be a closed subset of [0, ) and let f : C R be a convex functon. We use f(x) to denote the set of subgradents of f at x, and f (x) and f + (x) to denote the left and rght dervatves of f at x. 2 Model We consder a network wth I parallel lnks. Let I = {,..., I} denote the set of lnks. Let x denote the total flow on lnk, and x = [x,..., x I ] denote the vector of lnk flows. Each lnk n the network has a flow-dependent latency functon l (x ), whch measures the travel tme (or delay) as a functon of the total flow on lnk. We denote the prce per unt flow (bandwdth) of lnk by p. Let p = [p,..., p I ] denote the vector of prces. We are nterested n the problem of routng d unts of flow across the I lnks. We assume that ths s the aggregate flow of many small users and thus adopt the Wardrop s prncple (see [4]) n characterzng the flow dstrbuton n the network;.e., the flows are routed along paths wth mnmum effectve cost, defned as the sum of the latency at the gven flow and the prce of that path (see the defnton below). Wardrop s prncple s used extensvely n modellng traffc behavor n transportaton networks ([4], [9], [25]) and communcaton networks ([3], [8]). We also assume that the 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 2. 5 Defnton For a gven prce vector p 0, 6 a vector x W E R I + s a Wardrop 5 Ths smplfyng assumpton mples that all users are homogeneous n the sense that they have the same reservaton utlty, R. We dscuss potental ssues n extendng ths work to users wth elastc and heterogeneous requrements n the concludng secton. 6 Snce the reservaton utlty of users s equal to R, we can also restrct attenton to p R for all. Throughout the paper, we use p 0 and p [0, R] I nterchangeably. 4

6 u (x) Rd d x Fgure 2: Aggregate utlty functon. equlbrum (WE) f x W E arg max x 0 I= x d { = We denote the set of WE at a gven p by W (p). (R l (x W E ) p )x }. () We adopt the followng assumpton on the latency functons throughout the paper. Assumpton For each I, the latency functon l : [0, ) [0, ] s proper closed convex, nondecreasng, and satsfes l (0) = 0. Hence, we allow the latency functons to be extended real-valued (see [7]). Let C = {x [0, ) l (x) < } denote the effectve doman of l. By Assumpton, C s a closed nterval of the form [0, b] or [0, ). Let b C = sup x C x. Wthout loss of generalty, we can add the constrant x C n Eq. (). Usng the optmalty condtons for problem (), we see that a vector x W E R I + s a WE f and only f I = xw E d and there exsts some λ 0 such that λ( I = xw E d) = 0 and for all, R l (x W E ) p λ f x W E = 0, (2) = λ f 0 < x W E < b C, λ f x W E = b C. When the latency functons are real-valued [.e., C = [0, )], we obtan the followng characterzaton of a WE, whch s often used as the defnton of a WE n the lterature. Ths lemma states that n the WE, the effectve costs, defned as l (x W E ) + p, are equalzed on all lnks wth postve flows (proof omtted). Lemma Let Assumpton hold, and assume further that C = [0, ) for all I. Then a nonnegatve vector x W (p) f and only f l (x ) + p = mn {l (x ) + p }, wth x > 0, (3) 5

7 l (x ) + p R, wth x > 0, x d, = wth I = x = d f mn {l (x ) + p } < R. Example 2 below shows that condton (3) n ths lemma may not hold when the latency functons are not real-valued. Next we establsh the exstence of a WE. 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. Proof. Gven any p 0, consder the followng optmzaton problem maxmze x 0 subect to ( x ) (R p )x l (z)dz 0 x d. = = x C,. (4) In vew of Assumpton () (.e., l s nondecreasng for all ), t can be shown that the obectve functon of problem (4) s convex over the constrant set, whch s nonempty (snce 0 C ) and convex. Moreover, the frst order optmalty condtons of problem (4), whch are also suffcent condtons for optmalty, are dentcal to the WE optmalty condtons [cf. Eq. (2)]. Hence a flow vector x W E W (p) f and only f t s an optmal soluton of problem (4). Snce the obectve functon of problem (4) s contnuous and the constrant set s compact, ths problem has an optmal soluton, showng that W (p) s nonempty. The fact that W s an upper semcontnuous correspondence at every p follows by usng the Theorem of the Maxmum (see Berge [5], chapter 6) for problem (4). Q.E.D. WE flows satsfy ntutve monotoncty propertes gven n the followng proposton. The proof follows from the optmalty condtons [cf. Eq. (2)] and s omtted (see []). Proposton 2 (Monotoncty) Let Assumpton hold. For a gven p 0, let p = [p ]. (a) For some p p, let x W ( p) and x W (p). Then, I = x I = x. (b) For some p < p, let x W ( p, p ) and x W (p, p ). Then x x and x x, for all. (c) For some Ĩ I, suppose that p < p for all Ĩ and p = p for all / Ĩ, and let x W ( p) and x W (p). Then Ĩ x ( p) Ĩ x (p). 6

8 For a gven prce vector p, the WE need not be unque n general. The followng example llustrates some propertes of the WE. Example 2 Consder a two lnk network. Let the total flow be d = and the reservaton utlty be R =. Assume that the latency functons are gven by { 0 f 0 x 2 l (x) = l 2 (x) = 3 otherwse. At the prce vector (p, p 2 ) = (, ), the set of WE, W (p), s gven by the set of all vectors (x, x 2 ) wth 0 x 2/3 and x. At any prce vector (p, p 2 ) wth p > p 2 =, W (p) s gven by all (0, x 2 ) wth 0 x 2 2/3. Ths example also llustrates that Lemma need not hold when latency functons are not real-valued. Consder, for nstance, the prce vector (p, p 2 ) = ( ɛ, aɛ) for some scalar a >. In ths case, the unque WE s (x, x 2 ) = (/3, 2/3), and clearly effectve costs on the two routes are not equalzed despte the fact that they both have postve flows. Ths arses because the path wth the lower effectve cost s capacty constraned, so no more traffc can use that path. Under further restrctons on the l, we obtan the followng standard result n the lterature whch assumes strctly ncreasng latency functons. Proposton 3 (Unqueness) Let Assumpton hold. Assume further that l s strctly ncreasng over C. 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 obectve functon of problem (4) 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. Q.E.D. In general, under the Assumpton that l (0) = 0 for all I, we have the followng result, whch s essental n our analyss wth nonunque WE flows. Lemma 2 Let Assumpton hold. For a gven p 0, defne the set Ī = { I x, x W (p) wth x x }. (5) Then l (x ) = 0, Ī, x W (p), p = p,, Ī. 7

9 Proof. Consder some Ī and x W (p). Snce Ī, there exsts some ˆx W (p) such that x ˆx. Assume wthout loss of generalty that x > ˆx. There are two cases to consder: (a) If x k ˆx k for all k, then I x > I ˆx, whch mples that the WE optmalty condtons [cf. Eq. (2)] for ˆx hold wth ˆλ = 0. By Eq. (2) and x > ˆx, we have l (x ) + p R, l (ˆx ) + p R, whch together mply that l (x ) = l (ˆx ). By Assumpton (.e., l s convex and l (0) = 0), t follows that l (x ) = 0. (b) If x k < ˆx k for some k, by the WE optmalty condtons, we obtan l (x ) + p l k (x k ) + p k, l (ˆx ) + ˆp l k (ˆx k ) + p k. Combnng the above wth x > ˆx and x k < ˆx k, we see that l (x ) = l (ˆx ), and l k (x k ) = l k (ˆx k ). By Assumpton, ths shows that l (x ) = 0 (and also that p = p k ). Next consder some, Ī. We wll show that p = p. Snce x, ˆx W (p) such that x > ˆx. There are three cases to consder: Ī, there exst x < ˆx. Then a smlar argument to part (b) above shows that p = p. x > ˆx. If x k ˆx k for all k,, then m ˆx m < d, mplyng that the WE optmalty condtons hold wth ˆλ = 0. Therefore, we have l (x ) + p R, l (ˆx ) + p R, whch together wth l (x ) = l (ˆx ) = 0 mply that p = p. x = ˆx. Snce Ī, there exsts some x W (p) such that x x. Repeatng the above two steps wth x nstead of ˆx yelds the desred result. Q.E.D. We next defne the socal problem and the socal optmum, whch s the routng (flow allocaton) that would be chosen by a planner that has full nformaton and full control over the network. 8

10 Defnton 2 socal problem A flow vector x S s a socal optmum f t s an optmal soluton of the maxmze x 0 subect to = ( ) R l (x ) x (6) x d. = In vew of Assumpton, the socal problem has a contnuous obectve functon and a compact constrant set, guaranteeng the exstence of a socal optmum, x S. Moreover, usng the optmalty condtons for a convex program (see [7], Secton 4.7), we see that a vector x S R I + s a socal optmum f and only f I = xs d and there exsts a subgradent g l l (x S ) for each, and a λ S 0 such that λ S ( I = xs d) = 0 and for each, R l (x S ) x S g l λ S f x S = 0, (7) = λ S f 0 < x S < b C, λ S f x S = b C. For future reference, for a gven vector x R I +, we defne the value of the obectve functon n the socal problem, S(x) = (R l (x )) x, (8) = as the socal surplus,.e., the dfference between users wllngness to pay and the total latency. 3 Monopoly Equlbrum and Effcency In ths secton, we assume that a monopolst servce provder owns the I lnks and charges a prce of p per unt bandwdth on lnk. We consdered a related problem n Acemoglu and Ozdaglar [] for atomc users wth nelastc traffc (.e., the utlty functon of each of a fnte set of users s a step functon), and wth ncreasng, real-valued and dfferentable latency functons. Here we show that smlar results hold for the more general latency functons and the demand model consdered n Secton 2. The monopolst sets the prces to maxmze hs proft gven by Π(p, x) = p x, where x W (p). Ths defnes a two-stage dynamc prcng-congeston game, where the monopolst sets prces antcpatng the demand of users, and gven the prces (.e., n each subgame), users choose ther flow vectors accordng to the WE. 9 =

11 Defnton 3 A vector (p ME, x ME ) 0 s a Monopoly Equlbrum (ME) f x ME W (p ME ) and Π(p ME, x ME ) Π(p, x), p, x W (p). Our defnton of the ME s stronger than the standard subgame perfect Nash equlbrum concept for dynamc games. Wth a slght abuse of termnology, let us assocate a subgame perfect Nash equlbrum wth the on-the-equlbrum-path actons of the two-stage game. Defnton 4 A vector (p, x ) 0 s a subgame perfect equlbrum (SPE) of the prcng-congeston game f x W (p ) and for all p 0, there exsts x W (p) such that Π(p, x ) Π(p, x). The followng proposton shows that under Assumpton, the two soluton concepts concde. Snce the proof s not relevant for the rest of the argument, we provde t n Appendx A. Proposton 4 Let Assumpton hold. A vector (p ME, x ME ) s an ME f and only f t s an SPE of the prcng-congeston game. Snce an ME (p, x ) s an optmal soluton of the optmzaton problem maxmze p 0, x 0 subect to p x (9) = x W (p), t s slghtly easer to work wth than an SPE. Therefore, we use ME as the soluton concept n ths paper. The precedng problem has an optmal soluton, whch establshes the exstence of an ME. In the next proposton, we show that the flow allocaton at an ME and the socal optmum are the same. Proposton 5 Let Assumpton hold. A vector x s the flow vector at an ME f and only f t s a socal optmum. To prove ths result, we frst establsh the followng lemma. Lemma 3 have Let Assumpton hold. Let (p, x) be an ME. Then for all wth x > 0, we p = R l (x ). 0

12 Proof. Defne Ī = { I x > 0}. By the defnton of a WE, for all Ī, we have p + l (x ) R. We frst show that l (x ) + p = l (x ) + p,, Ī. Suppose l (x ) + p < l (x ) + p for some, Ī. Usng the optmalty condtons for a WE [cf. Eq. (2)], we see that x W (p + ɛ, p ) for suffcently small ɛ, contradctng the fact that (p, x) s an ME. Assume next that p + l (x ) < R for some Ī. Consder the prce vector p = p + ɛ Ī e, where e s the th unt vector and ɛ s such that p + ɛ + l (x ) < R, Ī. Hence, x s a WE at prce p, and therefore the vector ( p, x) s feasble for problem (9) and has a strctly hgher obectve functon value, contradctng the fact that (p, x) s an ME. Q.E.D. Proof of Proposton 5: In vew of Lemma 3, we can rewrte problem (9) as maxmze x 0 subect to = ( ) R l (x ) x x d. = Ths problem s dentcal to the socal problem [cf. problem (6)], completng the proof. Q.E.D. In addton to the socal surplus defned above, t s also useful to defne the consumer surplus, as the dfference between users wllngness to pay and effectve cost,.e., I = (R l (x ) p )x (See Mas-Colell, Wnston and Green, [2]). By Proposton 5 and Lemma 3, t s clear that even though the ME acheves the socal optmum, all of the surplus s captured by the monopolst, and users are ust ndfferent between sendng ther nformaton or not (.e., receve no consumer surplus). Our maor motvaton for the study of olgopolstc settngs s that they provde a better approxmaton to realty, where there s typcally competton among servce provders. A secondary motvaton s to see whether an olgopoly equlbrum wll acheve an effcent allocaton lke the ME, whle also transferrng some or all of the surplus to the consumers. 4 Olgopoly Equlbrum We suppose that there are S servce provders, denote the set of servce provders by S, and assume that each servce provder s S owns a dfferent subset I s of the lnks. Servce provder s charges a prce p per unt bandwdth on lnk I s. Gven the vector

13 of prces of lnks owned by other servce provders, p s = [p ] / Is, the proft of servce provder s s Π s (p s, p s, x) = I s p x, for x W (p s, p s ), where p s = [p ] Is. The obectve of each servce provder, lke the monopolst n the prevous secton, s to maxmze profts. Because ther profts depend on the prces set by other servce provders, each servce provder forms conectures about the actons of other servce provders, as well as the behavor of users, whch, we assume, 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 5 A vector (p OE, x OE ) 0 s a (pure strategy) Olgopoly Equlbrum (OE) f x OE W ( ) p OE s, p OE s and for all s S, Π s (p OE s, p OE s, x OE ) Π s (p s, p OE s, x), p s 0, x W (p s, p OE s ). (0) We refer to p OE as the OE prce. As for the monopoly case, there s a close relaton between a pure strategy OE and a pure strategy subgame perfect equlbrum. Agan assocatng the subgame perfect equlbrum wth the on-the-equlbrum-path actons, we have the followng standard defnton. Defnton 6 A vector (p, x ) 0 s a subgame perfect equlbrum (SPE) of the prce competton game f x W (p ) and there exsts a functon x(p) W (p) such that for all s S, Π s (p s, p s, x ) Π s (p s, p s, x ( p s, p s) ) ps 0. () The followng proposton generalzes Proposton 4 and enables us to work wth the OE defnton, whch s more convenent for the subsequent analyss. The proof parallels that of Proposton 4 and s omtted. Proposton 6 Let Assumpton hold. A vector (p OE, x OE ) s an OE f and only f t s an SPE of the prce competton game. The prce competton game s nether concave nor supermodular. Therefore, classcal arguments that are used to show the exstence of a pure strategy equlbrum do not hold (see [3], [37]). In the next proposton, we show that for lnear latency functons, there exsts a pure strategy OE. Proposton 7 Let Assumpton hold, and assume further that the latency functons are lnear. Then the prce competton game has a pure strategy OE. 2

14 Proof. For all I, let l (x) = a x. Defne the set I 0 = { I a = 0}. Let I 0 denote the cardnalty of set I 0. There are two cases to consder: I 0 2: Assume that there exst, I 0 such that I s and I s for some s s S. Then t can be seen that a vector (p OE, x OE ) wth p OE = 0 for all I 0 and x OE W (p OE ) s an OE. Assume next that for all I 0, we have I s for some s S. Then, we can assume wthout loss of generalty that provder s owns a sngle lnk wth a = 0 and consder the case I 0 =. I 0 : Let B s (p OE s ) be the set of p OE s Q.E.D. such that (p OE s, x OE ) arg max ps 0 x W (ps,p OE s ) I s p x. (2) Let B(p OE ) = [B s (p OE s )] s S. In vew of the lnearty of the latency functons, t follows that B(p OE ) s an upper semcontnuous and convex-valued correspondence. Hence, we can use Kakutan s fxed pont theorem to assert the exstence of a p OE such that B(p OE ) = p OE (see Berge [5]). To complete the proof, t remans to be shown that there exsts x OE W (p OE ) such that Eq. (0) holds. If I 0 =, we have by Proposton 3 that W (p OE ) s a sngleton, and therefore Eq. (0) holds and (p OE, W (p OE )) s an OE. Assume fnally that exactly one of the a s (wthout loss of generalty a ) s equal to 0. We show that for all x, x W (p OE ), we have x = x, for all. Let EC(x, p OE ) = mn {l (x ) + p OE }. If at least one of EC( x, p OE ) < R, or EC( x, p OE ) < R holds, then one can show that I = x = I = x = d. Substtutng x = d I, x n problem (4), we see that the obectve functon of problem (4) s strctly convex n x = [x ], thus showng that x = x. If both EC( x, p OE ) = R and EC( x, p OE ) = R, then x = x = l (R p OE ) for all, establshng our clam. For some x W (p OE ), consder the vector x OE = (d x, x ). Snce x s unquely defned and x s chosen such that the provder that owns lnk has no ncentve to devate, t follows that (p OE, x OE ) s an OE. The exstence result cannot be generalzed to pecewse lnear latency functons or to latency functons whch are lnear over ther effectve doman, as llustrated n the followng example. 3

15 Example 3 Consder a two lnk network. Let the total flow be d =. Assume that the latency functons are gven by { 0 f 0 x δ l (x) = 0, l 2 (x) = x δ x δ, ɛ for some ɛ > 0 and δ > /2, wth the conventon that when ɛ = 0, l 2 (x) = for x > δ. We frst show that there exsts no pure strategy olgopoly equlbrum for small ɛ (.e., there exsts no pure strategy subgame perfect equlbrum). The followng lst consders all canddate olgopoly prce equlbra (p, p 2 ) and proftable unlateral devatons for ɛ suffcently small, thus establshng the nonexstence of an OE:. p = p 2 = 0: A small ncrease n the prce of provder wll generate postve profts, thus provder has an ncentve to devate. 2. p = p 2 > 0: Let x be the flow allocaton at the OE. If x =, then provder 2 has an ncentve to decrease ts prce. If x <, then provder has an ncentve to decrease ts prce p < p 2 : Player has an ncentve to ncrease ts prce snce ts flow allocaton remans the same p 2 < p : For ɛ suffcently small, the proft functon of player 2, gven p, s strctly ncreasng as a functon of p 2, showng that provder 2 has an ncentve to ncrease ts prce. We next show that a mxed strategy OE always exsts. We defne a mxed strategy OE as a mxed strategy subgame perfect equlbrum of the prce competton game (see Dasgupta and Maskn, [0]). Let B n be the space of all (Borel) probablty measures on [0, R] n. Let I s denote the cardnalty of I s,.e., the number of lnks controlled by servce provder s. Let µ s B I s be a probablty measure, and denote the vector of these probablty measures by µ and the vector of these probablty measures excludng s by µ s. Defnton 7 (µ, x (p)) s a mxed strategy Olgopoly Equlbrum (OE) f the functon x (p) W (p) for every p [0, R] I and Π s (p s, p s, x (p s, p s )) d ( µ s (p s ) µ s (p s ) ) [0,R] I Π s (p s, p s, x (p s, p s ))d ( µ s (p s ) µ s (p s ) ) [0,R] I for all s and µ s B I s. Therefore, a mxed strategy OE smply requres that there s no proftable devaton to a dfferent probablty measure for each olgopolst. 4

16 Example 3 (contnued) We now show that the followng strategy profle s a mxed strategy OE for the above game when ɛ 0 (a mxed strategy OE also exsts when ɛ > 0, but ts structure s more complcated and less nformatve): 0 0 p R( δ), µ (p) = ( δ)r R( δ) p < R, p otherwse, 0 0 p R( δ), µ 2 (p) = ( δ)r R( δ) p R, δ δp otherwse. Notce that µ has an atom equal to δ at R. To verfy that ths profle s a mxed strategy OE, let µ be the densty of µ, wth the conventon that µ = when there s an atom at that pont. Let M = {p µ (p) > 0}. To establsh that (µ, µ 2 ) s a mxed strategy equlbrum, t suffces to show that the expected payoff to player s constant for all p M when the other player chooses p accordng to µ (see [24]). These expected payoffs are Π (p µ ) R 0 Π (p, p, x (p, p ))dµ (p ), whch are constant for all p M for =, 2. Ths establshes that (µ, µ 2 ) s a mxed strategy OE. The next proposton shows that a mxed strategy equlbrum always exsts. Proposton 8 Let Assumpton hold. Then the prce competton game has a mxed strategy OE, (µ OE, x OE (p)). The proof of ths proposton s long and not drectly related the rest of the argument, so t s gven n Appendx B. We next provde an explct characterzaton of pure strategy OE. Though of also ndependent nterest, these results are most useful for us to quantfy the effcency loss of olgopoly n the next secton. The followng lemma shows that an equvalent to Lemma (whch requred realvalued latency functons) also holds wth more general latency functons at the pure strategy OE. Lemma 4 Let Assumpton hold. If (p OE, x OE ) s a pure strategy OE, then l (x OE ) + p OE = mn {l (x OE ) + p OE }, wth x OE > 0, (3) l (x ) + p OE R, wth x OE > 0, (4) x OE d, (5) = 5

17 wth I = xoe = d f mn {l (x OE ) + p } < R. Proof. Let (p OE, x OE ) be an OE. Snce x OE W (p OE ), condtons (4) and (5) follow by the defnton of a WE. Consder condton (3). Assume that there exst some, I wth x OE > 0, x OE > 0 such that l (x OE ) + p OE < l (x OE ) + p OE. Usng the optmalty condtons for a WE [cf. Eq. (2)], ths mples that x OE = b C. Consder changng p OE to p OE + ɛ for some ɛ > 0. By checkng the optmalty condtons, we see that we can choose ɛ suffcently small such that x OE W (p OE + ɛ, p OE ). Hence servce provder that owns lnk can devate to p OE + ɛ and ncrease ts profts, contradctng the fact that (p OE, x OE ) s an OE. Fnally, assume to arrve at the contradcton that mn {l (x OE ) + p } < R and I = xoe < d. Usng the optmalty condtons for a WE [Eq. (2) wth λ = 0 snce I = xoe < d], ths mples that we must have x OE = b C for some. Wth a smlar argument to above, a devaton to p OE + ɛ keeps x OE as a WE, and s more proftable, completng the proof. Q.E.D. We need the followng addtonal assumpton for our prce characterzaton. Assumpton 2 Gven a pure strategy OE (p OE, x OE ), f for some I wth x OE > 0, we have l (x OE ) = 0, then I s = {}. Note that ths assumpton s automatcally satsfed f all latency functons are strctly ncreasng or f all servce provders own only one lnk. Lemma 5 Let (p OE, x OE ) be a pure strategy OE. Let Assumptons and 2 hold. Let Π s denote the proft of servce provder s at (p OE, x OE ). (a) If Π s > 0 for some s S, then Π s > 0 for all s S. (b) If Π s > 0 for some s S, then p OE x OE > 0 for all I s. Proof. (a) For some I s, defne K = p OE +l (x OE ), whch s postve snce Π s > 0. Assume Π s = 0 for some s. For k I s, consder the prce p k = K ɛ > 0 for some small ɛ > 0. It can be seen that at the prce vector ( p k, p OE k ), the correspondng WE lnk flow would satsfy x k > 0. Hence, servce provder s has an ncentve to devate to p k at whch he wll make postve proft, contradctng the fact that (p OE, x OE ) s a pure strategy OE. (b) Snce Π s > 0, we have p OE m x OE m > 0 for some m I s. By Assumpton 2, we can assume wthout loss of generalty that l m (x OE m ) > 0 (otherwse, we are done). Let I s and assume to arrve at a contradcton that p OE x OE = 0. The proft of servce provder s at the pure strategy OE can be wrtten as Π s = Π s + p OE m x OE m, 6

18 where Π s denotes the profts from lnks other than m and. Let p OE m for some K. Consder changng the prces p OE m Π s = Π s + (K l m (x OE m ɛ))(x OE m and p OE = K l m (x OE m ) such that the new proft s ɛ) + ɛ(k l (ɛ)). Note that ɛ unts of flow are moved from lnk m to lnk such that the flows of other lnks reman the same at the new WE. Hence, the change n the proft s Π s Π s = (l m (x OE m ) l m (x OE m ɛ))x OE m + ɛ(l m (x OE m ɛ) l (ɛ))). Snce l m (x OE m ) > 0, ɛ can be chosen suffcently small such that the above s strctly postve, contradctng the fact that (p OE, x OE ) s an OE. Q.E.D. The followng example shows that Assumpton 2 cannot be dspensed wth for part (b) of ths lemma. Example 4 Consder a three lnk network wth two provders, where provder owns lnks and 3 and provder 2 owns lnk 2. Let the total flow be d = and the reservaton utlty be R =. Assume that the latency functons are gven by l (x ) = 0, l 2 (x 2 ) = x 2, l 3 (x 3 ) = ax 3, for some a > 0. Any prce vector (p, p 2, p 3 ) = (2/3, /3, b) wth b 2/3 and (x, x 2, x 3 ) = (2/3, /3, 0) s a pure strategy OE, so p 3 x 3 = 0 contrary to part (b) of the lemma. To see why ths s an equlbrum, note that provder 2 s clearly playng a best response. Moreover, n ths allocaton Π = 4/9. We can represent any devaton of provder by (p, p 3 ) = (2/3 δ, 2/3 aɛ δ), for two scalars ɛ and δ, whch wll nduce a WE of (x, x 2, x 3 ) = (2/3 + δ ɛ, /3 δ, ɛ). The correspondng proft of provder at ths devaton s Π = 4/9 δ 2 < 4/9, establshng that provder s also playng a best response and we have a pure strategy OE. We next establsh that under an addtonal assumpton, a pure strategy OE wll never be at a pont of non-dfferentablty of the latency functons. Assumpton 3 There exsts some s S such that l s real-valued and contnuously dfferentable for all I s. Lemma 6 Let (p OE, x OE ) be an OE wth mn {p OE for some. Let Assumptons, 2 and 3 hold. Then + l (x OE l + (xoe ) = l (xoe ), I, )} < R and p OE x OE > 0 where l + (xoe ) and l (xoe ) are the rght and left dervatves of the functon l at x OE respectvely. Snce the proof of ths lemma s long, t s gven n Appendx C. Note that Assumpton 3 cannot be dspensed wth n ths lemma. Ths s llustrated n the next example. 7

19 Example 5 Consder a two lnk network. Let the total flow be d = and the reservaton utlty be R = 2. Assume that the latency functons are gven by { 0 f 0 x l (x) = l 2 (x) = 2 2 ( x 2) otherwse. It can be verfed that the vector (p OE, p OE 2 ) = (, ), wth (x OE, x OE 2 ) = (/2, /2) s a pure strategy OE, and s at a pont of non-dfferentablty for both latency functons. We next provde an explct characterzaton of the OE prces, whch s essental n our effcency analyss n Secton 5. The proof s gven n Appendx D. Proposton 9 Assumptons, 2, and 3 hold. a) Assume that mn {p OE p OE = Let (p OE, x OE ) be an OE such that p OE x OE l (x OE x OE b) Assume that mn {p OE x OE > 0 for some I. Let + l (x OE )} < R. Then, for all s S and I s, we have ), f l (x OE ) = 0 for some / I s, Is xoe, otherwse. (6) l (x OE ) + / Is l (xoe ) + l (x OE )} = R. Then, for all s S and I s, we have p OE x OE l (xoe ). (7) Moreover, f there exsts some I such that I s = {} for some s S, then p OE x OE l + (xoe ) + x OE l (xoe ). (8) If the latency functons l are all real-valued and contnuously dfferentable, then analyss of Karush-Kuhn-Tucker condtons for olgopoly problem [Eq. (78) n Appendx D] mmedately yelds the followng result: Corollary Let (p OE, x OE ) be an OE such that p OE x OE > 0 for some I. Let Assumptons and 2 hold. Assume also that l s real-valued and contnuously dfferentable for all. Then, for all s S and I s, we have x OE l { (x OE ), } f l (x OE ) = 0 for some / I s, p OE = mn R l (x OE ), x OE l (x OE ) + Is xoe / Is l (xoe ), otherwse. Ths corollary also mples that n the two lnk case wth real-valued and contnuously dfferentable latency functons and wth mnmum effectve cost less than R, the OE prces are p OE as clamed n the Introducton. (9) = x OE (l (x OE ) + l 2(x OE 2 )) (20) 8

20 5 Effcency of Olgopoly Equlbra In ths secton, we study the effcency propertes of olgopoly prcng. We take as our measure of effcency the rato of the socal surplus of the equlbrum flow allocaton to the socal surplus of the socal optmum, S(x )/S(x S ), where x refers to the monopoly or the olgopoly equlbrum [cf. Eq. (8)]. In Secton 3, we showed that the flow allocaton at a monopoly equlbrum s a socal optmum. Hence, n congeston games wth monopoly prcng, there s no effcency loss. The followng example shows that ths s not necessarly the case n olgopoly prcng. Example 6 Consder a two lnk network. Let the total flow be d = and the reservaton utlty be R =. The latency functons are gven by l (x) = 0, l 2 (x) = 3 2 x. The unque socal optmum for ths example s x S = (, 0). The unque ME (p ME, x ME ) s x ME = (, 0) and p ME = (, ). As expected, the flow allocatons at the socal optmum and the ME are the same. Next consder a duopoly where each of these lnks s owned by a dfferent provder. Usng Corollary and Lemma 4, t follows that the flow allocaton at the OE, x OE, satsfes l (x OE ) + x OE [l (x OE ) + l 2(x OE 2 )] = l 2 (x OE 2 ) + x OE 2 [l (x OE ) + l 2(x OE 2 )]. Solvng ths together wth x OE + x OE 2 = shows that the flow allocaton at the unque olgopoly equlbrum s x OE = (2/3, /3). The socal surplus at the socal optmum, the monopoly equlbrum, and the olgopoly equlbrum are gven by,, and 5/6, respectvely. Before provdng a more thorough analyss of the effcency propertes of the OE, the next proposton proves that, as clamed n the Introducton and suggested by Example 6, a change n the market structure from monopoly to duopoly n a two lnk network typcally reduces effcency. Proposton 0 Consder a two lnk network where each lnk s owned by a dfferent provder. Let Assumpton hold. Let (p OE, x OE ) be a pure strategy OE such that p OE x OE > 0 for some I and mn {p OE + l (x OE )} < R. If l (x OE )/x OE l 2(x OE 2 )/x OE 2, then S(x OE )/S(x S ) <. Proof. Combnng the OE prces wth the WE condtons, we have l (x OE ) + x OE (l (x OE ) + l 2(x OE 2 )) = l 2 (x OE 2 ) + x OE 2 (l (x OE ) + l 2(x OE 2 )), where we use the fact that mn {p OE + l (x OE )} < R. Moreover, we can use optmalty condtons (7) to prove that a vector (x S, x S 2 ) > 0 s a socal optmum f and only f l (x S ) + x S l (x S ) = l 2 (x S 2 ) + x S 2 l 2(x S 2 ), 9

21 (see also the proof of Lemma 4). Snce l (x OE )/x OE l 2(x OE 2 )/x OE 2, the result follows. Q.E.D. We next quantfy the effcency of olgopoly equlbra by provdng a tght bound on the effcency loss n congeston games wth olgopoly prcng. As we have shown n Secton 4, such games do not always have a pure strategy OE. In the followng, we frst provde bounds on congeston games that have pure strategy equlbra. We next study effcency propertes of mxed strategy equlbra. 5. Pure Strategy Equlbra We consder prce competton games that have pure strategy equlbra (ths set ncludes, but s larger than, games wth lnear latency functons, see Secton 4). We consder latency functons that satsfy Assumptons, 2, and 3. Let L I denote the set of latency functons {l } I such that the assocated congeston game has a pure strategy OE and the ndvdual l s satsfy Assumptons, 2, and 3. 7 Gven a parallel lnk network wth I lnks and latency functons {l } I L I, let OE({l }) denote the set of flow allocatons at an OE. We defne the effcency metrc at some x OE OE({l }) as r I ({l }, x OE ) = R I = xoe I )x OE = l (x OE R I = xs I = l (x S )xs, (2) where x S s a socal optmum gven the latency functons {l } I and R s the reservaton utlty. In other words, our effcency metrc s the rato of the socal surplus n an equlbrum relatve to the surplus n the socal optmum. Followng the lterature on the prce of anarchy, n partcular [9], we are nterested n the worst performance n an olgopoly equlbrum, so we look for a lower bound on nf nf {l } L r I ({l }, x OE ). I x OE OE({l }) We frst prove two lemmas, whch reduce the set of latency functons that need to be consdered n boundng the effcency metrc. The next lemma allows us to use the olgopoly prce characterzaton gven n Proposton 9 Lemma 7 Let (p OE, x OE ) be a pure strategy OE such that p OE x OE = 0 for all I. Then x OE s a socal optmum. Proof. We frst show that l (x OE ) = 0 for all I. Assume that l (x OE ) > 0 for some I. Ths mples that x OE > 0 and therefore p OE = 0. Snce l (x OE ) > 0, t follows by Lemma 2 that for all x W (p), we have x = x OE. Consder ncreasng p OE to some small ɛ > 0. By the upper semcontnuty of W (p), t follows that for all x W (ɛ, p OE ), we have x x OE < δ for some δ > 0. Moreover, by Lemma 2, we have, for all x W (ɛ, p OE ), x x OE for all. Hence, the proft of the provder that owns 7 More explctly, Assumpton 2 mples that f any OE (p OE, x OE ) assocated wth {l } I has x OE 0 and l (x OE ) = 0, then I s = {}. 20 >

22 lnk s strctly hgher at prce vector (ɛ, p OE ) than at p OE, contradctng the fact that (p OE, x OE ) s an OE. Clearly x OE > 0 for some and hence mn I {p OE whch mples by Lemma 4 that I xoe all, we have + l (x OE = d. Usng l (x OE R l (x OE ) x OE g l = R, I, )} = p OE + l (x OE ) = 0, ) for ) = 0, and 0 l (x OE for some g l l (x OE ). Hence, x OE satsfes the suffcent optmalty condtons for a socal optmum [cf. Eq. (7) wth λ S = R], and the result follows. Q.E.D. The next lemma allows us to assume wthout loss of generalty that R I I = l (x S )x S > 0 and I = xoe = d n the subsequent analyss. Lemma 8 Let {l } I L I. Assume that ether () I = l (x S )x S = R I = xs for some socal optmum x s, or () I = xoe < d for some x OE OE({l }). Then every x OE OE({l }) s a socal optmum, mplyng that r I ({l }, x OE ) =. = xs Proof. Assume that I = l (x S )x S = R I = xs. Snce x S s a socal optmum and every x OE OE({l }) s a feasble soluton to the socal problem [problem (6)], we have 0 = (R l (x S ))x S = = (R l (x OE ))x OE, x OE OE({l }). By the defnton of a WE, we have x OE 0 and R l (x OE ) p OE 0 (where p OE s the prce of lnk at the OE) for all. Ths combned wth the precedng relaton shows that x OE s a socal optmum. Assume next that I = xoe < d for some x OE OE({l }). Let p OE be the assocated OE prce. Assume that p OE x OE > 0 for some I (otherwse we are done by Lemma 7). Snce I = xoe < d, we have by Lemma 4 that mn I {p + l (x OE } = R. Moreover, by Lemma 5, t follows that p x OE > 0 for all I. Hence, for all s S, ((p OE ) Is, x OE ) s an optmal soluton of the problem maxmze ((p ) Is,x) Substtutng for (p ) Is p x I s subect to p + l (x ) = R, I s, p OE + l (x ) = R, / I s, x OE d. = n the above, we obtan maxmze x 0 (R l (x ))x I s 2

23 subect to x T, / I s, x OE d, = where T = {x p OE + l (x ) = R} s ether a sngleton or a closed nterval. Snce ths s a convex problem, usng the optmalty condtons, we obtan R l (x OE ) x OE g l = 0, I s, s S, where g l l (x OE ). By Eq. (7), t follows that x OE s a socal optmum. Q.E.D. Hence, n fndng a lower bound on the effcency metrc, we can restrct ourselves, wthout loss of generalty, to latency functons {l } L I such that I = l (x S )x S < R I = xs for some socal optmum x S, and I = xoe = d for all x OE OE({l }). By the followng lemma, we can also assume that I = xs = d. Lemma 9 For a set of latency functons {l } I, let Assumpton hold. Let (p OE, x OE ) be an OE and x S be a socal optmum. Then I x OE I Proof. Assume to arrve at a contradcton that I xoe > I xs. Ths mples that x OE > x S for some. We also have l (x OE ) > l (x S ). (Otherwse, we would have l (x S ) = l (x S ) = 0, whch yelds a contradcton by the optmalty condtons (7) and the fact that I xs < d). Usng the optmalty condtons (2) and (7), we obtan x S. R l (x OE ) p OE R l (x S ) x S g l, for some g l l (x S ). Combnng the precedng wth l (x OE ) (cf. Proposton 9), we see that x OE l (xoe ) > l (x S ) and p OE x OE l (xoe ) < x S g l, contradctng x OE > x S and completng the proof. Q.E.D. 5.. Two Lnks We frst consder a parallel lnk network wth two lnks owned by two servce provders. The next theorem provdes a tght lower bound on r 2 ({l }, x OE ) [cf. Eq. (2)]. In the followng, we assume wthout loss of generalty that d =. Also recall that latency functons n L 2 satsfy Assumptons, 2, and 3. Theorem Consder a two lnk network where each lnk s owned by a dfferent provder. Then r 2 ({l }, x OE ) 5 6, {l } =,2 L 2, x OE OE({l }), (22) 22

24 and the bound s tght,.e., there exsts {l } =,2 L 2 and x OE OE({l }) that attans the lower bound n Eq. (22). Proof. The proof follows a number of steps: Step : We are nterested n fndng a lower bound for the problem nf nf {l } L r 2 ({l }, x OE ). (23) 2 x OE OE({l }) Gven {l } L 2, let x OE OE({l }) and let x S be a socal optmum. By Lemmas 8 and 9, we can assume that 2 = xoe = 2 = xs =. Ths mples that there exsts some such that x OE < x S. Snce the problem s symmetrc, we can restrct ourselves to {l } L 2 such that x OE < x S. We clam where r OE 2,t = mnmze l S, (l S ) 0 l, l 0 y S, yoe 0 nf nf {l } L r 2 ({l }, x OE ) r2,t OE, (24) 2 x OE OE({l }) R l y OE l 2 y OE 2 R l S y S l S 2 y S 2 subect to l S y S (l S ), =, 2, (25) (E) l S 2 + y S 2 (l S 2 ) = l S + y S (l S ), (26) l S + y S (l S ) R, (27) 2 y S =, (28) = l l S, (29) l = 0, f l S = 0. (30) l y OE l, =, 2, (3) 2 y OE =, (32) = + {Olgopoly Equlbrum Constrants} t, t =, 2. Problem (E) can be vewed as a fnte dmensonal problem that captures the equlbrum and the socal optmum characterstcs of the nfnte dmensonal problem gven n Eq. (23). Ths mples that nstead of optmzng over the entre functon l, we optmze over the possble values of l ( ) and l ( ) at the equlbrum and the socal optmum, whch we denote by l, l, l S, (l S ) [.e., (l S ) s a varable that represents all possble values of g l l (y S )]. The constrants of the problem guarantee that these values satsfy the necessary optmalty condtons for a socal optmum and an OE. In partcular, condtons (25) and (3) capture the convexty assumpton on l ( ) by relatng the values l, l and l S, (l S ) [note that the assumpton l (0) = 0 s essental here]. Condton (26) s the 23

25 optmalty condton for the socal optmum. Condton (29) captures the nondecreasng assumpton on the latency functons; snce we are consderng {l } such that x OE < x S, we must have l l S. Condton (30) captures the relaton of l ( ) and l ( ) at x OE ; snce x OE < x S, the fact that l (x OE ) = l (x S ) = 0 mples, by Assumpton, that 0 s the unque element of l (x OE ). Fnally, the last set of constrants are the necessary condtons for a pure strategy OE. These are wrtten separately for t =, 2, for the two cases characterzed n Proposton 9, gvng us two bounds, whch we wll show to be equal. More explctly, the Olgopoly Equlbrum constrants are gven by: For t = : [correspondng to a lower bound for pure strategy OE, (p OE, y OE ), wth mn {p OE + l (y OE )} < R], where l = l (y OE ), l 2 = l 2(y OE 2 ) [cf. Eq. (6)]. l + y OE [l + l 2] = l 2 + y2 OE [l + l 2], (33) l + y OE [l + l 2] R, For t = 2: [correspondng to a lower bound for pure strategy OE, (p OE, y OE ), wth mn {p OE + l (y OE )} = R], where l = l + (y OE ) and l 2 = l 2 (y OE r OE 2, = r OE 2,2. R l 2 y2 OE l 2, (34) R l y OE [l + l 2], 2 ) [cf. Eqs. (7), (8)]. We wll show n Step 4 that Note that gven any feasble soluton of problem (23), we have a feasble soluton for problem (E) wth the same obectve functon value. Therefore, the optmum value of problem (E) s ndeed a lower bound on the optmum value of problem (23). Step 2: Let (l S, y S ) =,2 satsfy Eqs. (25)-(28). We show that Usng Eqs. (26), (27), and (28), we obtan l S y S + l S 2 y S 2 < R. (35) l S y S + l S 2 y S 2 + (y S ) 2 (l S ) + (y S 2 ) 2 (l S 2 ) R. If (y S ) 2 (l S ) + (y2 S ) 2 (l2 S ) > 0, then the result follows. If (y S ) 2 (l S ) + (y2 S ) 2 (l2 S ) = 0, then we have usng Eq. (25) that l S =0 for all, agan showng the result. Next, let (l, y OE ) =,2 satsfy Eq. (32) and one of the Olgopoly Equlbrum constrants [.e., Eqs. (33) or (34)]. Usng a smlar argument, we can show that l y OE + l 2 y OE 2 < R. (36) Step 3: To solve problem (E), we frst relax the last constrant [Eq. (30)]. Let ( l S, ( l S ), l, l, ȳ S, ȳ OE ) denote the optmal soluton of the relaxed problem [problem (E) wthout constrant (30)]. We show that l S = 0 for =, 2. 24