The Price of Anarchy in a Network Pricing Game

Transcription

1 The Prce of Anarchy n a Network Prcng Game John Muaccho and Shuang Wu Abtract We analyze a game theoretc model of competng network ervce provder that trategcally prce ther ervce n the preence of elatc uer demand Demand elatc n that t dmnhe both wth hgher prce and congeton The model we tudy baed on a model frt propoed and tuded by Acemoglu and Ozdaglar and later extended by Hayrapetyan, Tardo, and Wexler to conder elatc uer demand We conder the prce of anarchy, whch we defne a the rato of the ocal welfare of the ytem when a ocal planner chooe lnk prce veru the ocal welfare attaned when lnk owner chooe the lnk prce elfhly Ozdaglar ha recently hown that the prce of anarchy n the network prcng game wth elatc demand no more than 5 We have ndependently derved the ame reult In contrat to Ozdaglar proof baed on mathematcal programmng technque, our proof ue lnear algebra and motvated by makng an analogy to a network of retor Our technque ueful becaue t provde an ntutve explanaton for the reult, a well a provdng a framework from whch to derve extenon to the reult I MOTIVATION & INTRODUCTION We tudy a a network prcng model frt propoed and tuded by Acemoglu and Ozdaglar [ and later extended by Hayrapetyan, Tardo, and Wexler [ The model tude the prcng behavor of everal provder competng to offer uer connectvty between two node, and t ha the feature that a provder lnk become le attractve a t become congeted and that uer demand elatc uer wll chooe not to ue any lnk f the um of the prce and latency of the avalable lnk too hgh In the frt veron of the model tuded by Acemoglu and Ozdaglar, the uer elatcty modeled by aumng that all uer have a ngle reervaton utlty and that f the bet avalable prce plu latency exceed th level, uer do not ue any ervce In th ettng, the author fnd that the prce of anarchy the wort cae rato of ocal welfare acheved by a ocal planner choong prce to the ocal welfare arng when provder trategcally chooe ther prce ( + )/ [ (Or expreed the other way, a the rato of welfare n Nah equlbrum to ocal optmum, the rato ) Hayrapetyan, Tardo, and Wexler conder the model where uer demand elatc, whch the form of the model we tudy n the preent paper [ They derved the frt bound on the prce of anarchy of th form of the model, and fnd a bound of 5064 Recently, Ozdaglar ha proved that the bound actually 5, and furthermore that th bound tght [3 Ozdaglar dervaton ue technque of mathematcal programmng, Reearch upported n part by NSF Grant ANI J Muaccho and S Wu are wth the Technology and Informaton Management Program, Unverty of Calforna, Santa Cruz, Santa Cruz, CA 95064, USA {johnm,wu}@oeucccom and mlar to the technque ued n [ In the preent paper, we provde an ndependent dervaton of the ame reult In contrat to the proof of [3, our proof motvated by makng an analogy between the network prcng game and an electrcal crcut where each branch repreent a provder lnk and the current repreent flow, and ue technque from lnear algebra Our technque ueful becaue t provde an ntutve explanaton for the reult, a well a provdng a framework from whch to derve extenon to the reult The model of [ and [ are part of a tream of recent lterature tudyng the prce of anarchy for game of elfh routng Several work, ee for example [4, [5, [6, tudy game where uer are elfh, but the network pave, the owner of the edge are not trategcally choong ther prce Other work uch a [7 conder the problem of havng the network choong prce to nduce optmal routng among elfh uer, rather than havng part of the network elfhly chooe prce to maxmze revenue Another related work by Johar and Ttkl [8 In [8 the author tudy game where the uer requet a bt rate from the network, and n turn the network return to uer a prce that depend on the um of the requeted rate on each reource The model of [8 and the model of th paper tudy very dfferent tuaton In partcular n [8 the man trategc agent are the uer and n the model we tudy the trategc agent are lnk operator eekng to maxmze proft However, there are many mlarte between the two game n term of the tructure of the payoff functon of the player A n th paper, the author of [8 derve bound on the effcency lo A Model We conder a network contng of a ngle ource and detnaton node The node are connected by n lnk, each of whch owned by a dtnct elfh provder Provder charge a prce p per unt flow Each lnk ha a load dependent latency of l (f ) where f the flow on lnk and where l ( ) a convex functon We wll pay partcular attenton to the lnear cae where l (f ) = a f + b The dutlty of each lnk the um of latency and prce and therefore l (f ) + p Uer are nonatomc and are free to chooe the lnk that ha the lowet dutlty Therefore, n equlbrum, all ued lnk have a common dutlty value Uer have a lmt to how much dutlty they wll tolerate f all the lnk have a dutlty hgher than a uer tolerance or wllngne to uffer, that uer doe not ue any lnk Uer are dtrbuted n ther wllngne to uffer, o we may defne a functon U(x) to be the dutlty that would nduce a total flow of x acro all lnk To decrbe t another

2 way, uppoe each uer ha a wllngne to uffer that ndependent and dentcally dtrbuted lke the random varable W Let S be the total populaton of uer and let R(d) = SP(W > d) be S tme the complementary cumulatve dtrbuton functon of wllngne to uffer W Then U(x) the nvere functon of R(d) Clearly U(x) decreang We make the addtonal aumpton that U(x) concave The aumpton that U(x) concave a trong aumpton, but neceary to derve the bound n th work The author of [3 and [ make th aumpton a well We refer to an ntance of the network prcng game a G An ntance G pecfed by the collecton of latency functon {l ( )} and the dutlty functon U( ) We are ntereted n tudyng two confguraton of the ytem G In the frt confguraton, the lnk owner adjut ther prce non-cooperatvely untl Nah equlbrum acheved In the ocal optmum confguraton, a ocal planner chooe all prce to maxmze ocal welfare In a ene the ocal optmum confguraton a type of Nah equlbrum, n partcular a Wardrop equlbrum, where the non atomc uer are free to be trategc whle the lnk are agned prce by a ocal planner However to avod confuon, we wll refer to th tuaton a mply the ocal optmum confguraton The ocal welfare of the ytem defned to be the proft of provder plu the utlty gathered by uer The utlty gathered by uer found by ntegratng the dfference between each unt of flow wllngne to tolerate dutlty, and the dutlty the flow actually bear Therefore the utlty gathered by uer f 0 (U(x) d)dx where f the total flow carred by the ytem, and d the equlbrum dutlty found on all ued lnk II AFFINE LATENCY FUNCTIONS Throughout th ecton we wll aume l (f ) = a f +b and that a > 0 for all In the full paper we wll addre the cae where a = 0 by ung a contnuty argument A Prce-Flow Relatonhp In th ubecton we fnd relaton between the prce of each lnk and the flow on each lnk n both Nah equlbrum and the ocal optmum confguraton By ung thee relaton we are able to draw an analogy between the prcng game and a network of retor In the analogou retor network, provder chooe the retance on ther branch of the crcut In Nah equlbrum, provder pck a retance value that hgher than the ocal optmum value, caung current (flow) to be le than the ocally optmal value We tart by repeatng a reult found n [, that a Nah equlbrum ext Lemma : The network prcng game wth affne latency functon ha a pure trategy Nah equlbrum Proof: The proof found n [ The proof how that player bet repone functon are well defned, unque and contnuou and that therefore Brouwer fxed pont theorem guarantee the extence of a Nah equlbrum Next we characterze the Nah equlbrum, the reult mlar to Lemma of [ Lemma : Conder a game G wth lnear latency functon and where the dutlty functon U( ) contnuou, concave, and everywhere dfferentable Alo uppoe that all lnk are ued n the Nah equlbrum of G Equvalently f > 0 Then p = a + + f a j j where = U (f), and f the total flow n Nah equlbrum Proof: Suppoe the total flow at Nah equlbrum f We defne = U (f) We now derve condton o that no player (lnk owner) want to devate from her Nah equlbrum trategy Suppoe player unlaterally lower h prce o that the new dutlty d h We conder what effect th ha on the overall flow n the ytem, a well a the flow on each of the other player lnk The total flow n the the ytem ncreae by an amount not more than h becaue q(h) of the concavty aumpton Let h be the actual q(h) amount of the ncreae, and note that lm h 0 h Smlarly, the flow n each other lnk j decreae by h a j, provded that the orgnal flow on lnk j wa at leat h a j To cover the poblty that the orgnal flow were not th large y defne y j (h) = mn (f j a j, h) Then, lm j(h) h 0 h = and the flow n lnk j decreae by yj(h) Thu the flow on lnk y j(h) a j ncreae by j let θ(h) = j + q(h) a j y j (h)/h a j () For notatonal convenence, + q(h)/h The flow ncreae on lnk can now be expreed mply a hθ The new prce p can be found by takng the dfference between the new dutlty and the new latency on lnk Th dfference p = (d h) a f + y j (h) + q(h) b a j j = p h a h θ(h) where p the orgnal prce We wrte an expreon for the new proft π by takng the product of the new prce and flow whch π = (p h a hθ(h)) (f + hθ(h)) = h ( + a θ(h))θ(h) + h [(p a f )θ(h) f + π We would lke to fnd condton for π not greater than the old proft π for any h The frt order condton requre

3 etal that the lnear term of the above quadratc form n h have a coeffcent of 0 Th n turn requre that V f = p ( j a j + ) + a ( j a j + ) Note that th condton alo uffcent to nure that h = 0 a global maxmum, becaue under th condton, the lnear term n h vanhe and all that left the term that depend on h that ha a negatve coeffcent Reducng the condton further, we fnd that the condton equvalent to p = a + + () f a j j Now conder the ocally optmal prcng We frt gve an ntutve argument why the ocally optmal prce hould be a f where f the flow of lnk n ocal optmum The ocally optmal prcng hould prce the flow o that each uer bear the margnal cot to ocety of each addtonal unt of new flow The latency on each lnk a f + b However, the ocal cot the latency tme the amount of flow bearng that latency Therefore the cot a f +b f Thu the margnal cot a f + b The latency born by uer a f + b Therefore to make the dutlty born by uer reflect margnal ocal cot, the prce hould be et to a f We formalze the argument n the lemma and proof that follow Lemma 3: In the game G the ocal optmum prce vector atfe p = a f for all where f the flow n ocal optmum Proof: If a et of prce were to nduce a flow vector that maxmze ocal welfare, that et of prce would be ocally optmal One way to quantfy ocal welfare to ntegrate the area under the dutlty curve U(x) up to the amount of flow carred, and then ubtract the welfare lot due to the latency n each lnk Therefore the ocally optmal flow can be found by olvng the followng optmzaton problem: [ f max U(x) (a f + b f ) 0 t f f = 0, f 0 [ f Note that the functon 0 U(x) + (a f + b f ) convex and Slater contrant qualfcaton condton hold o there no dualty gap f we ue Lagrangan technque to fnd the optmal oluton [9 We therefore may expre the oluton to the above problem by wrtng the Lagrangan, evaluatng the frt order condton a well a the complementary lackne condton for the Lagrange multpler Fg p b a a P or dv e r P or tf cn y ol op ew r = op ew r = b a a A crcut analogou to the network prcng game aocated wth the nequalty contrant and mplfyng Th yeld { d a f b = 0 f > 0 d b < 0 f = 0 where d = U(f ) the dutlty of the ued lnk n the optmal oluton In our model the dfference between the dutlty and the lnk latency hould be the prce of the lnk The above expreon how that for each ued lnk, the dutlty mnu the latency a f +b equal to a f, therefore the prce that acheve a ocally optmal flow are a f The proof of Lemma 3 alo demontrate that the optmal prce acheve the optmal flow vector In other word, f we were to gve the ocal planner the power to agn uer route (chooe the flow vector) the planner could not acheve a better welfare than by merely choong the lnk prce B Crcut Analogy Before we make the analogy between the game and a network of retor, we ntroduce an llutraton lke the one ued n [ to vualze the relaton between flow f, prce p, latency functon lope a and offet b for each lnk The llutraton hown n panel () of Fgure The fgure how that the prce p plu latency a f + b of each lnk equal to the common value, U(f + f + f 3 ) = U(f) The fgure alo how the area that correpond to lnk owner proft and uer urplu b 3 3 a 3 a 3

4 ) ) d p - d b 3 ) d a f f f 3 v) d d* f b 3 a f f* Uer Surplu Provder Surplu Fg ) The Nah equlbrum of the game G ) The Nah equlbrum of the game G l, where the dutlty functon of G ha been lnearzed Note that the flow vector unchanged from the Nah equlbrum of G ) The Nah equlbrum of the game G t, where the dutlty functon of G ha been lnearzed and truncated v) The ocal optmum confguraton of the game G t Note that the flow f > f and that lnk not ued n the ocal optmum confguraton of th example We llutrate our analogy to a network of retor n Fgure In the analogy, each lnk one of n parallel branche n a crcut The current n the branch analogou to the flow on lnk and the latency analogou to the voltage drop acro a retor of a ohm and voltage ource of b volt connected n ere To enure that our magnary voltage ource doe not ever caue flow to go the wrong way (become negatve), we place a dode n each branch of the crcut The prce p found by takng the voltage drop acro a retor of δ + a ohm where δ = + a j j = p f a (3) The prce plu the latency on a lnk equal the common dutlty found acro all ued lnk In the crcut analogy, the voltage repreentng prce plu the voltage repreentng latency equal the common voltage d acro all the parallel branche where d = p + a f + b = (a + δ )f + b The value d hould alo correpond to the pont on the value U(f), and there a crcut analogy for th correpondence a well Suppoe we let V be the y-ntercept of the tangent to the dutlty curve at the pont U(f) = d Then V f = d Th analogou to havng a battery of V volt wth an nternal retance of ohm A the current drawn from the battery ncreae, the voltage output (analogou to the wllngne to tolerate dutlty) decreae Alternatvely, we could avod lnearzng by conderng a general power ource wth a nonlnear current to voltage charactertc In partcular the current to voltage functon hould matche the dutlty functon U( ) whch of coure relate flow to dutlty To model the ocal optmum confguraton, we keep the general power ource but replace the retor modellng the prce-to-flow relatonhp from a + δ to a on each branch Clearly, by decreang the retance n all branche, the total amount of current (flow) hould ncreae, and the voltage drop (dutlty d ) on all branche hould decreae However, the rato of the flow between branche can change, becaue the rato of the retance change between the two cae Therefore t poble that the flow on a partcular branch decreae n ocal optmum The flow on a branch can even become zero f d b We alo note that a branch wth a low enough b to carry a nonzero flow n the ocal optmum confguraton hould alo carry a nonzero flow n the Nah equlbrum confguraton In other word, a lnk that ued n ocal optmum hould be ued n Nah equlbrum Intuton ugget that f δ << a were mall for all, the dfference between ocal optmum and Nah equlbrum hould alo be mall Indeed δ maller than a j for all j However, for the mallet a, t could be that δ > a

5 Thu we cannot attempt to how that δ neglgbly mall In the next ecton, we wll decrbe the relaton between the δ and a, ung lnear algebra n order to develop our proof on the prce of anarchy bound C The Prce of Anarchy for Lnear Latency In th ecton we wll how that the prce of anarchy at mot 5 for lnear latency functon, where prce of anarchy defned to be the rato of welfare n ocal optmum to the welfare n Nah equlbrum We wll frt argue that we can retrct our attenton to example where the dutlty functon ha been lnearzed for value larger or equal to the equlbrum flow f and ha been truncated or et to be equal a contant for maller value of flow Th argument alo made n [, but preented agan here for completene We llutrate the progreon from the orgnal game, to a game wth a lnearzed dutlty functon, and fnally to a truncated dutlty functon n Fgure Once we have hown that we may retrct our analy to truncated lnear dutlty functon, we ue technque of lnear algebra to prove the fnal reult Before we begn to prove the 5 bound of the prce of anarchy, we gve an example where the prce of anarchy exactly 5 Conder a network contng of only one lnk where the dutlty functon { f 0 U(f) = f f and the latency functon l(f) = 0 It eay to verfy that the Nah equlbrum prce p = whch yeld a proft of for the provder and 0 uer urplu The ocally optmal prce p = 0 whch yeld a proft of 0 for the provder, but a uer urplu of 3/ Therefore the prce of anarchy n th example 5 The ret of th ecton wll prove that there not an example where the prce of anarchy larger than th Lemma 4: Let f be the total Nah equlbrum flow of the game G Let G l be a new game dentcal to G except that the dutlty functon of the new game a lne tangental to the dutlty functon of G at the pont U(f) = d Then both the Nah equlbrum flow and prce vector of the new game G l are the ame a the Nah flow and prce vector of the old game G Proof: The Nah flow vector of G atfe the neceary and uffcent condton n Lemma to be a Nah equlbrum Becaue the lope of the dutlty functon of G l the ame when the total flow f, the Lemma condton for G l evaluated at the Nah flow vector of G are atfed Thu the Nah flow vector of G alo a Nah flow for G l Lemma 5: Conder the game G t that cont of the ame latency functon a G and G l but ha a dutlty curve that modfed from the dutlty curve n G l by truncatng t n the followng way: { d x f U(x) = d (x f) x > f where = U (f) Then the Nah equlbrum flow and prce vector of the new game G t are the ame a the Nah flow and prce vector of the orgnal game G Proof: The argument eentally the ame a preented n [ Recall that Lemma found that relaton () hold n Nah equlbrum whenever the dutlty functon concave and everywhere dfferentable To prove th reult, we mut how that () tll hold for the truncated dutlty functon Whle the truncated dutlty functon no longer both-ded dfferentable at the equlbrum pont, t rght hand dfferentable The frt order condton on flow f found wth the rght hand dervatve almot the ame a wa found n Lemma but ntead t hold wth nequalty More precely f p ( j a j + ) + a ( j a j + ) for all the neceary and uffcent tet to dentfy a Nah equlbrum of game G t However, the Nah equlbrum flow vector for game G, mut atfy the above condton wth equalty, therefore t alo a Nah equlbrum of G t Lemma 6: Let N(G) and S(G) be the ocal welfare n the Nah equlbrum and ocal optmum confguraton of the game G Then S(G t ) N(G t ) > S(G) N(G) Proof: Let U(G) be the uer welfare n the Nah equlbrum of game G In the Nah equlbrum of game G t, Lemma 6 how that the flow vector wll be the ame a the Nah flow of G Thu the provder welfare wll be the ame n both cae However, the uer welfare wll be 0 Thu we have N(G t ) = N(G) U(G) Conder a new game G tx wth the ame latency functon a G but wth a dutlty functon that ha been truncated but not lnearzed Therefore { d x f U tx (x) = U(x) x > f where d and f are the Nah equlbrum dutlty and flow of G, U tx ( ) the dutlty functon of G tx, and U( ) the dutlty functon of G Becaue we know the dutlty d n ocal optmum le than d we can vew the optmzaton problem of fndng the optmal flow vector for game G tx a the ame problem a fndng the optmal flow vector for the game G but wth the contant U(G) = f (U(x) d)dx ubtracted off the objectve functon Thu, the ocal optmum 0 flow vector wll be dentcal for game G tx a for game G t Thu S(G tx ) = S(G) U(G) Now conder the game G t where we have both truncated and lnearzed the dutlty functon By convexty, the dutlty functon of G t never maller than that of G tx Thu the oluton to the optmzaton problem of fndng the optmal flow vector ha an objectve functon that never

6 maller than the objectve functon that we maxmzed to fnd S(G tx ) Conequently S(G t ) S(G tx ) = S(G) U(G) Combnng our obervaton, we have S(G t ) S(G) U(G) N(G t ) N(G) U(G) S(G) N(G) We defne the followng notaton The vector of flow n Nah equlbrum and ocal optmum are F = [f, f,, f n T, and F = [f, f,, f n T repectvely We defne V = [V, V, V T, and b = [b, b, b n T a an n dmenonal vector of all V and the vector of b repectvely The matrce A = dag(a, a,, a n ), and = dag(δ, δ,, δ n ) are dagonal matrce of the a and δ repectvely where recall δ defned by 3 It wll alo be convenent to defne M = to be a n n matrx of all one Wthout lo of generalty, we may renumber the lnk o that lnk ued both n Nah equlbrum and n ocal optmum are numbered m and the lnk ued n Nah equlbrum but not ocal optmum are numbered (m + )(m + k) = n There cannot ext lnk ued n ocal optmum that are not ued n Nah equlbrum, whle there could be lnk that are not ued n ocal optmum nor n Nah equlbrum, but we wll only conder example where uch lnk are removed Thu, we may defne Ā,, be the upper m m block of the matrce A and repectvely Thee block contan the a and δ aocated wth lnk that ued both n Nah equlbrum and ocal optmum Smlarly we defne A, to be the lower k k block of A and repectvely Thee block contan the a and δ aocated wth lnk that are ued both n Nah equlbrum but are undercut (not ued) n ocal optmum We therefore have A = [Ā, 0 0, A and = [, 0 0, Theorem : Conder the prcng game G t wth lnear latency functon, and a truncated lnear dutlty functon The dfference between three tme the Nah welfare W t = N(G t ) and two tme the ocal welfare Wt = S(G t) can be expreed a [ 3W t Wt = [ F T F T H H H H [ F F (4) where H = Ā + (Ā + M), H = (Ā + M) m k, H = H T, and H = 3A k k k m (Ā + M) m k Proof: We have that (A + M + )F = V b (5) We can wrte a mlar expreon for the flow n ocal optmum cae, wth the followng modfcaton In ocal optmum, the dutlty d may fall below b for ome, o that ome lnk that were ued n Nah equlbrum may become not ued n the ocal optmum cae Wthout lo of generalty, we renumber the lnk o that the the ued lnk number m and the unued lnk number (m + )(m + k) = n Let Ā, M,, be the upper m by m block of the matrce A, M and repectvely We therefore have [Ā + M, 0 m k F = [I m m, 0 m k (V b) or equvalently F = [ ( Ā + M) 0 m k (V b) 0 k m 0 k k By ubttutng (5) we fnd [ F ( Ā + = M) 0 m k 0 k m 0 k k [ Ā + M + m k F A + M + k m whch reduce to [ I + ( F Ā + = M) ( Ā + M) m k F 0 k m 0 k k Wth a lnear (not truncated) dutlty curve, the total ocal welfare gathered n Nah equlbrum gven by W = F T (A + M + )F whle the welfare gathered by the uer F T MF Therefore the Nah equlbrum ocal welfare for a truncated lnear demand curve W t = F T (A + )F (6) For a lnear (not truncated) dutlty functon the ocal welfare gathered n the optmal confguraton gven by W = F T (A + M)F Whle for a lnear truncated dutlty functon, the ocal welfare n the optmal confguraton W t = F T (Ā + M)F F T ( M)F = F T [ (Ā + M) +, m k F F T (M)F = [ F T G G F (7) G G

7 where G = Ā + + (Ā + M), G = (Ā + M) m k, G = G T, and G = k m (Ā + M) m k k k The Nah equlbrum ocal welfare, taken from expreon (6) but expreed ung the block matrx notaton [Ā + 0 W t = F T F (8) 0 A + We can ue expreon (7) and (8) to expre three tme the Nah welfare mnu two tme the ocal optmal welfare a [ [ 3W t Wt = [ F T F T H H F H H F where H = Ā + (Ā + M), H = (Ā + M) m k, H = H T, and H = 3A k k k m (Ā + M) m k Lemma 7: Defne β / + tr(ā ), α / + tr(a ), and e m Provded that a > 0 for all, the followng dentte are true: (Ā + M) = [Ā β Ā MĀ = (αā I) Ā (Ā + M) e = β (αā I) e e T ( A + M) e = tr(a ) β Proof: The Matrx Inveron Lemma tate that (F + UCV ) = F F U(C + V F U) V F where F, U, C, V are arbtrary matrce wth only the condton that the matrce are of approprate dmenon and all the nvere n the above expreon ext Let F = Ā, U = e, V = e T, C = / and note that M = ee T = m m Then we may apply the Matrx Inveron Lemma n the followng way: (Ā + M) = Ā 4Ā e + et Ā e e T Ā [ = [ Ā Ā e + tr(ā ) e T Ā = [Ā β Ā MĀ To derve the next dentty, recall δ = Therefore = [ tr(a ) + Ā = αi Ā [ j a + Thu Ā = αā I, and therefore Ā = (αā I) Note by contructon, α > / mn a o that (αā I) ext Fnally we may conclude that = (αā I) Ā To derve the next dentty we oberve that: (Ā + M) e = (αā I) e β [αā I ee T Ā e = [ (αā I) e tr(ā ) tr(ā ) + / (αā I) e = β (αā I) e The fnal dentty follow from the followng reaonng: e T (Ā + M) e = [ e T Ā e β et Ā ee T Ā e = [tr(a ) tr(a ) tr(a ) + / = tr(a ) β Wth the dentte of Lemma 7, we can now prove the man reult of the ecton Theorem : Conder the prcng game G wth affne latency functon The prce of anarchy, defned a S(G) N(G) at mot 5 Proof: By Lemma 6 t uffcent to how that the prce of anarchy for a game wth a truncated lnear dutlty functon 5 Wthout lo of generalty, we may aume that all lnk are ued n Nah equlbrum, becaue f otherwe, a lnk not ued n Nah equlbrum would not be ued n Socal Optmum Therefore one can dcard the lnk unued n Nah equlbrum from the game wthout changng the prce of anarchy Applyng the dentte n Lemma 7 to the bound of Theorem, we have H = (αā I) (α Ā 3 αā Ā + β H = β (αā I) e m e k H = H T H k k e k e T m( A + M) e m e T k = + tr(ā ) β = β k k k k tr(ā ) k k β M)(αĀ I) where recall that H, H, H, and H are the block of the matrc n expreon (4)

8 Let Q (αā I) F and Z F = e k F Note that Q > 0 becaue αa > for all By changng varable from F and F to Q and Z we obtan that 3W t Wt [ α [Q T, Z T Ā 3 αā Ā + β et m = β Q β Q Z + β Z + β M β e [ m Q Z β Q T (α Ā 3 αā (9) Ā)Q Note that the quadratc Q Q Z+Z nonnegatve Conder the followng two cae Cae : Suppoe α Ā 3 αā Ā 0, or equvalently αā j + 3 for each ā j on the dagonal of Ā Then 3W t Wt > 0 Cae : Suppoe that for ome j, αā j < + 3 Let x αā j Let j be the ndex of any uch ā j We therefore have αā j = x + < + 3, (0) We dvde both de of the equalty by ā j, and ubttute that α tr(a ) + = tr(ā ) + tr(a ) + to obtan that x + ā j = + ā a + where {a } are the dagonal entre of A Thu Defne x ā j = =j ā j ā =j + ā a a + [ a To extend our crcut analogy, ā j the equvalent retance of all of the retor {ā } =j connected n parallel whle a the equvalent retance of the retor {a } connected n parallel After ubttutng thee defnton, we have that and thu Recall that x ā j = ā j + a + ā j = x + ā j a + () δ j = + ā j a + From thee lat two relaton, we have that (ā j + δ j ) = (x + ) + ā j a + We wll now fnd an upper bound on the total Nah equlbrum flow on lnk that are not ued n ocal optmum Thu F = d b < [d mn(b a + δ ) a = [d mn(b a ) We alo note that Conequently, F j = d b j ā j + δ j F [d mn (b ) F j d b j ā j + δ j a We note that b > b j for any becaue lnk that are ued n ocal optmum mut have a lower b than lnk that are not ued n ocal optmum Therefore F F j We alo oberve that ā j + δ j a = x + a Q = e T (αā I) F = + ā j a + F αā Thu f we recall Z F, we obtan Let Z (x + )x Q a υ = (x + )x a ā j + a + F j αā j = F j x + ā j a + Then Z < υ Q If Z [0, Q then Q Q Z + Z decreae wth Z From the expreon above we ee that Υ < (x + )x, whle we alo have that x < 3 < 0367 from (0) Th mple υ < 3 < 5 and therefore Z mut le n the range where Q Q Z + Z decreae n Z Thu we have Q Q Z + Z > Q ( υ + υ ) = Q T ( ( υ + υ )M ) Q After we ubttute the above bound nto (9) we obtan 3W t W > Q T (( β υ β υ + β )M + α Ā 3 αā Ā)Q

9 We defne Λ to be the matrx between Q T and Q n the above expreon Therefore Λ (( β υ β υ + β )M + α Ā 3 αā Ā) Note that t uffcent to how that all entre of Λ are non-negatve becaue Q non-negatve The off-dagonal element of Λ are clearly potve Any dagonal element k atfyng αā k + 3 alo non-negatve becaue th condton make α ā 3 k αā k āk non-negatve and ( β υ β υ + β ) alway nonneagtve Thu we can focu on dagonal element of ndex j atfyng αā j < + 3 and recall we defned ndex j to be any arbtrary ndex for whch αā j < + 3 atfed Thu t uffcent to how that jth dagonal element whch λ jj = β υ β υ + β + α ā 3 j αā j āj () nonnegatve Equvalently, we need to how that λ jj β = υ υ + / + (x + x /)(βā j ) (3) potve Note that we have ued the fact that αa j = x+ We wll frt derve an expreon for the (x + x /)(βā j ) term Recall that β = ā j + ā j + We can ubttute relaton () to obtan β = a(x + ) + ā ja(x + ) + ā j, xā j a whle we alo note that Therefore ā j = x + ā j a + = xā j a ā j a + a + ā j βā j = a(x + ) + ā ja(x + ) + ā j ā j a + a + ā j Later on t wll be ueful to have th expreon wth a denomnator of (ā j a + a + ā j ) By multlplyng numerator and denomnator by a common expreon we obtan (a [x a[x + )ā j + ( a[x + + a [3x + )ā j + βā j = a [x + (ā j a + a + ā j ) We may now wrte the term of (3) we eek to evaluate a (x + x 5)βā j = ( ) [x + x 5 + a[x 3 + 4x + x + a [x 3 + 3x ā 5 j + ( ) a[x 3 + 3x + 5x + a [3x 3 + 5x ā + 05x j + a [x 3 + x + 5x 5 (ā j a + a + ā j ) (4) We now turn to dervng an expreon for the υ υ+/ term of (3) Recall υ = Th reduce to (x + )x a + ā j a + υ = (x + /)xā j ā j a + a + ā j (5) In order to have the denomnator the ame a that of expreon (4) we multply numerator and denomnator by a common expreon to obtan υ = (x + /)x( + a)ā j + a(x + /)xā j (ā j a + a + ā j ) (6) Whle quarng (5) yeld υ = [x 4 + x 3 + x /4ā j (ā j a + a + ā j ) (7) Ung expreon (6) and (7) we have that υ υ + 5 ( ) [x 4 + x 3 5x x+ a[ x ā x j + a( x x)ā j = (ā j a + a + ā j ) + 5 ( ) [x 4 + x 3 5x x + 5+ a[ x x + + 5a ā j + ( a[ x x + + a )ā j + 5 a = (ā j a + a + ā j ) (8)

10 By combnng (4) and (8) we obtan λ jj β = ( ) [x 4 + x 3 5x + a[x 3 + x + a [x 3 + 3x ā j + ( ) a[x 3 + x + 5x+ a [3x 3 + 5x ā + 05x j + a [x 3 + x + 5x (ā j a + a + ā j ) We ee that the only term that contan a mnu gn n the above expreon x 4 +x 3 5x, but th potve for x > 0 Therefore we may conclude that λ jj potve III CONVEX LATENCY FUNCTIONS In th ecton we extend the reult of Theorem to convex latency functon For convex latency functon, t poble that a pure trategy Nah equlbrum doe not ext The author of [ provde uch an example Therefore, our theorem apple only n the cae that a pure trategy Nah equlbrum ext Theorem 3: Conder the prcng game G wth convex and dfferentable latency functon Provded that G ha a pure trategy Nah equlbrum, the prce of anarchy defned a S(G) N(G), at mot 5 Proof: Let F be a Nah equlbrum flow vector for G If there are more than one pure trategy Nah equlbra, arbtrarly pck one Let a = d dx l (f ) for each,, n, then p and f mut atfy the frt order condton lad out n Lemma Thu equaton mut hold for all, n Let b = l(f ) a f,,, n Some b mght be negatve, o let B be the magntude of the larget negatve b term, and let b = b + B for all =,, n Now conder a new game G + derved from game G by takng l + (x) l (x) + B,, n and x R +, U + (x) U(x) + B x R + It traghtforward to ee that the Nah equlbrum flow, provder welfare, and uer welfare hould all be the ame n G + a n G Defne a new game G l+ by takng l l+ (x) a x + b,, n and x R + U + (x) U(x) + B x R + The flow F alo the Nah equlbrum of the game G l+ Th becaue the frt order condton that determne whether a F a Nah equlbrum n game G l+ mut alo be by atfed by F to be a Nah equlbrum of G + Furthermore, the provder proft and uer welfare are the ame a n the Nah equlbrum of G + The ocal optmum welfare of G l+ not wore than the ocal optmum welfare of G + Th becaue the convex latency functon of G l+ are never maller than ther tangent, whch were ued to make the latency functon of G + Thu the optmum welfare for G l+ cannot be le than for G + Thu the prce of anarchy of G the ame a that of G + Game G l+ ha the ame Nah welfare a G +, but ha a hgher (not maller) ocal optmal welfare Therefore the prce of anarchy of G l+ a large a the prce of anarchy of G Therefore 5, the wort cae prce of anarchy for game wth lnear latency functon, alo the wort cae prce of anarchy for game wth convex latency functon, provded the latter ha a pure trategy Nah equlbrum IV CONCLUSION We have developed a new technque for provng a bound on the prce of anarchy for a network prcng game wth competton, congeton, and elatc demand We beleve that th technque wll be ueful n tudyng a number of related model and extenon One partcular cae that we hope to tudy n future work one n whch uer vary n the relatve value they place on prce and latency In other word, n the model of the current paper, all uer only conder the um of prce and latency An extenon would be to tudy a model where uer look at a weghted um of the two, and the weght would be dfferent value for dfferent type of uer ACKNOWLEDGMENT We thank Jean Walrand for many ueful converaton about th work REFERENCES [ D Acemoglu and A Ozdaglar, Competton and Effcency n Congeted Market, Mathematc of Operaton Reearch, vol 3, no, pp 3, Feb 007 [ A Hayrapetyan, É Tardo and T Wexler, A Network Prcng Game for Selfh Traffc, to appear n Dtrbuted Computng [3 A Ozdaglar, Prce Competton wth Elatc Traffc, to appear n Network, 006 [4 T Roughgarden, The Prce of Anarchy Independent of the Network Topology, ACM Sympoum on Theory of Computng, pp , 00 [5 T Roughgarden, Stackelberg Schedulng Stratege, ACM Sympoum on Theory of Computng, pp 04-3, 00 [6 T Roughgarden and É Tardo, How Bad I Selfh Routng?, IEEE Sympoum on Foundaton of Computer Scence, pp 93-0, 000 [7 R Cole, Y Dod, T Roughgarden, How much can taxe help elfh routng?, vol 7, no 3, pp , 006 [8 R Johar and JN Ttkl, A Scalable Network Reource Allocaton Mechanm wth Bounded Effcency Lo IEEE Journal on Selected Area n Communcaton, vol 4, no 5, pp , 006 [9 S Boyd and L Vandenberghe, Convex Optmzaton, Cambrdge Unverty Pre, Cambrdge, UK, 004