The Effects of Monoolstc Prcng n a Network Resource Allocaton Game Danelle Hnton : Graduate Semnar n Area November 3, 2004 Introducton There s a recent body of lterature that seeks to quantfy the erformance degradaton of networks n the absence of centralzed control va game theoretc technques. Users are consdered as selfsh agents artcatng n a noncooeratve game who choose routes consstent wth the noton of a Nash Equlbrum. As we saw last week [Kh04], results by Johar and Tstslks have shown that equlbrum erformance degrades to ¾ of the socal otmum, whch could be acheved wth centralzed lannng [JoT03]. Other results by Roughgarden and Tardos [RoT00] show that f the latency of each edge s a lnear functon of ts congeston, the total latency of the routes chosen by selfsh network users s at most 4/3 tmes the mnmum ossble total latency. In the more general case where edge latency s assumed to be contnuous and non-decreasng n the edge congeston, they also rove that latency s no more than the total latency ncurred by otmally routng twce as much traffc. Whle the former models catures the decentralzed routng decsons of users, and the effects of those decsons on erformance; the lterature neglects that the networks are controlled by servce rovders whose obectve sn t oeratng at an effcent socal otmum, but rather s to mamze rofts. The ractcally motvated nuance ncororated n the work of D. Acemoglu and A. Ozdaglar s the consderaton of the effect of monoolstc rcng on congeston n unregulated networks. In Flow Control, Routng, and Performance from Servce Provder Vewont, the authors consder a two staged game where the sngle roft mamzng servce rovder sets rces antcatng the subsequent behavor of users, and then the users takng rces and latency as gven game selfshly to otmze ther own obectves. The authors develo results on the equlbrum behavor of such a network, characterze the equlbrum rces and develo nsghts nto why monooly rcng may mrove the erformance of the system. They also comare the erformance of the Monooly Equlbrum to the socal otmum (full nformaton and zero rces). In the summary whch follows, we frst outlne the network model n secton 2. In Secton 3, we characterze the Network Equlbrum. In Secton 4, we show the dervaton of the otmal monooly rcng scheme and elan ther effects on congeston. In Secton 5 we comare the erformance of the network under monoolzed rcng to the erformance n the centrally controlled, zero rced socal otmum. We conclude wth a summary of the key contrbutons of ths work. 2. Network and Game Theoretc Model
Danelle Hnton The network s modeled as havng I arallel lnks, and J users, where J >>. The followng arameters are used to descrbe the network and users: the flow of user on lnk l Γ I = = = ( ) = J the total flow rate of user on all lnks total flow on lnk l flow deendent latency functon whch secfes the delays n transmsson as a functon of the lnk load. -Contnuous and strctly ncreasng - l ( 0 ) = 0 as C Caacty constrant on the lnks u Γ : utlty functon ( ) ( ) rce of lnk The utlty functon s assumed to be strctly concave that s, that we are consderng elastc traffc (delay tolerant) where ncreased data rates has dmnshng returns for users. We also assume that the utlty functon s nondecreasng, and contnuously dfferentable whch allows the authors to characterze the equlbrum, and the equlbrum rces (shown here n sectons 3 and 4). For each user there s also an assocated ayoff functon whch deends not only on that user s own flows, but also on the flows of all the other users va the latency they all create on a gven lnk.. Each user acts as a rce taker and a lnk load taker, takng rces and lnk loads as gven and choosng flows to mamze ther ayoff functon: 3. Network Equlbrum I I v ( ; λ, ) = u l = = I ( ) To determne the equlbrum behavor f ths network, we consder the game between selfsh users who, gven rces, antcate the amount of congeston on all routes and choose ther own routes accordng to the noton of the Nash Equlbrum (NE). Recall that a Nash Equlbrum s a steady state of the lay of a strategc game n whch no layer can roftably devate gven the actons of the other layers. More formally: where a * A of actons, * * * ( a, a ) ( a, a ) a A = 2
Danelle Hnton No layer has an acton yeldng an outcome that he refers to that generated when he chooses a *, gven that every other layer chooses hs equlbrum * acton a. In the model descrbed above, we assumed that the users were both rce and latency takers that s, they take the latency of the lnks as gven. In the NE, users take nto account the effect of ther own flow on congeston. In the Wardro Equlbrum (WE), each user s flow s assumed to be small relatve the total flow on a gven lnk so that when that user swtches hs flow, there s no substantal change n the lnk latences (Wardro Prncle). As such, n the WE, users gnore the effects of ther own flow on congeston. Formally, the WE says that a vector r l, l =,2, KK, L s called a Wardro equlbrum f for each grou l, the followng holds: Let K l * K be the set of routes actually used, that s the set of classes such that: l l l ' l k K k K 0 such that B ( r) B ( r) k K k K : * l, k > k k ' *, So the rncles are fulflled because n ths state no grou can reduce the costs by swtchng from ts current aths to other ones connectng the same orgndestnaton ar. Followng HaM85, the authors rove that as the number of users becomes large, the NE converges to the WE (see 7-9). And so n ths work, the equlbrum of the network s characterzed n terms of the WE. For the model descrbed above, the Generalzed Wardro Equlbrum GWE -- generalzed because t s condtoned on a gven vector of rces -- s defned as the vector of flows of all the users =, K KK, J n the network such that for each : = arg ma v ( y ;, ), J 0 y C, The estence and the unqueness of the GWE s roved by consderng the otmzaton roblem: mamze u ( Γ ) l ( z) dz I 0 I subect to: Γ =, I = J 0, The rooston s roved as follows: Snce the obectve functon s contnuous and the feasble set s comact, ths roblem has an otmal soluton (estence of 3
Danelle Hnton the GWE). Snce u s strctly concave, and l s strctly ncreasng the obectve functon s a strctly concave functon of Γ and (unqueness of the flow rates and lnk loads at a GWE). Usng the essental unqueness of the equlbrum roved above, the authors are able to rove the contnuty and monotoncty of the flow rates and lnk loads whch are used later on n the characterzaton of monooly rces. The monotoncty results are qute ntutve. Flow rates beng monotonc n rces says essentally that when rces are hgher, users choose flow rates that are no hgher than the rates chosen at a lower rce. Smlarly, the monotoncty of lnk loads (Pro 2.4) n rces says formally lnk loads are monotoncally non-ncreasng n ther own rces, and monotoncally non-decreasng n other lnks rces. Intutvely, the monotoncty of lnk loads tells us that hgher rces reduce traffc on the lnk. And, a hgher rces for one lnk ncreases the traffc on the others (mlyng that for users, lnks are erfect substtutes). The authors go on to defne two lemmas whch wll be useful to n the later characterzaton of the GWE. The frst says that the effectve cost of all lnks wth ostve flow must be equal, suortng the ntutve noton from the monotncty of lnk loads that the lnks are erfect substtutes. The second characterzes the rce and roves the estence of a GWE and the corresondng flows at that rce vector. Lemma 2.: For a gven 0, let wth 0 : + l ( ) = mn + m I Proof: By the frst order necessary otmalty condtons, u ( Γ ( )) l ( ( )) 0 f u ( Γ ( )) l be the load of lnk at a GWE. Then for all ( ( )) = 0 m m m { l ( )} f 0 > 0 eqn 9 eqn0 so there ests some such that 0 and satsfes eqn 0. Snce eqn 9 also holds m, combnng eqn 9 and eqn 0 yelds: Lemma 2.2: For a gven 0,, then: m m m + l ( ) + l ( ) m let be a GWE. Let I = { > 0} and J = { Γ > 0} (a) For all I and J, : u ( Γ ) l ( ) = 0 (b) There ests a GWE ~ at ths rce vector that satsfes > 0 for all I and J. 4
Danelle Hnton Proof: s (a) Snce Γ > 0, then there ests some lnk s for whch > 0, and thus eqn m m m holds. By Lemma 2., we have that + l ( ) = + l ( ) snce > 0 s and > 0. Substtutng ths nto equaton roves art (a). (b) The estence of a GWE at ths rce vector s establshed by the argument that startng at any GWE, we can always construct an alternatve GWE wth the same ndvdual flow rates and lnk loads, n whch all ndvdual flows to all lnks wth mnmum effectve cost are ostve. As such, ths alternatve vector satsfes the frst order necessary and suffcent otmalty condtons and s thus a GWE at the rce vector. 4. Monoolstc Prcng The roft mamzng monoolsts sets rces on each lnk of the network wth the vector of rces whch solve the followng mamzaton roblem: I ma ( ) subect to: 0 The estence of ths otmal rce vector follows from the contnuty n of ( ). The vector * s termed the monooly equlbrum rce of the 2-stage game. The ar (*, *) defnes as the monooly equlbrum of the overall game. Some characterstcs of ths otmal rce vector are (roofs omtted):. > 0 otherwse, revenues would be decreased on that lnk, and reduce traffc and thus rofts on all other lnks. 2. > 0 otherwse, roft could be ncreased by addng traffc on ths lnk; and ths ncrease n rofts outweghs the decrease n rofts on the other lnks (see age 22 for the roof) To hel n the characterzaton of the equlbrum rces, establsh Lemma 4.3 whch shows the estence of an otmal soluton (, λ, Γ ) to the followng roblem where (,) s an ME: subect to ma I ' u ( Γ ) l ( ) = 0 u ( Γ ) l ( ) I = J Γ = 0 eqn I J {} 5
0, Γ 0 I, J Danelle Hnton Ths Lemma s roved by contradcton, and shows that the estence of some other feasble soluton wth a greater cost volates the noton of the soluton, λ, Γ ) corresondng to a ME (, ). (See ages 23-4 for the detaled roof). ( One of the key results of ths aer s the characterzaton of Equlbrum rces n Pro 4.8. We wll frst outlne the roof, then dscuss ts mlcatons. We assume that u () * s twce dfferentable for each, and l s contnuously dfferentable for each, and (, ) s an ME, and { Γ > 0} J. Then for I and, we have: = = ( l ) ( ) m m I + eqn 2 J u ( Γ ) '' The roof of ths Prooston uses Lagrange Multlers to solve equaton. From Lemma 4.3 (,, Γ ) s an otmal soluton to ths roblem. The Lagrangan functon for ths roblem assgns multlers λ I, µ J /, µ to each of the constrants resectvely: ' λ ( u ( Γ ) l ( ) ) + µ ( u ( Γ ) l ( ) ) + µ ( L(,, λ ) = I I J I J Γ ) Takng artal dervatves wth resect and λ yeld the followng equatons: = λ + eqn 3 µ J = λ eqn 4 λ l )'( ) ( u )( l )'( ) + = 0 eqn 5 ( µ J λ ( l )'( ) + u = 0 eqn 6 When combned, they reduce to : l )'( ) + µ = 0, ( I Takng the artal dervatve wth resect to Γ yelds: '' u ( Γ ) λ µ = 0 eqn 7 '' µ µ Γ ) = 0, J eqn 8 ( µ 6
Danelle Hnton Summng eqn 3 and eqn 4 overall, and combnng these wth eqn 8 and eqn 9 yelds: When these are substtuted n to eqn 6, we arrve at eqn 2. Equaton 2 shows that the roft mamzng rces for the monoolst conssts of two markus above the margnal cost of the monoolst (whch s zero n ths model). Its ths frst term ( l ) ( ) whch may work to mrove the allocaton of resources. Intutvely, snce the monoolst s aware that the user ayoff functon decreases wth both latency and rce, ths term can be nterreted as the monoolst s nterventon to reduce congeston. The second term n the equaton for the otmal monooly rcng s smlar to the standard marku term n the analyss of monooly rcng n ts consderaton of the elastcty of the utlty functon of users (whch can be translated nto a demand curve) 2. Ths term n standard economc analyss, s the source of neffcences, rasng cost and lowerng consumton below the socal otmum. 5 Performance of Network under Monoolzed Prcng versus Socal Otmum In the socal roblem, a network manager wth centralzed routng control and erfect knowledge of the user utlty functons and latency on the lnks would route traffc accordng to the otmal soluton of the followng roblem: ma J subect to: Γ = u ( Γ I = ) I, I, l ( ) I J 0, I, J Assumng the obectve roblem s concave, an equvalent characterzaton of the socal roblem s gven by the frst order condtons: 0 f 0 ' u ( Γ ) l ( ) ( l )'( ) = 0 f = 0 Recallng the frst order otmalty condtons n eqn 0, t can been seen that f each user s charged ( l ) ( ) n the case where routng s decentralzed, the allocaton n that GWE wll be equvalent to that of the socal roblem. The authors nterret ths as the margnal congeston cost whch s the rce charged to the users to nduce less selfsh behavor, modfyng users routng choces and flow selectons. In Economcs arlance, the margnal congeston cost encourages the users to nternalze the congeston eternaltes whch they gnore when actng In a comettve market, frms set rces to equal the margnal cost at some level of outut y. (See Varan, Chater 2). 2 See Varan C23.3 7
Danelle Hnton selfshly. Imlct n most economc models s the noton that agents make decsons wthout worryng about what the other agents are dong. All nteracton takes lace va the market, so that the agents only need to know the market rces and t s n ths case of the absence of eternalzes, that a gven market s caable of achevng effcent allocatons. The nternalzaton of an eternalty corresonds to some level of coordnaton whch should serve the nterests of the agents nvolved movng the market toward effcent allocatons. 3 The authors are then able to show the corresondence between user s total flow rate Γ under the three control/rcng schemes dscussed above: the monoolzed rcng scheme w/ decentralzed control, the GWE wth zero rces, and the socal otmum wth centralzed control. In Prooston 5.9, the authors rove (omtted here, see 35-8) that where corresonds to the GWE at 0 rces, ~ corresonds to the socal otmum, and denotes the ME, For all : Γ ~ Γ Γ Ths shows that when the rces are equal to 0, the users generate too much flow relatve to the socal otmum consstent wth the results from last week, and mentoned n the ntroducton. Monooly rcng mroves ths behavor, but t may ntroduce dstorton relatve to the socal otmum due to the flow control decsons of the users as a result of the monoolstc rcng. In the case of the routng roblem where the user has a fed amount of traffc to send, the authors demonstrate that ME acheves the socal otmum (see 29-33). 6. Conclusons Among the key contrbutons of ths work has been the develoment of a more realstc framework wthn whch the decentralzed routng choces of selfsh users as well as the roft mamzng rcng schemes of servce rovders can ontly be analyzed to characterze erformance of these networks, and to develo methods to mrove ther effcency. After develong that framework, Ozdaglar and Acemoglu then characterze the equlbrum behavor of such a network, and also characterze the monooly rces, and the flow rates and lnk loads whch result. In agreement wth earler results, when network equlbrum s analyzed wthout rcng, that network s roven to be neffcent relatve to the socal otmum. A key nsght develoed n ther work s that monoolstc rcng methods roduce an equlbrum that reduces flows below the socal otmum, but whch n some cases (routng roblem) may acheve an allocaton that s socally otmal. 3 See Varan (C3.5) 8
Danelle Hnton References [AcO03] Flow control, Routng and Performance from Servce Provder Vewont [BNO03] Bertsekas, D P, A Nedc and A E Ozdaglar, Conve Analyss and Otmzaton, Athena Scentfc: Belmont, 2003. [Ham85] Haure, A and Marcotte, P. On the relatonsh between Nash-Cournot and Wardro Equlbra, Networks, 25, 295-308, 985. [JoT03] Johar, R. and J. Tstskls, Network resource Allocaton and a congeston game: The sngle lnk case, Proceedngs of the 42 nd IEEE Conference on Decson and Control. Dec 2003. [Kh04] Khst, A. Effcency Loss n a Network Resource Allocaton Game. Summary. October 27, 2004. [OsR] Osborne, M J and A Rubnsten, A Course n Game Theory. MIT Press: Cambrdge, 2004. [Rot02] Roughgarden, T. and Tardos, E. How bad s selfsh routng? Journal of the ACM, vo 49, no. 2, 236-259, 2002. [Var93] Varan, H R. Intermedate Mcroeconomcs: A Modern Aroach. W. W. Norton & Comany: New York, 993. 9