Selfih Rouing Tim Roughgarden Cornell Univeriy Include join work wih Éva Tardo 1
Which roue would you chooe? Example: one uni of raffic (e.g., car) wan o go from o delay = 1 hour (no congeion effec) long + wide hor + narrow delay in hour = fracion of raffic on edge (depend on congeion) Queion: wha will elfih nework uer do? 2
Oucome of Selfihne Defining elfihne: all raffic wan o ge o a fa a poible [given wha oher do] Then: all raffic will ue he hor narrow road. For if no: hi flow i enviou! l(x)=1 Flow =? 1 1 l(x)=x Flow = 1-?? > 0 raffic on op i enviou? = 0 envy-free oucome 3
Can we improve over he elfih oucome? Noe: all raffic incur 1 hr of delay Flow = 0 x 1 Flow = 1 Conider inead: half of raffic ake 1 hour ame a before half of raffic ake 30 minue much improved! elfih rouing need no give he be-poible oucome Flow =.5 Flow =.5 1 x 4
Brae Paradox Beer nework, wore oucome: ½ ½ x 1 ½ ½ 1 x Delay = 1.5 hour x 1 0 1 x Delay = 2 hour All raffic experience addiional delay! [Brae 68] 5
The Agenda We know: elfih rouing lead o undeirable oucome. Our goal: prove wor-cae guaranee on elfih rouing v. be coordinaed oucome everiy of Brae Paradox (damage due o exra edge) 6
Traffic in Congeed The Model: Nework A direced graph G = (V,E) k ource-deinaion pair ( 1, 1 ),, ( k, k ) A rae r i of raffic from i o i For each edge e, a laency fn l e ( ) [coninuou, nondecreaing] Example: (k=1,r=4) l(x)=x+2 Flow = 2 1 1 l(x)=1 Flow = 2 7
Rouing of Traffic Traffic and Flow: f P = amoun of raffic roued on i - i pah P flow vecor f raffic rouing of Selfih rouing: wha flow arie a able equilibria among elfih nework uer? 8
Nah Flow Some aumpion: agen mall relaive o nework wan o minimize peronal laency Def: A flow i a Nah equilibrium (or i a Nah flow) if all flow i roued on min-laency pah [given curren edge congeion] Example: Flow = 1 Flow =.5 Flow =.5 1 x Flow = 0 x 1 hi flow i enviou! 9
Some Hiory raffic model, def of Nah flow due o [Wardrop 52] hiorically called a uer equilibrium Nah flow alway exi, are (eenially) unique due o [Beckmann e al. 56] 10
The Co of a Flow Our objecive funcion: l P (f) = um of laencie of edge of P (w.r.. he flow f) C(f) = co or oal laency of flow f: S P f P l P (f) Cenral queion #1: how good (or bad) are Nah flow? 11
The Inefficiency of Nah Flow Fac: Nah flow do no opimize oal laency [Pigou 1920] lack of coordinaion lead o inefficiency x 1 ½ 0 1 ½ Co of Nah flow = 1 1 + 0 1 = 1 Co of opimal (min-co) flow = ½ ½ +½ 1 = ¾ 12
How Bad i Selfih Rouing? Pigou example i imple x 1 ½ 0 1 ½ How inefficien are Nah flow: wih more realiic laency fn? in more realiic nework? Goal: prove ha Nah flow are near-opimal wan a laiez-faire approach o managing nework alo [Kououpia/Papadimiriou 99] 13
The Bad New Bad Example: (r = 1, d large) x d 1 1-? 0 1 Nah flow ha co 1, min co 0 Nah flow can co arbirarily more han he opimal (minco) flow even if laency funcion are polynomial? 14
A Bicrieria Bound Approach #1: ele for weaker ype of guaranee Theorem: [Roughgarden/Tardo 00] nework w/c, nondecreaing laency funcion co of Nah a rae r = co of op a rae 2r Corollary: M/M/1 delay fn (l(x)=1/(u-x), u = capaciy) Nah co w/ capaciie 2u = op co w/ capaciie u 15
Key Difficuly Sp f a Nah flow, f * an op flow a wice he rae. Noe: we can wrie C(f) = S e f e l e (f e ) um over edge inead of pah f e = amoun of flow on edge e Similarly: C(f * ) = S e f * l e (f * ) Problem: wha i he relaion beween l e (f e ) and l e (f * )? e e e 16
Key Trick Idea: lower bound co of f * uing a differen e of laency fn c wih he properie: eay o lower bound co of f * w.r.. laency fn c co of f * w.r.. laency fn c co of f * w.r.. laency fn l The conrucion: graph of l graph of c l e (f e ) 0 l e (f e ) 0 0 f e 0 f e 17
Lower Bounding OPT Aume: only one commodiy (mulicommodiy no harder). Key obervaion: laency of pah P w.r.. laency fn c wih no congeion i l P (f) [laency in Nah] l e (f e ) 0 0 f e Corollary: Sp in Nah, everyone ha laency L, o C(f) = rl. Then co of f * w.r.. laency fn c i 2rL. 18
Upper Bounding he Overeimae Thu: co of f * w.r.. c i 2C(f). Claim: (will finih proof of Thm) [co of f * w.r.. c] - C(f * ) = C(f). Reaon: difference in co on e i l e (f e ) 0 f e * 0 f e ypical value of c e (f e )f* e - * l e (f e )f* * e c e (f* e )f * e - l e (f* e )f * e = l e (f e )f e um over edge o ge Claim 19
Linear Laency Funcion Approach #2: reric cla of allowable laency funcion Def: a linear laency funcion i of he form l e (x)=a e x+b e Theorem: [Roughgarden/Tardo 00] nework w/linear laency fn co of Nah flow = 4/3 co of op flow aka price of anarchy [Papadimiriou 01] 20
Source of Inefficiency Corollary of main Theorem: For linear laency fn, wor Nah/OPT raio i realized in a wo-link nework! x 1 ½ 0 1 ½ Co of Nah = 1 Co of OPT = ¾ one ource of inefficiency: confroned w/wo roue, elfih uer overconge one of hem Corollary ha' all, folk! nework opology play no role 21
No Dependence on Nework Topology Thm: [Roughgarden 02] for any cla of convex laency fn including he conan fn, wor Nah/OPT raio occur in a wo-node, wo-link nework. inefficiency of Nah flow alway ha imple explanaion nework opology play no role Recall: wor raio may be (much) larger han 4/3 (modify Pigou ex) 22
Compuing he Price of Anarchy Applicaion: wor-cae example imple wor-cae raio i eay o calculae Example: polynomial wih degree = d, nonnegaive coeff price of anarchy T(d/log d) x d 1 Alo: M/M/1, M/G/1 queue delay fn, ec. 23
A Convex Program for he Opimal Flow Key Lemma: [BMW 56] he op flow i ju a Nah flow w.r.. a differen e of laency fn. Idea: he opimal flow i minimizer for a convex program: Min C(f) = S e f e l e (f e ).. f i a flow Queion: Wha doe hi buy u? 24
Opima of Convex Program Recall: A oluion i opimal for a convex program if and only if: a mall change in a locally feaible direcion canno decreae he co feaible direcion 25
Characerizing he Opimal Flow Direcion of change: moving a mall amoun of flow from one pah o anoher 1 x ½ 1 ½ 0 x a flow f i opimal if and only if i co canno be improved by moving a mall amoun of flow from one pah o anoher 26
Characerizing he Opimal Flow Co f e l e (f e ) marginal co of increaing flow on edge e i l e (f e ) + f e l e (f e ) laency of new flow Added laency for flow already uing edge Key Lemma: a flow f i opimal if and only if all flow ravel along pah wih minimum marginal co (w.r.. f). 27
Opimal Flow a a Socially Aware Nah Flow A flow f i opimal if and only if all flow ravel along pah wih minimum marginal co Marginal co: l e (f e ) + f e l e (f e ) A flow f i a Nah equilibrium if and only if all flow ravel along minimum laency pah Laency: l e (f e ) 28
Summary of Nah v. OPT Goal: prove ha lo in nework performance due o elfih rouing i no oo large. Problem: a Nah flow can co arbirarily more han an opimal flow. Soluion: prove a bicrieria bound inead reric cla of allowable edge laency funcion 29
Reul on Nah v. OPT Thm 1: co of Nah = co of OPT a wice he raffic rae. Key o analyi: lower-bound OPT uing modified laency funcion (eay o obain lower bound, bu alo cloe o original laency fn) Thm 2: wor-cae example for elfih rouing are imple. Wor-cae Nah/OPT raio mall unle laency fn highly nonlinear Key o analyi: OPT i ju a Nah flow w.r.. differen laency fn 30
Recall: Brae Paradox ½ ½ x 1 ½ ½ 1 x Co of Nah flow = 1.5 rae = 1 x 1 0 1 x Co of Nah flow = 2 Nah flow uffer due o exra edge 31
Generalizing Brae Paradox Queion: i Brae Paradox more evere in bigger nework? focu on ingle-commodiy nework Fac: w/linear fn, wor cae i x 1 1 x v. Reaon: wih linear laency fn, x 1 0 1 x co = 3/2 co = 2 co of Nah flow = 4/3 co of any oher flow 32
General Laency Fn A more evere verion: f(x) 1 1 f(x) where f(x) i: v. f(x) 1 0 1 f(x) co = 1 co = 2 1 0 0 ½ 1 exra edge caue facor 2 increae in delay 33
A Bigger Brae Paradox rae = 2 1 f g 1 g f 1 v. 1 f g 0 1 0 g f 1 common laency = 1 common laency = 3 where: 1 2 0 0 2/3 1 0 0 2/3 1 f(x) g(x) 34
An Infinie Family common laency = 1 common laency = 4 Thm: [Roughgarden 01] adding edge o an n-verex graph can increae common delay in he Nah flow by an n/2 facor. 35
A Maching Upper Bound Thm: [Roughgarden 01] adding edge o an n-verex graph increae common delay by a mo an n/2 facor. Noe: canno appeal o a reul relaing he oal laency of Nah and OPT flow no uch bound exi wih general laency fn even a a fn of he nework ize 36
Summary of Brae Paradox Brae Paradox: adding edge o a nework can make he Nah flow wore. Generalizaion: can add edge o an n-node graph o make Nah wore by an n/2 facor. Wor-cae guaranee: no wore example are poible. 37
Open Queion 1 Open: by adding only k edge, can you make he Nah flow wore by a facor of more han k+1? common laency = 1 common laency = 4 38
Brae Paradox and Embedded Subgraph Def: A graph i vulnerable if laency fn can be aigned o he edge o ha Brae Paradox occur. Fac: [Murchland 70] A graph i vulnerable if and only if i ha a ubgraph ha i a ubdiviion of: 39
Open Queion 2 Def: A graph i c-vulnerable if laency fn can be aigned o he edge o ha Nah i wore by a facor > c becaue of harmful exra edge. Prove or diprove: A graph i c-vulnerable if and only if i ha a ubgraph ha i a ubdiviion of: ize depend on c only 40