The P rice of Anarchy is I ndependent of t he Net work Topology - PDF document

The P rice of Anarchy is I ndependent of t he Net work Topology Tim Roughgarden Cornell Universit y 1

Traf f ic in Congest ed Net works The Model: • A dir ect ed gr aph G = (V,E) • k source-dest inat ion pair s (s 1 ,t 1 ), … , (s k ,t k ) • A rat e r i of t r af f ic f r om s i t o t i • For each edge e, a lat ency f n l e (•) [ct s, nondecreasing, convex] Example: (k,r=1) l (x)=x Flow = ½ s 1 t 1 l (x)=1 Flow = ½ 2

Self ish Rout ing Traf f ic and Flows: • f P = amount of t raf f ic rout ed on s i -t i pat h P • f low vect or f rout ing of t raf f ic s t Self ish rout ing: what f lows ar ise as t he rout es chosen by many noncooper at ive agent s? 3

Nash Flows Some assumpt ions: • agent s are small relat ive t o net work • want t o minimize personal lat ency Def : A f low is at Nash equilibrium (or is a Nash f low) if all f low is rout ed on min-lat ency pat hs Flow = 1 x x Flow = .5 s t s t 1 1 Flow = 0 Flow = .5 t his f low is envious! Fact : [Beckmann et al. 56] Nash f lows always exist 4

The Cost of a Flow Our obj ect ive f unct ion: • l P (f ) = sum of lat encies of edges on P (w.r.t . t he f low f ) • C(f ) = cost or t ot al lat ency of f low f : Σ P f P • l P (f ) s t Key quest ion: how good (or bad) ar e Nash f lows? 5

The I nef f iciency of Nash Flows Fact : Nash f lows do not opt imize t ot al lat ency [Pigou 1920] ⇒ lack of coordinat ion leads t o inef f iciency x • Cost of Nash = 1 1 ½ s t • Cost of OP T = ¾ 1 0 ½ Def : price of anarchy = wor st -case Nash/ OPT r at io • also coordinat ion rat io of [Kout soupias/ Papadimit riou 99] 6

Linear Lat ency Fns Def : a linear lat ency f unct ion is of t he f orm l e (x)=a e x+b e Thm : [Roughgarden/ Tardos 00] net wor k w/ linear lat ency f ns ⇒ cost of cost of = 4/ 3 × Nash f low opt f low Cor: pr ice of anar chy r ealized in a t wo-link net wor k! Point : worst -case Nash arises f rom overcongest ing one of t wo available rout es (and t hat ’s all) 7

No Dependence on Net work Topology Thm: f or any class of lat ency f ns including t he const ant f ns, worst Nash/ OPT rat io is in a t wo-link net wor k. • inef f iciency of Nash f lows always has simple explanat ion • net work t opology plays no role Not e: wor st r at io may be (much) lar ger t han 4/ 3 wit h nonlinear lat ency f ns (modif y Pigou’s ex) 8

Comparison t o Previous Work Remark: net wor ks of par allel links are not worst -case examples f or: • Appr oximat e Nash f lows, int egral Nash f lows [Roughgarden/ Tardos FOCS ‘00] • St ackelberg equilibria [Roughgarden STOC ‘01] • Braess’s par adox, maximum t r avel t ime obj f n [Roughgarden FOCS ‘01] 9

Charact erizing OPT Def : f is at Nash equilibrium if f all f low t ravels along pat hs wit h minimum lat ency Lat ency: l e (f e ) Lemma: [BMW 56] f is opt imal if f all f low t ravels along pat hs wit h minimum mar ginal cost ’ (f e ) Marginal cost : l e (f e ) + f e •l e lat ency of new f low added lat ency f or f low already on edge 10

Consequences f or Linear Lat ency Fns Observat ion: if l e (f e ) = a e f e +b e marginal cost of P w.r.t . f is: Σ 2a e f e +b e e ∈ P Corollary: [RT00] marginal lat encies = cost s of f / 2 of f 2a e (f e / 2) +b e = a e f e +b e • f a Nash f low at r at e r ⇒ f / 2 is opt imal wit h rat e r/ 2 11

Lower Bounding OPT (Linear Lat ency Fns) Goal: prove t hat cost of OPT is ≥ 3/ 4 t imes cost of Nash f low f I dea: break cost of OPT int o t wo pieces via previous Corollary Cost of Cost of Cost of = rat e r/ 2 + OPT at OPT at increasing rat e rat e r f rom r/ 2 t o r = cost of f / 2, a augment at ion w.r.t . “big chunk” of f large marginal cost s [ ≥ ¼ [ ≥ ½C(f )] C(f )] 12

Proof I dea f or Main Theorem Problem: wit h nonlinear lat ency f ns, f / 2 (or f / c, any c) is not opt imal! I dea: scale f low by dif f er ent f act ors on dif f erent edges • can scale edge-by edge so t hat new marginal cost s = old lat encies ⇒ equalizes marginal cost s ⇒ any “augment at ion” should be cost ly 13

Proof Sket ch (con’d) P roblem: scaling by dif f erent f act ors on dif f erent edges ⇒ violat es f low conservat ion! • lower -bounding cost of t he “augment at ion” is t r icky, must argue: – cut -by-cut (see pr oceedings) – edge-by-edge (simpler, see revision) [t hanks t o Amir Ronen] • gives bound on pr ice of anar chy; achieved in a Pigou-like example 14

Ext ensions Thm : f or any class of lat ency f ns • closed under scalar mult iplicat ion • including a f n l s.t . l (0) > 0 t he worst Nash/ OPT rat io is in a net wor k of par allel links. s t 15

Comput ing t he Price of Anarchy Applicat ion: worst -case examples simple ⇒ price of anar chy easy t o calculat e Example: polynomials wit h degree = d, nonnegat ive coef f s ⇒ price of anarchy T (d/ log d) x d s t 1 Also: M/ M/ 1, M/ G/ 1 queue delay f ns, et c. 16

When does t he price of self ishness have a succinct explanat ion? Example: Braess’s Paradox x 1 x 1 vs. s t 0 s t x x 1 1 bad Nash f low good Nash f low Quest ion: does t his example explain Br aess’s Par adox? • yes w/ linear lat ency f ns [RT 00] • no ot herwise (more complicat ed examples can be more severe) [R 01] 17