Self ish Rout ing and t he P rice of Anarchy Tim Roughgarden - PDF document

Self ish Rout ing and t he P rice of Anarchy Tim Roughgarden Cornell Universit y Includes joint work with Éva Tardos 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 unct ion l e (•) [ct s, nondecr easing] 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 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 [given cur r ent edge congest ion] Example: Flow = 1 x x Flow = .5 s t s t 1 1 Flow = 0 Flow = .5 t his f low is envious! 4

Some Hist ory • t raf f ic model, def of Nash f lows due t o [Wardrop 52] – hist or ically called user -opt imal/ user equilibr ium • Nash f lows always exist , are (essent ially) unique – due t o [Beckmann et al. 56] 5

The Cost of a Flow Our obj ect ive f unct ion: • l P (f ) = sum of lat encies of edges of P (w.r .t . t he f low f ) • C(f ) = cost or t ot al lat ency of Σ P f P • l P (f ) f low f : also: Σ e f e • l e (f e ) s t Cent ral quest ion: how good (or bad) ar e Nash f lows? 6

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 1 ½ s t 1 0 ½ Cost of Nash f low = 1•1 + 0•1 = 1 Cost of opt imal (min-cost ) f low = ½ •½ +½ •1 = ¾ 7

Braess’s Paradox Bet t er net wor k, wor se Nash f low: ½ ½ r at e = 1 x 1 s t ½ ½ x 1 Cost of Nash f low = 1.5 x 1 s t 0 x 1 Cost of Nash f low = 2 All t r af f ic incur s mor e lat ency! • due t o [Braess 68] • see also [Roughgarden 01] 8

How Bad is Self ish Rout ing? x 1 Pigou’s example ½ s t is simple… 1 0 ½ How inef f icient are Nash f lows: • wit h mor e r ealist ic lat ency f ns? • in more realist ic net wor ks? Goal : pr ove t hat Nash f lows ar e near-opt imal • want a laissez-f aire approach t o managing net works – also [Kout soupias/ Papadimit riou 99] 9

The Bad News Bad Example: (r = 1, d lar ge) x d 1 1- ? s t 1 0 ? Nash f low has cost 1, min cost ≈ 0 ⇒ Nash f low can cost ar bit r ar ily more t han t he opt imal (min- cost ) f low – even if lat ency f unct ions are polynomials 10

A Bicrit eria Bound Approach # 1: set t le f or weaker t ype of guar ant ee Theorem: [Roughgarden/ Tardos 00] net wor k w/ ct s, nondecr easing lat ency f unct ions ⇒ cost of Nash cost of opt = at rat e r at rat e 2r Corollary: M/ M/ 1 delay f ns (l (x)=1/ (u-x), u = capacit y) ⇒ Nash cost w/ opt cost w/ = capacit ies 2u capacit ies u 11

Linear Lat ency Funct ions Approach # 2: rest rict class of allowable lat ency f unct ions Def : a linear lat ency f unct ion is of t he f orm l e (x)=a e x+b e Theorem : [Roughgarden/ Tardos 00] net wor k w/ linear lat ency f ns ⇒ cost of cost of = 4/ 3 × Nash f low opt f low aka price of anarchy [Papadimit riou 01] 12

Sources of I nef f iciency Cor ollar y of main Theor em: • For linear lat ency f ns, worst Nash/ OPT rat io is realized in a t wo-link net wor k! x • Cost of Nash = 1 1 ½ s t • Cost of OP T = ¾ 1 0 ½ • one source of inef f iciency: – conf ront ed w/ t wo rout es, self ish users overcongest one of t hem • Corollary ⇒ t hat ' s all, f olks! – net work t opology plays no role 13

No Dependence on Net work Topology Thm: [Roughgarden 02] f or any class of convex lat ency f ns including t he const ant f ns, worst Nash/ OPT rat io occurs in a t wo-node, t wo-link net work. • inef f iciency of Nash f lows always has simple explanat ion • net work t opology plays no role Recall: wor st r at io may be (much) lar ger t han 4/ 3 (modif y Pigou’s ex) 14

Comput ing t he Price of Anarchy Applicat ion: worst -case examples simple ⇒ wor st -case rat io is 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. 15

Comparison t o Previous Work Remark: parallel links are not wor st -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] • Equilibria w/ explicit capacit ies [Schulz/ St ier SODA ‘03] 16

The Bicrit eria Bound Theorem: [Roughgarden/ Tardos 00] net wor k w/ ct s, nondecr easing lat ency f unct ions ⇒ cost of Nash cost of opt = at rat e r at rat e 2r Corollary: M/ M/ 1 delay f ns (l (x)=1/ (u-x), u = capacit y) ⇒ Nash cost w/ opt cost w/ = capacit ies 2u capacit ies u 17

Key Dif f icult y Sps f a Nash f low, f * an opt f low at t wice t he rat e. Recall: we can writ e C(f ) = Σ e f e • l e (f e ) • sum over edges inst ead of pat hs • f e = amount of f low on edge e Similarly: C(f * ) = Σ e f * • l e (f * ) e e Problem: what is t he relat ion bet ween l e (f e ) and l e (f * )? e 18

Key Trick I dea: lower bound cost of f * using a dif f er ent set of lat ency f ns c wit h t he propert ies: • easy t o lower bound cost of f * w.r .t . lat ency f ns c • cost of f * w.r .t . lat ency f ns c ≈ cost of f * w.r .t . lat ency f ns l The const r uct ion: graph of l graph of c l e (f e ) l e (f e ) 0 0 0 0 f e f e 19

Lower Bounding OPT Assume: only one commodit y (mult icommodit y no har der ). Key observat ion: lat ency of pat h P w.r .t . lat ency f ns c wit h no congest ion is l P (f ) [lat ency in Nash] l e (f e ) 0 0 f e Corollary: Suppose in Nash, everyone has lat ency L. Then: • cost of f * w.r .t . c is ≥ 2rL • C(f ) = r L. 20

Upper Bounding t he Overest imat e Thus: cost of f * w.r .t . c is ≥ 2C(f ). Claim: (will f inish proof of Thm) [cost of f * w.r.t . c] - C(f * ) = C(f ). Reason: dif f er ence in cost s on e is t ypical value of l e (f e ) * * * * c e (f e )f e - l e (f e )f e 0 * f e f e 0 ⇒ c e (f e )f e - l e (f e )f e = l e (f e )f e * * * * sum over edges t o get Claim 21

Summary Goal: prove t hat loss in net work per f or mance due t o self ish rout ing is not t oo large. Problem: a Nash f low can cost ar bit r ar ily mor e t han an opt imal f low. Solut ions: • prove a bicr it er ia bound inst ead • r est r ict class of allowable edge lat ency f unct ions 22

Nonat omic Congest ion Games Quest ion: in what ot her games is t he out come of self ishness near-opt imal? Thm: [Roughgarden/ Tardos 02] All r esult s f r om t his t alk generalize t o nonat omic congest ion games: • r eplace net wor k by gr ound set • s i -t i pat hs by set syst ems 23