Self ish Rout ing Tim Roughgarden Cornell Universit y Includes - PDF document

Self ish Rout ing 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 (•) Example: (k,r=1) l (x)=x Flow = ½ s 1 t 1 l (x)=1 Flow = ½ 2

Flows and t heir Cost 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 The Cost of a Flow: • 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 3

Self ish Rout ing • f low = rout es of many noncooper at ive agent s • Examples: – cars in a highway syst em [Wardrop 52] – packet s in a net work • cost (t ot al lat ency) of a f low as a measur e of social welf are • agent s ar e self ish – do not care about social welf are – want t o minimize personal lat ency 4

Flows at Nash Equilibr ium Def : A f low is at Nash equilibrium (is a Nash f low) if no agent can improve it s lat ency by changing it s pat h Flow = 1 x x Flow = .5 s t s t 1 1 Flow = 0 Flow = .5 t his f low is envious! Assumpt ion: edge lat ency f unct ions are cont inuous, nondecreasing Lemma: f is a Nash f low all f low on minimum-lat ency pat hs (w.r.t . f ) Fact : have exist ence, uniqueness 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 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 6

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/ P apadimit riou 99] 7

The Bad News Bad Example: (r = 1, d large) 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 8

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 9

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 10

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 11

No Dependence on Net work Topology Theor em: [Roughgar den 02] f or (almost ) any class of lat ency f ns including t he const ant f ns, worst Nash/ OPT rat io occurs in a t wo-link net wor k. Corollary: wor st -case f or bounded-degr ee polynomials is: x d 1 1- ? s t 1 0 ? 12

Coping wit h Self ishness Mot ivat ion: • Nash f lows inef f icient • cent ralized rout ing of t en inf easible Goal: design/ manage net wor ks s.t . self ish rout ing “not t oo bad” ⇒ adds new algor it hmic dimension Two Approaches: • net work design (next ) • St ackelberg rout ing (see t hesis) 13

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 exper iences addit ional lat ency! [Br aess 68] 14

Designing Net works f or Self ish Users The Problem: • given net wor k G = (V,E,l ) – assume single-commodit y • f ind subnet wor k minimizing lat ency exper ienced by all self ish users in a Nash f low 1 x 1 x 0 s t s t x 1 1 x ⇒ want t o avoid Braess’s Paradox 15

Generalizing Braess’s Par adox Quest ion: is Braess’s Paradox mor e sever e in bigger net wor ks? Fact : wit h linear lat ency f ns, worst case is x 1 x 1 vs. s t 0 s t x x 1 1 cost = 2 cost = 3/ 2 Reason: wit h linear lat ency f ns, average lat ency average lat ency = 4/ 3 × of Nash f low of any ot her f low 16

Braess’s Paradox wit h General Lat ency Fns A Bigger Br aess Par adox: t t s s Nash in whole graph Nash in opt subgraph common lat ency = 4 common lat ency = 1 ⇒ removing edges can improve Nash by a n/ 2 f act or (n=| V|) Thm: [R 01] t his is wor st possible. 17

The Tr ivial Algor it hm Def : The t r ivial algor it hm is t o build t he ent ir e net wor k. We know: t he t rivial algorit hm is • a 4/ 3-appr ox alg wit h linear lat ency f ns • an n/ 2-appr ox alg wit h gener al lat ency f ns Quest ion: what about mor e sophist icat ed algor it hms? 18

Designing Net works f or Self ish Users is Hard Thm: [R 01] For ? > 0, no (n/ 2 - ? )- appr oximat ion algor it hm exist s (unless P=NP). Thm: [R 01] For linear lat ency f unct ions, no (4/ 3 - ? )-appr ox algor it hm exist s (unless P=NP). Remar k: similar result s hold f or ot her classes of lat ency f ns. Corollary: Br aess’s Par adox eludes ef f icient algorit hms. 19

Direct ions f or Furt her Research Self ish Rout ing: many open quest ions, see t hesis Ot her Games: e.g., f low cont rol, compet it ive f acilit y locat ion, auct ions Par adigm f or st udying self ishness: • what is worst Nash/ OPT obj ect ive f n value rat io? • are ot her meaningf ul bounds (e.g., bicrit eria) possible? • sources of inef f iciency? • design/ management st rat egies f or coping wit h self ishness? 20