How Bad is Self ish Rout ing? Tim Roughgarden Cornell Universit y - PDF document

How Bad is Self ish Rout ing? Tim Roughgarden Cornell Universit y joint work with Éva Tardos November, 2000 Tim Roughgarden, Cornell University 1

Traf f ic in Congest ed Net works Given: • A dir ect ed gr aph G = (V,E) • A source s and a sink t • A rat e r of t raf f ic f rom s t o t • For each edge e, a lat ency f unct ion l e (•) Example: (r=1) x ½ s t 1 ½ November, 2000 Tim Roughgarden, Cornell University 2

Flows and t heir Cost Traf f ic and Flows: • f P = amount of t raf f ic rout ed on s-t pat h P • f low vect or f t raf f ic pat t ern at st eady-st at e 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 November, 2000 Tim Roughgarden, Cornell University 3

Flows and Game Theory • f low = rout es of many noncooper at ive agent s • Examples: – cars in a highway syst em – packet s in a net work • [at st eady-st at e] • 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 November, 2000 Tim Roughgarden, Cornell University 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 Assumpt ion: edge lat ency f unct ions are cont inuous, nondecreasing Lemma: f is a Nash f low if and only if all f low t ravels along minimum- lat ency pat hs (w.r.t . f ) November, 2000 Tim Roughgarden, Cornell University 5

Nash Flows and Social Welf ar e Cent ral Quest ion: To what ext ent does a Nash f low opt imize social welf are? What is t he cost of t he lack of coor dinat ion in a Nash f low? 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 November, 2000 Tim Roughgarden, Cornell University 6

Previous Work • [Beckmann et al. 56], … – Exist ence, uniqueness of f lows at Nash equilibrium • [Daf ermos/ Sparrow 69], … – Ef f icient ly comput ing Nash and opt imal f lows • [Braess 68], … – Net work design • [Kout soupias/ Papadimit riou 99] – Quant if ying t he cost of a lack of coordinat ion November, 2000 Tim Roughgarden, Cornell University 7

Braess’s Paradox Rate: r = 1 x 1 ½ s t ½ x 1 Cost of Nash flow = 1.5 x 1 0 s t x 1 Cost of Nash flow = 2 All flow experiences more latency! November, 2000 Tim Roughgarden, Cornell University 8

Our Result s f or Linear Lat ency Def : a linear lat ency f unct ion is of t he f orm l e (x)=a e x+b e Theorem 1: I n a net work wit h linear lat ency f unct ions, t he cost of a Nash f low is at most 4/ 3 t imes t hat of t he minimum- lat ency f low. November, 2000 Tim Roughgarden, Cornell University 9

General Lat ency Funct ions? Bad Example: (r = 1, k lar ge) x k 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 November, 2000 Tim Roughgarden, Cornell University 10

Our Result s f or General Lat ency All is not lost : t he previous example does not pr eclude int erest ing bicr it er ia result s. Theorem 2: I n any net work wit h cont inuous, nondecr easing lat ency f unct ions: The cost of a Nash f low wit h rat e r is at most t he cost of an opt imal f low wit h r at e 2r . November, 2000 Tim Roughgarden, Cornell University 11

Charact erizing t he Opt imal Flow Cost f e • l e (f e ) ⇒ marginal cost of incr easing f low on edge e is ’ (f e ) l e (f e ) + f e • l e Added lat ency lat ency of of f low already new f low on edge Key Lemma: a f low f is opt imal if and only if all f low t ravels along pat hs wit h minimum mar ginal cost (w.r .t . f ). November, 2000 Tim Roughgarden, Cornell University 12

The Opt imal Flow as a Socially Aware Nash A f low f is opt imal if and only if 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 A f low f is at Nash equilibrium if and only if all f low t ravels along minimum lat ency pat hs Lat ency: l e (f e ) November, 2000 Tim Roughgarden, Cornell University 13

Consequences f or Linear Lat ency Fns Observat ion: if l e (f e ) = a e f e +b e (lat ency f unct ions ar e linear ) ⇒ marginal cost of P w.r.t . f is: Σ 2a e f e +b e e ∈ P Corollary: f a Nash f low wit h r at e r in a net wor k wit h linear lat ency f ns ⇒ f / 2 is opt imal wit h rat e r / 2 November, 2000 Tim Roughgarden, Cornell University 14

Conclusions • Mult icommodit y analogues of bot h result s (can specif y rat e of t raf f ic bet ween each pair of nodes) • Approximat e versions assuming imprecise evaluat ion of pat h lat ency • Open: ext ension t o a model in which agent s may cont rol t he amount of t raf f ic (in addit ion t o t he rout es) – Problem: how t o avoid t he “t ragedy of t he commons”? November, 2000 Tim Roughgarden, Cornell University 15