How Unf air is Opt imal Rout ing? 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) • 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) l (x)=x Flow = ½ s t 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-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 3
Flows and Game Theory • 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 ar e • 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
Nash Flows and Social Welf ar e Fact : Nash f lows do not opt imize t ot al lat ency ⇒ 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? • [Roughgarden/ Tardos 00] – linear lat ency f unct ions ⇒ cost of Nash = 4/ 3 × cost of OP T – bicrit eria result f or arbit rary f ns • [Roughgarden 01,02]: ot her lat ency f ns • [Friedman 01]: includes f low cont rol • Dif f erent model, obj ect ive f n: – [Kout soupias/ P apadimit riou 99], [Mavronicolas/ Spirakas 01], [Czumaj / Vöcking 02] Quest ion: I s t he opt imal (min-cost ) rout ing really what we want ? – what about f airness? 7
Bad example r = 1, ? small x 1 1- ? s t 2(1- ? ) 0 ? ⇒ some “mart yrs” incur t wice as much lat ency in OPT as in Nash! Even wor se: r = 1, k large x k 1 1-d s t k+1- ? 0 d ⇒ some t raf f ic can be arbit rarily worse of f in OPT t han in Nash 8
How Unf air is Opt imal Rout ing? Def : Given a net wor k G, lat ency f ns l , t raf f ic rat e r: max lat ency in OPT unf airness := of (G,r , l ) common lat ency in Nash Examples: • Braess’s Paradox (unf airness = ¾ ) • bad example (unf airness ≈ k+1) Cent r al Quest ion: What is t he wor st -possible unf air ness? • f or a rest rict ed class of lat ency f ns 9
I nf ormal St at ement of Main Result s “Thm”: I n any net work wit h lat ency f ns t hat ar e “not t oo st eep”, unf airness is “small”. Special case: A net wor k wit h wit h polynomial lat ency f ns, max degree = k, has unf airness = k+1 Mat ching lower bound: x k 1 1- ? s t ≈ k+1 0 ? 10
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 ). 11
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 ) 12
Main Theorem Thm: For a net wor k G w/ lat ency f ns l , suppose wor st -case mar ginal cost vs. lat ency discr epancy is: ’ (x) l e (x) + x•l e max e , x = ?. l e (x) Then, unf airness of G is = ?. Example: if l e (x) = x k get x k + k•x k = k+1 x k 13
Proof Sket ch Lemma 1: Minimum-lat ency pat h in OPT =common lat ency in Nash. • ot herwise, Nash would have smaller t ot al lat ency t han OPT Lemma 2: Lat encies of OPT’s f low pat hs dif f er by =a ? f act or. • OPT f low pat hs have equal marginal cost • marginal cost s, lat encies dif f er only by a ? f act or 14
Conclusions Remar k: pr oof act ually shows t hat f or any f easible f low f , max-lat ency of max-lat ency = ? × an OP T pat h of an f pat h Fact : [Meyerson 01] False wit h mult iple commodit ies! • OPT can be arbit rarily less f air t han ot her f easible f lows – even wit h linear lat ency f unct ions – under many def init ions of f airness Open: is OPT almost as f air as Nash w/ many commodit ies? 15
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