How Unf air is Opt imal Rout ing? Tim Roughgarden Cornell - - PDF document

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How Unf air is Opt imal Rout ing?

Tim Roughgarden Cornell Universit y

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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(•)

s t l (x)=x Example: (r=1)

Flow = ½ Flow = ½

l (x)=1

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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 :

ΣP f P • l P(f ) s t

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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

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Flows at Nash Equilibr ium

Assumpt ion: edge lat ency f unct ions

are cont inuous, nondecreasing

Lemma: f is a Nash f low all f low

• n minimum-lat ency pat hs (w.r.t . f )

Fact : have exist ence, uniqueness

x

s t

1

Flow = .5 Flow = .5

s t

1

Flow = 0 Flow = 1

x

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

t his f low is envious!

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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 s t x 1

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

• ½

+½

• 1 = ¾

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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?

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Bad example

s t xk

1 1-d d

r = 1, ? small s t x 2(1-?)

1 1-?

?

r = 1, k large

⇒ some “mart yrs” incur t wice as

much lat ency in OPT as in Nash! k+1-?

Even wor se: ⇒ some t raf f ic can be arbit rarily

worse of f in OPT t han in Nash

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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: 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

unf airness

• f (G,r , l )

max lat ency in OPT

:=

common lat ency in Nash

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I nf ormal St at ement

• f 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:

s t xk

1 1-?

?

≈ k+1

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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

l e(f e) + f e • l e

’(f e)

lat ency of new f low Added lat ency

• f f low already
• n 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 ).

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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 Marginal cost : l e(f e) + f e•l e

’(f 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)

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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: Then, unf airness of G is = ?. Example: if l e(x) = xk get l e(x) + x•l e

’(x)

l e(x) maxe,x = ?. xk + k•xk xk = k+1

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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

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Conclusions

Remar k: pr oof act ually shows t hat f or any f easible f low f , Fact : [Meyerson 01] False wit h mult iple commodit ies!

• OPT can be arbit rarily less f air t han
• t 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?

max-lat ency of an OP T pat h max-lat ency

• f an f pat h

= ? ×