[PDF] - Self ish Rout ing and t he P rice of Anarchy Tim Roughgarden PDF Document

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Self ish Rout ing and t he P rice of Anarchy

Tim Roughgarden Cornell Universit y

Includes joint work with Éva Tardos

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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

(s1 ,t 1 ), … , (sk ,t k )

A rat e r i of t r af f ic f r om si t o t i
For each edge e, a lat ency

f unct ion l e(•) [ct s, nondecr easing]

s1 t 1 l (x)=x Example: (k,r=1)

Flow = ½ Flow = ½

l (x)=1

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Self ish Rout ing

Traf f ic and Flows:

f P = amount of t raf f ic rout ed
n si-t i pat h P
f low vect or f

rout ing of t raf f ic Self ish rout ing: what f lows ar ise as t he rout es chosen by many noncooper at ive agent s?

s t

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Nash Flows

Some assumpt ions:

agent s small relat ive t o net work
want t o minimize personal lat ency

x

s t

1

Flow = .5 Flow = .5

s t

1

Flow = 0 Flow = 1

x

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]

t his f low is envious!

Example:

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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]

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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

f low f : ΣP f P • l P(f ) also: Σe f e • l e(f e)

s t

Cent ral quest ion: how good

(or bad) ar e Nash f lows?

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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 s t x 1

1

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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Braess’s Paradox

Bet t er net wor k, wor se Nash f low: Cost of Nash f low = 1.5 Cost of Nash f low = 2

s t x 1 x 1 s t x 1 ½ x 1 ½ ½ ½

r at e = 1

All t r af f ic incur s mor e lat ency!

due t o [Braess 68]
see also [Roughgarden 01]

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How Bad is Self ish Rout ing?

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] s t x 1

1

½ ½

Pigou’s example is simple…

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The Bad News

Bad Example: (r = 1, d lar ge) 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 s t xd 1

1 1-?

?

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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 ⇒ = Corollary: M/ M/ 1 delay f ns (l (x)=1/ (u-x), u = capacit y) ⇒ = cost of Nash at rat e r cost of opt at rat e 2r Nash cost w/ capacit ies 2u

pt cost w/

capacit ies u

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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

f t he f orm l e(x)=aex+be

Theorem: [Roughgarden/ Tardos 00]

net wor k w/ linear lat ency f ns ⇒

= 4/ 3 × cost of Nash f low cost of

pt f low

aka price of anarchy

[Papadimit riou 01]

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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!

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 s t x 1

1

½ ½

Cost of Nash = 1
Cost of OP

T = ¾

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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)

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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)

Also: M/ M/ 1, M/ G/ 1 queue delay

f ns, et c.

s t xd 1

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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]

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The Bicrit eria Bound

Theorem: [Roughgarden/ Tardos 00]

net wor k w/ ct s, nondecr easing lat ency f unct ions ⇒ = Corollary: M/ M/ 1 delay f ns (l (x)=1/ (u-x), u = capacit y) ⇒ = cost of Nash at rat e r cost of opt at rat e 2r Nash cost w/ capacit ies 2u

pt cost w/

capacit ies u

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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 ) = Σe 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 ) = Σe f • l e(f ) Problem: what is t he relat ion bet ween l e(f e) and l e(f )?

e e e

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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:

l e(f e) f e

graph of l

l e(f e) f e

graph of c

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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] Corollary: Suppose in Nash, everyone has lat ency L. Then:

cost of f * w.r .t . c is ≥ 2rL
C(f ) = r L.

l e(f e) f e

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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

⇒ ce(f e)f e - l e(f e)f e = l e(f e)f e

l e(f e) f e

t ypical value of ce(f e)f e - l e(f e)f e

* * * *

f e

*

* * * *

sum over edges t o get Claim

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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

pt 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