The P rice of Anarchy is I ndependent of t he Net work Topology - - PDF document

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The P rice of Anarchy is I ndependent of t he Net work Topology

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)
• 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 n

l e(•) [ct s, nondecreasing, convex]

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 are 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

t his f low is envious!

Fact : [Beckmann et al. 56] Nash f lows always exist

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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 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

Key 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

Def : price of anarchy =

wor st -case Nash/ OPT r at io

• also coordinat ion rat io of

[Kout soupias/ Papadimit riou 99] s t x 1

1

½ ½

• Cost of Nash = 1
• Cost of OP

T = ¾

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Linear Lat ency Fns

Def : a linear lat ency f unct ion is

• f t he f orm l e(x)=aex+be

Thm: [Roughgarden/ Tardos 00]

net wor k w/ linear lat ency f ns ⇒ = 4/ 3 × Cor: pr ice of anar chy r ealized in a t wo-link net wor k! Point : worst -case Nash arises f rom

• vercongest ing one of t wo available

rout es (and t hat ’s all)

cost of Nash f low cost of

• pt f low

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No Dependence on Net work Topology

Thm: f or any class of lat ency f ns

including t he const ant f ns, worst Nash/ OPT rat io is in a t wo-link net wor k.

• inef f iciency of Nash f lows always

has simple explanat ion

• net work t opology plays no role

Not e: wor st r at io may be (much) lar ger t han 4/ 3 wit h nonlinear lat ency f ns (modif y Pigou’s ex)

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Comparison t o Previous Work

Remark: net wor ks of par allel

links are not worst -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]

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Charact erizing OPT

Def : f is at Nash equilibrium if f all f low t ravels along pat hs wit h minimum lat ency Lat ency: l e(f e) Lemma: [BMW 56] f is opt imal if f 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)

lat ency of new f low added lat ency f or f low already

• n edge

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Consequences f or Linear Lat ency Fns

Observat ion: if l e(f e) = ae f e +be marginal cost of P w.r.t . f is: Σ 2ae f e +be Corollary: [RT00] =

• f a Nash f low at r at e r

⇒ f / 2 is opt imal wit h rat e r/ 2

e∈P

marginal cost s of f / 2 lat encies

• f f

2ae(f e/ 2) +be = aef e +be

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Lower Bounding OPT (Linear Lat ency Fns)

Goal: prove t hat cost of OPT is

≥ 3/ 4 t imes cost of Nash f low f Cost of OPT at rat e r

=

Cost of increasing rat e f rom r/ 2 t o r Cost of OPT at rat e r/ 2 +

I dea: break cost of OPT int o t wo pieces via previous Corollary

= cost of f / 2, a “big chunk” of f [≥ ¼ C(f )] augment at ion w.r.t . large marginal cost s [≥ ½C(f )]

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Proof I dea f or Main Theorem

Problem: wit h nonlinear lat ency

f ns, f / 2 (or f / c, any c) is not

• pt imal!

I dea: scale f low by dif f er ent

f act ors on dif f erent edges

• can scale edge-by edge so t hat

new marginal cost s = old lat encies ⇒ equalizes marginal cost s ⇒ any “augment at ion” should be cost ly

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Proof Sket ch (con’d)

P roblem: scaling by dif f erent

f act ors on dif f erent edges ⇒ violat es f low conservat ion!

• lower -bounding cost of t he

“augment at ion” is t r icky, must argue: – cut -by-cut (see pr oceedings) – edge-by-edge (simpler, see revision) [t hanks t o Amir Ronen]

• gives bound on pr ice of anar chy;

achieved in a Pigou-like example

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Ext ensions

Thm: f or any class of lat ency f ns

• closed under scalar mult iplicat ion
• including a f n l s.t . l (0) >

t he worst Nash/ OPT rat io is in a net wor k of par allel links.

s t

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Comput ing t he Price of Anarchy

Applicat ion: worst -case

examples simple ⇒ price of anar chy 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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When does t he price of self ishness have a succinct explanat ion?

Example: Braess’s Paradox

Quest ion: does t his example explain Br aess’s Par adox?

• yes w/ linear lat ency f ns [RT 00]
• no ot herwise (more complicat ed

examples can be more severe) [R 01] s t x 1 x 1 s t x 1 x 1

vs.

good Nash f low bad Nash f low