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

November, 2000 Tim Roughgarden, Cornell University 1

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

joint work with Éva Tardos

November, 2000 Tim Roughgarden, Cornell University 2

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(•)

s t x 1 ½ ½ Example: (r=1)

November, 2000 Tim Roughgarden, Cornell University 3

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 :

ΣPf P• l P(f ) s t

November, 2000 Tim Roughgarden, Cornell University 4

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 5

Flows at Nash Equilibr ium

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 )

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

November, 2000 Tim Roughgarden, Cornell University 6

Nash Flows and Social Welf ar e

Cent ral Quest ion: To what

ext ent does a Nash f low

• pt imize social welf are? What

is t he cost of t he lack of coor dinat ion in a Nash f low?

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

November, 2000 Tim Roughgarden, Cornell University 7

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

• pt 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 8

Braess’s Paradox

s t x 1 x 1 Rate: r = 1 Cost of Nash flow = 1.5 Cost of Nash flow = 2

All flow experiences more latency!

s t x 1 x ½ ½ 1

November, 2000 Tim Roughgarden, Cornell University 9

Our Result s f or Linear Lat ency

Def : a linear lat ency f unct ion is

• f t he f orm l e(x)=aex+be

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 10

General Lat ency Funct ions?

Bad Example: (r = 1, k 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 xk 1

1 1-?

?

November, 2000 Tim Roughgarden, Cornell University 11

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

• pt imal f low wit h r at e 2r .

November, 2000 Tim Roughgarden, Cornell University 12

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 ).

November, 2000 Tim Roughgarden, Cornell University 13

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)

November, 2000 Tim Roughgarden, Cornell University 14

Consequences f or Linear Lat ency Fns

Observat ion: if l e(f e) = ae f e +be (lat ency f unct ions ar e linear ) ⇒ marginal cost of P w.r.t . f is: Σ 2ae f e +be 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

e∈P

November, 2000 Tim Roughgarden, Cornell University 15

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”?