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

How Bad is Self ish Rout ing?

Tim Roughgarden Cornell Universit y

joint work with Éva Tardos

December, 2000 Tim Roughgarden, Cornell University 2

Traf f ic in Congest ed Net works

Mat hemat ical model:

• A dir ect ed gr aph G = (V,E)
• sour ce–sink pair s si,t i f or i=1,..,k
• rat e r i ≥ 0 of t r af f ic bet ween si

and t i f or each i=1,..,k

• For each edge e, a lat ency

f unct ion l e(•)

s1 t 1 x+1 2

r 1 =1

December, 2000 Tim Roughgarden, Cornell University 3

Example

Traf f ic rat e: r = 1, one source-sink x s t 1

Flow = ½ Flow = ½

Tot al lat ency = ½

• ½+ ½
• 1 =¾

But t raf f ic on lower edge is envious. An envy f ree f low: x s t 1

Flow = 0 Flow = 1

Tot al lat ency = 1

December, 2000 Tim Roughgarden, Cornell University 4

Flows

Traf f ic and Flows:

– f P = amount rout ed on si-t i pat h P f low vect or f ⇔ t raf f ic pat t ern at st eady-st at e

f e = ½+ ½=1

l e(f ) =1

½ s t x 1 ½ x 1 edge e

December, 2000 Tim Roughgarden, Cornell University 5

Cost of a Flow

Lat ency along pat h P :

• l P(f ) = sum of lat encies of edges in P

The Cost of a Flow f : = t ot al lat ency

• C(f ) = ΣP f P • l P(f )

l P(f ) = .5 + 0 + 1

. 5

P s t x 1

. 5

x 1

December, 2000 Tim Roughgarden, Cornell University 6

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

December, 2000 Tim Roughgarden, Cornell University 7

Flows at Nash Equilibr ium

Assumpt ion: edge lat ency f unct ions are cont inuous, nondecreasing

Lemma: a f low f is a Nash f low if and

• nly if all f low t ravels along

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

Def n: A f low is at Nash equilibrium (or

is a Nash f low) if no agent can improve it s lat ency by changing it s pat h

x

s t

1

Flow = .5 Flow = .5

s t

1

Flow = 0 Flow = 1

x

December, 2000 Tim Roughgarden, Cornell University 8

Nash Flows and Social Welf are

Cent ral Quest ion:

• 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

Analogous t o I P versus ATM:

• ATM ≈ cent r al cont r ol ≈ min cost
• I P

≈ no cent ral cont rol ≈ self ish

• Cost of Nash = 1
• min-cost

= ½

• ½

+ ½

• 1 =¾

December, 2000 Tim Roughgarden, Cornell University 9

What I s Know About Nash?

Flow at Nash equilibrium exist s and is essent ially unique [Beckmann et al. 56], … Nash and opt imal f lows can be comput ed ef f icient ly [Daf er mos/ Spar r ow 69], … Net wor k design: what net wor ks admit “good” Nash f lows? [Br aess 68], …

December, 2000 Tim Roughgarden, Cornell University 10

The Braess Par adox

Bet t er net wor k, wor se delays:

• Cost of Nash f low = 1.5
• Cost of Nash f low = 2

All t he f low has increased delay!

s t x 1 x 1 s t x 1 ½ x 1 ½ ½ ½

r at e = 1

December, 2000 Tim Roughgarden, Cornell University 11

Our Result s f or Linear Lat ency

lat ency f unct ions of t he f orm l e(x)=aex+be 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

s t x 1 ½ x 1 ½ s t x 1 x 1 Delay = 1.5 Delay = 2

December, 2000 Tim Roughgarden, Cornell University 12

General Lat ency Funct ions?

Bad Example: (r = 1, i large)

s t xi 1

1 1-e e

Nash f low cost =1, min cost ≈ 0 ⇒ Nash f low can cost ar bit r ar ily more t han t he opt imal f low

December, 2000 Tim Roughgarden, Cornell University 13

Our Result s f or General Lat ency

I n any net wor k wit h lat ency f unct ions t hat ar e

• cont inuous,
• non-decr easing

t he cost of a Nash f low wit h r at es r i f or i=1,..,k is at most t he cost of a minimum cost f low wit h rat es 2r i f or i=1,..,k

December, 2000 Tim Roughgarden, Cornell University 14

Mor ale f or I P versus ATM?

I P t oday no worse t han ATM a year f rom now … I nst ead of

• building cent ral cont rol
• build net works t hat

support t wice as much t raf f ic

December, 2000 Tim Roughgarden, Cornell University 15

What I s t he Minimum- cost Flow Like?

Minimize C(f ) = Σe f e• l e(f e) – by summing over edges rat her t han pat hs – f e amount of f low on edge e Cost C(f ) usually convex – e.g., if l e(f e) convex – if l e(f e) = ae f e +be

⇒ C(f ) = Σe f e • (ae f e +be)

convex quadrat ic

December, 2000 Tim Roughgarden, Cornell University 16

Why I s Convexit y Good?

A solut ion is opt imal f or a convex cost if and only if – t iny change in a locally f easible dir ect ion cannot decr ease t he cost

f easible direct ions

December, 2000 Tim Roughgarden, Cornell University 17

Charact erizing t he Opt imal Flow

Direct ion of change: moving a t iny f low f r om one pat h t o anot her f low f is minimum cost if and only if cost cannot be impr oved by moving a t iny f low f r om one pat h t o anot her

½ s t x 1 ½ x 1

December, 2000 Tim Roughgarden, Cornell University 18

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 ).

December, 2000 Tim Roughgarden, Cornell University 19

Min-cost I s a Socially Aware Nash

f low f is minimum cost if and only if all f low t r avels along pat hs wit h minimum mar ginal cost Marginal cost : l e(f e) + f e•l e

’(f e)

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)

December, 2000 Tim Roughgarden, Cornell University 20

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

Corollaries

• if ae = 0 f or all e, Nash and opt imal

f lows coincide (obvious)

• if be = 0 f or all e, Nash and opt imal

f lows coincide (not as obvious)

e∈P

December, 2000 Tim Roughgarden, Cornell University 21

Example

• Nash f low of rat e 1, lat ency L=2
• Not e: Same f low f or r at e ½

, – All pat hs have marginal cost = 2

⇒ it is min-cost f or r at e ½

,

s t x 1 x 1 Edge cost = x2 ⇒ marginal cost = 2x

December, 2000 Tim Roughgarden, Cornell University 22

Key Observat ion

Nash f low f f or rat e r –all f low pat hs have lat ency L

⇒ C(f ) = r L

⇒ f / 2 is opt imal wit h rat e r/ 2 and –all f low pat hs have marginal cost L

December, 2000 Tim Roughgarden, Cornell University 23

Bound f or Nash: Linear Lat ency

Goal: prove t hat cost of opt f low is at least 3/ 4 t imes t he cost

• f a Nash f low f

Cost of

• pt at

rat e r

=

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

• pt at

rat e r/ 2

+

• pt is f / 2

C(f / 2) ≥ ¼ C(f ) At least (r/ 2)•L ≥ ½ C(f )

December, 2000 Tim Roughgarden, Cornell University 24

Nonlinear Lat ency

Goal: cost of a Nash f low wit h rat e r is at most t he cost of t he opt imal f low wit h r at e 2r Analogous pr oof sket ch??

Tr oubles:

Can be close t o zer o What is opt at rat e r? and what is it s marginal cost ?

= +

Cost of

• pt at

r at e 2r Cost of

• pt at

rat e r Cost of augment ing opt f low at rat e r t o

• pt at rat e 2r

December, 2000 Tim Roughgarden, Cornell University 25

Ot her Models?

• An approximat e version of Theorem

f or non-linear lat ency wit h imprecise evaluat ion of pat h lat ency

• Analogue f or t he case of f init ely

many agent s (split t able f low)

• I mpossibilit y result s f or f init ely

many agent s, unsplit t able f low, i.e., – if each agent i cont rols a posit ive amount of f low r i ≥ 0 – f low of a single agent has t o be rout ed on a single pat h

December, 2000 Tim Roughgarden, Cornell University 26

Ot her Games?

Kout soupias & Papadimit r iou STACS’99

– scheduling wit h t wo parallel machines – Negat ive result s f or more machines First paper t o propose quant if ying t he cost of a lack of coordinat ion

– What ot her games have good Nash equilibr ium?

December, 2000 Tim Roughgarden, Cornell University 27

More Open Quest ions

• I s t her e any model in which

posit ive r esult s ar e possible f or unsplit t able f low?

• Consider models in which

agent s may cont r ol t he amount

• f t r af f ic (in addit ion t o t he

rout es) – Problem: how t o avoid t he “t r agedy of t he commons”?