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

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 s i ,t i f or i=1,..,k • rat e r i ≥ 0 of t r af f ic bet ween s i and t i f or each i=1,..,k • For each edge e, a lat ency f unct ion l e (•) r 1 =1 x+1 s 1 t 1 2 December, 2000 Tim Roughgarden, Cornell University 2

Example Traf f ic rat e: r = 1, one source-sink x Flow = ½ Tot al lat ency = s t 1 • ½ + ½ •1 =¾ ½ Flow = ½ But t raf f ic on lower edge is envious. An envy f ree f low: x Flow = 1 Tot al lat ency = 1 s t 1 Flow = 0 December, 2000 Tim Roughgarden, Cornell University 3

Flows Traf f ic and Flows: – f P = amount rout ed on s i -t i pat h P ⇔ t raf f ic pat t ern at f low vect or f st eady-st at e ½ x 1 s t f e = ½+ ½=1 0 x l e (f ) =1 1 ½ edge e December, 2000 Tim Roughgarden, Cornell University 4

Cost of a Flow P . 5 x 1 s t 0 x 1 l P (f ) = .5 + 0 + 1 . 5 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 ) = Σ P f P • l P (f ) December, 2000 Tim Roughgarden, Cornell University 5

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 6

Flows at Nash Equilibr ium 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 Flow = 1 x x Flow = .5 s t s t 1 1 Flow = 0 Flow = .5 Assumpt ion: edge lat ency f unct ions are cont inuous, nondecreasing Lemma: a f low 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 ). December, 2000 Tim Roughgarden, Cornell University 7

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? x • Cost of Nash = 1 ½ 1 s t • min-cost 1 = ½ •½ + ½ •1 =¾ 0 ½ Analogous t o I P versus ATM: • ATM ≈ cent r al cont r ol ≈ min cost ≈ no cent ral cont rol ≈ self ish • I P December, 2000 Tim Roughgarden, Cornell University 8

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 9

The Braess Par adox Bet t er net wor k, wor se delays: ½ ½ r at e = 1 x 1 s t ½ ½ x 1 • Cost of Nash f low = 1.5 x 1 s t 0 x 1 • Cost of Nash f low = 2 All t he f low has increased delay! December, 2000 Tim Roughgarden, Cornell University 10

Our Result s f or Linear Lat ency lat ency f unct ions of t he f orm l e (x)=a e x+b e 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 ½ x 1 x 1 s t s t 0 ½ x 1 1 x Delay = 2 Delay = 1.5 December, 2000 Tim Roughgarden, Cornell University 11

General Lat ency Funct ions? Bad Example: (r = 1, i large) x i 1- e 1 s t 1 0 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 12

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 13

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 14

What I s t he Minimum- cost Flow Like? Minimize C(f ) = Σ e 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 ) = a e f e +b e ⇒ C(f ) = Σ e f e • (a e f e +b e ) convex quadrat ic December, 2000 Tim Roughgarden, Cornell University 15

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 16

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 ½ 1 x s 0 t x 1 ½ 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 December, 2000 Tim Roughgarden, Cornell University 17

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 ’ (f e ) l e (f e ) + f e • l e Added lat ency lat ency of of f low already new f low on 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 18

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 ’ (f e ) Marginal cost : l e (f e ) + f e •l 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 19

Consequences f or Linear Lat ency Fns Observat ion: if l e (f e ) = a e f e +b e ⇒ marginal cost of P w.r.t . f is: Σ 2a e f e +b e e ∈ P Corollaries • if a e = 0 f or all e, Nash and opt imal f lows coincide (obvious) • if b e = 0 f or all e, Nash and opt imal f lows coincide (not as obvious) December, 2000 Tim Roughgarden, Cornell University 20

Example x 1 Edge cost = x 2 ⇒ 0 s t marginal cost = 2x 1 x • 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 , December, 2000 Tim Roughgarden, Cornell University 21

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 22

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 of a Nash f low f Cost of Cost of Cost of increasing rat e = + opt at opt at f rom rat e r/ 2 rat e r rat e r/ 2 t o rat e r opt is f / 2 At least (r/ 2)•L C(f / 2) ≥ ¼ ≥ ½ C(f ) C(f ) December, 2000 Tim Roughgarden, Cornell University 23

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?? Cost of Cost of Cost of augment ing opt = + opt at opt at f low at rat e r t o r at e 2r rat e r opt at rat e 2r Can be What is opt at rat e r? Tr oubles: close t o and what is it s zer o marginal cost ? December, 2000 Tim Roughgarden, Cornell University 24

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 25

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 26

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 of 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”? December, 2000 Tim Roughgarden, Cornell University 27