Selfish Routing and the Price of Anarchy. Tim Roughgarden Cornell University

Transcription

1 Selfih Rouing and he Price of Anarchy Tim Roughgarden Cornell Univeriy 1

2 Algorihm for Self-Inereed Agen Our focu: problem in which muliple agen (people, compuer, ec.) inerac Moivaion: he Inerne decenralized operaion and ownerhip Tradiional algorihmic approach: agen claified a obedien or adverarial example: diribued algorihm, crypography 2

3 Algorihm and Game Theory Recen rend: agen have own independen objecive (and ac accordingly) New goal: algorihm ha accoun for raegic behavior by elf-inereed agen Naural ool: game heory heory of raional behavior in compeiive, collaboraive eing [von Neumann/Morgenern 44] 3

4 Objecive Thi alk: underand conequence of noncooperaive behavior when i he co of elfih behavior evere? he price of anarchy [Kououpia/Papadimiriou 99] wha can we do abou i? deign raegie, economic incenive Our eing: rouing in a congeed nework will focu on [Roughgarden/Tardo FOCS 00/JACM 02] and alo [Roughgarden STOC 02/JCSS o appear] 4

5 Moivaing Example Example: one uni of raffic wan o go from o l(x)=x delay depend on congeion l(x)=1 no congeion effec Queion: wha will elfih nework uer do? aume everyone wan malle-poible delay 5

6 Moivaing Example Claim: all raffic will ake he op link. l(x)=x Flow = 1-Є hi flow i enviou! l(x)=1 Flow = Є Reaon: Є > 0 raffic on boom i enviou Є = 0 envy-free oucome all raffic incur one uni of delay 6

7 Can We Do Beer? Conider inead: raffic pli equally l(x)=x Flow = ½ Improvemen: l(x)=1 half of raffic ha delay 1 (ame a before) half of raffic ha delay ½ (much improved!) Flow = ½ 7

8 Brae Paradox Iniial Nework: ½ ½ x 1 ½ ½ 1 x Delay = 1.5 8

9 Brae Paradox Iniial Nework: Augmened Nework: ½ ½ x 1 ½ ½ 1 x ½ ½ x 1 ½ 0 ½ 1 x Delay = 1.5 Now wha? 9

10 Brae Paradox Iniial Nework: Augmened Nework: ½ ½ x 1 ½ ½ 1 x x x Delay = 1.5 Delay = 2 10

11 Brae Paradox Iniial Nework: Augmened Nework: ½ ½ x 1 ½ ½ 1 x x x Delay = 1.5 Delay = 2 All raffic incur more delay! [Brae 68] alo ha phyical analog [Cohen/Horowiz 91] 11

12 The Mahemaical Model a direced graph G = (V,E) k ource-deinaion pair ( 1, 1 ),, ( k, k ) a rae r i of raffic from i o i for each edge e, a laency funcion l e ( ) aumed coninuou and nondecreaing Example: (k,r=1) l(x)=x Flow = ½ 1 1 l(x)=1 Flow = ½ 12

13 Rouing of Traffic Traffic and Flow: f P = amoun of raffic roued on i - i pah P flow vecor f rouing of raffic Selfih rouing: wha flow arie a he roue choen by many noncooperaive agen? 13

14 Nah Flow Some aumpion: agen mall relaive o nework wan o minimize peronal laency Def: A flow i a Nah equilibrium (or i a Nah flow) if all flow i roued on min-laency pah [given curren edge congeion] Example: x 1 Flow =.5 Flow =.5 x 1 Flow = 1 Flow = 0 14

15 Some Hiory raffic model, definiion of Nah flow given by [Wardrop 52] hiorically called uer-opimal/uer equilibrium Nah flow exi, are (eenially) unique due o [Beckmann e al. 56] Nah flow alo arie via diribued hore-pah proocol (e.g., OSPF, BGP) a long a laency ued for edge weigh convergence udied in [Tiikli/Bereka 86] 15

16 The Co of a Flow Def: he co C(f) of flow f = um of all delay incurred by raffic (aka oal laency) x 1 ½ ½ Co = ½ ½ +½ 1 = ¾ 16

17 The Co of a Flow Def: he co C(f) of flow f = um of all delay incurred by raffic (aka oal laency) x 1 ½ ½ Co = ½ ½ +½ 1 = ¾ Formally: if l P (f) = um of laencie of edge of P (w.r.. he flow f), hen: C(f) = Σ P f P l P (f) 17

18 Inefficiency of Nah Flow Noe: Nah flow do no minimize oal laency oberved informally by [Pigou 1920] lack of coordinaion lead o inefficiency x 1 ½ 0 1 ½ Co of Nah flow = = 1 Co of opimal (min-co) flow = ½ ½ +½ 1 = ¾ 18

19 How Bad I Selfih Rouing? x Pigou example i imple 1 ½ 1 0 ½ Cenral queion: How inefficien are Nah flow in more realiic nework? Goal: prove ha Nah flow are near-opimal wan laiez-faire approach o managing nework 19

20 The Bad New Bad Example: x d 1 1-Є 0 1 Є (r = 1, d large) Nah flow ha co 1, min co 0 Nah flow can co arbirarily more han he opimal (min-co) flow even if laency funcion are polynomial 20

21 Hardware Offe Selfihne Approach #1: ue differen ype of guaranee Theorem: [Roughgarden/Tardo 00] for every nework: Nah co a rae r op co a rae 2r 21

22 Hardware Offe Selfihne Approach #1: ue differen ype of guaranee Theorem: [Roughgarden/Tardo 00] for every nework: Nah co a rae r op co a rae 2r Alo: M/M/1 fn (l(x)=1/(u-x), u = capaciy) Nah w/capaciie 2u op w/capaciie u 22

23 Linear Laency Funcion Approach #2: reric cla of allowable laency funcion Def: linear laency fn i of form l e (x)=a e x+b e Theorem: [Roughgarden/Tardo 00] for every nework wih linear laency fn: co of Nah flow 4/3 co of op flow 23

24 Source of Inefficiency Corollary of previou Theorem: For linear laency fn, wor Nah/OPT raio i realized in a wo-link nework! x 1 ½ 0 1 ½ Co of Nah = 1 Co of OPT = ¾ imple explanaion for wor inefficiency confroned w/wo roue, elfih uer overconge one of hem 24

25 Simple Wor-Cae Nework Theorem: [Roughgarden 02] fix any cla of laency fn, and he wor Nah/OPT raio occur in a wo-node, wo-link nework. under mild aumpion (convexiy, richne) inefficiency of Nah flow alway ha imple explanaion; imple nework are wor example 25

26 Simple Wor-Cae Nework Theorem: [Roughgarden 02] fix any cla of laency fn, and he wor Nah/OPT raio occur in a wo-node, wo-link nework. under mild aumpion (convexiy, richne) inefficiency of Nah flow alway ha imple explanaion; imple nework are wor example Proof Idea: Nah flow olve a cerain minimizaion problem no quie oal laency, bu cloe elecrical curren i phyical analog 26

27 Compuing he Price of Anarchy Applicaion: wor-cae example imple wor-cae raio i eay o calculae Example: polynomial wih degree d, nonnegaive coeff price of anarchy Θ(d/log d) x d 1 27

28 Hardware Offe Selfihne Theorem: [Roughgarden/Tardo 00] for every nework: Nah co a rae r op co a rae 2r Corollary: nework wih M/M/1 delay fn Nah w/capaciie 2u op w/capaciie u 28

29 Key Difficuly Suppoe f a Nah flow, f * an op flow a wice he rae. Wan o how ha C(f * ) C(f). Noe: co of f can be wrien a C(f) = Σ e f e l e (f e ) Similarly: C(f * ) = Σ e f * l e (f * ) Problem: wha i he relaion beween l e (f e ) and l e (f * )? e e e 29

30 Key Trick Idea: lower bound co of f * uing a differen e of laency fn c uch ha: eay o lower bound co of f * w.r.. laency fn c co of f * w.r.. fn c co of f * w.r.. fn l 30

31 Key Trick Idea: lower bound co of f * uing a differen e of laency fn c uch ha: eay o lower bound co of f * w.r.. laency fn c co of f * w.r.. fn c co of f * w.r.. fn l The conrucion: graph of l graph of c l e (f e ) 0 l e (f e ) 0 0 f e 0 f e 31

32 Lower Bounding OPT Aume for impliciy: only one commodiy. all raffic in Nah flow ha ame laency, ay L co of Nah flow eay o compue: C(f) = rl 32

33 Lower Bounding OPT Aume for impliciy: only one commodiy. all raffic in Nah flow ha ame laency, ay L co of Nah flow eay o compue: C(f) = rl Key obervaion: laency of pah P w.r.. laency fn c wih no congeion i l P (f) l e (f e ) 0 pah laency in Nah flow 0 f e 33

34 Lower Bounding OPT Aume for impliciy: only one commodiy. all raffic in Nah flow ha ame laency, ay L co of Nah flow eay o compue: C(f) = rl Key obervaion: laency of pah P w.r.. laency fn c wih no congeion i l P (f) l e (f e ) 0 pah laency in Nah flow 0 f e co of f * w.r.. c i a lea 2rL = 2C(f) 34

35 Bounding he Overeimae So far: co of f * w.r.. c i 2C(f). Claim: (will finih proof of Thm) [co of f * w.r.. c] -C(f * ) C(f). 35

36 Bounding he Overeimae So far: co of f * w.r.. c i 2C(f). Claim: (will finih proof of Thm) [co of f * w.r.. c] -C(f * ) C(f). Reaon: difference in co on edge e i l e (f e ) ypical value of c e (f* e )f* e - l e (f* e )f* e 0 f e * 0 f e 36

37 Bounding he Overeimae So far: co of f * w.r.. c i 2C(f). Claim: (will finih proof of Thm) [co of f * w.r.. c] -C(f * ) C(f). Reaon: difference in co on edge e i l e (f e ) ypical value of c e (f* e )f* e - l e (f* e )f* e 0 f e * 0 f e c e (f* e )f * e - l e (f* e )f * l e e (f e )f e um over edge o ge Claim 37

38 Summary Goal: prove ha lo in nework performance due o elfih rouing i no oo large. Problem: a Nah flow can co far more han an opimal flow. Soluion: compare Nah o op flow wih exra raffic reric cla of allowable edge laency funcion, obain bounded price of anarchy x d 1 1-Є 0 1 Є 38

39 Coping wih Selfihne Goal: deign/manage nework o ha elfih rouing no oo bad add algorihmic dimenion Approach #1: Nework deign wan o avoid Brae Paradox Reul: [Roughgarden FOCS 01] Brae Paradox can be arbirarily evere in larger nework, hard o efficienly deec alo [Lin/Roughgarden/Tardo, in prep] 39

40 Coping wih Selfihne Approach #2: Sackelberg rouing ome raffic roued cenrally, elfih uer reac o congeion accordingly [Roughgarden STOC 01]: Sackelberg rouing can dramaically improve over he Nah flow Approach #3: Edge pricing ue economic incenive (axe) o influence elfih behavior [Cole/Dodi/Roughgarden EC 03 + STOC 03]: explore hi idea for elfih rouing 40

41 Fuure Reearch Explore oher game-heoreic environmen uing an approximaion framework [Czumaj/Krya/Voecking STOC 02], [Vea FOCS 02], ec. Approximaion algorihm for nework deign alo inereing wihou game-heoreic conrain [Kumar/Gupa/Roughgarden FOCS 02] [Gupa/Kumar/Roughgarden STOC 03] Algorihm for key game-heoreic concep Nah/marke equilibria (e.g., [Devanur e al FOCS 02]) 41

42 Exenion Fac: poiive reul coninue o hold for: approximae Nah flow [RT00] uer roue on approximaely min-laency pah finiely many agen, pliable flow [RT00] weaken aumpion ha agen are mall nonaomic congeion game, game wihou combinaorial rucure of a nework [RT02] 42