How Much Can Taxes Help Selfish Routing?

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1 How Much Can Taxe Help Selfih Rouing? Tim Roughgarden (Cornell) Join wih Richard Cole (NYU) and Yevgeniy Dodi (NYU)

2 Selfih Rouing a direced graph G = (V,E) a ource and a deinaion one uni of raffic from o for each edge e, a laency funcion l e ( ) aumed coninuou, nondecreaing Example: l(x)=x Flow = ½ l(x)=1 Flow = ½ 2

3 Rouing of Traffic Traffic and Flow: f P = fracion of raffic roued on - pah P flow vecor f rouing of raffic Selfih rouing: wha flow arie a he roue choen by many noncooperaive agen? 3

4 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] have exience, uniquene [Wardrop, Beckmann e al 50] Example: x 1 Flow =.5 Flow =.5 x 1 Flow = 1 Flow = 0 4

5 Inefficiency of Nah Flow Our objecive funcion: average laency Nah flow need no be opimal oberved informally by [Pigou 1920] x 1 ½ 0 1 ½ Average laency of Nah flow = = 1 of opimal flow = ½ ½ +½ 1 = ¾ 5

6 Brae Paradox Iniial Nework: ½ ½ x 1 ½ ½ 1 x Delay = 1.5 6

7 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? 7

8 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] 8

9 Marginal Co Taxe Goal: do beer wih axe (one per edge) no addreing implemenaion 9

10 Marginal Co Taxe Goal: do beer wih axe (one per edge) no addreing implemenaion Aume: all raffic minimize ime + money ee STOC 03 paper for relaxing hi Def: he marginal co ax of an edge (w.r.. a flow) i he exra delay o exiing raffic caued by a marginal increae in raffic 10

11 Marginal Co Taxe Goal: do beer wih axe (one per edge) no addreing implemenaion Aume: all raffic minimize ime + money ee STOC 03 paper for relaxing hi Def: he marginal co ax of an edge (w.r.. a flow) i he exra delay o exiing raffic caued by a marginal increae in raffic Thm: [folklore] marginal co axe w.r.. he op flow induce he op flow a a Nah eq. 11

12 Are Taxe a Social Lo? Problem wih MCT: min delay i holy grail; exorbian axe ignored 12

13 Are Taxe a Social Lo? Problem wih MCT: min delay i holy grail; exorbian axe ignored Ever reaonable?: ye, iff axe can be refunded (direcly or indirecly) 13

14 Are Taxe a Social Lo? Problem wih MCT: min delay i holy grail; exorbian axe ignored Ever reaonable?: ye, iff axe can be refunded (direcly or indirecly) New Goal: minimize oal diuiliy wih nonrefundable axe (delay + axe paid) call new objecive fn he co marginal co axe now no a good idea, e.g.: Thm: w/linear laency fn, MCT never help. 14

15 Taxe v. Edge Removal Noe: axe a lea a good a edge removal can effec edge deleion wih large ax are hey ricly more powerful? 15

16 Taxe v. Edge Removal Noe: axe a lea a good a edge removal can effec edge deleion wih large ax are hey ricly more powerful? Thm: axe can improve co by a facor of n/2 (n = V ), bu no more. ame for edge removal [Roughgarden FOCS 01] alo ame a edge removal for rericed clae of laency fn 16

17 Taxe v. Edge Removal Queion: axe no beer han edge removal in be cae, how abou in pecific nework? 17

18 Taxe v. Edge Removal Queion: axe no beer han edge removal in be cae, how abou in pecific nework? Thm: (a) axe can improve he Nah flow co by an n/2 facor more han edge removal ue ep funcion-like laency fn variaion of Brae graph from [Roughgarden FOCS 01] (b) axe are never more powerful han edge removal in nework w/linear laency fn 18

19 Taxe v. Edge Removal General Laency Fn n/2 Nah co afer axe Linear Laency Fn 0 Nah co afer axe Nah co afer edge removal n/2 Nah co afer edge removal 4/3 n/2 4/3 original Nah co original Nah co 19

20 Proof Skech for Linear Cae Fir: aume fale, look a minimal counerexample. Look a counerexample ax on hi nework ha minimize co and ha malle um. Technical Lemma: hi minimum exi (ue minimaliy). Underand how Nah flow change under local perurbaion of he ax (minimaliy, lineariy). Perurbing o a maller ax mu increae co. Oppoie perurbaion lower co (conradicion). 20

21 Taxe Are Powerful bu Eluive Recall: axe can improve co by a facor of n/2 (n = V ), bu no more. powerful, bu can we compue hem? Thm: opimal axe NP-hard o approximae wihin facor of o(n/log n). complexiy ca doub on poenial for axe ha minimize co baed on [Roughgarden FOCS 01] 21

22 Some Fuure Direcion Improve model convergence iue, imperfec info oher noion of incenive-compaibiliy e.g., robu o maliciou uer oher objecive fn Beer reul in hi model mulicommodiy flow nework 22

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