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Selfish Routing in Non-cooperative Networks

Selfish Routing in Non-cooperative Networks. Balázs Sziklai. Historical Outline. J.G. Wardrop (1952): Some theoretical aspects of road traffic research , Proc . of the Institute of Civil Engineers D. Braess (1968): Über ein Paradox der Verkehrsplanung , Unternehmensforschung

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Selfish Routing in Non-cooperative Networks

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  1. SelfishRoutinginNon-cooperativeNetworks Balázs Sziklai

  2. HistoricalOutline • J.G. Wardrop (1952): Sometheoreticalaspects of roadtrafficresearch, Proc. of the Institute of Civil Engineers • D. Braess (1968): Überein Paradox der Verkehrsplanung, Unternehmensforschung • E. Koutsoupias and C. Papadimitriou (1999): WorstCaseEquilibria, Proc. Of the 16th International SymposiumonTheoreticalAspects of Computer Science • Feldmann et al. (2003): SelfishRoutinginNon-cooperativeNetworks: A Survey, LectureNotesin Computer Science

  3. The WardropModel • A probleminstanceconsists of a triple • a directedgraph • a set of routingtasks • and a set of edgelatencyfunctions • For a fixed flow f thelatency of an edgee is definedastheproduct of thelatencyfunctiononewhenroutingtrafficftimesthetrafficfitself. • The socialcost of a flow is definedto be the sum of theedgelatencies. x+ 1 s t

  4. Pigou’sexample • Wewouldliketoroute a total of one unit of flow fromstot. • Toobtaintheoptimalsolutionwehavetocomputethe minimum of thefollowingquadraticpolinomial: • Simplecalculationshowsthatthesocial optimum is toroute ½ of thetrafficontheupperedge and ½ of thetrafficontheloweredge. • Howeverthelatencycost is lowerontheupperedgeforanyuser, whichmeansitcant be a Nashequilibrium. • The unique NE point is toroute allthetrafficontheupperedge. x s t 1

  5. Braess Paradox • Wewouldliketoroute 6 unit of trafficonthefollowinggraph: s 10x x + 50 x + 50 10x t

  6. Braess Paradox • The socialcost of the flow attheNashEquilibrium: 2 ∙[3 ∙(10∙3+(3+50))] = 6 ∙ 83 = 496 s 3 3 3 3 t

  7. Braess Paradox • Nowweconnectthetwomiddlepointswith an edge. s 10x x + 50 x +10 x + 50 10x t

  8. Braess Paradox • Usingthenewedgesome of theuserscan be betteroffbychangingtheirpathfromstot. s 3 3 83 = (3 ∙10) + (3+50) > (3 ∙10) + (1+10) + (4 ∙10) = 81 1 2 4 t

  9. Braess Paradox • Usingthenewedgesome of theuserscan be betteroffbychangingtheirpathfromstot. s 4 2 93 = (3+50) + (4 ∙10) > (4 ∙10) + (2+10) + (4 ∙10) = 92 2 2 4 t

  10. Braess Paradox • Again wearrivedto a NE point. No usercanimproveitsprivatelatencybyunilaterallychangingitsroute. s The newsocialcost is 4 ∙(10∙4+(2+50))+ 2 ∙(10∙4+(2+10)+ 10∙4) = 6 ∙ 92 = 558 > 496 4 2 2 2 4 t

  11. HowtoresolvetheBraess Paradox • Tofindthe ‘bad’ edgesin a network is NP-hard (Roughgarden 2002). • Capacityshould be addedacrossthenetworkratherthanon a local scale (Korilis et al. 1995). • Stackelbergapproach: controllingjust a smallportion of thetrafficthesystemcan be drivencloseintothenetwork optimum (Korilis et al. 1995, Roughgarden 2001).

  12. A physicalrepresentation of theBraess Paradox 1 kg 1 kg

  13. Price of Anarchy • IntroducedbyPapadimitriouin a conferencepaper (2001). • Bydefinitionsit is the ratio of the (worstcase) Nashequilibriumsocialcost and theoptimalsocialcost. • IntheWardropmodelthecoordination ratio cannot be boundedfromaboveby a constant, whenarbitraryedgelatencyfunctionsareallowed. • Howeverwhenwerestrictourselvestolinearedgelatencyfunctionsthecoordination ratio is boundedfromaboveby 4/3. • Thisbound is tight (seePigou’sexample).

  14. KP-model • NamedafterKoutsoupias and Papadimitriou (1999). • Simplenetworkwithmparalelllinks and nusers. • Eachuseri has a weightwi and thesetrafficsareunsplittable. • A purestrategy of a user is a specific link, and a mixed strategy is a probabilitydistribution over theset of links. s t

  15. KP-model • The socialcost is theexpected maximum latencyon a link, wheretheexpectation is taken over all random choices of theusers. • Basic model – identicallinks. • General model – relatedlinks (differentlinkshavedifferentcapacities).

  16. FMNE conjecture • Considertheinstance of threeidenticallinks and threeuserswithtrafficweight of 2. A B s t C

  17. FMNE conjecture • Fully Mixed NashEquilibrium is a NE whereeveryuserroutesalongeverypossibleedge. • Considerthemodel of arbitrarytraffics and relatedlinks. Thenanytrafficvectorw suchthatthefully mixed NashEquilibriumFexists, and foranyNashequilibriumP, SC(w, P) ≤ SC(w, F).

  18. Differencesbetweenthetwomodels • Wardropmodeldefinesthesocialcostasthe sum of theedgelatencieswhiletheKP-modelastheexpected maximum edgelatencyon a link. • WhileintheWardrop-modeltrafficcan be splittedintoinfinitelytinypieces, trafficratesareunsplittableintheKP-model. • AllNashEquilibriaarepure and havethesamesocialcostintheWardrop-model.

  19. Unsolvedproblems • Giveupper and lowerboundsforthe Price of Anarchy. • Howto design a network free fromBraess paradox? • Is the FMNE conjecturetrue? • Findbetteralgorithmstocompute NE points.

  20. Thankyouforyourattention!

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