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Navigating Selfish Networks: Understanding Nash Equilibria and Coordination Ratios

Explore theoretical concepts from historical works by Wardrop and Braess on traffic flow research. Discover the Wardrop model and Braess Paradox, analyzing network equilibria. Delve into Price of Anarchy and KP-model implications with unsolved problems in network design.

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Navigating Selfish Networks: Understanding Nash Equilibria and Coordination Ratios

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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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