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Sensitivity of Wardrop Equilibria

  • Matthias Englert
  • Thomas Franke
  • Lars Olbrich
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4997)

Abstract

We study the sensitivity of equilibria in the well-known game theoretic traffic model due to Wardrop. We mostly consider single-commodity networks. Suppose, given a unit demand flow at Wardrop equilibrium, one increases the demand by ε or removes an edge carrying only an ε-fraction of flow. We study how the equilibrium responds to such an ε-change.

Our first surprising finding is that, even for linear latency functions, for every ε> 0, there are networks in which an ε-change causes every agent to change its path in order to recover equilibrium. Nevertheless, we can prove that, for general latency functions, the flow increase or decrease on every edge is at most ε.

Examining the latency at equilibrium, we concentrate on polynomial latency functions of degree at most p with nonnegative coefficients. We show that, even though the relative increase in the latency of an edge due to an ε-change in the demand can be unbounded, the path latency at equilibrium increases at most by a factor of (1 + ε) p . The increase of the price of anarchy is shown to be upper bounded by the same factor. Both bounds are shown to be tight.

Let us remark that all our bounds are tight. For the multi-commodity case, we present examples showing that neither the change in edge flows nor the change in the path latency can be bounded.

Keywords

Latency Function Path Latency Global Instability Unit Demand Wardrop Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Beckmann, M., McGuire, C.B., Winston, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)Google Scholar
  2. 2.
    Braess, D.: Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1968)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dafermos, S., Nagurney, A.: Sensitivity Analysis for the Asymmetric Network Equilibrium Problem. Mathematical Programming 28, 174–184 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dafermos, S., Sparrow, F.T.: The Traffic Assignment Problem for a General Network. Journal of Research of the National Bureau of Standards 73(2), 91–118 (1969)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fisk, C.: More Paradoxes in the Equilibrium Assignment Problem. Transportation Research, Series B 13(4), 305–309 (1979)CrossRefGoogle Scholar
  6. 6.
    Hall, M.A.: Properties of the Equilibrium State in Transportation Networks. Transportation Science 12, 208–216 (1978)CrossRefGoogle Scholar
  7. 7.
    Josefsson, M., Patriksson, M.: Sensitivity Analysis of Separable Traffic Equilibria with Application to Bilevel Optimization in Network Design. Transportation Research Series B 41(1), 4–31 (2007)CrossRefGoogle Scholar
  8. 8.
    Koutsoupias, E., Papadimitriou, C.: Worst-Case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  9. 9.
    Patriksson, M.: Sensitivity Analysis of Traffic Equilibria. Transportation Research 38, 258–281 (2004)Google Scholar
  10. 10.
    Pigou, A.C.: The Economics of Welfare. Macmillan, Basingstoke (1920)Google Scholar
  11. 11.
    Roughgarden, T., Tardos, É.: How Bad is Selfish Routing. Journal of the ACM 49(2), 236–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Roughgarden, T.: How Unfair is Selfish Routing. In: Roughgarden, T. (ed.) Proc. of th 13th Annual Symposium on Discrete Algorithms (SODA), pp. 203–204 (2002)Google Scholar
  13. 13.
    Roughgarden, T.: The Price of Anarchy is Independent of the Network Topology. In: Proc. of th 34th Annual Symposium on Theory of Computing Discrete Algorithms (STOC), pp. 428–437 (2002)Google Scholar
  14. 14.
    Roughgarden, T.: On the Severity of Braess’s Paradox: Designing Networks for Selfish Users is Hard. Journal of Computer and System Sciences 72(5), 922–953 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Wardrop, J.G.: Some Theoretical Aspects of Road Traffic Research. In: Proc. of the Institute of Civil Engineers Pt. II, pp. 325–378 (1952)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Matthias Englert
    • 1
  • Thomas Franke
    • 1
  • Lars Olbrich
    • 1
  1. 1.Dept. of Computer ScienceRWTH Aachen UniversityGermany

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