Braess’s Paradox in Large Sparse Graphs

  • Fan Chung
  • Stephen J. Young
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6484)

Abstract

Braess’s paradox, in its original context, is the counter-intuitive observation that, without lessening demand, closing roads can improve traffic flow. With the explosion of distributed (selfish) routing situations understanding this paradox has become an important concern in a broad range of network design situations. However, the previous theoretical work on Braess’s paradox has focused on “designer” graphs or dense graphs, which are unrealistic in practical situations. In this work, we exploit the expansion properties of Erdős-Rényi random graphs to show that Braess’s paradox occurs when np ≥ c log(n) for some c > 1.

Keywords

Braess’s paradox price of anarchy random graphs selfish routing 

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References

  1. 1.
    Beckmann, M., McGuire, C.B., Winsten, C.B.: Studies in the economics of transportation. Yale University Press, London (1956)Google Scholar
  2. 2.
    Fisk, C., Pallottino, S.: Empirical evidence for equilibrium paradoxes with implications for optimal planning strategies. Transportation Research Part A: General 15(3), 245–248 (1981)CrossRefGoogle Scholar
  3. 3.
    Frank, M.: The Braess paradox. Math. Programming 20(3), 283–302 (1981)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Hall, M.A.: Properties of the equilibrium state in transportation networks. Transportation Science 12(3), 208–216 (1978)CrossRefGoogle Scholar
  5. 5.
    Kameda, H.: How harmful the paradox can be in the braess/cohen-kelly-jeffries networks. In: Proceedings of IEEE Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, INFOCOM 2002, vol. 1, pp. 437–445 (2002)Google Scholar
  6. 6.
    Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Inequalities, III (Proc. Third Sympos., Univ. California, Los Angeles, Calif. (1969); dedicated to the memory of Theodore S. Motzkin), pp. 159–175. Academic Press, New York (1972) Google Scholar
  7. 7.
    Kolata, G.: What if they closed 42d street and nobody noticed? The New York Times (December 25, 1990)Google 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.
    Lin, H., Roughgarden, T., Tardos, É.: A stronger bound on Braess’s paradox. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms (electronic), pp. 340–341. ACM, New York (2004)Google Scholar
  10. 10.
    Papadimitriou, C.: Algorithms, games, and the internet. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing (electronic), pp. 749–753. ACM, New York (2001)Google Scholar
  11. 11.
    Pas, E.I., Principio, S.L.: Braess’ paradox: Some new insights. Transportation Research Part B: Methodological 31(3), 265–276 (1997)CrossRefGoogle Scholar
  12. 12.
    Penchina, C.M.: Braess paradox: Maximum penalty in a minimal critical network. Transportation Research Part A: Policy and Practice 31(5), 379–388 (1997)Google Scholar
  13. 13.
    Roughgarden, T.: On the severity of Braess’s Paradox: designing networks for selfish users is hard. J. Comput. System Sci. 72(5), 922–953 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM (electronic) 49(2), 236–259 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Spielman, D.A., Teng, S.-H.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM (electronic) 51(3), 385–463 (2004)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Valiant, G., Roughgarden, T.: Braess’s paradox in large random graphs. Random Structures Algorithms (to appear)Google Scholar
  17. 17.
    Wardrop, J.: Some theoretical aspects of road traffic research. Proceedings of the Institution of Civil Engineers, Part II 1(36), 352–362 (1952)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Fan Chung
    • 1
  • Stephen J. Young
    • 1
  1. 1.Department of MathematicsUniversity of CaliforniaLa Jolla

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