Braess’s Paradox, Fibonacci Numbers, and Exponential Inapproximability

  • Henry Lin
  • Tim Roughgarden
  • Éva Tardos
  • Asher Walkover
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3580)


We give the first analyses in multicommodity networks of both the worst-case severity of Braess’s Paradox and the price of anarchy of selfish routing with respect to the maximum latency. Our first main result is a construction of an infinite family of two-commodity networks, related to the Fibonacci numbers, in which both of these quantities grow exponentially with the size of the network. This construction has wide implications, and demonstrates that numerous existing analyses of selfish routing in single-commodity networks have no analogues in multicommodity networks, even in those with only two commodities. This dichotomy between single- and two-commodity networks is arguably quite unexpected, given the negligible dependence on the number of commodities of previous work on selfish routing.

Our second main result is an exponential upper bound on the worst-possible severity of Braess’s Paradox and on the price of anarchy for the maximum latency, which essentially matches the lower bound when the number of commodities is constant.

Finally, we use our family of two-commodity networks to exhibit a natural network design problem with intrinsically exponential (in)approximability: while there is a polynomial-time algorithm with an exponential approximation ratio, subexponential approximation is unachievable in polynomial time (assuming PNP).


Nash Equilibrium Latency Function Average Latency Network Design Problem Maximum Latency 
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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  1. 1.
    Anshelevich, E., Dasgupta, A., Tardos, É., Wexler, T.: Near-optimal network design with selfish agents. In: Proceedings of the 35th Annual ACM Symposium on the Theory of Computing (STOC), pp. 511–520 (2003)Google Scholar
  2. 2.
    Beckmann, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press (1956)Google Scholar
  3. 3.
    Braess, D.: Über ein Paradoxon aus der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chau, C.K., Sim, K.M.: The price of anarchy for non-atomic congestion games with symmetric cost maps and elastic demands. Operations Research Letters 31(5), 327–335 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Correa, J.R., Schulz, A.S., Stier Moses, N.E.: Computational complexity, fairness, and the price of anarchy of the maximum latency problem. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 59–73. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Correa, J.R., Schulz, A.S., Stier Moses, N.E.: Selfish routing in capacitated networks. Mathematics of Operations Research 29(4), 961–976 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Czumaj, A.: Selfish routing on the Internet. In: Leung, J. (ed.) Handbook of Scheduling: Algorithms, Models, and Performance Analysis, ch. 42, CRC Press, Boca Raton (2004)Google Scholar
  8. 8.
    Devanur, N., Garg, N., Khandekar, R., Pandit, V., Saberi, A.: Price of anarchy, locality gap, and a network service provider game (2003) (Unpublished manuscript)Google Scholar
  9. 9.
    Fabrikant, A., Luthra, A., Maneva, E., Papadimitriou, C.H., Shenker, S.J.: On a network creation game. In: Proceedings of the 22nd ACM Symposium on Principles of Distributed Computing (PODC), pp. 347–351 (2003)Google Scholar
  10. 10.
    Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Selfish routing in non-cooperative networks: A survey. In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 21–45. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979)zbMATHGoogle Scholar
  12. 12.
    Johari, R., Tsitsiklis, J.N.: Efficiency loss in a network resource allocation game. Mathematics of Operations Research 29(3), 407–435 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science (STACS), pp. 404–413 (1999)Google Scholar
  14. 14.
    Lin, H., Roughgarden, T., Tardos, É.: A stronger bound on braess’s paradox. In: Proceedings of the 15th Annual Symposium on Discrete Algorithms (SODA), pp. 333–334 (2004)Google Scholar
  15. 15.
    Papadimitriou, C.H.: Algorithms, games, and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on the Theory of Computing (STOC), pp. 749–753 (2001)Google Scholar
  16. 16.
    Perakis, G.: The price of anarchy when costs are non-separable and asymmetric. In: Bienstock, D., Nemhauser, G.L. (eds.) IPCO 2004. LNCS, vol. 3064, pp. 46–58. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Roughgarden, T.: Designing networks for selfish users is hard. In: Proceedings of the 42nd Annual Symposium on Foundations of Computer Science, pp. 472–481 (2001)Google Scholar
  18. 18.
    Roughgarden, T.: The price of anarchy is independent of the network topology. Journal of Computer and System Sciences 67(2), 341–364 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Roughgarden, T.: The maximum latency of selfish routing. In: Proceedings of the 15th Annual Symposium on Discrete Algorithms (SODA), pp. 973–974 (2004)Google Scholar
  20. 20.
    Roughgarden, T.: Selfish Routing and the Price of Anarchy. MIT Press, Cambridge (2005)Google Scholar
  21. 21.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? Journal of the ACM 49(2), 236–259 (2002)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Roughgarden, T., Tardos, É.: Bounding the inefficiency of equilibria in nonatomic congestion games. Games and Economic Behavior 47(2), 389–403 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Smith, M.J.: The existence, uniqueness and stability of traffic equilibria. Transportation Research 13B, 295–304 (1979)Google Scholar
  24. 24.
    Vetta, A.: Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In: Proceedings of the 43rd Annual Symposium on Foundations of Computer Science (FOCS), pp. 416–425 (2002)Google Scholar
  25. 25.
    Weitz, D.: The price of anarchy (2001) (Unpublished manuscript)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Henry Lin
    • 1
  • Tim Roughgarden
    • 2
  • Éva Tardos
    • 3
  • Asher Walkover
    • 4
  1. 1.Computer Science DivisionUC BerkeleyBerkeley
  2. 2.Department of Computer ScienceStanford UniversityStanford
  3. 3.Department of Computer ScienceCornell UniversityIthaca
  4. 4.Google Inc.Mountain View

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