Selfish Routing in Non-cooperative Networks: A Survey

  • R. Feldmann
  • M. Gairing
  • Thomas Lücking
  • Burkhard Monien
  • Manuel Rode
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2747)


We study the problem of n users selfishly routing traffics through a shared network. Users route their traffics by choosing a path from their source to their destination of the traffic with the aim of minimizing their private latency. In such an environment Nash equilibria represent stable states of the system: no user can improve its private latency by unilaterally changing its strategy.

In the first model the network consists only of a single source and a single destination which are connected by m parallel links. Traffics are unsplittable. Users may route their traffics according to a probability distribution over the links. The social optimum minimizes the maximum load of a link. In the second model the network is arbitrary, but traffics are splittable among several paths leading from their source to their destination. The goal is to minimize the sum of the edge latencies.

Many interesting problems arise in such environments: A first one is the problem of analyzing the loss of efficiency due to the lack of central regulation, expressed in terms of the coordination ratio. A second problem is the Nashification problem, i.e. the problem of converting any given non-equilibrium routing into a Nash equilibrium without increasing the social cost. The Fully Mixed Nash Equilibrium Conjecture (FMNE Conjecture) states that a Nash equilibrium, in which every user routes along every possible edge with probability greater than zero, is a worst Nash equilibrium with respect to social cost. A third problem is to exactly specify the sub-models in which the FMNE Conjecture is valid. The well-known Braess’s Paradox shows that there exist networks, such that strict sub-networks perform better when users are selfish. A natural question is the following network design problem: Given a network, which edges should be removed to obtain the best possible Nash equilibrium.

We present complexity results for various problems in this setting, upper and lower bounds for the coordination ratio, and algorithms solving the problem of Nashification. We survey results on the validity of the FMNE Conjecture in the model of unsplittable flows, and for the model of splittable flows we survey results for the network design problem.


Nash Equilibrium Problem Instance Social Cost Pure Strategy Edge 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.


  1. 1.
    Beckmann, M.J.: On the theory of traffic flow in networks. Traffic Quart 21, 109–116 (1967)Google Scholar
  2. 2.
    Brucker, P., Hurink, J., Werner, F.: Improving local search heuristics for some scheduling problems. part II. Discrete Applied Mathematics 72, 47–69 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Beckmann, M., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven and London (1956)Google Scholar
  4. 4.
    Braess, D.: Über ein Paradoxon der Verkehrsplanung. Unternehmensforschung 12, 258–268 (1968)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cho, Y., Sahni, S.: Bounds for list schedules on uniform processors. SIAM Journal on Computing 9(1), 91–103 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. In: Proc. of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002), pp. 413–420 (2002)Google Scholar
  7. 7.
    Dafermos, S.C., Sparrow, F.T.: The traffic assignment problem for a general network. Journal of Research of the National Bureau of Standards, Series B 73B(2), 91–118 (1969)MathSciNetGoogle Scholar
  8. 8.
    Even-Dar, E., Kesselmann, A., Mansour, Y.: Convergence time to nash equilibria. In: Proc. of the 30th International Colloquium on Automata, Languages, and Programming, ICALP 2003 (2003)Google Scholar
  9. 9.
    Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Nashification and the coordination ratio for a selish routing game. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  10. 10.
    Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT 19, 312–320 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of nash equilibria for a selish routing game. In: Widmayer, P., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 123–134. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  12. 12.
    Feigenbaum, J., Papdimitriou, C., Shenker, S.: Sharing the cost of multicast transmissions. In: Proc. of the 32nd Annual ACM Symposium on the Theory of Computing, pp. 218–227 (2000)Google Scholar
  13. 13.
    Friesen, D.K.: Tighter bounds for lpt scheduling on uniform processors. SIAM Journal on Computing 16(3), 554–560 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  15. 15.
    Gairing, M., Lüocking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Extreme nash equilibria. Technical report, FLAGS-TR-03-10, University of Paderborn (2002) Google Scholar
  16. 16.
    Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM Journal of Applied Mathematics 17(2), 416–429 (1969)zbMATHCrossRefGoogle Scholar
  17. 17.
    Haurie, A., Marcotte, P.: On the relatonship between nash-cournot and wardrop equilibria. Networks 15, 295–308 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Hochbaum, D.S., Shmoys, D.: Using dual approximation algorithms for scheduling problems: Theoretical and practical results. Journal of the ACM 34(1), 144–162 (1987)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Hochbaum, D.S., Shmoys, D.: A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM Journal on Computing 17(3), 539–551 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Jahn, O., Möhring, R.H., Schulz, A.S., Stier Moses, N.E.: System-Optimal Routing of Traffic Flows With User Constraints in Networks With Congestion. MIT Sloan School of Management Working Paper No. 4394-02 (2002)Google Scholar
  21. 21.
    Jain, K., Vazirani, V.: Applications of approximation algorithms to cooperative games. In: Proc. of the 33rd Annual ACM Symposium on Theory of Computing (STOC 2001), pp. 364–372 (2001)Google Scholar
  22. 22.
    Korilis, Y.A., Lazar, A.A., Orda, A.: Architecting noncooperative networks. IEEE Journal on Selected Areas in Communications 13(7), 1241–1251 (1995)CrossRefGoogle Scholar
  23. 23.
    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
  24. 24.
    Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate Equilibria and Ball Fusion. In: Proc. of the 9th International Colloquium on Structural Information and Communication Complexity, SIROCCO 2002 (2002) (accepted for TOCS)Google Scholar
  25. 25.
    Lücking, T., Mavronicolas, M., Monien, B., Rode, M., Spirakis, P., Vrto, I.: Which is the worst-case nash equilibrium? In: Rovan, B., Vojtáš, P. (eds.) MFCS 2003. LNCS, vol. 2747, pp. 551–561. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  26. 26.
    Lenstra, J.K., Shmoys, D.B., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. In: Proc. ofthe 28th Annual Symposium on Foundations of Computer Science (FOCS 1987), pp. 217–224 (1987)Google Scholar
  27. 27.
    McKelvey, R.D., McLennan, A.: Computation of equilibria in inite games. In: Amman, H., Kendrick, D., Rust, J. (eds.) Handbook of Computational Economics (1996)Google Scholar
  28. 28.
    Mavronicolas, M., Spirakis, P.: The price of selish routing. In: Proc. of the 33rd Annual ACM Symposium on Theory of Computing (STOC 2001), pp. 510–519 (2001)Google Scholar
  29. 29.
    Nash, J.: Non-cooperative games. Annals ofMathematics 54(2), 286–295 (1951)CrossRefMathSciNetGoogle Scholar
  30. 30.
    Nisan, N.: Algorithms for selish agents. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 1–15. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  31. 31.
    Nisan, N., Ronen, A.: Algorithmic mechanism design. In: Andersson, S.I. (ed.) Summer University of Southern Stockholm 1993. LNCS, vol. 888, pp. 129–140. Springer, Heidelberg (1999)Google Scholar
  32. 32.
    Osborne, M.J., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge (1994)zbMATHGoogle Scholar
  33. 33.
    Papadimitriou, C.H.: On the complexity of the parity argument and other inefficient proofs of existence. Journal of Computer and System Science 48(3), 498–532 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Papadimitriou, C.H.: Algorithms, games, and the internet. In: Andersson, S.I. (ed.) Summer University of Southern Stockholm 1993. LNCS, vol. 888, pp. 749–753. Springer, Heidelberg (2001)Google Scholar
  35. 35.
    Pigou, A.C.: The economics of ’welfare. Macmillan, Basingstoke (1920)Google Scholar
  36. 36.
    Roughgarden, T., Tardos, E.: How bad is selfish routing? Journal of the ACM 49(2), 236–259 (2002)CrossRefMathSciNetGoogle Scholar
  37. 37.
    Roughgarden, T.: Designing Networks for Selfish Users is Hard. In: Proc. of the 42nd Annual ACM Symposium on Foundations of Computer Science (FOCS 2001), pp. 472–481 (2001)Google Scholar
  38. 38.
    Roughgarden, T.: The Price of Anarchy is Independent of the Network Topology. In: Proc. of the 34th Annual ACM Symposium on Theory of Computing (STOC 2002), pp. 428–437 (2002)Google Scholar
  39. 39.
    Schulz, A.S., Stier Moses, N.E.: On The Performance of User Equilibria in Traffic Networks. MIT Sloan School of Management Working Paper No. 4274-02 (2002) Google Scholar
  40. 40.
    Schuurman, P., Vredeveld, T.: Performance guarantees of load search for multiprocessor scheduling. In: Aardal, K., Gerards, B. (eds.) IPCO 2001. LNCS, vol. 2081, pp. 370–382. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  41. 41.
    Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proc. of the Institute of Civil Engineers, Pt. II, vol. 1, pp. 325–378 (1952)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • R. Feldmann
    • 1
  • M. Gairing
    • 1
  • Thomas Lücking
    • 1
  • Burkhard Monien
    • 1
  • Manuel Rode
    • 1
  1. 1.Department of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany

Personalised recommendations