Theory of Computing Systems

, Volume 42, Issue 1, pp 91–130

Selfish Routing with Incomplete Information

Article
  • 145 Downloads

Abstract

In his seminal work, Harsanyi (Manag. Sci. 14, 159–182, 320–332, 468–502, 1967) introduced an elegant approach to study non-cooperative games with incomplete information. In our work, we use this approach to define a new selfish routing game with incomplete information that we call Bayesian routing game. Here, each of n selfish users wishes to assign its traffic to one of m parallel links. However, users do not know each other’s traffic. Following Harsanyi’s approach, we introduce, for each user, a set of possible types. In our model, each type of a user corresponds to some traffic and the players’ uncertainty about each other’s traffic is described by a probability distribution over all possible type profiles.

We present a comprehensive collection of results about our Bayesian routing game. Our main findings are as follows:

  1. Using a potential function, we prove that every Bayesian routing game has a pure Bayesian Nash equilibrium. More precisely, we show this existence for a more general class of games that we call weighted Bayesian congestion games. For Bayesian routing games with identical links and independent type distribution, we give a polynomial time algorithm to compute a pure Bayesian Nash equilibrium.

     
  2. We study structural properties of fully mixed Bayesian Nash equilibria for the case of identical links and show that they maximize Individual Cost. In general, there is more than one fully mixed Bayesian Nash equilibrium. We characterize fully mixed Bayesian Nash equilibria for the case of independent type distribution.

     
  3. We conclude with bounds on Coordination Ratio for the case of identical links and for three different Social Cost measures: Expected Maximum Latency, Sum of Individual Costs and Maximum Individual Cost. For the latter two, we are able to give (asymptotically) tight bounds using the properties of fully mixed Bayesian Nash equilibria we proved.

     

Keywords

Selfish routing Incomplete information Bayesian Nash equilibria Coordination ratio 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Awerbuch, B., Azar, Y., Epstein, A.: The price of routing unsplittable flow. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 57–66 (2005) Google Scholar
  2. 2.
    Beier, R., Czumaj, A., Krysta, P., Vöcking, B.: Computing equilibria for congestion games with (im)perfect information. In: Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 746–755 (2004) Google Scholar
  3. 3.
    Berenbrink, P., Goldberg, L.A., Goldberg, P.W., Martin, R.: Utilitarian resource assignment. J. Discret. Algorithms 4(4), 567–587 (2006) MATHCrossRefGoogle Scholar
  4. 4.
    Berger, J.O.: Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer, New York (1980) Google Scholar
  5. 5.
    Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 67–73 (2005) Google Scholar
  6. 6.
    Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1), Article 4 (2007). Special Issue of SODA’02 Google Scholar
  7. 7.
    Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure Nash equilibria. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 604–612 (2004) Google Scholar
  8. 8.
    Facchini, G., van Megan, F., Borm, P., Tijs, S.: Congestion models and weighted Bayesian potential games. Theory Decis. 42, 193–206 (1997) MATHCrossRefGoogle Scholar
  9. 9.
    Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Nashification and the coordination ratio for a selfish routing game. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) Proceedings of the 30th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 2719, pp. 514–526. Springer, New York (2003) CrossRefGoogle Scholar
  10. 10.
    Feldmann, R., Gairing, M., Lücking, T., Monien, B., Rode, M.: Selfish routing in non-cooperative networks: a survey. In: Rovan, B., Vojtás, P. (eds.) Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 2747, pp. 21–45. Springer, New York (2003) Google Scholar
  11. 11.
    Fischer, S., Vöcking, B.: On the structure and complexity of worst-case equilibria. In: Deng, X., Ye, Y. (eds.) Proceedings of the 1st International Workshop on Internet and Network Economics. Lecture Notes in Computer Science, vol. 3828, pp. 151–160. Springer, New York (2005) Google Scholar
  12. 12.
    Fotakis, D., Kontogiannis, S., Koutsoupias, E., Mavronicolas, M., Spirakis, P.: The structure and complexity of Nash equilibria for a selfish routing game. In: Widmayer, P., Ruiz, F.T., Bueno, R.M., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) Proceedings of the 29th International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 2380, pp. 123–134. Springer, New York (2002) CrossRefGoogle Scholar
  13. 13.
    Fotakis, D., Kontogiannis, S., Spirakis, P.: Selfish unsplittable flows. In: Diaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) Proceedings of the 31st International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 3142, pp. 593–605. Springer, New York (2004) Google Scholar
  14. 14.
    Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: Nash equilibria in discrete routing games with convex latency functions. In: Diaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) Proceedings of the 31st International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 3142, pp. 645–657. Springer, New York (2004) Google Scholar
  15. 15.
    Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Structure and complexity of extreme Nash equilibria. Theor. Comput. Sci. 343, 133–157 (2005) MATHCrossRefGoogle Scholar
  16. 16.
    Gairing, M., Lücking, T., Monien, B., Tiemann, K.: Nash equilibria, the price of anarchy and the fully mixed Nash equilibrium conjecture. In: Proceedings of the 32nd International Colloquium on Automata, Languages, and Programming. Lecture Notes in Computer Science, vol. 3580, pp. 51–65. Springer, New York (2005) Google Scholar
  17. 17.
    Georgiou, C., Pavlides, T., Philippou, A.: Network uncertainty in selfish routing. In: Proc. of the 20th IEEE International Parallel & Distributed Processing Symposium, p. 105 (2006) Google Scholar
  18. 18.
    Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1969) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Harsanyi, J.C.: Games with incomplete information played by Bayesian players, I, II, III. Manag. Sci. 14, 159–182, 320–332, 468–502 (1967) Google Scholar
  20. 20.
    Harsanyi, J.C.: Games with randomly disturbed payoffs. Int. J. Game Theory 21, 1–23 (1973) CrossRefMathSciNetGoogle Scholar
  21. 21.
    Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate equilibria and ball fusion. Theory Comput. Syst. 36(6), 683–693 (2003) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) Proceedings of the 16th International Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 1563, pp. 404–413. Springer, New York (1999) Google Scholar
  23. 23.
    Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: A new model for selfish routing. In: Diekert, V., Habib, M. (eds.) Proceedings of the 21st International Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 2996, pp. 547–558. Springer, New York (2004) Google Scholar
  24. 24.
    Lücking, T., Mavronicolas, M., Monien, B., Rode, M., Spirakis, P., Vrto, I.: Which is the worst-case Nash equilibrium?. In: Rovan, B., Vojtás, P. (eds.) Proceedings of the 28th International Symposium on Mathematical Foundations of Computer Science. Lecture Notes in Computer Science, vol. 2747, pp. 551–561. Springer, New York (2003) Google Scholar
  25. 25.
    Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995) Google Scholar
  26. 26.
    Mavronicolas, M., Spirakis, P.: The price of selfish routing. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 510–519 (2001) Google Scholar
  27. 27.
    Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econ. Behav. 13(1), 111–124 (1996) MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14(1), 124–143 (1996) MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press (1997) Google Scholar
  30. 30.
    Papadimitriou, C.H.: Algorithms, games, and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, pp. 749–753 (2001) Google Scholar
  31. 31.
    Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973) MATHCrossRefGoogle Scholar
  32. 32.
    Voorneveld, M., Borm, P., van Megan, F., Tijs, S., Facchini, G.: Congestion games and potentials reconsidered. Int. Game Theory Rev. 1, 283–299 (1999) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Martin Gairing
    • 1
  • Burkhard Monien
    • 1
  • Karsten Tiemann
    • 2
  1. 1.Faculty of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.International Graduate School of Dynamic Intelligent SystemsUniversity of PaderbornPaderbornGermany

Personalised recommendations