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Selfish Routing with Incomplete Information

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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.

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References

  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)

  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)

  3. Berenbrink, P., Goldberg, L.A., Goldberg, P.W., Martin, R.: Utilitarian resource assignment. J. Discret. Algorithms 4(4), 567–587 (2006)

    Article  MATH  Google Scholar 

  4. Berger, J.O.: Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer, New York (1980)

    Google Scholar 

  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)

  6. Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. ACM Trans. Algorithms 3(1), Article 4 (2007). Special Issue of SODA’02

  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)

  8. Facchini, G., van Megan, F., Borm, P., Tijs, S.: Congestion models and weighted Bayesian potential games. Theory Decis. 42, 193–206 (1997)

    Article  MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. 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. 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)

    Chapter  Google Scholar 

  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. 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. Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Structure and complexity of extreme Nash equilibria. Theor. Comput. Sci. 343, 133–157 (2005)

    Article  MATH  Google Scholar 

  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. 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)

  18. Graham, R.L.: Bounds on multiprocessing timing anomalies. SIAM J. Appl. Math. 17(2), 416–429 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  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. Harsanyi, J.C.: Games with randomly disturbed payoffs. Int. J. Game Theory 21, 1–23 (1973)

    Article  MathSciNet  Google Scholar 

  21. Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate equilibria and ball fusion. Theory Comput. Syst. 36(6), 683–693 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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. Mas-Colell, A., Whinston, M.D., Green, J.R.: Microeconomic Theory. Oxford University Press, Oxford (1995)

    Google Scholar 

  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)

  27. Milchtaich, I.: Congestion games with player-specific payoff functions. Games Econ. Behav. 13(1), 111–124 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  28. Monderer, D., Shapley, L.S.: Potential games. Games Econ. Behav. 14(1), 124–143 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  29. Myerson, R.B.: Game Theory: Analysis of Conflict. Harvard University Press (1997)

  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)

  31. Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65–67 (1973)

    Article  MATH  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Fürstenallee 11, 33102, Paderborn, Germany

    Martin Gairing & Burkhard Monien

  2. International Graduate School of Dynamic Intelligent Systems, University of Paderborn, Paderborn, Germany

    Karsten Tiemann

Authors
  1. Martin Gairing
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  2. Burkhard Monien
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  3. Karsten Tiemann
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Correspondence to Martin Gairing.

Additional information

This work has been partially supported by the DFG-SFB 376 and by the European Union within the 6th Framework Programme under contract 001907 ( \(\mathsf{DELIS}\) ). A preliminary version of this paper appeared in the Proceedings of the 17th ACM Symposium on Parallelism in Algorithms and Architectures, pp. 203–212, July 2005.

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Gairing, M., Monien, B. & Tiemann, K. Selfish Routing with Incomplete Information. Theory Comput Syst 42, 91–130 (2008). https://doi.org/10.1007/s00224-007-9015-8

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  • DOI: https://doi.org/10.1007/s00224-007-9015-8

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