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The speed of convergence in congestion games under best-response dynamics

Published: 24 July 2012 Publication History

Abstract

We investigate the speed of convergence of best response dynamics to approximately optimal solutions in congestion games with linear delay functions. In Ackermann et al. [2008] it has been shown that the convergence time of such dynamics to Nash equilibrium may be exponential in the number of players n. Motivated by such a negative result, we focus on the study of the states (not necessarily being equilibria) reached after a limited number of players' selfish moves, and we show that Θ(n log log n) best responses are necessary and sufficient to achieve states that approximate the optimal solution by a constant factor, under the assumption that every O(n) steps each player performs a constant (and nonnull) number of best responses. We show that such result is tight also for the simplest case of singleton congestion games.

References

[1]
Ackermann, H., Röglin, H., and Vöcking, B. 2008. On the impact of combinatorial structure on congestion games. J. ACM 55, 6.
[2]
Awerbuch, B., Azar, Y., and Epstein, A. 2005. Large the price of routing unsplittable flow. In Proceedings of the 37th ACM Symposium on Theory of Computing (STOC). H. N. Gabow and R. Fagin, Eds. ACM, 57--66.
[3]
Awerbuch, B., Azar, Y., Epstein, A., Mirrokni, V. S., and Skopalik, A. 2008. Fast convergence to nearly optimal solutions in potential games. In Proceedings of the 9th ACM Conference on Electronic Commerce (EC). L. Fortnow, J. Riedl and T. Sandholm, Eds., ACM, 264--273.
[4]
Bhalgat, A., Chakraborty, T., and Khanna, S. 2009. Nash dynamics in congestion games with similar resources. In Proceedings of the 5th International Workshop on Internet and Network Economics (WINE). S. Leonardi, Ed., Lecture Notes in Computer Science, vol. 5929, Springer, 362--373.
[5]
Chien, S. and Sinclair, A. 2007. Convergence to approximate Nash equilibria in congestion games. In Proceedings of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). N. Bansal, K. Pruhs, and C. Stein, Eds., SIAM, 169--178.
[6]
Christodoulou, G. and Koutsoupias, E. 2005. The price of anarchy of finite congestion games. In Proceedings of the 37th ACM Symposium on Theory of Computing (STOC). H. N. Gabow and R. Fagin, Eds., ACM, 67--73.
[7]
Christodoulou, G., Mirrokni, V. S., and Sidiropoulos, A. 2006. Convergence and approximation in potential games. In Proceedings of the 23rd Annual Symposium on Theoretical Aspects of Computer Science (STACS). B. Durand and W. Thomas, Eds., Lecture Notes in Computer Science, vol. 3884, Springer. 349--360.
[8]
Christodoulou, G., Mirrokni, V. S., and Sidiropoulos, A. 2007. Convergence and approximation in potential games. http://people.csail.mit.edu/mirrokni/potentia106.ps.
[9]
D. S. Johnson, C. H. Papadimitriou, M. Y. 1988. How easy is local search? J. Comput. Syst. Sci. 37, 79--100.
[10]
Fabrikant, A., Papadimitriou, C. H., and Talwar, K. 2004. The complexity of pure Nash equilibria. In Proceedings of the 36th ACM Symposium on Theory of Computing (STOC). L. Babai, Ed., ACM, 604--612.
[11]
Goemans, M. X., Mirrokni, V. S., and Vetta, A. 2005. Sink equilibria and convergence. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE, 142--154.
[12]
Mirrokni, V. S. and Vetta, A. 2004. Convergence issues in competitive games. In Proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and the 8th International Workshop on Randomization and Computation (APPROX-RANDOM). K. Jansen, S. Khanna, J. D. P. Rolim and D. Ron, Eds., Lecture Notes in Computer Science, vol. 3122, Springer, 183--194.
[13]
Monderer, D. and Shapley, L. S. 1996. Potential games. Games Econ. Behav. 14, 1, 124--143.
[14]
Nash, J. F. 1950. Equilibrium points in n-person games. Proc. Nat. Acad. Sci. 36, 48--49.
[15]
Rosenthal, R. W. 1973. A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory 2, 65--67.
[16]
Skopalik, A. and Vöcking, B. 2008. Inapproximability of pure Nash equilibria. In Proceedings of the 40th ACM Symposium on Theory of Computing (STOC). R. E. Ladner and C. Dwork, Eds., ACM, 355--364.
[17]
Vetta, A. 2002. Nash equilibria in competitive societies, with applications to facility location, traffic routing and auctions. In Proceedings of the 43th Annual IEEE Symposium on Foundations of Computer Science (FOCS). IEEE, 416--.

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    cover image ACM Transactions on Algorithms
    ACM Transactions on Algorithms  Volume 8, Issue 3
    July 2012
    257 pages
    ISSN:1549-6325
    EISSN:1549-6333
    DOI:10.1145/2229163
    Issue’s Table of Contents
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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    Publication History

    Published: 24 July 2012
    Accepted: 01 April 2010
    Revised: 01 March 2010
    Received: 01 August 2009
    Published in TALG Volume 8, Issue 3

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    Author Tags

    1. Congestion games
    2. best response dynamics

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