Abstract
We investigate the speed of convergence of congestion games with linear latency functions under best response dynamics. Namely, we estimate the social performance achieved after a limited number of rounds, during each of which every player performs one best response move. In particular, we show that the price of anarchy achieved after k rounds, defined as the highest possible ratio among the total latency cost, that is the sum of all players latencies, and the minimum possible cost, is \(O(\sqrt[2^{k-1}] {n})\), where n is the number of players. For constant values of k such a bound asymptotically matches the \(\Omega(\sqrt[2^{k-1}] {n}/k)\) lower bound that we determine as a refinement of the one in [7]. As a consequence, we prove that order of loglogn rounds are not only necessary, but also sufficient to achieve a constant price of anarchy, i.e. comparable to the one at Nash equilibrium. This result is particularly relevant, as reaching an equilibrium may require a number of rounds exponential in n. We then provide a new lower bound of \(\Omega(\sqrt[2^k-1] {n})\) for load balancing games, that is congestion games in which every strategy consists of a single resource, thus showing that a number of rounds proportional to loglogn is necessary and sufficient also under such a restriction.
Our results thus solve the important left open question of the polynomial speed of convergence of linear congestion games to constant factor solutions.
This work was partially supported by the Future and Emerging Technologies Unit of EC (IST priority - 6th FP), under contract no. FP6-021235-2 (project ARRIVAL).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. In: FOCS, pp. 613–622. IEEE Computer Society, Los Alamitos (2006)
Awerbuch, B., Azar, Y., Epstein, A.: Large the price of routing unsplittable flow. In: STOC, pp. 57–66. ACM, New York (2005)
Awerbuch, B., Azar, Y., Epstein, A., Mirrokni, V.S., Shopalik, A.: Fast convergence to nearly optimal solutions in potential games. In: EC 2008 (to appear, 2008)
Caragiannis, I., Flammini, M., Kaklamanis, C., Kanellopoulos, P., Moscardelli, L.: Tight bounds for selfish and greedy load balancing. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 311–322. Springer, Heidelberg (2006)
Chien, S., Sinclair, A.: Convergence to approximate nash equilibria in congestion games. In: SODA, pp. 169–178. SIAM, Philadelphia (2007)
Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: STOC, pp. 67–73. ACM, New York (2005)
Christodoulou, G., Mirrokni, V.S., Sidiropoulos, A.: Convergence and approximation in potential games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 349–360. Springer, Heidelberg (2006)
Christodoulou, G., Mirrokni, V.S., Sidiropoulos, A.: Convergence and approximation in potential games. Personal Communication (2007)
Yannakakis, M., Johnson, D.S., Papadimitriou, C.H.: How easy is local search? Journal of Computer and System Sciences 37, 79–100 (1988)
Fabrikant, A., Papadimitriou, C.H., Talwar, K.: The complexity of pure equilibria. In: Proceedings of the 36th ACM Symposium on Theory of Computing (STOC), pp. 604–612. ACM, New York (2004)
Fanelli, A., Flammini, M., Melideo, G., Moscardelli, L.: Multicast transmissions in non-cooperative networks with a limited number of selfish moves. In: Královič, R., Urzyczyn, P. (eds.) MFCS 2006. LNCS, vol. 4162, pp. 363–374. Springer, Heidelberg (2006)
Fanelli, A., Flammini, M., Moscardelli, L.: On the convergence of multicast games in directed networks. In: SPAA, pp. 330–338. ACM, New York (2007)
Goemans, M.X., Mirrokni, V.S., Vetta, A.: Sink equilibria and convergence. In: FOCS, pp. 142–154. IEEE Computer Society, Los Alamitos (2005)
Koutsoupias, E., Papadimitriou, C.H.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)
Milchtaich, I.: Congestion games with player-specific payoff functions. Games and Economic Behavior 13, 111–124 (1996)
Mirrokni, V.S., Vetta, A.: Convergence issues in competitive games. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 183–194. Springer, Heidelberg (2004)
Nash, J.F.: Equilibrium points in n-person games. Proceedings of the National Academy of Sciences 36, 48–49 (1950)
Rosenthal, R.W.: A class of games possessing pure-strategy nash equilibria. International Journal of Game Theory 2, 65–67 (1973)
Skopalik, A., Vöcking, B.: Inapproximability of convergence in congestion games (manuscript, 2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fanelli, A., Flammini, M., Moscardelli, L. (2008). The Speed of Convergence in Congestion Games under Best-Response Dynamics. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds) Automata, Languages and Programming. ICALP 2008. Lecture Notes in Computer Science, vol 5125. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70575-8_65
Download citation
DOI: https://doi.org/10.1007/978-3-540-70575-8_65
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70574-1
Online ISBN: 978-3-540-70575-8
eBook Packages: Computer ScienceComputer Science (R0)