Abstract
In this paper we consider the influence of link restrictions on the price of anarchy for several social cost functions in the following model of selfish routing. Each of n players in a network game seeks to send a message with a certain length by choosing one of m parallel links. Each player is restricted to transmit over a certain subset of links and desires to minimize his own transmission-time (latency). We study Nash equilibria of the game, in which no player can decrease his latency by unilaterally changing his link. Our analysis of this game captures two important aspects of network traffic: the dependency of the overall network performance on the total traffic t and fluctuations in the length of the respective message-lengths. For the latter we use a probabilistic model in which message lengths are random variables.
We evaluate the (expected) price of anarchy of the game for two social cost functions. For total latency cost, we show the tight result that the price of anarchy is essentially \({\it \Theta}({n\sqrt{m}/t})\). Hence, even for congested networks, when the traffic is linear in the number of players, Nash equilibria approximate the social optimum only by a factor of \({\it \Theta}({\sqrt{m}})\). This efficiency loss is caused by link restrictions and remains stable even under message fluctuations, which contrasts the unrestricted case where Nash equilibria achieve a constant factor approximation. For maximum latency the price of anarchy is at most 1 + m 2/t. In this case Nash equilibria can be (almost) optimal solutions for congested networks depending on the values for m and t. In addition, our analyses yield average-case analyses of a polynomial time algorithm for computing Nash equilibria in this model.
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References
Awerbuch, B., Azar, Y., Epstein, A.: The price of routing unsplittable flow. In: Proc. 37th STOC, pp. 57–66 (2005)
Awerbuch, B., Azar, Y., Richter, Y., Tsur, D.: Tradeoffs in worst-case equilibria. In: Proc. 1st WAOA, pp. 41–52 (2003)
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)
Christodoulou, G., Koutsoupias, E.: The price of anarchy of finite congestion games. In: Proc. 37th STOC, pp. 67–73 (2005)
Czumaj, A., Krysta, P., Vöcking, B.: Selfish traffic allocation for server farms. In: Proc. 34th STOC, pp. 287–296 (2002)
Czumaj, A., Vöcking, B.: Tight bounds for worst-case equilibria. In: Proc. 13th SODA, pp. 413–420 (2002)
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B.: Computing nash equilibria for scheduling on restricted parallel links. In: Proc. 36th STOC, pp. 613–622 (2004)
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Rode, M.: Nash equilibria in discrete routing games with convex latency functions. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) ICALP 2004. LNCS, vol. 3142, pp. 645–657. Springer, Heidelberg (2004)
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P.: Structure and complexity of extreme Nash equilibria. Theoretical Computer Science 343(1-2), 133–157 (2005)
Hoefer, M., Souza, A.: Tradeoffs and average-case equilibria in selfish routing. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 63–74. Springer, Heidelberg (2007)
Kolliopoulos, S., Stein, C.: Approximation algorithms for single-source unsplittable flow. SIAM Journal on Computing 31, 919–946 (2002)
Kontogiannis, S., Spirakis, P.: Atomic selfish routing in networks: A survey. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 989–1002. Springer, Heidelberg (2005)
Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)
Lenstra, J.K., Shmoys, D., Tardos, E.: Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming 46, 259–271 (1990)
Mavronicolas, M., Panagopoulou, P., Spirakis, P.: A cost mechanism for fair pricing of resource usage. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 210–224. Springer, Heidelberg (2005)
Scharbrodt, M., Schickinger, T., Steger, A.: A new average case analysis for completion time scheduling. In: Proc. 34th STOC, pp. 170–178 (2002)
Souza, A., Steger, A.: The expected competitive ratio for weighted completion time scheduling. In: Diekert, V., Habib, M. (eds.) STACS 2004. LNCS, vol. 2996, pp. 620–631. Springer, Heidelberg (2004)
Suri, S., Toth, C., Zhou, Y.: Selfish load balancing and atomic congestion games. Algorithmica 47(1), 79–96 (2007)
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Hoefer, M., Souza, A. (2008). The Influence of Link Restrictions on (Random) Selfish Routing. In: Monien, B., Schroeder, UP. (eds) Algorithmic Game Theory. SAGT 2008. Lecture Notes in Computer Science, vol 4997. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79309-0_4
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DOI: https://doi.org/10.1007/978-3-540-79309-0_4
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