Abstract
We study the combinatorial structure and computational complexity of extreme Nash equilibria, ones that maximize or minimize a certain objective function, in the context of a selfish routing game. Specifically, we assume a collection of nusers, each employing a mixed strategy, which is a probability distribution over m parallel links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user routes its traffic on links that minimize its expected latency cost.
Our structural results provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria and that under a certain condition, the social cost of any Nash equilibrium is within a factor of 6 + ε, of that of the fully mixed Nash equilibrium, assuming that link capacities are identical.
Our complexity results include hardness, approximability and inapproximability ones. Here we show, that for identical link capacities and under a certain condition, there is a randomized, polynomial-time algorithm to approximate the worst social cost within a factor arbitrarily close to 6 + ε. Furthermore, we prove that for any arbitrary integer k > 0, it is \(\mathcal{NP}\)-hard to decide whether or not any given allocation of users to links can be transformed into a pure Nash equilibrium using at most k selfish steps. Assuming identical link capacities, we give a polynomial-time approximation scheme (PTAS) to approximate the best social cost over all pure Nash equilibria. Finally we prove, that it is \(\mathcal{NP}\)-hard to approximate the worst social cost within a multiplicative factor 2 - \(\frac{\rm 2}{m+1} - \epsilon\). The quantity 2 - \(\frac{\rm 2}{m+1}\) is the tight upper bound on the ratio of the worst social cost and the optimal cost in the model of identical capacities.
This work has been partially supported by the IST Program of the European Union under contract numbers IST-1999-14186 (ALCOM-FT) and IST-2001-33116 (FLAGS), by funds from the Joint Program of Scientific and Technological Collaboration between Greece and Cyprus, and by research funds from the University of Cyprus.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Brucker, P., Hurink, J., Werner, F.: Improving Local Search Heuristics for Some Scheduling Problems. Part II. Discrete Applied Mathematics 72(1-2), 47–69 (1997)
Czumaj, A., Vöcking, B.: Tight Bounds forWorst-Case Equilibria. In: Proceedings of the 13th Annual ACM Symposium on Discrete Algorithms, January 2002, pp. 413–420 (2002)
Deng, X., Papadimitriou, C., Safra, S.: On the Complexity of Equilibri. In: Proceedings of the 34th Annual ACM Symposium on Theory of Computing, May 2002, pp. 67–71 (2002)
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.) ICALP 2003. LNCS, vol. 2719, pp. 514–526. Springer, Heidelberg (2003)
Finn, G., Horowitz, E.: A linear time approximation algorithm for multiprocessor scheduling. BIT 19, 312–320 (1979)
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., Triguero, F., Morales, R., Hennessy, M., Eidenbenz, S., Conejo, R. (eds.) ICALP 2002. LNCS, vol. 2380, pp. 123–134. Springer, Heidelberg (2002)
Garey, M.R., Johnson, D.S.: Complexity Results for Multiprocessor Scheduling Under Resoiurce Constraints. SIAM Journal on Computing 4, 397–411 (1975)
Garey, M.R., Johnson, D.S.: Computers and intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)
Gonnet, G.H.: Expected Length of the Longest Probe Sequence in Hash Code Searching. Journal of the ACM 28(2), 289–304 (1981)
Hochbaum, D.S., Shmoys, D.: Using Dual Approximation Algorithms for Scheduling Problems: Theoretical and Practical Results. Journal of the ACM 34(1), 144–162 (1987)
Horowitz, E., Sahni, S.: Exact and Approximate Algorithms for Scheduling Non-Identical Processors. Journal of the ACM 23(2), 317–327 (1976)
Karp, R.M.: Reducibility among Combinatorial Problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)
Kolchin, V.F., Chistiakov, V.P., Sevastianov, B.A.: Random Allocations. V. H. Winston, New York (1978)
Koutsoupias, E., Mavronicolas, M., Spirakis, P.: Approximate Equilibria and Ball Fusion. In: Proceedings of the 9th International Colloquium on Structural Information and Communication Complexity, Andros, Greece (June 2002) (Accepted to Theory of Computing Systems); Earlier version appeared as A Tight Bound on Coordination Ratio, Technical Report 0100229, Department of Computer Science, University of California at Los Angeles (April 2001)
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)
Lücking, T., Mavronicolas, M., Monien, B., Rode, M., Spirakis, P., Vrto, I.: Which is theWorst-case Nash Equilibrium? In: 26th International Symposium on Mathematical Foundations of Computer Science (August 2003) (to appear)
Marshall, A., Olkin, I.: Theory of Majorization and Its Applications. Academic Press, Orlando (1979)
Mavronicolas, M., Spirakis, P.: The Price of Selfish Routing. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, July 2001, pp. 510–519 (2001)
McDiarmid, C.: “Concentration. In: Habib, M., McDiarmidt, C., Ramires-Alfonsin, J., Reed, B. (eds.) Probabilistic Methods for Algorithmic Discrete Mathematics, vol. ch. 9, Springer, Heidelberg (1998)
Moulin, H., Vial, L.: Strategically Zero-Sum Games: The Class of Games whose Completely Mixed Equilibria Cannot be Improved Upon. International Journal of Game Theory 7(3/4), 201–221 (1978)
Nash, J.F.: Equilibrium Points in N-Person Games. Proceedings of the National Academy of Sciences 36, 48–49 (1950)
Nash, J.F.: Non-cooperative Games. Annals of Mathematics 54(2), 286–295 (1951)
Osborne, M.J., Rubinstein, A.: A Course in Game Theory. The MIT Press, Cambridge (1994)
Papadimitriou, C.H.: Algorithms, Games and the Internet. In: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, July 2001, pp. 749–753 (2001)
Raghavan, T.E.S.: Completely Mixed Strategies in Bimatrix Games. Journal of London Mathematical Society 2(2), 709–712 (1970)
Ross, S.M.: Stochastic Processes, 2nd edn. John Wiley & Sons, Inc., Chichester (1996)
Schuurman, P., Vredeveld, T.: Performance Guarantees of Load Search for Multiprocessor Scheduling. In: Proceedings of the 8th Conference on Integer Programming and Combinatorial Optimization, June 2001, pp. 370–382 (2001)
Shaked, M., Shanthikumar, J.G.: Stochastic Orders and Their Applications. Academic Press, San Diego (1994)
Vetta, A.: Nash Equilibria in Competitive Societies, with Applications to Facility Location, Traffic Routing and Auctions. In: Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, October 2002, pp. 416–425 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gairing, M., Lücking, T., Mavronicolas, M., Monien, B., Spirakis, P. (2003). Extreme Nash Equilibria. In: Blundo, C., Laneve, C. (eds) Theoretical Computer Science. ICTCS 2003. Lecture Notes in Computer Science, vol 2841. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45208-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-45208-9_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-20216-5
Online ISBN: 978-3-540-45208-9
eBook Packages: Springer Book Archive