Abstract
We prove \(\mathcal{NP}\)-hardness of pure Nash equilibrium for some problems of scheduling games and connection games. The technique is standard: first, we construct a gadget without the desired property and then embed it to a larger game which encodes a \(\mathcal{NP}\)-hard problem in order to prove the complexity of the desired property in a game. This technique is very efficient in proving \(\mathcal{NP}\)-hardness for deciding the existence of Nash equilibria. In the paper, we illustrate the efficiency of the technique in proving the \(\mathcal{NP}\)-hardness of deciding the existence of pure Nash equilibria of Matrix Scheduling Games and Weighted Connection Games. Moreover, using the technique, we can settle the complexity not only of the existence of equilibrium but also of the existence of good cost-sharing protocol.
This is a preview of subscription content, log in via an institution to check access.
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
Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, E., Wexler, T., Roughgarden, T.: The price of stability for network design with fair cost allocation. In: Proceeding of the 45th IEEE Symposium on Foundations of Computer Science (FOCS), pp. 295–304 (2004)
Anshelevich, E., Dasgupta, A., Tardos, É., Wexler, T.: Near-optimal network design with selfish agents. In: Proceedings of the 35th Annual ACM Symposium on Theory of Computing (STOC), pp. 511–520 (2003)
Blair, J.R.S., Mutchler, D., Van Lent, M.: Perfect recall and pruning in games with imperfect information. Computational Intelligence 12, 131–154 (1996)
Chekuri, C., Chuzhoy, J., Lewin-Eytan, L., Naor, J., Orda, A.: Non-Cooperative Multicast and Facility Location Games. In: Proceedings of the 7th ACM Conference on Electronic Commerce (EC), pp. 72–81 (2006)
Chen, H.-L., Roughgarden, T.: Network Design with Weighted Players. In: Proceedings of the 18th Annual ACM Symposium on Parallel Algorithms and Architectures (SPAA), pp. 29–38 (2006)
Chen, H.-L., Roughgarden, T., Valiant, G.: Designing Networks with Good Equilibria. In: Proceedings of the 19th ACM-SIAM Symposium on Discrete Algorithms (SODA) (2008)
Dunkel, J., Schulz, A.S.: On the complexity of pure-strategy nash equilibria in congestion and local-effect games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 62–73. Springer, Heidelberg (2006)
Dürr, C., Thang, N.K.: Nash equilibria in voronoi games on graphs. In: Arge, L., Hoffmann, M., Welzl, E. (eds.) ESA 2007. LNCS, vol. 4698, pp. 17–28. Springer, Heidelberg (2007)
Even-Dar, E., Kesselman, A., Mansour, Y.: Convergence time to Nash equilibrium in load balancing. ACM Transactions on Algorithms 3(3) (2007)
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 (STOC), pp. 604–612 (2004)
Garey, M.R., Johnson, D.S.: Computers and Intractability; A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1990)
Monderer, D., Shapley, L.S.: Potential games. Games and Economic Behavior 14, 124–143 (1996)
Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, New York (2007)
Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. International Journal of Game Theory 2, 65–67 (1973)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Thang, N.K. (2009). \(\mathcal{NP}\)-Hardness of Pure Nash Equilibrium in Scheduling and Connection Games. In: Nielsen, M., Kučera, A., Miltersen, P.B., Palamidessi, C., Tůma, P., Valencia, F. (eds) SOFSEM 2009: Theory and Practice of Computer Science. SOFSEM 2009. Lecture Notes in Computer Science, vol 5404. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-95891-8_38
Download citation
DOI: https://doi.org/10.1007/978-3-540-95891-8_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-95890-1
Online ISBN: 978-3-540-95891-8
eBook Packages: Computer ScienceComputer Science (R0)