Abstract
Bottleneck congestion games properly model the properties of many real-world network routing applications. They are known to possess strong equilibria – a strengthening of Nash equilibrium to resilience against coalitional deviations. In this paper, we study the computational complexity of pure Nash and strong equilibria in these games. We provide a generic centralized algorithm to compute strong equilibria, which has polynomial running time for many interesting classes of games such as, e.g., matroid or single-commodity bottleneck congestion games. In addition, we examine the more demanding goal to reach equilibria in polynomial time using natural improvement dynamics. Using unilateral improvement dynamics in matroid games pure Nash equilibria can be reached efficiently. In contrast, computing even a single coalitional improvement move in matroid and single-commodity games is strongly NP-hard. In addition, we establish a variety of hardness results and lower bounds regarding the duration of unilateral and coalitional improvement dynamics. They continue to hold even for convergence to approximate equilibria.
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References
Ackermann, H., Röglin, H., Vöcking, B.: On the impact of combinatorial structure on congestion games. J. ACM 55(6), 1–22 (2008)
Ackermann, H., Röglin, H., Vöcking, B.: Pure Nash equilibria in player-specific and weighted congestion games. Theor. Comput. Sci. 410(17), 1552–1562 (2009)
Aumann, R.: Acceptable points in general cooperative n-person games. In: Contributions to the Theory of Games, vol. 4 (1959)
Banner, R., Orda, A.: Bottleneck routing games in communication networks. IEEE J. Selected Areas Comm. 25(6), 1173–1179 (2007)
Chien, S., Sinclair, A.: Convergence to approximate Nash equilibria in congestion games. In: Proc. 18th Symp. Disc. Algorithms (SODA), pp. 169–178 (2007)
Cole, R., Dodis, Y., Roughgarden, T.: Bottleneck links, variable demand, and the tragedy of the commons. In: Proc. 17th Symp. Disc. Algorithms (SODA), pp. 668–677 (2006)
Cunningham, W.H.: Improved bounds for matroid partition and intersection algorithms. SIAM J. Comput. (15), 948–957 (1986)
Edmonds, J.: Matroid partition. In: Dantzig, G.B., Veinott, A.F. (eds.) Mathematics of the Decision Sciences, pp. 335–345 (1968)
Fabrikant, A., Papadimitriou, C., Talwar, K.: The complexity of pure Nash equilibria. In: Proc. 36th Symp. Theory of Computing (STOC), pp. 604–612 (2004)
Fortune, S., Hopcroft, J.E., Wyllie, J.C.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10, 111–121 (1980)
Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. J. Comput. Syst. Sci. 50(2), 259–273 (1995)
Harks, T., Klimm, M., Möhring, R.H.: Strong Nash equilibria in games with the lexicographical improvement property. In: Leonardi, S. (ed.) WINE 2009. LNCS, vol. 5929, pp. 463–470. Springer, Heidelberg (2009)
Holzman, R., Law-Yone, N.: Strong equilibrium in congestion games. Games Econ. Behav. 21(1-2), 85–101 (1997)
Keshav, S.: An engineering approach to computer networking: ATM networks, the Internet, and the telephone network. Addison-Wesley, Reading (1997)
Konishi, H., Le Breton, M., Weber, S.: Equilibria in a model with partial rivalry. J. Econ. Theory 72(1), 225–237 (1997)
Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms. Springer, Heidelberg (2002)
Mazalov, V., Monien, B., Schoppmann, F., Tiemann, K.: Wardrop equilibria and price of stability for bottleneck games with splittable traffic. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 331–342. Springer, Heidelberg (2006)
Nash-Williams, C.: An application of matroids to graph theory. In: Rosenstiehl, P. (ed.) Theory of Graphs; Proc. Intl. Symp. Rome 1966, pp. 263–265 (1967)
Qiu, L., Yang, Y.R., Zhang, Y., Shenker, S.: On selfish routing in internet-like environments. IEEE/ACM Trans. Netw. 14(4), 725–738 (2006)
Rosenthal, R.W.: A class of games possessing pure-strategy Nash equilibria. Intl. J. Game Theory 2(1), 65–67 (1973)
Roughgarden, T.: Routing games. In: Nisan, N., Tardos, É., Roughgarden, T., Vazirani, V. (eds.) Algorithmic Game Theory, ch. 18, Cambridge University Press, Cambridge (2007)
Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency. Springer, Heidelberg (2003)
Skopalik, A., Vöcking, B.: Inapproximability of pure Nash equilibria. In: Proc. 40th Symp. Theory of Computing (STOC), pp. 355–364 (2008)
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Harks, T., Hoefer, M., Klimm, M., Skopalik, A. (2010). Computing Pure Nash and Strong Equilibria in Bottleneck Congestion Games. In: de Berg, M., Meyer, U. (eds) Algorithms – ESA 2010. ESA 2010. Lecture Notes in Computer Science, vol 6347. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15781-3_3
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DOI: https://doi.org/10.1007/978-3-642-15781-3_3
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