Abstract
We present a general technique, based on a primal-dual formulation, for analyzing the quality of self-emerging solutions in weighted congestion games. With respect to traditional combinatorial approaches, the primal-dual schema has at least three advantages: first, it provides an analytic tool which can always be used to prove tight upper bounds for all the cases in which we are able to characterize exactly the polyhedron of the solutions under analysis; secondly, in each such a case the complementary slackness conditions give us an hint on how to construct matching lower bounding instances; thirdly, proofs become simpler and easy to check. For the sake of exposition, we first apply our technique to the problems of bounding the prices of anarchy and stability of exact and approximate pure Nash equilibria, as well as the approximation ratio of the solutions achieved after a one-round walk starting from the empty strategy profile, in the case of affine latency functions and we show how all the known upper bounds for these measures (and some of their generalizations) can be easily reobtained under a unified approach. Then, we use the technique to attack the more challenging setting of polynomial latency functions. In particular, we obtain the first known upper bounds on the price of stability of pure Nash equilibria and on the approximation ratio of the solutions achieved after a one-round walk starting from the empty strategy profile for unweighted players in the cases of quadratic and cubic latency functions.
This work was partially supported by the PRIN 2008 research project COGENT “Computational and game-theoretic aspects of uncoordinated networks” funded by the Italian Ministry of University and Research and by the “Progetto 5 per mille per la ricerca”: “Collisioni fra vortici puntiformi e fra filamenti di vorticità: singolarità, trasporto e caos” at the University of Salento.
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References
Aland, S., Dumrauf, D., Gairing, M., Monien, B., Schoppmann, F.: Exact Price of Anarchy for Polynomial Congestion Games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 218–229. Springer, Heidelberg (2006)
Anshelevich, E., Dasgupta, A., Kleinberg, J.M., Tardos, E., Wexler, T., Roughgarden, T.: The Price of Stability for Network Design with Fair Cost Allocation. SIAM Journal on Computing 38(4), 1602–1623 (2008)
Awerbuch, B., Azar, Y., Epstein, A.: The Price of Routing Unsplittable Flow. In: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 57–66. ACM Press (2005)
Bhawalkar, K., Gairing, M., Roughgarden, T.: Weighted Congestion Games: Price of Anarchy, Universal Worst-Case Examples, and Tightness. In: de Berg, M., Meyer, U. (eds.) ESA 2010, Part II. LNCS, vol. 6347, pp. 17–28. Springer, Heidelberg (2010)
Bilò, V., Fanelli, A., Flammini, M., Moscardelli, L.: Performances of One-Round Walks in Linear Congestion Games. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 311–322. Springer, Heidelberg (2009)
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: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pp. 67–73. ACM Press (2005)
Christodoulou, G., Koutsoupias, E.: On the Price of Anarchy and Stability of Correlated Equilibria of Linear Congestion Games. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 59–70. Springer, Heidelberg (2005)
Christodoulou, G., Koutsoupias, E., Spirakis, P.G.: On the Performance of Approximate Equilibria in Congestion Games. Algorithmica (to appear)
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)
Fotakis, D., Kontogiannis, S.C., Spirakis, P.G.: Symmetry in Network Congestion Games: Pure Equilibria and Anarchy Cost. In: Erlebach, T., Persinao, G. (eds.) WAOA 2005. LNCS, vol. 3879, pp. 161–175. Springer, Heidelberg (2006)
Harks, T., Klimm, M.: On the Existence of Pure Nash Equilibria in Weighted Congestion Games. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 79–89. Springer, Heidelberg (2010)
Harks, T., Klimm, M., Möhring, R.H.: Characterizing the Existence of Potential Functions in Weighted Congestion Games. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 97–108. Springer, Heidelberg (2009)
Koutsoupias, E., Papadimitriou, C.: Worst-Case Equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999)
Mirrokni, V.S., Vetta, A.: Convergence Issues in Competitive Games. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) APPROX and RANDOM 2004. LNCS, vol. 3122, pp. 183–194. Springer, Heidelberg (2004)
Nadav, U., Roughgarden, T.: The Limits of Smoothness: A Primal-Dual Framework for Price of Anarchy Bounds. In: Saberi, A. (ed.) WINE 2010. LNCS, vol. 6484, pp. 319–326. Springer, Heidelberg (2010)
Panagopoulou, P.N., Spirakis, P.G.: Algorithms for Pure Nash Equilibria in Weighted Congestion Games. Journal of Experimental Algorithmics 11(2.7) (2006)
Roughgarden, T.: Intrinsic Robustness of the Price of Anarchy. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 513–522. ACM Press (2009)
Rosenthal, R.W.: A Class of Games Possessing Pure-Strategy Nash Equilibria. International Journal of Game Theory 2, 65–67 (1973)
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Bilò, V. (2013). A Unifying Tool for Bounding the Quality of Non-cooperative Solutions in Weighted Congestion Games. In: Erlebach, T., Persiano, G. (eds) Approximation and Online Algorithms. WAOA 2012. Lecture Notes in Computer Science, vol 7846. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38016-7_18
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DOI: https://doi.org/10.1007/978-3-642-38016-7_18
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