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Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

  • George Christodoulou
  • Kurt Mehlhorn
  • Evangelia Pyrga
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6942)

Abstract

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms.

We increase the latency functions of the edges in the network, i.e., if ℓ e (x) is the latency function of an edge e, we replace it by \(\hat{\ell}_e(x)\) with \(\ell_e(x) \le \hat{\ell}_e(x)\) for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if \(\hat{C}_N(r)\) denotes the cost of the worst Nash flow in the modified network for rate r and C opt (r) denotes the cost of the optimal flow in the original network for the same rate then
$$ \mathit{ePoA} = \max_{r \ge 0} \frac{\hat{C}_N(r)}{C_{\mathit{opt}}(r)}.$$

We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.

Keywords

Nash Equilibrium Latency Function Coordination Mechanism Original Network User Equilibrium 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Angel, E., Bampis, E., Pascual, F.: Truthful algorithms for scheduling selfish tasks on parallel machines. Theor. Comput. Sci. 369(1-3), 157–168 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angel, E., Bampis, E., Pascual, F., Tchetgnia, A.-A.: On truthfulness and approximation for scheduling selfish tasks. J. Scheduling 12(5), 437–445 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Azar, Y., Jain, K., Mirrokni, V.S.: (Almost) optimal coordination mechanisms for unrelated machine scheduling. In: SODA, pp. 323–332 (2008)Google Scholar
  4. 4.
    Bernstein, D., Smith, T.E.: Equilibria for networks with lower semicontinuous costs: With an application to congestion pricing. Transportation Science 28(3), 221–235 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bonifaci, V., Salek, M., Schäfer, G.: On the efficiency of restricted tolls in network routing games. In: SAGT (2011) (to appear)Google Scholar
  6. 6.
    Caragiannis, I.: Efficient coordination mechanisms for unrelated machines scheduling. In: SODA, pp. 815–824 (2009)Google Scholar
  7. 7.
    Christodoulou, G., Gourvès, L., Pascual, F.: Scheduling selfish tasks: About the performance of truthful algorithms. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 187–197. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Christodoulou, G., Koutsoupias, E., Nanavati, A.: Coordination mechanisms. Theor. Comput. Sci. 410(36), 3327–3336 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cole, R., Correa, J.R., Gkatzelis, V., Mirrokni, V., Olver, N.: Inner product spaces for minsum coordination mechanisms. In: STOC (2011)Google Scholar
  10. 10.
    Cole, R., Dodis, Y., Roughgarden, T.: Pricing network edges for heterogeneous selfish users. In: STOC, pp. 521–530 (2003)Google Scholar
  11. 11.
    Cole, R., Dodis, Y., Roughgarden, T.: How much can taxes help selfish routing? J. Comput. Syst. Sci. 72(3), 444–467 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Correa, J.R., Schulz, A.S., Stier-Moses, N.E.: A geometric approach to the Price of Anarchy in nonatomic congestion games. Games and Economic Behavior 64, 457–469 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Dafermos, S.C., Sparrow, F.T.: The traffic assignment problem for a general network. Journal of Research of the National Bureau of Standards, Series B 73B(2), 91–118 (1969)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Dafermos, S.: An extended traffic assignment model with applications to two-way traffic. Transportation Science 5, 366–389 (1971)CrossRefGoogle Scholar
  15. 15.
    de Palma, A., Nesterov, Y.: Optimization formulations and static equilibrium in congested transportation networks. In: CORE Discussion Paper 9861, pp. 12–17. Université Catholique de Louvain, Louvain-la-Neuve (1998)Google Scholar
  16. 16.
    Fleischer, L.: Linear tolls suffice: New bounds and algorithms for tolls in single source networks. Theor. Comput. Sci. 348(2-3), 217–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Fleischer, L., Jain, K., Mahdian, M.: Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games. In: FOCS, pp. 277–285 (2004)Google Scholar
  18. 18.
    Immorlica, N., Li, L., Mirrokni, V.S., Schulz, A.: Coordination mechanisms for selfish scheduling. In: Deng, X., Ye, Y. (eds.) WINE 2005. LNCS, vol. 3828, pp. 55–69. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  19. 19.
    Karakostas, G., Kolliopoulos, S.G.: Edge pricing of multicommodity networks for heterogeneous selfish users. In: FOCS 2004, pp. 268–276 (2004)Google Scholar
  20. 20.
    Karakostas, G., Kolliopoulos, S.G.: The efficiency of optimal taxes. In: López-Ortiz, A., Hamel, A.M. (eds.) CAAN 2004. LNCS, vol. 3405, pp. 3–12. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  21. 21.
    Kollias, K.: Non-preemptive coordination mechanisms for identical machine scheduling games. In: Shvartsman, A.A., Felber, P. (eds.) SIROCCO 2008. LNCS, vol. 5058, pp. 197–208. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  22. 22.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. Computer Science Review 3(2), 65–69 (2009)CrossRefzbMATHGoogle Scholar
  23. 23.
    Marcotte, P., Patriksson, M.: Traffic equilibrium. In: Transportation, Handbooks in Operations Research and Management Science. ch.10, vol. 14, pp. 623–713. North-Holland, Amsterdam (2007)Google Scholar
  24. 24.
    Nisan, N., Roughgarden, T., Tardos, E., Vazirani, V.V.: Algorithmic Game Theory. Cambridge University Press, Cambridge (2007)CrossRefzbMATHGoogle Scholar
  25. 25.
    Patriksson, M.: The Traffic Assignment Problem: Models and Methods. V.S.P. Intl. Science (1994)Google Scholar
  26. 26.
    Roughgarden, T.: Designing networks for selfish users is hard. In: FOCS (2001)Google Scholar
  27. 27.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J.ACM 49, 236–259 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Part II, vol. 1, pp. 325–378 (1952)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • George Christodoulou
    • 1
  • Kurt Mehlhorn
    • 2
  • Evangelia Pyrga
    • 3
  1. 1.University of LiverpoolUnited Kingdom
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany
  3. 3.Technische Universität MünchenGermany

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