Advertisement

Price of Anarchy for Highly Congested Routing Games in Parallel Networks

  • Riccardo Colini-Baldeschi
  • Roberto Cominetti
  • Marco Scarsini
Article
  • 368 Downloads
Part of the following topical collections:
  1. Special Issue on Algorithmic Game Theory (SAGT 2016)

Abstract

We consider nonatomic routing games with one source and one destination connected by multiple parallel edges. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we prove that under suitable conditions on the costs the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case, and that these counterexamples already occur in simple networks with only 2 parallel links.

Keywords

Nonatomic routing games Price of Anarchy Regularly varying functions Wardrop equilibrium Parallel networks High congestion 

Notes

Acknowledgments

Riccardo Colini-Baldeschi is a member of GNAMPA-INdAM. Roberto Cominetti gratefully acknowledges the support and hospitality of LUISS during a visit in which this research was initiated. Roberto Cominetti’s research is also supported by FONDECYT 1130564 and Núcleo Milenio ICM/FIC RC130003 “Información y Coordinación en Redes”. Marco Scarsini is a member of GNAMPA-INdAM. He gratefully acknowledges the support and hospitality of FONDECYT 1130564 and Núcleo Milenio “Información y Coordinación en Redes”.

References

  1. 1.
    Attouch, H.: Variational Convergence for Functions and Operators. Pitman, Boston (1984)zbMATHGoogle Scholar
  2. 2.
    Beckmann, M.J., McGuire, C.B., Winsten, C.B.: Studies in the Economics of Transportation. Yale University Press, New Haven (1956)Google Scholar
  3. 3.
    Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation, volume 27 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  4. 4.
    Cole, R., Tao, Y.: Large market games with near optimal efficiency. In: Conitzer, V., Bergemann, D., Chen, Y. (eds.) Proceedings of the 2016 ACM Conference on Economics and Computation, EC ’16, July 24–28, 2016, pp. 791–808. ACM, Maastricht (2016)Google Scholar
  5. 5.
    Correa, J.R., Schulz, A.S., Stier-Moses, N.E.: Selfish routing in capacitated networks. Math. Oper. Res. 29(4), 961–976 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Correa, J.R., Schulz, A.S., Stier-Moses, N.E.: Fast, fair, and efficient flows in networks. Oper. Res. 55(2), 215–225 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Correa, J.R., Schulz, A.S., Stier-Moses, N.E.: A geometric approach to the price of anarchy in nonatomic congestion games. Games Econ. Behav. 64(2), 457–469 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    de Haan, L.: On Regular Variation and its Application to the Weak Convergence of Sample Extremes, volume 32 of Mathematical Centre Tracts. Mathematisch Centrum, Amsterdam (1970)Google Scholar
  9. 9.
    Dumrauf, D., Gairing, M.: Price of anarchy for polynomial Wardrop games. In: Spirakis, P., Mavronicolas, M., Kontogiannis, S. (eds.) Internet and Network Economics: Second International Workshop, WINE 2006, Patras, Greece, December 15–17, 2006. Proceedings, pp. 978–3-540-68141-0. Springer, Berlin (2006)Google Scholar
  10. 10.
    Englert, M., Franke, T., Olbrich, L.: Sensitivity of Wardrop equilibria. Theory Comput. Syst. 47(1), 3–14 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Feldman, M., Immorlica, N., Lucier, B., Roughgarden, T., Syrgkanis, V.: The price of anarchy in large games. In: Wichs, D., Mansour, Y. (eds.) Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, June 18–21, vol. 2016, pp. 963–976. ACM, Cambridge (2016)Google Scholar
  12. 12.
    Florian, M., Hearn, D.: Network equilibrium and pricing. In: Hall, R.W. (ed.) Handbook of Transportation Science, pp. 373–411. Springer US, Boston (2003)Google Scholar
  13. 13.
    González Vayá, M., Grammatico, S., Andersson, G., Lygeros, J.: On the price of being selfish in large populations of plug-in electric vehicles. In: 2015 54th IEEE Conference on Decision and Control (CDC), pp. 6542–6547 (2015)Google Scholar
  14. 14.
    Josefsson, M., Patriksson, M.: Sensitivity analysis of separable traffic equilibrium equilibria with application to bilevel optimization in network design. Transp. Res. B Methodol. 41(1), 4–31, 1 (2007)CrossRefGoogle Scholar
  15. 15.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: STACS 99 (Trier), volume 1563 of Lecture Notes in Computer Science, pp. 404–413. Springer, Berlin (1999)Google Scholar
  16. 16.
    Law, L.M., Huang, J., Liu, M.: Price of anarchy for congestion games in cognitive radio networks. IEEE Trans. Wirel. Commun. 11(10), 3778–3787 (2012)CrossRefGoogle Scholar
  17. 17.
    Mas-Colell, A.: On a theorem of Schmeidler. J. Math. Econ. 13(3), 201–206 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Milchtaich, I.: Generic uniqueness of equilibrium in large crowding games. Math. Oper. Res. 25(3), 349–364 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Milchtaich, I.: Social optimality and cooperation in nonatomic congestion games. J. Econ. Theory 114(1), 56–87 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    O’Hare, S.J., Connors, R.D., Watling, D.P.: Mechanisms that govern how the price of anarchy varies with travel demand. Transp. Res. B Methodol. 84, 55–80, 2 (2016)CrossRefGoogle Scholar
  21. 21.
    Panageas, I., Piliouras, G.: Approximating the geometry of dynamics inppotential games. Technical report, arXiv:1403.3885v5 (2015)
  22. 22.
    Papadimitriou, C.: Algorithms, games, and the Internet. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 749–753. ACM, New York (2001)Google Scholar
  23. 23.
    Patriksson, M.: Sensitivity analysis of traffic equilibria. Transp. Sci. 38(3), 258–281 (2004)CrossRefGoogle Scholar
  24. 24.
    Pigou, A.C.: The Economics of Welfare, 1st edn. Macmillan and Co., London (1920)Google Scholar
  25. 25.
    Piliouras, G., Nikolova, E., Shamma, J.S.: Risk sensitivity of price of anarchy under uncertainty. In: Proceedings of the Fourteenth ACM Conference on Electronic Commerce, EC ’13, pp. 715–732. ACM, New York (2013)Google Scholar
  26. 26.
    Roughgarden, T.: The price of anarchy is independent of the network topology. J. Comput. Syst. Sci. 67(2), 341–364 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Roughgarden, T.: Routing games. In: Algorithmic Game Theory, pp. 461–486. Cambridge University Press, Cambridge (2007)Google Scholar
  28. 28.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49(2), 236–259 (electronic) (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Roughgarden, T., Tardos, É.: Bounding the inefficiency of equilibria in nonatomic congestion games. Games Econom. Behav. 47(2), 389–403 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Roughgarden, T., Tardos, É.: Introduction to the inefficiency of equilibria. In: Algorithmic Game Theory, pp. 443–459. Cambridge University Press, Cambridge (2007)Google Scholar
  31. 31.
    Schmeidler, D.: Equilibrium points of nonatomic games. J. Statist. Phys. 7, 295–300 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institute of Civil Engineers, Pt. II, vol. 1, pp. 325–378 (1952)Google Scholar
  33. 33.
    Youn, H., Gastner, M.T., Jeong, H.: Price of anarchy in transportation networks: efficiency and optimality control. Phys. Rev. Lett. 101, 128701 (2008)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di Economia e FinanzaLUISSRomaItaly
  2. 2.Facultad de Ingeniería y CienciasUniversidad Adolfo IbáñezSantiagoChile

Personalised recommendations