On the Price of Anarchy of Highly Congested Nonatomic Network Games

  • Riccardo Colini-Baldeschi
  • Roberto Cominetti
  • Marco Scarsini
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9928)


We consider nonatomic network games with one source and one destination. We examine the asymptotic behavior of the price of anarchy as the inflow increases. In accordance with some empirical observations, we show that, under suitable conditions, the price of anarchy is asymptotic to one. We show with some counterexamples that this is not always the case. The counterexamples occur in simple parallel graphs.



Riccardo Colini-Baldeschi is a member of GNCS-INdAM. Roberto Cominetti gratefully acknowledges the support and hospitality of LUISS during a visit in which this research was initiated. His research is also supported by Núcleo Milenio Información y Coordinación en Redes ICM/FIC P10-024F. Marco Scarsini is a member of GNAMPA-INdAM. His work is partially supported by PRIN and MOE2013-T2-1-158.

The authors thank three referees for their insightful comments.


  1. 1.
    Attouch, H.: Variational Convergence for Functions and Operators. Pitman, Boston (1984)zbMATHGoogle Scholar
  2. 2.
    Beckmann, M.J., McGuire, C., 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, Encyclopedia of Mathematics and its Applications, vol. 27. Cambridge University Press, Cambridge (1989)zbMATHGoogle Scholar
  4. 4.
    Cole, R., Tao, Y.: The price of anarchy of large Walrasian auctions. Technical report, arXiv:1508.07370v4 (2015).
  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). MathSciNetzbMATHCrossRefGoogle 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). MathSciNetzbMATHCrossRefGoogle 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 Econom. Behav. 64(2), 457–469 (2008). MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dumrauf, D., Gairing, M.: Price of anarchy for polynomial Wardrop games. In: Spirakis, P.G., Mavronicolas, M., Kontogiannis, S.C. (eds.) WINE 2006. LNCS, vol. 4286, pp. 319–330. Springer, Heidelberg (2006). CrossRefGoogle Scholar
  9. 9.
    Englert, M., Franke, T., Olbrich, L.: Sensitivity of Wardrop equilibria. Theory Comput. Syst. 47(1), 3–14 (2010). MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Feldman, M., Immorlica, N., Lucier, B., Roughgarden, T., Syrgkanis, V.: The price of anarchy in large games. Technical report, arXiv:1503.04755 (2015)
  11. 11.
    Florian, M., Hearn, D.: Network equilibrium and pricing. In: Hall, R.W. (ed.) Handbook of Transportation Science, pp. 373–411. Springer, US, 978-0-306-48058-4 (2003).
  12. 12.
    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
  13. 13.
    Josefsson, M., Patriksson, M.: Sensitivity analysis of separable traffic equilibrium equilibria with application to bilevel optimization in network design. Transp. Res. Part B: Methodol. 41(1), 4–31 (2007). CrossRefGoogle Scholar
  14. 14.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999). CrossRefGoogle Scholar
  15. 15.
    Law, L.M., Huang, J., Liu, M.: Price of anarchy for congestion games in cognitive radio networks. IEEE Trans. Wireless Commun. 11(10), 3778–3787 (2012)CrossRefGoogle Scholar
  16. 16.
    Mas-Colell, A.: On a theorem of Schmeidler. J. Math. Econom. 13(3), 201–206 (1984). MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Milchtaich, I.: Generic uniqueness of equilibrium in large crowding games. Math. Oper. Res. 25(3), 349–364 (2000). MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Milchtaich, I.: Social optimality and cooperation in nonatomic congestion games. J. Econom. Theory 114(1), 56–87 (2004). MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    O’Hare, S.J., Connors, R.D., Watling, D.P.: Mechanisms that govern how the price of anarchy varies with travel demand. Transp. Res. Part B: Methodol. 84, 55–80 (2016). CrossRefGoogle Scholar
  20. 20.
    Panageas, I., Piliouras, G.: Approximating the geometry of dynamics in potential games. Technical report, arXiv:1403.3885v5 (2015).
  21. 21.
    Papadimitriou, C.: Algorithms, games, and the Internet. In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pp. 749–753. (2001).
  22. 22.
    Patriksson, M.: Sensitivity analysis of traffic equilibria. Transp. Sci. 38(3), 258–281 (2004). MathSciNetCrossRefGoogle Scholar
  23. 23.
    Pigou, A.C.: The Economics of Welfare, 1st edn. Macmillan and Co., London (1920)Google Scholar
  24. 24.
    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 2013, pp. 715–732. ACM, New York (2013).
  25. 25.
    Roughgarden, T.: The price of anarchy is independent of the network topology. J. Comput. System Sci. 67(2), 341–364 (2003). MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Roughgarden, T.: Routing games. In: Algorithmic Game Theory, pp. 461–486. Cambridge Univ. Press, Cambridge (2007)Google Scholar
  27. 27.
    Roughgarden, T., Tardos, É.: How bad is selfish routing? J. ACM 49(2), 236–259 (2002). (electronic) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Roughgarden, T., Tardos, É.: Bounding the inefficiency of equilibria in nonatomic congestion games. Games Econom. Behav. 47(2), 389–403 (2004). MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Roughgarden, T., Tardos, É.: Introduction to the inefficiency of equilibria. In: Algorithmic Game Theory, pp. 443–459. Cambridge Univ. Press, Cambridge (2007)Google Scholar
  30. 30.
    Schmeidler, D.: Equilibrium points of nonatomic games. J. Statist. Phys. 7, 295–300 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    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).
  32. 32.
    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-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Riccardo Colini-Baldeschi
    • 1
  • Roberto Cominetti
    • 2
  • Marco Scarsini
    • 1
  1. 1.Dipartimento di Economia e FinanzaLUISSRomeItaly
  2. 2.Facultad de Ingeniería y CienciasUniversidad Adolfo IbáñezSantiagoChile

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