Path Deviations Outperform Approximate Stability in Heterogeneous Congestion Games

  • Pieter Kleer
  • Guido SchäferEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10504)


We consider non-atomic network congestion games with heterogeneous players where the latencies of the paths are subject to some bounded deviations. This model encompasses several well-studied extensions of the classical Wardrop model which incorporate, for example, risk-aversion, altruism or travel time delays. Our main goal is to analyze the worst-case deterioration in social cost of a deviated Nash flow (i.e., for the perturbed latencies) with respect to an original Nash flow.

We show that for homogeneous players deviated Nash flows coincide with approximate Nash flows and derive tight bounds on their inefficiency. In contrast, we show that for heterogeneous populations this equivalence does not hold. We derive tight bounds on the inefficiency of both deviated and approximate Nash flows for arbitrary player sensitivity distributions. Intuitively, our results suggest that the negative impact of path deviations (e.g., caused by risk-averse behavior or latency perturbations) is less severe than approximate stability (e.g., caused by limited responsiveness or bounded rationality).

We also obtain a tight bound on the inefficiency of deviated Nash flows for matroid congestion games and homogeneous populations if the path deviations can be decomposed into edge deviations. In particular, this provides a tight bound on the Price of Risk-Aversion for matroid congestion games.



We thank the anonymous referees for their very useful comments, and one reviewer for pointing us to Lemma 1 [4].


  1. 1.
    Chen, P.-A., Keijzer, B.D., Kempe, D., Schäfer, G.: Altruism and its impact on the price of anarchy. ACM Trans. Econ. Comput. 2(4), 17:1–17:45 (2014)CrossRefGoogle Scholar
  2. 2.
    Christodoulou, G., Koutsoupias, E., Spirakis, P.G.: On the performance of approximate equilibria in congestion games. Algorithmica 61(1), 116–140 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Englert, M., Franke, T., Olbrich, L.: Sensitivity of Wardrop equilibria. In: Monien, B., Schroeder, U.-P. (eds.) SAGT 2008. LNCS, vol. 4997, pp. 158–169. Springer, Heidelberg (2008). doi: 10.1007/978-3-540-79309-0_15CrossRefGoogle Scholar
  4. 4.
    Fleischer, L.: Linear tolls suffice: new bounds and algorithms for tolls in single source networks. Theoret. Comput. Sci. 348(2), 217–225 (2005)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fujishige, S., Goemans, M.X., Harks, T., Peis, B., Zenklusen, R.: Matroids are immune to braess paradox. CoRR, abs/1504.07545 (2015)Google Scholar
  6. 6.
    Kleer, P., Schäfer, G.: The impact of worst-case deviations in non-atomic network routing games. CoRR, abs/1605.01510 (2016)Google Scholar
  7. 7.
    Kleer, P., Schäfer, G.: Path deviations outperform approximate stability in heterogeneous congestion games. CoRR, abs/1707.01278 (2017)Google Scholar
  8. 8.
    Koutsoupias, E., Papadimitriou, C.: Worst-case equilibria. In: Meinel, C., Tison, S. (eds.) STACS 1999. LNCS, vol. 1563, pp. 404–413. Springer, Heidelberg (1999). doi: 10.1007/3-540-49116-3_38CrossRefGoogle Scholar
  9. 9.
    Lianeas, T., Nikolova, E., Stier-Moses, N.E.: Asymptotically tight bounds for inefficiency in risk-averse selfish routing. In: Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence. IJCAI 2016, NY, USA, New York, pp. 338–344 (2016)Google Scholar
  10. 10.
    Lin, H., Roughgarden, T., Tardos, É., Walkover, A.: Stronger bounds on Braess’s paradox and the maximum latency of selfish routing. SIAM J. Discrete Math. 25(4), 1667–1686 (2011)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Meir, R., Parkes, D.C.: Playing the wrong game: smoothness bounds for congestion games with risk averse agents. CoRR, abs/1411.1751 (2017)Google Scholar
  12. 12.
    Milchtaich, I.: Network topology and the efficiency of equilibrium. Games Econ. Behav. 57(2), 321–346 (2006)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Nikolova, E., Stier-Moses, N.E.: The burden of risk aversion in mean-risk selfish routing. In: Proceedings of the Sixteenth ACM Conference on Economics and Computation, EC 2015, pp. 489–506. ACM, New York (2015)Google Scholar
  14. 14.
    Roughgarden, T.: On the severity of Braess’s paradox: designing networks for selfish users is hard. J. Comput. Syst. Sci. 72(5), 922–953 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Roughgarden, T.: Intrinsic robustness of the price of anarchy. J. ACM 62(5), 32 (2015)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency Vol. B Matroids Trees Stable Sets Algorithms and Combinatorics. Springer, Heidelberg (2003). Chaps. 39–69Google Scholar
  17. 17.
    Wardrop, J.G.: Some theoretical aspects of road traffic research. In: Proceedings of the Institution of Civil Engineers, vol. 1, pp. 325–378 (1952)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Centrum Wiskunde & Informatica (CWI), Networks and Optimization GroupAmsterdamThe Netherlands
  2. 2.Department of Econometrics and Operations ResearchVrije Universiteit AmsterdamAmsterdamThe Netherlands

Personalised recommendations