Improving Selfish Routing for Risk-Averse Players

  • Dimitris Fotakis
  • Dimitris Kalimeris
  • Thanasis Lianeas
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9470)


We investigate how and to which extent one can exploit risk-aversion and modify the perceived cost of the players in selfish routing so that the Price of Anarchy (\(\mathrm {PoA}\)) is improved. We introduce small random perturbations to the edge latencies so that the expected latency does not change, but the perceived cost of the players increases due to risk-aversion. We adopt the model of \(\gamma \)-modifiable routing games, a variant of routing games with restricted tolls. We prove that computing the best \(\gamma \)-enforceable flow is \(\mathrm {NP}\)-hard for parallel-link networks with affine latencies and two classes of heterogeneous risk-averse players. On the positive side, we show that for parallel-link networks with heterogeneous players and for series-parallel networks with homogeneous players, there exists a nicely structured \(\gamma \)-enforceable flow whose \(\mathrm {PoA}\) improves fast as \(\gamma \) increases. We show that the complexity of computing such a \(\gamma \)-enforceable flow is determined by the complexity of computing a Nash flow of the original game. Moreover, we prove that the \(\mathrm {PoA}\) of this flow is best possible in the worst-case, in the sense that there are instances where (i) the best \(\gamma \)-enforceable flow has the same \(\mathrm {PoA}\), and (ii) considering more flexible modifications does not lead to any further improvement.


  1. 1.
    Angelidakis, H., Fotakis, D., Lianeas, T.: Stochastic congestion games with risk-averse players. In: Vöcking, B. (ed.) SAGT 2013. LNCS, vol. 8146, pp. 86–97. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  2. 2.
    Bonifaci, V., Salek, M., Schäfer, G.: Efficiency of restricted tolls in non-atomic network routing games. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 302–313. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  3. 3.
    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
  4. 4.
    Fiat, A., Papadimitriou, C.: When the players are not expectation maximizers. In: Kontogiannis, S., Koutsoupias, E., Spirakis, P.G. (eds.) SAGT 2010. LNCS, vol. 6386, pp. 1–14. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  5. 5.
    Fleischer, L.: Linear tolls suffice: new bounds and algorithms for tolls in single source networks. Theoret. Comput. Sci. 348, 217–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Fleischer, L., Jain, K., Mahdian, M.: Tolls for heterogeneous selfish users in multicommodity networks and generalized congestion games. In: Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS 2004), pp. 277–285 (2004)Google Scholar
  7. 7.
    Fotakis, D.A., Spirakis, P.G.: Cost-balancing tolls for atomic network congestion games. In: Deng, X., Graham, F.C. (eds.) WINE 2007. LNCS, vol. 4858, pp. 179–190. Springer, Heidelberg (2007) CrossRefGoogle Scholar
  8. 8.
    Hall, M.A.: Properties of the equilibrium state in transportation networks. Transp. Sci. 12(3), 208–216 (1978)CrossRefGoogle Scholar
  9. 9.
    Hoefer, M., Olbrich, L., Skopalik, A.: Taxing subnetworks. In: Papadimitriou, C., Zhang, S. (eds.) WINE 2008. LNCS, vol. 5385, pp. 286–294. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  10. 10.
    Jelinek, T., Klaas, M., Schäfer, G.: Computing optimal tolls with arc restrictions and heterogeneous players. In: Proceedings of the 31st Symposium on Theoretical Aspects of Computer Science (STACS 2014), LIPIcs 25, pp. 433–444 (2014)Google Scholar
  11. 11.
    Karakostas, G., Kolliopoulos, S.: Edge pricing of multicommodity networks for heterogeneous users. In: Proceedings of the 45th IEEE Symposium on Foundations of Computer Science (FOCS 2004), pp. 268–276 (2004)Google Scholar
  12. 12.
    Nikolova, E., Stier-Moses, N.E.: Stochastic selfish routing. In: Persiano, G. (ed.) SAGT 2011. LNCS, vol. 6982, pp. 314–325. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  13. 13.
    Nikolova, E., Stier-Moses, N.: The burden of risk aversion in mean-risk selfish routing. In: Proceedings of the 16th ACM Conference on Electronic Commerce (EC 2015), pp. 489–506 (2015)Google Scholar
  14. 14.
    Ordóñez, F., Stier-Moses, N.: Wardrop equilibria with risk-averse users. Transp. Sci. 44(1), 63–86 (2010)CrossRefGoogle Scholar
  15. 15.
    Piliouras, G., Nikolova, E., Shamma, J.S.: Risk Sensitivity of price of anarchy under uncertainty. In: Proceedings of the 14th ACM Conference on Electronic Commerce (EC 2013), pp. 715–732 (2013)Google Scholar
  16. 16.
    Roughgarden, T.: Stackelberg scheduling strategies. SIAM J. Comput. 33(2), 332–350 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Roughgarden, T.: Selfish Routing and The Price of Anarchy. MIT press, Cambridge (2005) zbMATHGoogle Scholar
  18. 18.
    Tversky, A., Kahneman, D.: Prospect theory: an analysis of decision under risk. Econometrica 47(2), 263–291 (1979)CrossRefzbMATHGoogle Scholar
  19. 19.
    Valdez, J., Tarjan, R.E., Lawler, E.L.: The recognition of series-parallel digraphs. SIAM J. Comput. 11(2), 298–313 (1982)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Open Access This chapter is distributed under the terms of the Creative Commons Attribution Noncommercial License, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  • Dimitris Fotakis
    • 1
  • Dimitris Kalimeris
    • 2
  • Thanasis Lianeas
    • 3
  1. 1.School of Electrical and Computer EngineeringNational Technical University of AthensAthensGreece
  2. 2.Department of Informatics and TelecommunicationsNational and Kapodistrian University of AthensAthensGreece
  3. 3.Department of Electrical and Computer EngineeringUniversity of Texas at AustinAustinUSA

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