Nashification and the Coordination Ratio for a Selfish Routing Game

  • Rainer Feldmann
  • Martin Gairing
  • Thomas Lücking
  • Burkhard Monien
  • Manuel Rode
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2719)


We study the problem of n users selfishly routing traffic through a network consisting of m parallel related links. Users route their traffic by choosing private probability distributions over the links with the aim of minimizing their private latency. In such an environment Nash equilibria represent stable states of the system: no user can improve its private latency by unilaterally changing its strategy.

Nashification is the problem of converting any given non-equilibrium routing into a Nash equilibrium without increasing the social cost. Our first result is an O(nm 2) time algorithm for Nashification. This algorithm can be used in combination with any approximation algorithm for the routing problem to compute a Nash equilibrium of the same quality. In particular, this approach yields a PTAS for the computation of a best Nash equilibrium. Furthermore, we prove a lower bound of \( \Omega \left( {2^{\sqrt n } } \right) \) and an upper bound of O(2n) for the number of greedy selfish steps for identical link capacities in the worst case.

In the second part of the paper we introduce a new structural parameter which allows us to slightly improve the upper bound on the coordination ratio for pure Nash equilibria in [3]. The new bound holds for the individual coordination ratio and is asymptotically tight. Additionally, we prove that the known upper bound of \( \frac{{1 + \sqrt {4m - 3} }} {2} \) on the coordination ratio for pure Nash equilibria also holds for the individual coordination ratio in case of mixed Nash equilibria, and we determine the range of m for which this bound is tight.


Nash Equilibrium Social Cost Mixed Strategy Pure Strategy Link Capacity 
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Rainer Feldmann
    • 1
  • Martin Gairing
    • 1
  • Thomas Lücking
    • 1
  • Burkhard Monien
    • 1
  • Manuel Rode
    • 1
  1. 1.Department of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany

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