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Automata, Languages and Programming

Volume 2719 of the series Lecture Notes in Computer Science pp 514-526

Date:

Nashification and the Coordination Ratio for a Selfish Routing Game

  • Rainer FeldmannAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
  • , Martin GairingAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
  • , Thomas LückingAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
  • , Burkhard MonienAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn
  • , Manuel RodeAffiliated withDepartment of Computer Science, Electrical Engineering and Mathematics, University of Paderborn

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Abstract

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.