Extreme Nash Equilibria

  • Martin Gairing
  • Thomas Lücking
  • Marios Mavronicolas
  • Burkhard Monien
  • Paul Spirakis
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2841)


We study the combinatorial structure and computational complexity of extreme Nash equilibria, ones that maximize or minimize a certain objective function, in the context of a selfish routing game. Specifically, we assume a collection of nusers, each employing a mixed strategy, which is a probability distribution over m parallel links, to control the routing of its own assigned traffic. In a Nash equilibrium, each user routes its traffic on links that minimize its expected latency cost.

Our structural results provide substantial evidence for the Fully Mixed Nash Equilibrium Conjecture, which states that the worst Nash equilibrium is the fully mixed Nash equilibrium, where each user chooses each link with positive probability. Specifically, we prove that the Fully Mixed Nash Equilibrium Conjecture is valid for pure Nash equilibria and that under a certain condition, the social cost of any Nash equilibrium is within a factor of 6 + ε, of that of the fully mixed Nash equilibrium, assuming that link capacities are identical.

Our complexity results include hardness, approximability and inapproximability ones. Here we show, that for identical link capacities and under a certain condition, there is a randomized, polynomial-time algorithm to approximate the worst social cost within a factor arbitrarily close to 6 + ε. Furthermore, we prove that for any arbitrary integer k > 0, it is \(\mathcal{NP}\)-hard to decide whether or not any given allocation of users to links can be transformed into a pure Nash equilibrium using at most k selfish steps. Assuming identical link capacities, we give a polynomial-time approximation scheme (PTAS) to approximate the best social cost over all pure Nash equilibria. Finally we prove, that it is \(\mathcal{NP}\)-hard to approximate the worst social cost within a multiplicative factor 2 - \(\frac{\rm 2}{m+1} - \epsilon\). The quantity 2 - \(\frac{\rm 2}{m+1}\) is the tight upper bound on the ratio of the worst social cost and the optimal cost in the model of identical capacities.


Nash Equilibrium Problem Instance Social Cost Pure Strategy Link Capacity 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Martin Gairing
    • 1
  • Thomas Lücking
    • 1
  • Marios Mavronicolas
    • 2
  • Burkhard Monien
    • 1
  • Paul Spirakis
    • 3
    • 4
  1. 1.Faculty of Computer Science, Electrical Engineering and MathematicsUniversity of PaderbornPaderbornGermany
  2. 2.Department of Computer ScienceUniversity of CyprusNicosiaCyprus
  3. 3.Computer Technology InstitutePatrasGreece
  4. 4.Department of Computer Engineering and InformaticsUniversity of Patras, RionPatrasGreece

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