Facets of the Fully Mixed Nash Equilibrium Conjecture

  • Authors:

  • Rainer Feldmann;Marios Mavronicolas;Andreas Pieris

  • Affiliations:

  • Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, Paderborn, Germany 33102;Department of Computer Science, University of Cyprus, Nicosia CY-1678, Cyprus, Currently visiting Faculty of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, 3310 ...;Computing Laboratory, University of Oxford, Oxford, United Kingdom OX1 3QD

  • Venue:
  • SAGT '08 Proceedings of the 1st International Symposium on Algorithmic Game Theory
  • Year:
  • 2008

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Abstract

In this work, we continue the study of the many facets of the Fully Mixed Nash Equilibrium Conjecture, henceforth abbreviated as the FMNEConjecture, in selfish routing for the special case of nidentical usersover two (identical) parallel links. We introduce a new measure of Social Cost, defined to be the expectation of the square of the maximum congestionon a link; we call it Quadratic Maximum Social Cost. A Nash equilibrium(NE) is a stable state where no user can improve her (expected) latency by switching her mixed strategy; a worst-caseNEis one that maximizes Quadratic Maximum Social Cost. In the fully mixedNE, all mixed strategiesachieve full support.Formulated within this framework is yet another facet of the FMNEConjecture, which states that the fully mixed Nash equilibrium is the worst-case NE. We present an extensive proof of the FMNEConjecture; the proof employs a mixture of combinatorial arguments and analytical estimations. Some of these analytical estimations are derived through some new bounds on generalized mediansof the binomial distribution [22] we obtain, which are of independent interest.