Facets of the Fully Mixed Nash Equilibrium Conjecture

AbstractIn this work, we continue the study of the many facets of the Fully Mixed Nash Equilibrium Conjecture, henceforth abbreviated as the $\mathsf{FMNE}$Conjecture, in selfish routing for the special case of n identical users over two (identical) parallel links. We introduce a new measure of Social Cost, defined as the expectation of the square of the maximum congestion on a link; we call it Quadratic Maximum Social Cost. A Nash equilibrium is a stable state where no user can improve her (expected) latency by switching her mixed strategy; a worst-case Nash equilibrium is one that maximizes Quadratic Maximum Social Cost. In the fully mixed Nash equilibrium, all mixed strategies achieve full support.Formulated within this framework is yet another facet of the $\mathsf{FMNE}$Conjecture, which states that the fully mixed Nash equilibrium is the worst-case Nash equilibrium. We present an extensive proof of the $\mathsf{FMNE}$Conjecture; the proof employs a combination of combinatorial arguments and analytical estimations.

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