In this work, we continue the study of the many facets of the Fully Mixed Nash Equilibrium Conjecture, henceforth abbreviated as the 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 to be the expectation of the square of the maximum congestion on 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-case NE is one that maximizes Quadratic Maximum Social Cost. In the fully mixed NE, all mixed strategies achieve full support. Formulated within this framework is yet another facet of the FMNE Conjecture, which states that the fully mixed Nash equilibrium is the worst-case NE. We present an extensive proof of the FMNE Conjecture; the proof employs a mixture of combinatorial arguments and analytical estimations. Some of these analytical estimations are derived through some new bounds on generalized medians of the binomial distribution [22] we obtain, which are of independent interest.
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