Some Development on the Fully Mixed Nash Equilibrium Conjecture∗

A Nash equilibrium of a routing network represents a stable state of the network where no user finds it beneficial to unilaterally deviate from its routing strategy. In this work, we investigate the structure of such equilibria within the context of a certain game that models selfish routing for a set of n users each shipping its traffic over a network consisting of m parallel links. In particular, we are interested in identifying the worst-case Nash equilibrium the one that maximizes social cost. Worst-case Nash equilibria were first introduced and studied in the pioneering work of Koutsoupias and Papadimitriou. More specifically, we consider a variation of the selfish routing game of Koutsoupias and Papadimitriou (KP model), with quadratic social cost, and study the Conjecture of the Fully Mixed Nash Equilibrium, henceforth abbreviated as FMNE Conjecture, which asserts that the fully mixed Nash equilibrium, when existing, is the worst-case Nash equilibrium. (In the fully mixed Nash equilibrium, the mixed strategy of each user assigns (strictly) positive probability to every link.) We report substantial progress towards identifying the validity and methodologies to establish the FMNE Conjecture. Our result provides strong evidence for the validity of the FMNE Conjecture by establishing it in an interesting instance of the routing game we consider. This instance falls with in the model of related links, where there is a capacity c for each link j, and the individual delay of user i (with traffic wi) on link j is wi cj . Specifically, we prove that the FMNE Conjecture is valid when there are just two links with identical capacities and an arbitrary number of users with identical traffics. ∗This work was partially supported by the European Union under IST FET Integrated Project 015964 AEOLUS †Department of Computer Science, Electrical Engineering and Mathematics, University of Paderborn, 33102 Paderborn, Germany. Email: [email protected] ‡Department of Computer Science, University of Cyprus, Nicosia CY-1678, Cyprus. Email: {mavronic,cs03pa2}@ucy.ac.cy