Stability of Nash equilibria in the congestion game under Replicator dynamics

We consider the single commodity non-atomic congestion game, in which the player population is assumed to obey the replicator dynamics. We study the resulting rest points, and relate them to the Nash equilibria of the one-shot congestion game. The rest points of the replicator dynamics, also called evolutionary stable points, are known to coincide with a superset of Nash equilibria, called restricted equilibria. By studying the spectrum of the linearized system around rest points, we show that Nash equilibria are locally asymptotically stable stationary points. We also show that under the additional assumption of strictly increasing congestion functions, Nash equilibria are exactly the set of exponentially stable points. We illustrate these results on numerical examples.