Follow-the-Regularized-Leader Routes to Chaos in Routing Games

TL;DR

This paper investigates how Follow-the-Regularized-Leader (FoReL) learning dynamics can become chaotic in congestion games as population size or costs increase, revealing complex behaviors and coexistence of equilibrium and chaos.

Contribution

It generalizes chaos results in FoReL dynamics to broader classes and identifies conditions under which chaos and stable equilibria coexist in congestion games.

Findings

Chaos emerges in simple congestion games with increasing population or costs.

Stable equilibria can coexist with chaotic dynamics in the same game.

Time averages of dynamics still converge to Nash equilibria despite chaos.

Abstract

We study the emergence of chaotic behavior of Follow-the-Regularized Leader (FoReL) dynamics in games. We focus on the effects of increasing the population size or the scale of costs in congestion games, and generalize recent results on unstable, chaotic behaviors in the Multiplicative Weights Update dynamics to a much larger class of FoReL dynamics. We establish that, even in simple linear non-atomic congestion games with two parallel links and any fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to become unstable and eventually chaotic, in the sense of Li-Yorke and positive topological entropy. Furthermore, we show the existence of novel non-standard phenomena such as the coexistence of stable Nash equilibria and chaos in the same game. We also observe the simultaneous creation of a chaotic attractor as another chaotic attractor gets destroyed. Lastly, although FoReL dynamics can be strange and non-equilibrating, we prove that the time average still converges to an exact equilibrium for any choice of learning rate and any scale of costs.