The route to chaos in routing games: When is Price of Anarchy too optimistic?

TL;DR

This paper reveals that in routing games, increasing demand can lead to chaotic dynamics and inefficiencies despite classic equilibrium properties, challenging the assumption that Price of Anarchy remains low.

Contribution

It demonstrates the emergence of chaos and instability in routing games under multiplicative weights update, even when equilibria are theoretically optimal.

Findings

Chaos and bifurcations occur as demand increases.

Time-average social cost can reach worst-case levels.

Results extend to various game classes and cost functions.

Abstract

Routing games are amongst the most studied classes of games. Their two most well-known properties are that learning dynamics converge to equilibria and that all equilibria are approximately optimal. In this work, we perform a stress test for these classic results by studying the ubiquitous dynamics, Multiplicative Weights Update, in different classes of congestion games, uncovering intricate non-equilibrium phenomena. As the system demand increases, the learning dynamics go through period-doubling bifurcations, leading to instabilities, chaos and large inefficiencies even in the simplest case of non-atomic routing games with two paths of linear cost where the Price of Anarchy is equal to one. Starting with this simple class, we show that every system has a carrying capacity, above which it becomes unstable. If the equilibrium flow is a symmetric $50-50\%$ split, the system exhibits one period-doubling bifurcation. A single periodic attractor of period two replaces the attracting fixed point. Although the Price of Anarchy is equal to one, in the large population limit the time-average social cost for all but a zero measure set of initial conditions converges to its worst possible value. For asymmetric equilibrium flows, increasing the demand eventually forces the system into Li-Yorke chaos with positive topological entropy and periodic orbits of all possible periods. Remarkably, in all non-equilibrating regimes, the time-average flows on the paths converge exactly to the equilibrium flows, a property akin to no-regret learning in zero-sum games. These results are robust. We extend them to routing games with arbitrarily many strategies, polynomial cost functions, non-atomic as well as atomic routing games and heteregenous users. Our results are also applicable to any sequence of shrinking learning rates, e.g., $1/\sqrt{T}$, by allowing for a dynamically increasing population size.