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

Thiparat Chotibut, Fryderyk Falniowski+2 more|arXiv (Cornell University)|Jun 6, 2019

Game Theory and Applications80 references5 citations

TL;DR

This paper demonstrates that even in simple non-atomic routing games with linear costs—where the Price of Anarchy is exactly 1—Multiplicative Weights Update (MWU) dynamics can become unstable and chaotic as system demand increases. Despite the equilibrium being socially optimal, time-average social costs can reach their worst possible value due to period-doubling bifurcations and Li-Yorke chaos, undermining the predictive power of equilibrium metrics like PoA.

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.

Motivation & Objective

• To investigate the robustness of equilibrium-based efficiency guarantees, such as the Price of Anarchy, in learning dynamics of congestion games.
• To examine whether the convergence of learning dynamics to Nash equilibria holds under increasing system demand.
• To identify conditions under which MWU leads to non-equilibrating, chaotic behavior even in simple routing games.
• To explore the relationship between time-average performance, regret, and social cost in non-equilibrating regimes.
• To extend findings to atomic, non-atomic, polynomial-cost, and heterogeneous-user congestion games.

Proposed method

• Analyzing the dynamics of Multiplicative Weights Update (MWU) in two-strategy non-atomic congestion games with linear cost functions.
• Using bifurcation theory and dynamical systems analysis to identify transitions from stable equilibria to limit cycles and chaos.
• Deriving a critical parameter $ a = (α + β)N \ln(1/(1-\epsilon)) $ that determines whether the system remains in a chaotic regime despite diminishing step-sizes.
• Applying the Feigenbaum route to chaos via period-doubling bifurcations to characterize instability onset.
• Extending results to multi-strategy games, polynomial cost functions, atomic games, and heterogeneous users through analytical and numerical validation.
• Demonstrating that time-average flows converge to equilibrium values even in chaotic regimes, a property akin to no-regret learning in zero-sum games.

Experimental results

Research questions

• RQ1 Under what conditions does the Multiplicative Weights Update algorithm fail to converge to equilibrium in non-atomic routing games?
• RQ2 How does increasing system demand affect the stability of learning dynamics in congestion games with a Price of Anarchy of 1?
• RQ3 Can chaotic dynamics emerge in simple congestion games even when all equilibria are socially optimal?
• RQ4 What is the relationship between time-average social cost and the system's dynamical regime (equilibrium vs. chaos)?
• RQ5 To what extent do time-average flows and costs remain close to equilibrium values in non-equilibrating, chaotic regimes?

Key findings

• For symmetric equilibrium flows (50-50 split), increasing demand triggers a single period-doubling bifurcation, leading to a stable limit cycle of period two.
• In the large population limit, the time-average social cost for almost all initial conditions converges to the worst possible value, despite PoA = 1.
• For asymmetric equilibrium flows, increasing demand leads to Li-Yorke chaos with positive topological entropy and periodic orbits of all possible periods.
• Even in chaotic regimes, the time-average flows on each path converge exactly to the Nash equilibrium flows, a property reminiscent of no-regret learning.
• The system has a finite carrying capacity: above this threshold, dynamics become non-equilibrating and inefficient regardless of diminishing learning rates.
• A slowly increasing population size (e.g., sublinear in time) can sustain the system in a chaotic regime indefinitely, even with shrinking step-sizes like $1/\sqrt{n}$.