Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos

2026-07-23

AI Summary

This study addresses the challenge that multiplicative weights update algorithms in games often fail to converge to Nash equilibria and exhibit unpredictable long-term behavior due to Li-Yorke chaos. To overcome this, we introduce for the first time the natural invariant measure from ergodic theory into the analysis of game dynamics. This framework not only characterizes strategy frequencies but also precisely computes long-run time averages of economically relevant observables—such as payoffs, social cost, and regret—even in the absence of pointwise convergence. Focusing on two-strategy congestion games, we rigorously establish that the system retains statistical predictability and provide a unified description of its full dynamical spectrum, ranging from periodic attractors to coexisting chaotic regimes, thereby revealing the algorithm’s capacity to replicate canonical behaviors of one-dimensional dynamical systems.

Abstract

We study the long-term behavior of the Multiplicative Weights Update (MWU) algorithm in game settings where learning dynamics frequently fail to converge to Nash equilibria and instead exhibit Li-Yorke chaos. While such chaos precludes the prediction of specific long-term strategy profiles, it does not imply a lack of statistical structure. We demonstrate that natural invariant measures - a fundamental concept from ergodic theory - provide the rigorous framework necessary to find order within this chaos. Focusing on a two-strategy congestion game, we prove that these measures allow for a comprehensive statistical characterization of the dynamics. Crucially, we show that this framework extends beyond simple strategy frequencies to \emph{general observables}, enabling the precise calculation of long-term time averages for broad classes of economic metrics - including payoffs, social cost, and regret - despite chaos. Our results reveal that this simple learning algorithm captures the full spectrum of behaviors found in one-dimensional dynamical systems, from unique or multiple absolutely continuous invariant measures to complex periodic attractors as well as coexisting chaotic and stable (periodic) behaviors. By bridging game theory and dynamical systems, we show that statistical predictability is attainable even in the absence of pointwise convergence.

Problem

• chaos
• invariant measures
• game dynamics
• statistical predictability
• Multiplicative Weights Update

Innovation

• natural invariant measures
• Multiplicative Weights Update
• Li-Yorke chaos
• ergodic theory
• statistical predictability