aiqbench analysis
This research investigates the long-term statistical behavior of the Multiplicative Weights Update algorithm in game-theoretic settings where traditional convergence to Nash equilibria fails. By applying concepts from ergodic theory, the study demonstrates that even chaotic dynamics possess invariant measures allowing for precise calculation of economic metrics like payoffs and regret over time. The work bridges theoretical computer science with dynamical systems to show how statistical predictability can be achieved despite a lack of pointwise convergence in learning algorithms.
Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos
arXiv:2607.21805v1 Announce Type: cross 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.
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.
Dynamical Systems (math.DS); Machine Learning (cs.LG); Theoretical Economics (econ.TH)
arXiv:2607.21805 [math.DS]
(or arXiv:2607.21805v1 [math.DS] for this version)
https://doi.org/10.48550/arXiv.2607.21805 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
https://doi.org/10.1145/3821539.3827778 Focus to learn more DOI(s) linking to related resources
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