Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos
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Mathematics > Dynamical Systems
arXiv:2607.21805 (math)
[Submitted on 23 Jul 2026]
Title:Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos
Authors:Jakub Bielawski, Thiparat Chotibut, Fryderyk Falniowski, Michał Misiurewicz, Georgios Piliouras
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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.
| Subjects: | Dynamical Systems (math.DS); Machine Learning (cs.LG); Theoretical Economics (econ.TH) |
| Cite as: | arXiv:2607.21805 [math.DS] |
| (or arXiv:2607.21805v1 [math.DS] for this version) | |
| https://doi.org/10.48550/arXiv.2607.21805
arXiv-issued DOI via DataCite (pending registration)
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| Related DOI: | https://doi.org/10.1145/3821539.3827778
DOI(s) linking to related resources
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Submission history
From: Fryderyk Falniowski [view email][v1] Thu, 23 Jul 2026 20:45:08 UTC (1,626 KB)
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View a PDF of the paper titled Natural Invariant Measures for Chaotic Game Dynamics: Finding Order in Chaos, by Jakub Bielawski and 4 other authors
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