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

arXiv: https://arxiv.org/abs/2607.21805 · HTML: https://mlsanity.com/paper/2607.21805/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 56 references that are in the map.
- First posted on arXiv 23 Jul 2026.

## Builds on (most cited first, 12 of 12)

- [Multi-agent Performative Prediction: From Global Stability and Optimality to Chaos](https://mlsanity.com/paper/2201.10483/) (2022)
- [Multiplicative Weights Update with Constant Step-Size in Congestion Games: Convergence, Limit Cycles and Chaos](https://mlsanity.com/paper/1703.01138/) (2017)
- [Chaos, Extremism and Optimism: Volume Analysis of Learning in Games](https://mlsanity.com/paper/2005.13996/) (2020)
- [The route to chaos in routing games: When is Price of Anarchy too optimistic?](https://mlsanity.com/paper/1906.02486/) (2019)
- [Learning in Matrix Games can be Arbitrarily Complex](https://mlsanity.com/paper/2103.03405/) (2021)
- [Follow-the-Regularized-Leader Routes to Chaos in Routing Games](https://mlsanity.com/paper/2102.07974/) (2021)
- [No-Regret Learning and Equilibrium Computation in Quantum Games](https://mlsanity.com/paper/2310.08473/) (2023)
- [Dynamical Analysis of the EIP-1559 Ethereum Fee Market](https://mlsanity.com/paper/2102.10567/) (2021)
- [Optimality Despite Chaos in Fee Markets](https://mlsanity.com/paper/2212.07175/) (2022)
- [No-Regret Learning in Games is Turing Complete](https://mlsanity.com/paper/2202.11871/) (2022)
- [On the discrete-time origins of the replicator dynamics: From convergence to instability and chaos](https://mlsanity.com/paper/2402.09824/) (2024)
- [Chaos in Autobidding Auctions](https://mlsanity.com/paper/2602.09118/) (2026)

## Abstract (arXiv)

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 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.