发表机构
Krakow University of Economics; Chulalongkorn University; Indiana University Indianapolis; Google DeepMind（克拉科夫经济大学; 朱拉隆功大学; 印第安纳大学印第安纳波利斯分校; 谷歌DeepMind）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究乘法权重更新算法在博弈中呈现李 - 约克混沌时的长期行为，利用自然不变测度提供框架，以双策略拥塞博弈为例，能对动态全面统计表征，涵盖多种行为谱，实现统计可预测性。
AI 中文摘要
我们研究了乘法权重更新（MWU）算法在博弈环境中的长期行为，其中学习动态常常无法收敛到纳什均衡，而是呈现李 - 约克混沌。虽然这种混沌排除了对特定长期策略配置的预测，但并不意味着缺乏统计结构。我们证明自然不变测度——遍历理论中的一个基本概念——为在这种混沌中找到秩序提供了必要的严格框架。以双策略拥塞博弈为例，我们证明这些测度允许对动态进行全面的统计表征。关键的是，我们表明这个框架不仅适用于简单策略频率，还适用于一般可观测量，能够精确计算包括收益、社会成本和遗憾等广泛经济指标的长期时间平均值。我们的结果表明，这个简单的学习算法涵盖了一维动力系统中发现的全部行为谱，从唯一或多个绝对连续不变测度到复杂周期吸引子，以及共存的混沌和稳定（周期）行为。通过将博弈论与动力系统联系起来，我们表明即使在没有逐点收敛的情况下，统计可预测性也是可以实现的。
英文摘要
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.
