We study the emergence of chaotic behavior of Follow-the-Regularized Leader (FoReL) dynamics in games. We focus on the effects of increasing the population size or the scale of costs in congestion games, and generalize recent results on unstable, chaotic behaviors in the Multiplicative Weights Update dynamics to a much larger class of FoReL dynamics. We establish that, even in simple linear non-atomic congestion games with two parallel links and any fixed learning rate, unless the game is fully symmetric, increasing the population size or the scale of costs causes learning dynamics to become unstable and eventually chaotic, in the sense of Li-Yorke and positive topological entropy. Furthermore, we show the existence of novel non-standard phenomena such as the coexistence of stable Nash equilibria and chaos in the same game. We also observe the simultaneous creation of a chaotic attractor as another chaotic attractor gets destroyed. Lastly, although FoReL dynamics can be strange and non-equilibrating, we prove that the time average still converges to an exact equilibrium for any choice of learning rate and any scale of costs.

中文翻译：

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在路由游戏中遵循正则表达式的领导者路线走向混乱

我们研究了在游戏中遵循常规化领导者（FoReL）动力学的混沌行为的出现。我们关注于拥挤游戏中增加人口规模或成本规模的影响，并在乘性权重更新动态中将关于不稳定，混乱行为的最新结果推广到更大的FoReL动态中。我们确定，即使在具有两个平行链接且具有固定学习率的简单线性非原子拥塞游戏中，除非该游戏是完全对称的，否则增加人口规模或成本规模也会导致学习动力变得不稳定，最终变得混乱。 Li-Yorke的感觉和正拓扑熵。此外，我们证明了在同一游戏中存在新颖的非标准现象，例如稳定的纳什均衡和混沌并存。我们还观察到同时产生了一个混沌吸引子，因为另一个混沌吸引子被破坏了。最后，尽管FoReL动力学可能很奇怪且不平衡，但我们证明，对于任何选择的学习率和成本规模，时间平均仍会收敛到精确的平衡。