As autobidding systems increasingly dominate online advertising auctions, characterizing their long-term dynamical behavior is brought to the fore. In this paper, we examine the dynamics of autobidders who optimize value subject to a return-on-spend (RoS) constraint under uniform bid scaling. Our main set of results show that simple autobidding dynamics can exhibit formally chaotic behavior. This significantly strengthens the recent results of Leme, Piliouras, Schneider, Spendlove, and Zuo (EC '24) that went as far as quasiperiodicity. Our proof proceeds by establishing that autobidding dynamics can simulate -- up to an arbitrarily small error -- a broad class of continuous-time nonlinear dynamical systems. This class contains as a special case Chua's circuit, a classic chaotic system renowned for its iconic double scroll attractor. Our reduction develops several modular gadgets, which we anticipate will find other applications going forward. Moreover, in discrete time, we show that different incarnations of mirror descent can exhibit Li-Yorke chaos, topological transitivity, and sensitivity to initial conditions, connecting along the way those dynamics to classic dynamical systems such as the logistic map and the Ricker population model. Taken together, our results reveal that the long-term behavior of ostensibly simple second-price autobidding auctions can be inherently unpredictable and complex.

翻译：随着自动竞价系统日益主导在线广告拍卖，刻画其长期动态行为变得至关重要。本文研究了在统一出价缩放机制下，受投资回报率约束的自动竞价系统优化价值时的动态特性。我们的主要结果表明，简单的自动竞价动态可能展现出形式化的混沌行为。这显著强化了Leme、Piliouras、Schneider、Spendlove和Zuo（EC '24）近期仅证明准周期性的研究成果。我们的证明通过建立自动竞价动态能够以任意小误差模拟一大类连续时间非线性动力系统来实现。该类系统特例包含蔡氏电路——一个以标志性双涡卷吸引子闻名的经典混沌系统。我们的归约方法开发了多个模块化工具组件，预计这些组件将在未来研究中获得其他应用。此外，在离散时间框架下，我们证明了镜像下降算法的不同变体可能呈现李-约克混沌、拓扑传递性及初始条件敏感性，并将这些动态与逻辑斯蒂映射、里克种群模型等经典动力系统建立联系。综合而言，我们的研究揭示了表面简单的次高价自动竞价拍卖的长期行为可能具有内在的不可预测性与复杂性。