Chaos in Autobidding Auctions

TL;DR

This paper demonstrates that autobidding auction dynamics can exhibit chaotic behavior, revealing inherent unpredictability in seemingly simple online advertising systems through mathematical modeling and simulation.

Contribution

It shows that autobidding systems can simulate complex nonlinear dynamical systems, including chaos, advancing understanding of their long-term unpredictable behavior.

Findings

Autobidding dynamics can exhibit chaos similar to Chua's circuit.

Mirror descent in autobidding can show Li-Yorke chaos and sensitivity.

Autobidding systems can simulate complex nonlinear systems with small errors.

Abstract

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.