Complex Dynamics in Autobidding Systems

TL;DR

This work develops a dynamical-systems framework for autobidding in auctions, modeling bidder multipliers with dm/dt = U(m) under a return-over-spend constraint and uniform bid-scaling. It reveals rich dynamical behavior, including convergence in two-bidder markets, oscillations in three-bidder setups, and motif-based phenomena such as repressilators leading to bistability, limit cycles, and quasi-periodicity. The authors prove that ROS dynamics can simulate arbitrary linear systems and discrete boolean circuits, linking autobidding to broader dynamical-systems and computational paradigms. These results highlight the potential for complex, non-equilibrium market behavior under autobidding and provide a bridge between economics, synthetic biology motifs, and computation. The findings have implications for understanding market stability, designing auction formats, and exploring computational capabilities embedded in automated bidding ecosystems.

Abstract

It has become the default in markets such as ad auctions for participants to bid in an auction through automated bidding agents (autobidders) which adjust bids over time to satisfy return-over-spend constraints. Despite the prominence of such systems for the internet economy, their resulting dynamical behavior is still not well understood. Although one might hope that such relatively simple systems would typically converge to the equilibria of their underlying auctions, we provide a plethora of results that show the emergence of complex behavior, such as bi-stability, periodic orbits and quasi periodicity. We empirically observe how the market structure (expressed as motifs) qualitatively affects the behavior of the dynamics. We complement it with theoretical results showing that autobidding systems can simulate both linear dynamical systems as well logical boolean gates.

Figures (20)

• Figure 1: The function $U_i$ is non-decreasing for $m_i \leq 1$ and non-increasing for $m_i \geq 1$.
• Figure 2: Vector field for two bidders [left]. The vector $(U_1(m), U_2(m))$ gives the direction of the dynamics at point $m$. Orbits of the dynamical system [right].
• Figure 3: The evolution of multipliers $m_1(t), m_2(t), m_3(t)$ over time.
• Figure 4: Left: The orbit of the multipliers projected on the first two multipliers $(m_1(t), m_2(t))$. Right: three orbits of the same dynamics with three different starting points. All the orbits are attracted by the same periodic cycle.
• Figure 5: Examples of bidding repressilators with two through five bidders.

Theorems & Definitions (31)

• Theorem 3.1: Poincare-Bendixson Refinement
• Lemma 4.1
• Lemma 4.2
• Theorem 4.3
• proof
• Lemma 4.4
• Theorem 5.1
• Lemma 5.2
• Lemma 5.3
• proof