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Chaos in Autobidding Auctions

Ioannis Anagnostides, Ian Gemp, Georgios Piliouras, Kelly Spendlove

TL;DR

The paper shows that autobidding dynamics under return-on-spend constraints can exhibit formal chaos in both continuous and discrete time, significantly strengthening prior results that only showed quasiperiodicity. It achieves this through a general simulation framework that can replicate a broad class of nonlinear systems, including Chua's circuit, via a sequence of autobidding gadgets (notably continuous negation and nonlinear-simulation constructs). In discrete time, simple mirror-descent updates already give rise to Li-Yorke chaos and connections to the Ricker and logistic maps, revealing how large learning rates induce complex dynamics even in modest market settings. The findings imply that long-horizon forecasting in autobidding markets may be inherently intractable and underscore the importance of analyzing statistical or ergodic properties rather than relying on convergent equilibria.

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.

Chaos in Autobidding Auctions

TL;DR

The paper shows that autobidding dynamics under return-on-spend constraints can exhibit formal chaos in both continuous and discrete time, significantly strengthening prior results that only showed quasiperiodicity. It achieves this through a general simulation framework that can replicate a broad class of nonlinear systems, including Chua's circuit, via a sequence of autobidding gadgets (notably continuous negation and nonlinear-simulation constructs). In discrete time, simple mirror-descent updates already give rise to Li-Yorke chaos and connections to the Ricker and logistic maps, revealing how large learning rates induce complex dynamics even in modest market settings. The findings imply that long-horizon forecasting in autobidding markets may be inherently intractable and underscore the importance of analyzing statistical or ergodic properties rather than relying on convergent equilibria.

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.
Paper Structure (51 sections, 22 theorems, 53 equations, 17 figures)

This paper contains 51 sections, 22 theorems, 53 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Our results: the onset of chaos
  3. Continuous-time dynamics
  4. Proof of our general simulation result
  5. Discrete-time dynamics
  6. Related work
  7. Autobidding systems
  8. Chua's circuit
  9. Chaos in multiagent settings
  10. Competitve systems
  11. Complexity and dynamical systems
  12. Preliminaries
  13. Value maximization with RoS constraints
  14. RoS objective
  15. Uniform bid scaling
  16. ...and 36 more sections

Key Result

Lemma 3.1

Let $h: [1.1, 1.9] \to \mathbb{R}$ be any continuous and differentiable almost everywhere function such that $h'(\cdot) < 0$. There is an instance in which the utility of the autobidder is equal to $h$, modulo an additive constant.

Figures (17)

  • Figure 1: Trajectories of autobidding dynamics.
  • Figure 2: The key steps in our main simulation result (\ref{['theorem:simulation-general']})
  • Figure 3: An illustrative utility plotted as a function of the multiplier. Under VCG, selecting a multiplier $m = 1$ can always guarantee the maximum utility, but that is typically not the optimal choice for value maximization under an RoS constraint. Increasing the multiplier beyond 1---which corresponds to overbidding---can decrease its quasi-linear utility since it can pay for an item more than its value for it. The goal is to select the largest possible multiplier that guarantees nonnegative utility; this corresponds to the red dot in the figure.
  • Figure 4: \ref{['lemma:continuousnegation']} in action concerning \ref{['eq:augmented-Chua']}. As $\lambda$ increases, $x(t) + \widebar{x}(t)$ approaches 3.
  • Figure 5: Verification that the approximate version of Chua's circuit given in \ref{['eq:augmented-Chua']} satisfies the precondition of \ref{['theorem:deformed']}. A detailed, self-contained overview of the significance of this geometric condition and its relation to the iconic (deformed) horseshoe map is given in \ref{['app:galias_framework']}.
  • ...and 12 more figures

Theorems & Definitions (40)

  • Remark 2.1: Continuum of items
  • Remark 2.2: Reserve prices
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.2
  • proof
  • Lemma 3.2
  • Definition 3.3: Approximate simulation
  • Theorem 3.4
  • Claim 4.1: Entropic mirror descent
  • ...and 30 more