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The Invisible Handshake: Tacit Collusion between Adaptive Market Agents

Luigi Foscari, Emanuele Guidotti, Nicolò Cesa-Bianchi, Tatjana Chavdarova, Alfio Ferrara

TL;DR

The paper investigates how simple learning dynamics among adaptive market participants—specifically a market maker and a market taker—can drive tacit collusion in a stochastic market with endogenous price formation. It models the interaction as a two-player Markov game, derives a parameterized family of strategy profiles, and shows that gradient-ascent–like learning drives agents toward collusive equilibria where price impact accumulates and prices drift supra-competitively. Key contributions include a price-positivity and feasibility framework, a collusion criterion based on the positive drift parameter $\mu_\eta$, and finite-time convergence results with explicit time bounds, complemented by simulations illustrating collusive vs anti-collusive regimes. The findings imply that even simple, wealth-maximizing learning rules can generate tacit coordination among AI-driven traders, highlighting regulatory and design concerns for automated markets and motivating future work on long-horizon and risk-aware strategies.

Abstract

We study the emergence of tacit collusion between adaptive trading agents in a stochastic market with endogenous price formation. Using a two-player repeated game between a market maker and a market taker, we characterize feasible and collusive strategy profiles that raise prices beyond competitive levels. We show that, when agents follow simple learning algorithms (e.g., gradient ascent) to maximize their own wealth, the resulting dynamics converge to collusive strategy profiles, even in highly liquid markets with small trade sizes. By highlighting how simple learning strategies naturally lead to tacit collusion, our results offer new insights into the dynamics of AI-driven markets.

The Invisible Handshake: Tacit Collusion between Adaptive Market Agents

TL;DR

The paper investigates how simple learning dynamics among adaptive market participants—specifically a market maker and a market taker—can drive tacit collusion in a stochastic market with endogenous price formation. It models the interaction as a two-player Markov game, derives a parameterized family of strategy profiles, and shows that gradient-ascent–like learning drives agents toward collusive equilibria where price impact accumulates and prices drift supra-competitively. Key contributions include a price-positivity and feasibility framework, a collusion criterion based on the positive drift parameter , and finite-time convergence results with explicit time bounds, complemented by simulations illustrating collusive vs anti-collusive regimes. The findings imply that even simple, wealth-maximizing learning rules can generate tacit coordination among AI-driven traders, highlighting regulatory and design concerns for automated markets and motivating future work on long-horizon and risk-aware strategies.

Abstract

We study the emergence of tacit collusion between adaptive trading agents in a stochastic market with endogenous price formation. Using a two-player repeated game between a market maker and a market taker, we characterize feasible and collusive strategy profiles that raise prices beyond competitive levels. We show that, when agents follow simple learning algorithms (e.g., gradient ascent) to maximize their own wealth, the resulting dynamics converge to collusive strategy profiles, even in highly liquid markets with small trade sizes. By highlighting how simple learning strategies naturally lead to tacit collusion, our results offer new insights into the dynamics of AI-driven markets.
Paper Structure (18 sections, 12 theorems, 44 equations, 3 figures, 1 algorithm)

This paper contains 18 sections, 12 theorems, 44 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related works
  3. Online learning for market making.
  4. Online learning for market taking.
  5. Multi-Agent Learning Dynamics and algorithmic collusion.
  6. Model
  7. Two-Player game
  8. Strategy profiles
  9. Parameterization
  10. Learning dynamics
  11. Learnability of collusive strategy profiles
  12. Simulation
  13. Future works and conclusion
  14. Technical Appendix
  15. Proof of Theorem \ref{['th:collusive']}
  16. ...and 3 more sections

Key Result

Lemma 1

[Price positivity characterization] A strategy profile $\pi$ is price-positive if and only if for all $t \ge 1$ such that $Q_t < 0$ it holds

Figures (3)

  • Figure 1: Ask and bid parameters of the two learning agents for non-collusive strategy profiles (dotted line) and collusive strategy profiles (solid line). The experiment used $\varphi = 1/2$ and projected gradient ascent. The shaded region denoted the feasibility region as defined in \ref{['th:feasible']}.
  • Figure 2: Market impact of a fixed collusive strategy profile with $\varphi = 0.7$ and $k_\alpha = k_\beta = v_\alpha = v_\beta = 1/2$.
  • Figure 3: Market impact of a fixed non-collusive strategy profile with $\varphi = 0.3$ and $k_\alpha = k_\beta = v_\alpha = v_\beta = 1/2$.

Theorems & Definitions (15)

  • Definition 1: Price positivity
  • Lemma 1
  • Definition 2: Feasible strategy profile
  • Lemma 2
  • Definition 3: Collusion
  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • ...and 5 more