LLM Collusion

TL;DR

The paper develops a theoretical framework to study LLM-driven pricing in a duopoly, revealing that shared latent preferences and data-driven retraining can produce collusion via a phase transition controlled by the output fidelity parameter $\rho$. Using a stochastic approximation with IPW-based estimation, it shows a deterministic mean-field limit in the large-batch regime and a noise-driven, path-dependent regime for finite batches; a critical fidelity threshold $\rho_c(r)$ delineates regimes of unique competition from bistable collusion, with perfect fidelity yielding guaranteed collusion. Finite batches introduce random trajectory selection between equilibria, with larger batch sizes increasing collusion probability when initial propensity is favorable, and the indeterminacy region shrinking as $O(1/\sqrt{b})$. The results carry regulatory implications: practitioners should be mindful that robustness-focused configuration and infrequent retraining can unintentionally foster tacit collusion through correlated outputs and data-driven feedback loops. The framework advances understanding of monoculture-enabled coordination and provides quantitative benchmarks for mitigating LLM-induced price coordination.

Abstract

We study how delegating pricing to large language models (LLMs) can facilitate collusion in a duopoly when both sellers rely on the same pre-trained model. The LLM is characterized by (i) a propensity parameter capturing its internal bias toward high-price recommendations and (ii) an output-fidelity parameter measuring how tightly outputs track that bias; the propensity evolves through retraining. We show that configuring LLMs for robustness and reproducibility can induce collusion via a phase transition: there exists a critical output-fidelity threshold that pins down long-run behavior. Below it, competitive pricing is the unique long-run outcome. Above it, the system is bistable, with competitive and collusive pricing both locally stable and the realized outcome determined by the model's initial preference. The collusive regime resembles tacit collusion: prices are elevated on average, yet occasional low-price recommendations provide plausible deniability. With perfect fidelity, full collusion emerges from any interior initial condition. For finite training batches of size $b$, infrequent retraining (driven by computational costs) further amplifies collusion: conditional on starting in the collusive basin, the probability of collusion approaches one as $b$ grows, since larger batches dampen stochastic fluctuations that might otherwise tip the system toward competition. The indeterminacy region shrinks at rate $O(1/\sqrt{b})$.

Figures (3)

• Figure 1: Comparison of pricing mechanisms. (a) Under reinforcement learning, each seller deploys an independent algorithm that learns pricing strategies from scratch through repeated market interactions. Collusion emerges slowly over millions of iterations. (b) Under LLM-based pricing, both sellers query a shared pretrained model with a latent high-price preference $\theta$. The shared knowledge infrastructure creates correlated recommendations, while data sharing during retraining aggregates seller feedback, forming a self-reinforcing loop that drives rapid convergence to collusive outcomes.
• Figure 2: Phase diagram of LLM pricing dynamics in the large batch limit ($b\to\infty$) for $r=1.5$. The shaded regions represent basins of attraction: the blue region is the competitive basin where the propensity converges to $\theta=0$, while the pink region is the collusive basin where the propensity converges to the stable equilibrium $\theta_+(\rho,r)$. The solid red curve shows the convergence frontier $\theta_+(\rho,r)$, the long-run collusive equilibrium for each $\rho\ge\rho_c(r)$. The dashed gray curves mark the boundaries $\theta_\pm(\rho,r)$: the lower boundary $\theta_-$ separates the two basins, while the region between $\theta_-$ and $\theta_+$ is where $\Delta(\theta)>0$ (high-price strategy $H$ outperforms $L$). The vertical line indicates the critical threshold $\rho_c(1.5)\approx 0.789$, separating the competitive regime (left) from the bistable regime (right). The horizontal dashed line shows the initial condition $\theta_0=1/2$. Arrows indicate the direction of propensity evolution: red upward arrows where $\Delta(\theta)>0$, purple downward arrows where $\Delta(\theta)<0$.
• Figure 3: Convergence trajectories of propensity $\theta_n$ for different batch sizes. Each panel displays 20 independent replications (faint lines) and their sample mean (bold line) for the batch size $b\in\{1,4,16,64\}$. The horizontal dashed line indicates the theoretical collusive equilibrium $\theta_+(\rho,r)\approx 0.846$. Parameters: $\rho=0.85$, $r=1.5$, $\theta_0=1/2$. Larger batch sizes reduce trajectory variance and increase the probability of converging to the collusive equilibrium.

Theorems & Definitions (9)

• Lemma 1: Unbiased Estimation
• Lemma 2: ODE Tracking
• Proposition 1: Conditions for Price Propensity Drift
• Proposition 2: Collusive Behavior in the Large-Batch Limit
• Example 1: Illustrative Visualization of Collusion Dynamics
• Proposition 3: Convergence for Finite Batch Size
• Remark 1
• Proposition 4: Equilibrium Selection and the Effect of Batch Size
• Example 2: Batch Size and Collusion Probability