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 $O(1/\sqrt{b})$.

翻译：我们研究当双寡头卖家均依赖同一预训练模型时，将定价权委托给大型语言模型如何促进合谋。大型语言模型具有以下两个特征：（i）倾向参数，反映其内在偏向高价推荐的倾向；（ii）输出保真度参数，衡量输出追踪该倾向的精确程度；该倾向通过再训练演化。研究表明，为追求鲁棒性和可复现性而配置大型语言模型会通过相变诱发合谋：存在一个决定长期行为的临界输出保真度阈值。低于该阈值时，竞争性定价是唯一的长期结果。高于该阈值时，系统呈现双稳态，竞争性定价与合谋定价均为局部稳定状态，实际结果由模型初始偏好决定。合谋机制类似于默契合谋：价格整体抬高，但偶发的低价推荐提供了可推诿性。在完全保真度下，从任意内部初始条件出发均会形成完全合谋。对于大小为$b$的有限训练批次，由计算成本驱动的低频再训练将进一步放大合谋效应：若初始状态位于合谋吸引域内，随着$b$增大，合谋概率趋近于1，因为更大批次抑制了可能使系统滑向竞争状态的随机波动。不确定区域以$O(1/\sqrt{b})$的速率收缩。