LLM Collusion

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

Best Arm Identification in Batched Multi-armed Bandit Problems

Recently multi-armed bandit problem arises in many real-life scenarios where arms must be sampled in batches, due to limited time the agent can wait for the feedback. Such applications include biological experimentation and online marketing. The problem is further complicated when the number of arms is large and the number of batches is small. We consider pure exploration in a batched multi-armed bandit problem. We introduce a general linear programming framework that can incorporate objectives of different theoretical settings in best arm identification. The linear program leads to a two-stage algorithm that can achieve good theoretical properties. We demonstrate by numerical studies that the algorithm also has good performance compared to certain UCB-type or Thompson sampling methods.