LLM providers learn to price cold-start services efficiently
Cold-Start Pricing of LLM Services under Two-Sided Learning
Computer Science and Game Theory
Summary
Pricing new AI language services is hard because providers lack data and users change how much they want to pay over time. The authors model this as a two-way learning process: users learn from using the service, while providers adjust prices based only on whether users buy or not. They find two scenarios depending on service costs; with low costs, pricing can improve at a certain rate, and with high costs, a different, slower rate applies. Their algorithms match these rates and outperform common approaches in experiments.
What this means in practice
- •For cloud service providers: Optimize pricing strategies for newly launched AI services without relying on user data upfront to maximize profits.$Commercial implications: Enables selling AI service subscriptions with adaptive pricing from launch, improving competitive advantage.
- •For e-commerce platform managers: Adjust prices dynamically when introducing new digital products that customers learn to value over time.
Authors
Zhicheng Du, Pinyan Lu, Keyu Wang, Zihe Wang, Xiang Yan, Chunxue Yang
Abstract
Online large language model (LLM) services face a pricing cold start: providers initially lack the historical data and user information needed for precise pricing, while users' willingness to pay evolves with repeated use. We formulate this challenge as a \emph{two-sided learning} problem between a provider and a user. The user learns from private usage experience, whereas the provider incurs a serving cost and dynamically adapts prices using only purchase decisions to maximize cumulative profit. The provider's central challenge is to learn to price against an evolving willingness to pay without observing the private experiences driving that evolution. We establish matching upper and lower bounds on the minimax regret over horizon $T$, identifying two cost-dependent regimes. When serving costs are sufficiently low, our Calibration-and-Tracking (CAT) algorithm achieves the optimal rate $Θ(T^{1/3})$. In the higher-cost regime, where the provider must learn whether continued service is profitable, Calibration-and-Tracking with Retirement (CAT-R) attains the optimal rate $Θ(\sqrt{T})$. This phase transition shows that serving costs affect not only profit margins but also the fundamental difficulty of learning to price LLM services. Simulations support the predicted regret scaling and show that our algorithms outperform standard online learning baselines in the evaluated settings.