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Pricing Online LLM Services with Data-Calibrated Stackelberg Routing Game

Zhendong Guo, Wenchao Bai, Jiahui Jin

TL;DR

PriLLM tackles real-time, data-driven pricing for LLM routing by modeling the market as a Stackelberg routing game with providers as leaders and users as followers, incorporating both objective QoS metrics and subjective user preferences. It overcomes computational intractability through a differentiable, data-calibrated game abstraction that preserves the user NE, and a deep aggregation network that reduces rival complexity while maintaining equilibrium behavior. The approach includes a principled initialization of subjective biases, learnable parameterization of user preferences, and a loss that aligns profits between the original and abstracted games, enabling end-to-end training. Empirical results on OpenRouter data show PriLLM achieves near-optimal profits (over 95%) with substantial speedups (under 5% of brute-force solver time) and strong predictive fit (e.g., $R^2=0.8982$), indicating strong practical potential for scalable, interpretable dynamic pricing in LLM service markets.

Abstract

The proliferation of Large Language Models (LLMs) has established LLM routing as a standard service delivery mechanism, where users select models based on cost, Quality of Service (QoS), among other things. However, optimal pricing in LLM routing platforms requires precise modeling for dynamic service markets, and solving this problem in real time at scale is computationally intractable. In this paper, we propose \PriLLM, a novel practical and scalable solution for real-time dynamic pricing in competitive LLM routing. \PriLLM models the service market as a Stackelberg game, where providers set prices and users select services based on multiple criteria. To capture real-world market dynamics, we incorporate both objective factors (\eg~cost, QoS) and subjective user preferences into the model. For scalability, we employ a deep aggregation network to learn provider abstraction that preserve user-side equilibrium behavior across pricing strategies. Moreover, \PriLLM offers interpretability by explaining its pricing decisions. Empirical evaluation on real-world data shows that \PriLLM achieves over 95\% of the optimal profit while only requiring less than 5\% of the optimal solution's computation time.

Pricing Online LLM Services with Data-Calibrated Stackelberg Routing Game

TL;DR

PriLLM tackles real-time, data-driven pricing for LLM routing by modeling the market as a Stackelberg routing game with providers as leaders and users as followers, incorporating both objective QoS metrics and subjective user preferences. It overcomes computational intractability through a differentiable, data-calibrated game abstraction that preserves the user NE, and a deep aggregation network that reduces rival complexity while maintaining equilibrium behavior. The approach includes a principled initialization of subjective biases, learnable parameterization of user preferences, and a loss that aligns profits between the original and abstracted games, enabling end-to-end training. Empirical results on OpenRouter data show PriLLM achieves near-optimal profits (over 95%) with substantial speedups (under 5% of brute-force solver time) and strong predictive fit (e.g., ), indicating strong practical potential for scalable, interpretable dynamic pricing in LLM service markets.

Abstract

The proliferation of Large Language Models (LLMs) has established LLM routing as a standard service delivery mechanism, where users select models based on cost, Quality of Service (QoS), among other things. However, optimal pricing in LLM routing platforms requires precise modeling for dynamic service markets, and solving this problem in real time at scale is computationally intractable. In this paper, we propose \PriLLM, a novel practical and scalable solution for real-time dynamic pricing in competitive LLM routing. \PriLLM models the service market as a Stackelberg game, where providers set prices and users select services based on multiple criteria. To capture real-world market dynamics, we incorporate both objective factors (\eg~cost, QoS) and subjective user preferences into the model. For scalability, we employ a deep aggregation network to learn provider abstraction that preserve user-side equilibrium behavior across pricing strategies. Moreover, \PriLLM offers interpretability by explaining its pricing decisions. Empirical evaluation on real-world data shows that \PriLLM achieves over 95\% of the optimal profit while only requiring less than 5\% of the optimal solution's computation time.

Paper Structure

This paper contains 30 sections, 2 theorems, 22 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Work
  3. Stackelberg Game in Service Pricing
  4. LLM Service Evaluation
  5. Model Relaxation in MPEC
  6. Stackelberg Routing Game
  7. Preliminaries
  8. Game Formulation
  9. Data-Driven Game Calibration
  10. Learnability of Game Parameters
  11. Initialize and Learn Game Parameters
  12. Dynamic Pricing with Game Abstraction
  13. Rival Ranking and Abstraction
  14. Loss Function and Model Solving
  15. Evaluation
  16. ...and 15 more sections

Key Result

Theorem 1

A unique NE exists in the user-side game.

Figures (3)

  • Figure 1: An example of an LLM routing system that dynamically matches user requests to the most suitable providers based on individual preferences. In this example, users receive updates on key metrics of service providers and then adjust their allocation strategies accordingly (Steps 1--4). Their subjective preferences will be updated after providers accept and fulfill the requests (Steps 5--6).
  • Figure 2: Overview of $$PriLLM.
  • Figure 3: Experiment of $$PriLLM's Game Calibration.

Theorems & Definitions (7)

  • Definition 1: Nash Equilibrium of the user-side game
  • Theorem 1
  • proof : Proof Sketch
  • Theorem 2
  • proof : Proof Sketch
  • proof
  • proof