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Pay for The Second-Best Service: A Game-Theoretic Approach Against Dishonest LLM Providers

Yuhan Cao, Yu Wang, Sitong Liu, Miao Li, Yixin Tao, Tianxing He

TL;DR

We address the vulnerability of dishonest LLM API providers by casting user-provider interactions as a sequential delegation game and designing a mechanism that mitigates deceit through incentive alignment. The authors prove an impossibility result for achieving first-best user utility and provide an $O(T^{1-\epsilon}\log T)$-approximate incentive-compatible mechanism that delivers near second-best user utility, implemented via four phases: exploration, exploitation, and two blind-trust stages. The theoretical results are complemented by simulations using real-world API settings, demonstrating robustness and near-optimal user utilities across scenarios. This work introduces a principled economic framework for securing LLM service markets and suggests practical directions for future research, including multi-user extensions and budget constraints.

Abstract

The widespread adoption of Large Language Models (LLMs) through Application Programming Interfaces (APIs) induces a critical vulnerability: the potential for dishonest manipulation by service providers. This manipulation can manifest in various forms, such as secretly substituting a proclaimed high-performance model with a low-cost alternative, or inflating responses with meaningless tokens to increase billing. This work tackles the issue through the lens of algorithmic game theory and mechanism design. We are the first to propose a formal economic model for a realistic user-provider ecosystem, where a user can iteratively delegate $T$ queries to multiple model providers, and providers can engage in a range of strategic behaviors. As our central contribution, we prove that for a continuous strategy space and any $ε\in(0,\frac12)$, there exists an approximate incentive-compatible mechanism with an additive approximation ratio of $O(T^{1-ε}\log T)$, and a guaranteed quasi-linear second-best user utility. We also prove an impossibility result, stating that no mechanism can guarantee an expected user utility that is asymptotically better than our mechanism. Furthermore, we demonstrate the effectiveness of our mechanism in simulation experiments with real-world API settings.

Pay for The Second-Best Service: A Game-Theoretic Approach Against Dishonest LLM Providers

TL;DR

We address the vulnerability of dishonest LLM API providers by casting user-provider interactions as a sequential delegation game and designing a mechanism that mitigates deceit through incentive alignment. The authors prove an impossibility result for achieving first-best user utility and provide an -approximate incentive-compatible mechanism that delivers near second-best user utility, implemented via four phases: exploration, exploitation, and two blind-trust stages. The theoretical results are complemented by simulations using real-world API settings, demonstrating robustness and near-optimal user utilities across scenarios. This work introduces a principled economic framework for securing LLM service markets and suggests practical directions for future research, including multi-user extensions and budget constraints.

Abstract

The widespread adoption of Large Language Models (LLMs) through Application Programming Interfaces (APIs) induces a critical vulnerability: the potential for dishonest manipulation by service providers. This manipulation can manifest in various forms, such as secretly substituting a proclaimed high-performance model with a low-cost alternative, or inflating responses with meaningless tokens to increase billing. This work tackles the issue through the lens of algorithmic game theory and mechanism design. We are the first to propose a formal economic model for a realistic user-provider ecosystem, where a user can iteratively delegate queries to multiple model providers, and providers can engage in a range of strategic behaviors. As our central contribution, we prove that for a continuous strategy space and any , there exists an approximate incentive-compatible mechanism with an additive approximation ratio of , and a guaranteed quasi-linear second-best user utility. We also prove an impossibility result, stating that no mechanism can guarantee an expected user utility that is asymptotically better than our mechanism. Furthermore, we demonstrate the effectiveness of our mechanism in simulation experiments with real-world API settings.

Paper Structure

This paper contains 39 sections, 11 theorems, 10 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. The User-Provider Delegation Game Model.
  4. Impossibility Result and Utility Guarantee Mechanism.
  5. Simulation experiments with real-world API settings.
  6. Related Work
  7. Auditing Large Language Models.
  8. Game Theory Meets Large Language Models.
  9. Online Learning in Large Language Model Scenario.
  10. The Model
  11. The User-Provider Delegation Game
  12. Model Components and Interaction Protocol.
  13. Provider's Strategy Space and Truthful Behavior.
  14. Objectives and Utilities.
  15. Goals of Mechanism Design
  16. ...and 24 more sections

Key Result

Theorem 5.1

For any $\xi\in(0,1)$, there is no $o(T)$-approximate incentive compatible mechanism can guarantee an expected user utility of $\xi u_{FB}+(1-\xi)u_{SB}$, where $u_{FB}$ and $u_{SB}$ are the first-best and the second-best user utility (defined in Section subsec:goals).

Figures (4)

  • Figure 1: An illustration figure of our proposed mechanism, in which the provider's optimal strategy would guarantee the user a second-best overall utility. The mechanism splits the user's queries among four sequential phases. In the first phase (exploration), a batch of sample outputs from each service provider is collected. They need to compete to get selected in the second phase (exploitation), where the best-performing provider is asked to deliver the second-best user utility for the majority of queries. This is followed by the final two blind trust phases (detailed in Section \ref{['sec:mech']}), which serve as incentives for providers to behave well in the first two phases.
  • Figure 2: Simulation experiment where each provider enumerates strategies listed in Table \ref{['tab:phase_strategy_mapping']}. provider 1's expected provider utility, user utility, and number of delegations, averaged under permutations. Our proposed strategy gets the highest utility. The average delegations of ours, honest, ours-honest-token are the same.
  • Figure 3: Average provider utility, user utility (from each provider), and number of delegations for provider 2 and provider 3 adopting different strategies when provider 1 adopts our proposed strategy. Since provider 1 has the best-performing LLM, they can not achieve a decent utility.
  • Figure 4: User utility of provider 1 as $T$ increases from 1 million to 2 million. We also plot a theoretical reference, $u_{SB}$ (defined in Section \ref{['sec:model']}).

Theorems & Definitions (15)

  • Remark 1
  • Definition 1: Dominant Strategy
  • Definition 2: $o(T)$-Dominant Strategy
  • Theorem 5.1: The impossibility of the first-best user utility
  • Theorem 5.2: $O(T^{1-\epsilon}\log T)$-dominant strategy of all model providers
  • Corollary 1
  • Theorem 5.3: Second-best user utility guarantee
  • Definition 3: Type and Type Set
  • Theorem A.1: The impossibility of the first-best user utility
  • lemma 1: Hoeffding's inequality
  • ...and 5 more