Keep, Customize, or Exit: Default Design and Token Pricing in LLM Reasoning Services
On the map: LLM reasoning → Test-time compute and efficient thinking
Abstract (arXiv)
We study a large language model (LLM) service in which a provider chooses a per-token price and a default reasoning-token allocation, while a user may accept the default, customize the allocation, or exit. Larger allocations can improve accuracy but increase token cost and latency. We model this interaction as a Stackelberg game and derive the user's unique optimal customized allocation in closed form. For any price, the acceptable defaults form either an empty set or a compact interval. We characterize the provider's optimal default through a three-regime rule, reduce equilibrium computation to a one-dimensional price optimization, and prove the existence of the equilibrium. We further show that defaults affect the implemented reasoning allocation only when users value the convenience of avoiding customization; otherwise, every service-providing outcome implements the user's optimal customized allocation. Experiments with two compact open-weight reasoning models on five mathematics and science benchmarks support the accuracy-token model and show how model and task characteristics determine equilibrium prices, defaults, and reasoning allocations.
Key facts. No citing papers in the map yet (data through 14 Oct 2026). Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning. Builds on 24 of its 32 references that are in the map. First posted on arXiv 13 Aug 2026.
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24 of 32 references.
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- 2026Budget-Aware Anytime Reasoning with LLM-Synthesized Preference Data16 Jan
- 2025An Empirical Study of LLM Reasoning Ability Under Strict Output Length Constraint19 Apr
- 2025s1: Simple test-time scaling31 Jan
- 2024Token-Budget-Aware LLM Reasoning24 Dec
- 2024Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Models1 Aug
- 2024Reasoning in Token Economies: Budget-Aware Evaluation of LLM Reasoning Strategies10 Jun
- 2023Tree of Thoughts: Deliberate Problem Solving with Large Language Models17 May
Other references · 17
- 2022Chain-of-Thought Prompting Elicits Reasoning in Large Language Modelscited by 12,459
- 2021Training Verifiers to Solve Math Word Problemscited by 10,109
- 2025DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learningcited by 6,330
- 2023Efficient Memory Management for Large Language Model Serving with PagedAttentioncited by 5,944
- 2022Self-Consistency Improves Chain of Thought Reasoning in Language Modelscited by 5,708
- 2023Let's Verify Step by Stepcited by 3,789
- 2023GPQA: A Graduate-Level Google-Proof Q&A Benchmarkcited by 2,879
- 2023FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performancecited by 853
- 2024RouteLLM: Learning to Route LLMs with Preference Datacited by 682
- 2024RouterBench: A Benchmark for Multi-LLM Routing Systemcited by 238
- 2024GraphRouter: A Graph-based Router for LLM Selectionscited by 165
- 2025MixLLM: Dynamic Routing in Mixed Large Language Modelscited by 59
5 more
- 2026Don't Always Pick the Highest-Performing Model: An Information Theoretic View of LLM Ensemble Selectioncited by 11
- 2024LMaaS: Exploring Pricing Strategy of Large Model as a Service for Communicationcited by 2
- 2025Balancing Information Accuracy and Response Timeliness in Networked LLMscited by 2
- 2025Pricing Online LLM Services with Data-Calibrated Stackelberg Routing Gamecited by 2
- 2026MORES: Mobile Reasoning-as-a-Service via Distributed LLM Inference-Time Scalingcited by 1
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- 2026The Shadow Price of Reasoning: Economic Perspective on Optimal Budget Allocation for LLMs
- 2026Test-Time Compute Games
- 2025BAR Conjecture: the Feasibility of Inference Budget-Constrained LLM Services with Authenticity and Reasoning
- 2026The Reasoning Tax: Token Economics of LLM Reasoning Across Task Types and Deployment Contexts
- 2025Matching Markets Meet LLMs: Algorithmic Reasoning with Ranked Preferences
- 2025SelfBudgeter: Adaptive Token Allocation for Efficient LLM Reasoning
- 2026Queueing-Aware Optimization of Reasoning Tokens for Accuracy-Latency Trade-offs in LLM Servers
- 2025In-Token Rationality Optimization: Towards Accurate and Concise LLM Reasoning via Self-Feedback