# Keep, Customize, or Exit: Default Design and Token Pricing in LLM Reasoning Services

arXiv: https://arxiv.org/abs/2608.13315 · HTML: https://mlsanity.com/paper/2608.13315/

## 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.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 24 of 24)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Tree of Thoughts: Deliberate Problem Solving with Large Language Models](https://mlsanity.com/paper/2305.10601/) (2023)
- [GPQA: A Graduate-Level Google-Proof Q&A Benchmark](https://mlsanity.com/paper/2311.12022/) (2023)
- [s1: Simple test-time scaling](https://mlsanity.com/paper/2501.19393/) (2025)
- [FrugalGPT: How to Use Large Language Models While Reducing Cost and Improving Performance](https://mlsanity.com/paper/2305.05176/) (2023)
- [RouteLLM: Learning to Route LLMs with Preference Data](https://mlsanity.com/paper/2406.18665/) (2024)
- [Inference Scaling Laws: An Empirical Analysis of Compute-Optimal Inference for Problem-Solving with Language Models](https://mlsanity.com/paper/2408.00724/) (2024)
- [Token-Budget-Aware LLM Reasoning](https://mlsanity.com/paper/2412.18547/) (2024)
- [RouterBench: A Benchmark for Multi-LLM Routing System](https://mlsanity.com/paper/2403.12031/) (2024)
- [GraphRouter: A Graph-based Router for LLM Selections](https://mlsanity.com/paper/2410.03834/) (2024)
- [MixLLM: Dynamic Routing in Mixed Large Language Models](https://mlsanity.com/paper/2502.18482/) (2025)
- [Reasoning in Token Economies: Budget-Aware Evaluation of LLM Reasoning Strategies](https://mlsanity.com/paper/2406.06461/) (2024)
- [An Empirical Study of LLM Reasoning Ability Under Strict Output Length Constraint](https://mlsanity.com/paper/2504.14350/) (2025)
- [Don't Always Pick the Highest-Performing Model: An Information Theoretic View of LLM Ensemble Selection](https://mlsanity.com/paper/2602.08003/) (2026)
- [LMaaS: Exploring Pricing Strategy of Large Model as a Service for Communication](https://mlsanity.com/paper/2401.02675/) (2024)
- [Balancing Information Accuracy and Response Timeliness in Networked LLMs](https://mlsanity.com/paper/2508.02209/) (2025)
- [Pricing Online LLM Services with Data-Calibrated Stackelberg Routing Game](https://mlsanity.com/paper/2511.09062/) (2025)
- [Budget-Aware Anytime Reasoning with LLM-Synthesized Preference Data](https://mlsanity.com/paper/2601.11038/) (2026)
- [MORES: Mobile Reasoning-as-a-Service via Distributed LLM Inference-Time Scaling](https://mlsanity.com/paper/2607.08116/) (2026)

## 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.