Test-Time Compute Games

TL;DR

This paper examines the inefficiency in test-time compute pricing for LLMs and proposes a reverse second-price auction mechanism to align provider incentives with social welfare.

Contribution

It introduces a novel auction-based mechanism to reduce social inefficiency in test-time compute pricing for large language models.

Findings

Market of LLM-as-a-service is socially inefficient due to provider incentives.

Proposed auction mechanism aligns provider incentives with social welfare.

Experimental results with Llama, Qwen, and DeepSeek-R1 models demonstrate effectiveness.

Abstract

Test-time compute has emerged as a promising strategy to enhance the reasoning abilities of large language models (LLMs). However, this strategy has in turn increased how much users pay cloud-based providers offering LLM-as-a-service, since providers charge users for the amount of test-time compute they use to generate an output. In our work, we show that the market of LLM-as-a-service is socially inefficient: providers have a financial incentive to increase the amount of test-time compute, even if this increase contributes little to the quality of the outputs. To address this inefficiency, we introduce a reverse second-price auction mechanism where providers bid their offered price and (expected) quality for the opportunity to serve a user, and users pay proportionally to the marginal value generated by the winning provider relative to the second-highest bidder. To illustrate and complement our theoretical results, we conduct experiments with multiple instruct models from the $\texttt{Llama}$ and $\texttt{Qwen}$ families, as well as reasoning models distilled from $\texttt{DeepSeek-R1}$, on math and science benchmark datasets.