Test-Time Compute Games

Abstract: Test-time compute has emerged as a promising strategy to enhance the reasoning abilities of 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.

• The paper introduces a game-theoretic model of LLM compute allocation, revealing that providers escalate compute to maximize profits at the cost of social welfare.
• It demonstrates that such strategic behavior can lead to up to 19% inefficiency as measured by the price-of-anarchy in real-world LLM markets.
• A reverse second-price auction mechanism is proposed, effectively aligning provider incentives and eliminating welfare loss under ideal conditions.

Test-Time Compute Games: Strategic Provider Behavior and Market Inefficiency in LLM-as-a-Service

Introduction and Market Motivation

LLM inference increasingly relies on cloud providers offering LLM-as-a-service. A central factor in production cost and model efficacy is test-time compute (TTC): the computational effort spent during inference, involving techniques such as chain-of-thought (CoT), best-of-n sampling, or tree-of-thought methods. Economically, providers directly tie user payments to the volume of compute expended, with billing models sensitive to the number of generated tokens—an established practice in leading commercial APIs.

The paper "Test-Time Compute Games" (2601.21839) initiates an economic and game-theoretic analysis of strategic TTC deployment, demonstrating how provider incentives in market equilibrium diverge from socially optimal compute allocation. When providers compete for users, they are incentivized to inflate compute (and thus costs) regardless of whether increased TTC yields substantial improvements in output quality or user utility. Crucially, this setting exposes a price-of-anarchy (PoA): the degree to which market competition, absent regulatory or incentive-alignment interventions, leads to outcomes that are suboptimal for aggregate social welfare.

Game-Theoretic Modeling of Provider Competition

The model frames provider competition for user tasks as an N-player normal-form game, with each provider choosing level $\theta \in \Theta$ of TTC for each task/query $\mathcal{T}$. Output quality functions $q_i(\theta)$ typically increase with $\theta$, but so do generation costs $c_i(\theta)$ and consequently the user-facing prices $p_i(\theta)$.

User value for provider $i$ at compute level $\theta$ is $V_i(\theta) = q_i(\theta) - p_i(\theta)$. Users allocate demand probabilistically (softmax, bounded rationality) or deterministically to the provider with the highest value (Bertrand competition). Provider utility is proportional to market share and profit: $$U_i(\theta_i; \boldsymbol{\theta}_{-i}) = s_i(V_i(\theta_i); \mathbf{V}_{-i}(\boldsymbol{\theta}_{-i})) \cdot (p_i(\theta_i) - c_i(\theta_i))$$.

The global welfare function aggregates provider profit and user value, excluding monetary transfers which cancel:

$$\mathrm{SW}(\boldsymbol{\theta}) = \sum_{i=1}^N s_i(V_i(\theta_i); \mathbf{V}_{-i}(\boldsymbol{\theta}_{-i})) \cdot (q_i(\theta_i) - c_i(\theta_i))$$

Equilibrium Analysis and Social Inefficiency

Providers act strategically to maximize individual utility (profit), resulting in a Nash equilibrium for TTC selection. Theoretical analysis establishes:

• Existence and structure: The game is a generalized ordinal potential game, guaranteeing pure Nash equilibria and convergence under better-response dynamics.
• Suboptimality of equilibrium: Providers select compute to maximize profit, not welfare; this systematically diverges from the compute allocation that would maximize $\mathrm{SW}$.

Quantitatively, the price-of-anarchy (PoA) lower bounds the ratio of optimal social welfare to equilibrium welfare. Even with high rationality parameter $\beta$, the market can exhibit nontrivial inefficiency.

Figure 1: Dynamics of a test-time compute game using majority-voting illustrate strategic escalation of compute and divergence from optimal configuration.

Reverse Second-Price Auction: Welfare Alignment

To mitigate inefficiency, the paper introduces a reverse second-price auction mechanism:

• Providers submit bids specifying TTC, expected output quality, and price.
• The user pays a price linked to the marginal value generated by the winning provider over the second-best bid.
• Winning providers serve queries at the compute maximizing their individual welfare contribution.

Formally, for provider $i$, dominant strategy is to bid price equal to cost and choose the welfare-maximizing compute level. This mechanism implements the social optimum, eliminating PoA (i.e., PoA = 1):

$$\pi_1 = \arg\max_{i} \max_{\theta} (q_i(\theta) - c_i(\theta))$$

Market dynamics under auction guarantee non-negative provider profit, truth-telling, and sharp welfare alignment.

Empirical Evaluation

Experiments validate theory using Llama and Qwen families (instruct and distilled reasoning models) on STEM benchmarks (GSM8K, GPQA, AIME). For each model and TTC setting, accuracy and inference costs are measured, enabling calculation of equilibrium market shares, user value, provider utility, global welfare, and PoA.

Figure 2: Accuracy of Llama and Qwen models on GSM8K under majority voting and best-of-n.

Figure 3: Accuracy of reasoning models distilled from DeepSeek-R1 on GSM8K using chain-of-thought, showing solution quality as TTC increases.

Results:

• Equilibrium in pay-for-compute markets induces up to 19% inefficiency as measured by PoA.
• Welfare loss manifests due to strategic escalation of TTC by dominant providers, evident in empirical better-response dynamics.
• Auction mechanism achieves higher user value (up to 29% increase), provider profit, and total welfare, fully eliminating inefficiency under ideal assumptions.

Figure 4: Better-response dynamics of compute allocation by providers illustrate market convergence and inefficiency at equilibrium.

Figure 5: Potential function evolution under best-of-n sampling; providers jointly optimize profit and second-highest user value among competitors.

Practical and Theoretical Implications

The analysis rigorously demonstrates that current LLM pricing models inherently incentivize providers to over-allocate TTC, generating significant deadweight welfare loss. As LLMs scale and TTC becomes a major cost driver (including externalities such as energy consumption [Fernandez et al., (Fernandez et al., 24 Apr 2025)]), aligning provider incentives with social welfare is both an economic and societal imperative.

Mechanism design via auctions (multi-attribute, reverse second-price) provides a robust provable framework for efficiency, but introduces vulnerabilities to collusion and manipulation [Mailath & Zemsky 1991; Chen & Tauman 2006]. Practical deployment requires careful scrutiny of auction implementation, quality verification, and applicability to open-ended or non-verifiable tasks.

Future directions involve auction mechanism robustness, integration with carbon- or energy-aware prompting [Adamska et al., (Adamska et al., 9 Mar 2025)], and empirical analysis with heterogeneous user preferences and provider service attributes.

Conclusion

"Test-Time Compute Games" establishes a rigorous model for the strategic interaction of LLM providers in TTC markets. The work introduces fundamental inefficiency results for real-world LLM-as-a-service ecosystems and provides an actionable, theoretically grounded auction mechanism for welfare maximization. As model inference costs escalate and AI markets mature, insights from this paper will be instrumental for regulatory design, platform engineering, and sustainable AI deployment.