Circumventing the Computational Hardness of Computational Arbitrage

Introduction

AI model marketplaces bring together providers offering inference services at varying costs and capabilities. This heterogeneity creates a natural opportunity for computational arbitrage: an intermediary purchases inference from a curated portfolio of models and resells the results to end‑users at a profit. While small‑scale experiments have demonstrated that such arbitrage can be profitable, scaling the approach to large, realistic marketplaces raises two key challenges: (1) the underlying allocation problem is computationally difficult, and (2) identifying profitable allocations requires costly performance estimation.

The authors formalize the arbitrage task as an allocation problem, prove its general NP‑hardness, and then identify conditions under which optimal allocations can be computed efficiently. They also propose a practical, sample‑efficient technique for estimating model performance, reducing the upfront data collection burden.

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1. Formalizing Computational Arbitrage

The arbitrage problem is cast as a resource‑allocation model. An intermediary has a budget \(B\) to spend on inference across a set of models \(\{Mi\}\). Each model \(Mi\) has a cost per inference \(ci\) and delivers a utility (e.g., accuracy, latency improvement) that depends on the amount of inference budget allocated to it, denoted \(ui(xi)\) where \(xi\) is the spending on \(Mi\). The goal is to choose \(\{xi\}\) that maximize total utility subject to \(\sumi xi \le B\).

Under this formulation, the authors prove that the decision version of the problem is NP‑complete by reduction from a classic knapsack variant. This result confirms that, without additional structure, finding the optimal arbitrage strategy is computationally intractable for large model pools.

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2. Tractability via Diminishing‑Returns

A central insight of the paper is that many real‑world model performance curves exhibit diminishing returns: the marginal utility per additional dollar spent decreases as more budget is allocated to the same model. Formally, the utility functions satisfy \(ui'(x) \ge ui'(y)\) for \(x < y\).

When all utility functions are concave and satisfy this diminishing‑returns property, the allocation problem becomes a convex optimization task. The authors show that the optimal solution can be found in polynomial time using a greedy algorithm that equalizes the marginal utility per dollar across all funded models. This condition aligns with empirical observations in AI inference markets, where scaling inference on a single model yields progressively smaller performance gains.

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3. Sample‑Efficient Performance Estimation

Even with tractable optimization, the intermediary needs reliable estimates of \(u_i(x)\). Directly measuring utility across the full budget range for every model would be prohibitively expensive. The paper introduces a sample‑efficient estimator that leverages a small set of pilot runs to fit a parametric diminishing‑returns curve for each model. By exploiting the concave structure, the estimator requires far fewer data points than a naïve grid search, dramatically lowering the upfront cost of strategy discovery.

The method proceeds in three steps:

1. Pilot Sampling – Run each model on a limited set of inference budgets.
2. Curve Fitting – Fit a concave function (e.g., a logarithmic or power‑law form) to the observed utility‑cost pairs.
3. Validation – Use cross‑validation on the pilot data to ensure the fitted curve generalizes.

Empirical evaluations on synthetic and real‑world marketplace data demonstrate that the estimator predicts utility within a small error margin while using an order of magnitude fewer samples than baseline approaches.

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4. Economic Insights from the Optimal Allocation

The analysis yields several intuitive economic conclusions:

• Equalized Marginal Utility: At the optimum, every funded model delivers the same marginal utility per dollar, mirroring the principle of price‑taking behavior in competitive markets.
• Revenue Reallocation: Models that provide low utility relative to their price receive little to no budget, effectively shifting revenue toward higher‑value providers.
• Complementary Specialization: The optimal portfolio often includes models that excel on different subsets of tasks, highlighting the value of diversity rather than simply selecting the single best performer.

These insights suggest that computational arbitrage can improve overall market efficiency by directing spending toward models that offer the greatest incremental benefit.

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Conclusion

The paper establishes a rigorous foundation for computational arbitrage in AI model marketplaces. By proving NP‑hardness in the general case and identifying a realistic diminishing‑returns condition that restores tractability, it bridges the gap between theoretical difficulty and practical feasibility. The proposed sample‑efficient performance estimator further lowers the barrier to deploying arbitrage strategies at scale. Together, these contributions provide a roadmap for intermediaries seeking to profit from, and simultaneously enhance, the efficiency of AI inference markets.