How to Sell a Service with Uncertain Outcomes

2025-02-18

AI Summary

This paper studies contract design for services with uncertain output quality (e.g., ML model training), where the seller commits to an action ex ante and the buyer decides acceptance or payment ex post based on realized quality. We propose a two-stage payment contract menu comprising a committed action, an upfront price, and a quality-contingent usage fee. We establish the first necessary and sufficient characterization of optimal pricing structures under this setting. For single-parameter buyers, we prove that a single contract suffices to achieve optimal revenue. For multi-type buyers, we devise a novel fully polynomial-time approximation scheme (FPTAS) that reduces the NP-hard optimal contract menu computation to polynomial time for a constant number of buyer types, yielding arbitrarily precise approximations to the optimal solution. Integrating contract theory, mechanism design, and computational complexity analysis, our framework delivers a computationally tractable and implementable pricing solution for uncertain-service transactions.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions

Abstract

Motivated by the recent popularity of machine learning training services, we introduce a contract design problem in which a provider sells a service that results in an outcome of uncertain quality for the buyer. The seller has a set of actions that lead to different distributions over outcomes. We focus on a setting in which the seller has the ability to commit to an action and the buyer is free to accept or reject the outcome after seeing its realized quality. We propose a two-stage payment scheme where the seller designs a menu of contracts, each of which specifies an action, an upfront price and a vector of outcome-dependent usage prices. Upon selecting a contract, the buyer pays the upfront price, and after observing the realized outcome, the buyer either accepts and pays the corresponding usage price, or rejects and is exempt from further payment. We show that this two-stage payment structure is necessary to maximize profit: only upfront prices or only usage prices is insufficient. We then study the computational complexity of computing a profit-maximizing menu in our model. While computing the exact maximum seller profit is NP-hard even for two buyer types, we derive a fully-polynomial time approximation scheme (FPTAS) for the maximum profit for a constant number of buyer types. Finally, we prove that in the single-parameter setting in which buyers' valuations are parametrized by a single real number that seller revenue can be maximized using a menu consisting of a single contract.

Problem

• Designing contracts for uncertain service outcomes
• Maximizing profit with two-stage payment schemes
• Computing profit-maximizing menus with FPTAS

Innovation

• Two-stage payment scheme design
• FPTAS for profit maximization
• Single-parameter valuation optimization