How to Sell a Service with Uncertain Outcomes

TL;DR

This paper introduces a contract design framework for selling services with uncertain outcomes, proposing a two-stage payment scheme and analyzing its computational complexity and optimality.

Contribution

It develops a two-stage payment scheme for uncertain outcome services, proves its necessity for profit maximization, and analyzes the computational complexity of optimal menu design.

Findings

Two-stage payment scheme is necessary for profit maximization.

Computing the exact optimal menu is NP-hard with multiple buyer types.

An FPTAS exists for a constant number of buyer types.

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.