Selling Multiple Items to a Unit-Demand Buyer via Automated Mechanism Design

TL;DR

This paper introduces a new automated approach using deep learning and duality verification to determine optimal selling mechanisms for multiple items to a single buyer, revealing that uniform pricing is often optimal.

Contribution

It develops a general methodology for automated mechanism design and proves the optimality of uniform pricing under broad conditions, including for multiple items and various value distributions.

Findings

Uniform pricing is optimal for any number of items with certain value distributions.

Necessary and sufficient conditions for uniform pricing optimality are established for uniform distributions.

The approach simplifies verifying optimality compared to traditional analytical methods.

Abstract

Finding the optimal (revenue-maximizing) mechanism to sell multiple items has been a prominent and notoriously difficult open problem. Existing work has mainly focused on deriving analytical results tailored to a particular class of problems (for example, Giannakopoulos, 2015; Yang, 2025). The present paper explores the possibility of a generally applicable methodology of the Automated Mechanism Design (AMD). We first employ the deep learning algorithm developed by D\"utting et al. (2023) to numerically solve small-sized problems, and the results are then generalized by educated guesswork and finally rigorously verified through duality. By focusing on a single buyer who can consume one item, our approach leads to two key contributions: establishing a much simpler way to verify the optimality of a wide range of problems and discovering a completely new result about the optimality of uniform pricing. First, we show that selling each item at an identical price (or equivalently, selling the grand bundle of all items) is optimal for any number of items when the value distributions belong to a class that includes the uniform distribution as a special case. Different items are allowed to have different distributions. Second, for each number of items, we established necessary and sufficient conditions that $c$ must satisfy for uniform pricing to be optimal when the value distribution is uniform over an interval $[c, c+1]$. This latter model does not satisfy the previously known sufficient conditions for the optimality of grand bundling (Haghpanah & Hartline, 2021). Our results are in contrast to the only known results for $n$ items (for any $n$), Giannakopoulos (2015) and Daskalakis et al. (2017), which consider a single buyer with additive preferences, where the values of items are narrowly restricted to i.i.d. according to a uniform or exponential distribution.