The Unit-Demand Envy-Free Pricing Problem

TL;DR

This paper investigates the computational complexity of the unit-demand envy-free pricing problem, introduces four new mixed-integer programming formulations, and explores approximation bounds for a geometric series pricing variant.

Contribution

It presents four novel MIP formulations for the problem, compares them experimentally, and analyzes a geometric series pricing variant with provable approximation bounds.

Findings

Four new MIP formulations outperform previous models in experiments

A geometric series pricing variant has a provable approximation ratio

The problem remains NP-hard and unlikely to be in APX

Abstract

We consider the unit-demand envy-free pricing problem, which is a unit-demand auction where each bidder receives an item that maximizes his utility, and the goal is to maximize the auctioneer's profit. This problem is NP-hard and unlikely to be in APX. We present four new MIP formulations for it and experimentally compare them to a previous one due to Shioda, Tun\c{c}el, and Myklebust. We describe three models to generate different random instances for general unit-demand auctions, that we designed for the computational experiments. Each model has a nice economic interpretation. Aiming approximation results, we consider the variant of the problem where the item prices are restricted to be chosen from a geometric series, and prove that an optimal solution for this variant has value that is a fraction (depending on the series used) of the optimal value of the original problem. So this variant is also unlikely to be in APX.