Scalable Fair Division for 'At Most One' Preferences

TL;DR

This paper develops a scalable algorithm for fair division under 'at most one' preferences, leveraging convex programming and market equilibrium concepts, with practical validation on real datasets.

Contribution

It introduces a polynomial-time method for approximate CEEI solutions in AMO preferences, extending fair division theory to more realistic preference models.

Findings

MNW solution remains convex and is a CEEI in large, low-rank instances

Algorithm scales to large problems with practical efficiency

Empirical analysis shows approximate properties hold in real datasets

Abstract

Allocating multiple scarce items across a set of individuals is an important practical problem. In the case of divisible goods and additive preferences a convex program can be used to find the solution that maximizes Nash welfare (MNW). The MNW solution is equivalent to finding the equilibrium of a market economy (aka. the competitive equilibrium from equal incomes, CEEI) and thus has good properties such as Pareto optimality, envy-freeness, and incentive compatibility in the large. Unfortunately, this equivalence (and nice properties) breaks down for general preference classes. Motivated by real world problems such as course allocation and recommender systems we study the case of additive `at most one' (AMO) preferences - individuals want at most 1 of each item and lotteries are allowed. We show that in this case the MNW solution is still a convex program and importantly is a CEEI solution when the instance gets large but has a `low rank' structure. Thus a polynomial time algorithm can be used to scale CEEI (which is in general PPAD-hard) for AMO preferences. We examine whether the properties guaranteed in the limit hold approximately in finite samples using several real datasets.