Fairly Allocating Many Goods with Few Queries

TL;DR

This paper explores the query complexity of fairly allocating indivisible goods among agents, showing efficient algorithms for certain cases and proving high complexity for others, advancing practical fair division methods.

Contribution

It introduces algorithms achieving logarithmic query complexity for EF1 fairness in specific cases and proves linear complexity for EFX fairness, highlighting the limits of efficient fair allocation.

Findings

Logarithmic query complexity for EF1 with two agents.

Logarithmic query complexity for three agents with additive utilities.

Linear query complexity for EFX even with two agents with identical utilities.

Abstract

We investigate the query complexity of the fair allocation of indivisible goods. For two agents with arbitrary monotonic utilities, we design an algorithm that computes an allocation satisfying envy-freeness up to one good (EF1), a relaxation of envy-freeness, using a logarithmic number of queries. We show that the logarithmic query complexity bound also holds for three agents with additive utilities, and that a polylogarithmic bound holds for three agents with monotonic utilities. These results suggest that it is possible to fairly allocate goods in practice even when the number of goods is extremely large. By contrast, we prove that computing an allocation satisfying envy-freeness and another of its relaxations, envy-freeness up to any good (EFX), requires a linear number of queries even when there are only two agents with identical additive utilities.