EFx Budget-Feasible Allocations with High Nash Welfare

TL;DR

This paper investigates fair and efficient allocations of indivisible items under budget constraints, focusing on EFx fairness and Nash welfare approximation, with new procedures for two and three agents.

Contribution

It introduces procedures that ensure EFx fairness while approximating Nash social welfare in budget-constrained allocations, addressing cases with two and three agents.

Findings

Two-agent EFx allocations may not be Pareto efficient.

A procedure achieves optimal Nash welfare approximation for two agents.

A constant-factor approximation is possible for three or more agents.

Abstract

We study the problem of allocating indivisible items to budget-constrained agents, aiming to provide fairness and efficiency guarantees. Specifically, our goal is to ensure that the resulting allocation is envy-free up to any item (EFx) while minimizing the amount of inefficiency that this needs to introduce. We first show that there exist two-agent problem instances for which no EFx allocation is Pareto efficient. We, therefore, turn to approximation and use the Nash social welfare maximizing allocation as a benchmark. For two-agent instances, we provide a procedure that always returns an EFx allocation while achieving the best possible approximation of the optimal Nash social welfare that EFx allocations can achieve. For the more complicated case of three-agent instances, we provide a procedure that guarantees EFx, while achieving a constant approximation of the optimal Nash social welfare for any number of items.