The Fairness of Maximum Nash Social Welfare Under Matroid Constraints and Beyond

TL;DR

This paper investigates the fairness of maximum Nash social welfare allocations under matroid and related constraints, establishing tight bounds for envy-freeness up to one item and Pareto optimality, and extending results to special valuation types.

Contribution

It proves that Max-NSW allocations under matroid constraints are 1/2-EF1 and PO, resolving an open question, and extends these results to p-extendible systems and specific valuation models.

Findings

Max-NSW achieves 1/2-EF1 under matroid constraints.

Max-NSW guarantees max{1/a^2, 1/2}-EF1 for 2-valued valuations.

Under p-extendible systems, Max-NSW guarantees max{1/p, 1/4}-EF1 for binary valuations.

Abstract

We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under matroid constraints and two generalizations: $p$-extendible system and independence system constraints. The objective is to find fair and efficient allocations in which the subset of items assigned to every agent satisfies the given constraint. We focus on a common fairness notion of envy-freeness up to one item (EF1) and a well-known efficient (and fair) notion of the maximum Nash social welfare (Max-NSW). By using properties of matroids, we demonstrate that the Max-NSW allocation, implying Pareto optimality (PO), achieves a tight $1/2$-EF1 under matroid constraints. This result resolves an open question proposed in prior literature [26]. In particular, if agents have 2-valued ($\{1, a\}$) valuations, we prove that the Max-NSW allocation admits $\max\{1/a^2, 1/2\}$-EF1 and PO. Under strongly $p$-extendible system constraints, we show that the Max-NSW allocation guarantees $\max\{1/p, 1/4\}$-EF1 and PO for identical binary valuations. Indeed, the approximation of $1/4$ is the ratio for independence system constraints and additive valuations. Additionally, for lexicographic preferences, we study possibly feasible allocations other than Max-NSW admitting exactly EF1 and PO under the above constraints.