Fairness under Equal-Sized Bundles: Impossibility Results and Approximation Guarantees

TL;DR

This paper investigates fair allocation of indivisible goods with fixed bundle sizes, revealing fundamental limitations and proposing approximation algorithms under certain valuation assumptions.

Contribution

It provides both impossibility results and approximation algorithms for flip-based envy-freeness notions, contrasting with classic EFX concepts.

Findings

Standard techniques fail to guarantee EFFX approximations.

A constant factor approximation is achievable with shared item rankings.

Nash welfare maximization guarantees a 1/2-EFF1 allocation, which is tight.

Abstract

We study the fair allocation of indivisible goods under cardinality constraints, where each agent must receive a bundle of fixed size. This models practical scenarios, such as assigning shifts or forming equally sized teams. Recently, variants of envy-freeness up to one/any item (EF1, EFX) were introduced for this setting, based on flips or exchanges of items. Namely, one can define envy-freeness up to one/any flip (EFF1, EFFX), meaning that an agent $i$ does not envy another agent $j$ after performing one or any one-item flip between their bundles that improves the value of $i$. We explore algorithmic aspects of this notion, and our contribution is twofold: we present both algorithmic and impossibility results, highlighting a stark contrast between the classic EFX concept and its flip-based analogue. First, we explore standard techniques used in the literature and show that they fail to guarantee EFFX approximations. On the positive side, we show that we can achieve a constant factor approximation guarantee when agents share a common ranking over item values, based on the well-known envy cycle elimination technique. This idea also leads to a generalized algorithm with approximation guarantees when agents agree on the top $n$ items and their valuation functions are bounded. Finally, we show that an algorithm that maximizes the Nash welfare guarantees a 1/2-EFF1 allocation, and that this bound is tight.