MMS Approximations Under Additive Leveled Valuations

TL;DR

This paper investigates fair division of indivisible goods with additive leveled valuations, correcting a previous proof and establishing a stronger MMS approximation for EFX allocations.

Contribution

It corrects a flawed proof regarding MMS approximation and provides a new, stronger MMS bound for EFX allocations under additive leveled valuations.

Findings

Counter-example to previous MMS bound proof

Stronger MMS approximation established

Algorithm achieves EFX and improved MMS bounds

Abstract

We study the problem of fairly allocating indivisible goods to a set of agents with additive leveled valuations. A valuation function is called leveled if and only if bundles of larger size have larger value than bundles of smaller size. The economics literature has well studied such valuations. We use the maximin-share (MMS) and EFX as standard notions of fairness. We show that an algorithm introduced by Christodoulou et al. ([11]) constructs an allocation that is EFX and $\frac{\lfloor \frac{m}{n} \rfloor}{\lfloor \frac{m}{n} \rfloor + 1}\text{-MMS}$. In the paper, it was claimed that the allocation is EFX and $\frac{2}{3}\text{-MMS}$. However, the proof of the MMS-bound is incorrect. We give a counter-example to their proof and then prove a stronger approximation of MMS.