Fair and Truthful Allocations Under Leveled Valuations

TL;DR

This paper investigates fair division of indivisible goods among agents with leveled valuation functions, providing new approximation guarantees for MMS and EFX fairness notions, and designing truthful mechanisms.

Contribution

It introduces new approximation bounds for MMS under leveled valuations, proves existence of exact EFX allocations, and develops truthful protocols for fair division.

Findings

Existence of 2/3-MMS allocations for leveled, submodular valuations.

Existence of 2/3-MMS allocations for two submodular agents.

Existence of exact EFX allocations under general leveled valuations.

Abstract

We study the problem of fairly allocating indivisible goods among agents which are equipped with {\em leveled} valuation functions. Such preferences, that have been studied before in economics and fair division literature, capture a simple and intuitive economic behavior; larger bundles are always preferred to smaller ones. We provide a fine-grained analysis for various subclasses of leveled valuations focusing on two extensively studied notions of fairness, (approximate) MMS and EFX. In particular, we present a general positive result, showing the existence of $2/3$-MMS allocations under valuations that are both leveled and submodular. We also show how some of our ideas can be used beyond the class of leveled valuations; for the case of two submodular (not necessarily leveled) agents we show that there always exists a $2/3$-MMS allocation, complementing a recent impossibility result. Then, we switch to the case of subadditive and fractionally subadditive leveled agents, where we are able to show tight (lower and upper) bounds of $1/2$ on the approximation factor of MMS. Moreover, we show the existence of exact EFX allocations under general leveled valuations via a simple protocol that in addition satisfies several natural economic properties. Finally, we take a mechanism design approach and we propose protocols that are both truthful and approximately fair under leveled valuations.