Fair and Truthful Allocations Under Leveled Valuations

2024-07-08

arXiv.org

AI Summary

This paper studies fair allocation of indivisible goods among agents with hierarchical valuation functions, focusing on two fairness criteria: approximate Maximin Share (MMS) and Envy-Free up to any Good (EFX). Using techniques from fair division theory, submodular/subadditive function analysis, and constructive protocol design, we establish the following contributions: (i) For hierarchical + submodular valuations, a 2/3-MMS allocation always exists—and this bound is tight; (ii) We devise the first exact EFX construction protocol for arbitrary hierarchical valuations, proving EFX is always attainable; (iii) We extend MMS approximation guarantees to the non-hierarchical two-agent submodular setting; (iv) We design a polynomial-time mechanism that is both truthful and approximately fair, achieving tight MMS approximations—2/3 for hierarchical submodular valuations and 1/2 for general subadditive valuations.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Fair allocation of indivisible goods among agents.
• Analysis of leveled and submodular valuation functions.
• Existence of truthful and approximately fair protocols.

Innovation

• Leveled valuation functions analysis
• 2/3-MMS allocations existence
• Truthful fair protocols design