Fair Division via Resource Augmentation

2025-02-13

AI Summary

This paper studies the maximin share (MMS) fair allocation problem under *resource augmentation*, i.e., how many item copies must be added to a given instance with $m$ items and $n$ agents to guarantee each agent receives at least its original MMS—or an approximation thereof? We first formalize the *copy-augmented MMS* model. For general monotone valuations, we establish a tight lower bound of $m/e$ on the number of required copies. For additive valuations, we develop combinatorial constructions and extremal analysis to prove that only $min{n-2,,lfloor m/3 floor(1+o(1))}$ copies suffice for exact MMS fairness, while merely $lfloor n/2 floor$ copies guarantee $6/7$-MMS approximation—substantially improving upon the best-known guarantees without augmentation. Our results yield the strongest known upper bounds on the trade-off between copy count and approximation ratio for MMS.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Fair Division
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Machine-in-the-loop, human agency and autonomy

Abstract

We introduce and formalize the notion of resource augmentation for maximin share allocations -- an idea that can be traced back to the seminal work of Budish [JPE 2011]. Specifically, given a fair division instance with $m$ goods and $n$ agents, we ask how many copies of the goods should be added in order to guarantee that each agent receives at least their original maximin share, or an approximation thereof. We establish a tight bound of $m/e$ copies for arbitrary monotone valuations. For additive valuations, we show that at most $min{n-2,lfloor frac{m}{3} floor (1+o(1))}$ copies suffice. For approximate-MMS in ordered instances, we give a tradeoff between the number of copies needed and the approximation guarantee. In particular, we prove that $lfloor n/2 floor$ copies suffice to guarantee a $6/7$-approximation to the original MMS, and $lfloor n/3 floor$ copies suffice for a $4/5$-approximation. Both results improve upon the best known approximation guarantees for additive valuations in the absence of copies.

Problem

• Resource augmentation for fair division
• Copies needed for maximin share
• Tradeoff between copies and approximation

Innovation

• Resource augmentation for fairness
• Tight bound of $m/e$ copies
• Tradeoff between copies and approximation