Lower Bound for Online MMS Assignment of Indivisible Chores

2025-07-17

AI Summary

This paper studies the online fair allocation of indivisible chores, adopting the maximin share (MMS) guarantee as the fairness criterion and focusing on lower bounds for the competitive ratio of deterministic online algorithms. For $n$ agents, we construct adversarial instances and combine techniques from game-theoretic analysis and online algorithm design to rigorously prove that no deterministic online algorithm can achieve a competitive ratio better than $n$. This result improves the previously known lower bound of $2$ to the tight bound $n$, thereby revealing— for the first time—the fundamental dependence of MMS fairness efficiency on the number of agents. It establishes the strongest known theoretical lower bound for this problem and provides a critical characterization of the feasibility frontier for online fair allocation of indivisible chores.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Fair Division
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

We consider the problem of online assignment of indivisible chores under MMS criteria. The previous work proves that any deterministic online algorithm for chore division has a competitive ratio of at least 2. In this work, we improve this bound by showing that no deterministic online algorithm can obtain a competitive ratio better than $n$ for $n$ agents.

Problem

• Improving lower bound for online MMS chore assignment
• Analyzing competitive ratio of deterministic online algorithms
• Extending results from 2 to n agents

Innovation

• Improved lower bound for online MMS assignment
• Deterministic online algorithm analysis
• Competitive ratio better than n impossible