Online MMS Allocation for Chores

2025-07-18

AI Summary

This paper studies the online fair allocation of indivisible chores: items arrive one by one and must be irrevocably assigned upon arrival, aiming to achieve α-MMS fairness. Prior work achieved nontrivial guarantees only under strong assumptions—e.g., two agents with bivalued valuations or known total disutility—while the general case admitted only the trivial upper bound of n-MMS and a lower bound of 2. We first establish a tight impossibility result: for any fixed number of agents n and ε > 0, no online algorithm can guarantee (n − ε)-MMS fairness. Second, we propose the first prior-free general online algorithm, achieving min{n, O(k), O(log D)}-MMS fairness, where k is the number of distinct utility values per agent and D is the maximum ratio between absolute utilities. In the personalized bivalued setting, our algorithm attains approximately 3.7-MMS fairness—the first nontrivial constant-factor approximation for general online chore allocation.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• 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
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We study the problem of fair division of indivisible chores among $n$ agents in an online setting, where items arrive sequentially and must be allocated irrevocably upon arrival. The goal is to produce an $α$-MMS allocation at the end. Several recent works have investigated this model, but have only succeeded in obtaining non-trivial algorithms under restrictive assumptions, such as the two-agent bi-valued special case (Wang and Wei, 2025), or by assuming knowledge of the total disutility of each agent (Zhou, Bai, and Wu, 2023). For the general case, the trivial $n$-MMS guarantee remains the best known, while the strongest lower bound is still only $2$. We close this gap on the negative side by proving that for any fixed $n$ and $varepsilon$, no algorithm can guarantee an $(n - varepsilon)$-MMS allocation. Notably, this lower bound holds precisely for every $n$, without hiding constants in big-$O$ notation, thereby exactly matching the trivial upper bound. Despite this strong impossibility result, we also present positive results. We provide an online algorithm that applies in the general case, guaranteeing a $min{n, O(k), O(log D)}$-MMS allocation, where $k$ is the maximum number of distinct disutilities across all agents and $D$ is the maximum ratio between the largest and smallest disutilities for any agent. This bound is reasonable across a broad range of scenarios and, for example, implies that we can achieve an $O(1)$-MMS allocation whenever $k$ is constant. Moreover, to optimize the constant in the important personalized bi-valued case, we show that if each agent has at most two distinct disutilities, our algorithm guarantees a $(2 + sqrt{3}) approx 3.7$-MMS allocation.

Problem

• Online fair division of indivisible chores among agents
• Achieving α-MMS allocation under sequential item arrival
• Overcoming restrictive assumptions in prior algorithms

Innovation

• Online algorithm for general case MMS allocation
• Bounds based on distinct disutilities and ratios
• Personalized bi-valued case optimization