Guaranteeing MMS for All but One Agent When Allocating Indivisible Chores

TL;DR

This paper introduces a new fairness guarantee called $ ext{all-but-one MMS}$ for allocating indivisible chores among agents, ensuring most agents meet their MMS and providing bounds on the approximation for the remaining agent.

Contribution

The paper proposes the $ ext{all-but-one MMS}$ fairness notion and establishes existence results with specific approximation ratios for different numbers of agents.

Findings

Existence of $ ext{all-but-one MMS}$ allocations for 3, 4, and n agents.

Approximation ratios: 9/8 for 3 agents, 4/3 for 4 agents, and $(n+1)^2/4n$ for n ≥ 5.

The new fairness concept strengthens previous MMS guarantees for indivisible chores.

Abstract

We study the problem of allocating $m$ indivisible chores to $n$ agents with additive cost functions under the fairness notion of maximin share (MMS). In this work, we propose a notion called $\alpha$-approximate all-but-one maximin share ($\alpha$-AMMS) which is a stronger version of $\alpha$-approximate MMS. An allocation is called $\alpha$-AMMS if $n-1$ agents are guaranteed their MMS values and the remaining agent is guaranteed $\alpha$-approximation of her MMS value. We show that there exist $\alpha$-AMMS allocations, with $\alpha = 9/8$ for three agents; $\alpha = 4/3$ for four agents; and $\alpha = (n+1)^2/4n$ for $n\geq 5$ agents.