Fairness Criteria for Allocating Indivisible Chores: Connections and Efficiencies

TL;DR

This paper explores fairness criteria for allocating indivisible chores with additive and submodular costs, analyzing their relationships, guarantees, and efficiency trade-offs in different settings.

Contribution

It establishes connections and differences among fairness notions for chores, especially highlighting the limitations in the submodular case and analyzing efficiency costs.

Findings

Strong links between fairness criteria in additive chores allocation.

Intrinsic differences between goods and chores allocations.

Limited guarantees for fairness in submodular chores allocation.

Abstract

We study several fairness notions in allocating indivisible chores (i.e., items with non-positive values) to agents who have additive and submodular cost functions. The fairness criteria we are concern with are envy-free up to any item (EFX), envy-free up to one item (EF1), maximin share (MMS), and pairwise maximin share (PMMS), which are proposed as relaxations of envy-freeness in the setting of additive cost functions. For allocations under each fairness criterion, we establish their approximation guarantee for other fairness criteria. Under the additive setting, our results show strong connections between these fairness criteria and, at the same time, reveal intrinsic differences between goods allocation and chores allocation. However, such strong relationships cannot be inherited by the submodular setting, under which PMMS and MMS are no longer relaxations of envy-freeness and, even worse, few non-trivial guarantees exist. We also investigate efficiency loss under these fairness constraints and establish their prices of fairness.