Almost (Weighted) Proportional Allocations for Indivisible Chores

TL;DR

This paper introduces a new fair allocation concept called PROPX for indivisible chores, proves its existence and efficiency, and explores its implications for fairness and social welfare in various settings.

Contribution

It establishes the existence and efficient computation of PROPX allocations, introduces APS fairness for chores, and analyzes the price of fairness under this framework.

Findings

PROPX allocations always exist and can be computed efficiently.

PROPX guarantees a 2-approximation of MMS and APS fairness.

The price of fairness is Theta(n) for symmetric agents and unbounded for asymmetric agents.

Abstract

In this paper, we study how to fairly allocate m indivisible chores to n (asymmetric) agents. We consider (weighted) proportionality up to any item (PROPX) and show that a (weighted) PROPX allocation always exists and can be computed efficiently. For chores, we argue that PROPX might be a more reliable relaxation for proportionality by the facts that any PROPX allocation ensures 2-approximation of maximin share (MMS) fairness [Budish, 2011] for symmetric agents and of anyprice share (APS) fairness [Babaioff et al, 2021] for asymmetric agents. APS allocations for chores have not been studied before the current work, and our result implies a 2-approximation algorithm. Another by-product result is that an EFX and a weighted EF1 allocation for indivisible chores exist if all agents have the same ordinal preference, which might be of independent interest. We then consider the partial information setting and design algorithms that only use agents' ordinal preferences to compute approximately PROPX allocations. Our algorithm achieves a 2-approximation for both symmetric and asymmetric agents, and the approximation ratio is optimal. Finally, we study the price of fairness (PoF), i.e., the loss in social welfare by enforcing allocations to be (weighted) PROPX. We prove that the tight ratio for PoF is Theta(n) for symmetric agents and unbounded for asymmetric agents.