Existence of Fair and Efficient Allocation of Indivisible Chores

TL;DR

This paper proves the existence of allocations of indivisible chores that are both envy-free up to one chore and Pareto optimal, using novel methods, and extends results to weighted fairness scenarios.

Contribution

It establishes the existence of EF1 and PO allocations for indivisible chores, introduces a new proof technique, and extends results to weighted fairness.

Findings

Existence of EF1 and PO allocations for indivisible chores.

Polynomial-time computation for a fixed number of agents.

Extension of results to weighted EF1 scenarios.

Abstract

We study the problem of allocating indivisible chores among agents with additive cost functions in a fair and efficient manner. A major open question in this area is whether there always exists an allocation that is envy-free up to one chore (EF1) and Pareto optimal (PO). Our main contribution is to provide a positive answer to this question by proving the existence of such an allocation for indivisible chores under additive cost functions. This is achieved by a novel combination of a fixed point argument and a discrete algorithm, providing a significant methodological advance in this area. Our additional key contributions are as follows. We show that there always exists an allocation that is EF1 and fractional Pareto optimal (fPO), where fPO is a stronger efficiency concept than PO. We also show that an EF1 and PO allocation can be computed in polynomial time when the number of agents is constant. Finally, we extend all of these results to the more general setting of weighted EF1 (wEF1), which accounts for the entitlements of agents.