Weighted EF1 and PO Allocations with Few Types of Agents or Chores
Jugal Garg, Aniket Murhekar, John Qin
TL;DR
This paper addresses fair and efficient allocation of indivisible chores among weighted, asymmetric agents by proving the existence and computability of allocations that are both weighted envy-free up to one chore ($wEF1$) and fractionally Pareto-optimal ($fPO$). It presents polynomial-time algorithms for two structured instance classes: three-agent-type and two-chore-type, by integrating a weighted picking sequence (WPS) with competitive equilibrium (CE) techniques and carefully orchestrated chore transfers and payment adjustments. The core results extend known EF1/PO results to weighted settings, generalize several prior unweighted and special-case findings, and provide a potentially broadly applicable algorithmic toolkit for fair division with asymmetries. The techniques emphasize maintaining MPB within a CE while resolving cross-group envy through targeted transfers, offering practical relevance for workloads and liabilities with heterogeneous entitlements. The work opens avenues for applying the CE+WPS approach to broader structured instances and possibly to goods with weights as well.
Abstract
We investigate the existence of fair and efficient allocations of indivisible chores to asymmetric agents who have unequal entitlements or weights. We consider the fairness notion of weighted envy-freeness up to one chore (wEF1) and the efficiency notion of Pareto-optimality (PO). The existence of EF1 and PO allocations of chores to symmetric agents is a major open problem in discrete fair division, and positive results are known only for certain structured instances. In this paper, we study this problem for a more general setting of asymmetric agents and show that an allocation that is wEF1 and PO exists and can be computed in polynomial time for instances with: - Three types of agents, where agents with the same type have identical preferences but can have different weights. - Two types of chores, where the chores can be partitioned into two sets, each containing copies of the same chore. For symmetric agents, our results establish that EF1 and PO allocations exist for three types of agents and also generalize known results for three agents, two types of agents, and two types of chores. Our algorithms use a weighted picking sequence algorithm as a subroutine; we expect this idea and our analysis to be of independent interest.