On the Incompatibility of Weighted PROPX and Pareto Optimality for Indivisible Chores

2026-08-17

AI Summary

This study addresses the compatibility between weighted PROPX and Pareto optimality (PO) in the allocation of indivisible chores. By constructing minimal counterexamples, we refute the conjecture that these properties are compatible under positive costs, demonstrating incompatibility with just two agents and four items and establishing a general counterexample for n agents and n+1 items. Conversely, we delineate the compatibility boundary by proving that weighted PROPX and PO remain compatible when the number of items does not exceed the number of agents. These findings reveal fundamental distinctions between weighted PROPX and EF1, precisely characterizing the theoretical limits between fairness and efficiency in the allocation of indivisible bads.

Abstract

Proportionality (PROP) is one of the simplest fairness criteria for allocating items among agents with additive preferences. With indivisible chores, however, PROP is not always satisfiable. We study proportionality up to any item (PROPX), which requires every agent to satisfy proportionality after any chore is removed from her bundle. Under strictly positive costs, we settle the weighted compatibility question negatively: weighted PROPX and Pareto optimality are incompatible already for two agents and four chores. Moreover, for every $n\geq3$, we give an $n$-agent, $(n+1)$-chore counterexample whose shares can be arbitrarily close to equal. These counterexamples are item-minimal: under strictly positive costs, weighted PROPX and Pareto optimality are always compatible when the number of chores is at most the number of agents, and they are compatible for two agents with at most three chores. Our impossibility result contrasts with the compatibility theorem of Mahara (2026) for weighted envy-freeness up to one item (EF1) and Pareto optimality .

Problem

• Weighted PROPX
• Pareto Optimality
• Indivisible Chores
• Incompatibility

Innovation

• Weighted PROPX
• Pareto Optimality
• Indivisible Chores
• Incompatibility
• Fair Division