Position Fair Mechanisms Allocating Indivisible Goods

TL;DR

This paper introduces a new fairness criterion called position envy-freeness up to one good (PEF1) for mechanisms allocating indivisible goods, and proposes polynomial-time mechanisms that satisfy PEF1 and produce fair, Pareto optimal allocations.

Contribution

It defines PEF1 as a novel fairness criterion, and develops polynomial-time mechanisms that satisfy PEF1 and achieve envy-freeness and Pareto optimality for indivisible goods.

Findings

Proposed a scale-invariant, polynomial-time mechanism satisfying PEF1.

Established that maximum Nash welfare allocations eliminate envy based on positions for two agents.

Presented a polynomial-time mechanism based on the adjusted winner procedure satisfying PEF1, EF1, and Pareto optimality.

Abstract

Fair division mechanisms for indivisible goods require agent orderings to deterministically select one allocation when running the algorithm in practice. We introduce position envy-freeness up to one good (PEF1) as a fairness criterion for mechanisms: a mechanism is said to satisfy PEF1 if for any pair of agent orderings, no agent prefers their bundle determined under one ordering to that under another ordering by more than the utility of a single good. First, we propose a scale-invariant, polynomial-time mechanism that satisfies PEF1 and yields an envy-freeness up to one good (EF1) allocation. For the case of two agents, we establish that any mechanism producing a maximum Nash welfare allocation eliminates envy based on positions by removing one good, provided that utilities are positive. Additionally, we present a polynomial-time mechanism based on the adjusted winner procedure, which satisfies PEF1 and produces an EF1 and Pareto optimal allocation for two agents. In contrast, we demonstrate that well-known mechanisms such as round-robin and envy-cycle elimination do not generally satisfy PEF1.