Position Fair Mechanisms Allocating Indivisible Goods

2024-09-10

arXiv.org

AI Summary

This paper addresses fairness violations in indivisible item allocation arising from dependence on agent reporting order. We introduce *position-envy-freeness up to k items* (PEFk), a novel fairness criterion requiring that the utility difference for any agent pair across different input orders be bounded by the value of at most *k* items—extending the cake-cutting notion of meta-envy-freeness to discrete settings. We formally define PEFk and its relaxations. Our contributions include: (i) the first deterministic mechanism satisfying both PEF1 and EF1; (ii) a proof that maximum Nash welfare (NSW) allocations are inherently PEF1 for two agents; (iii) an enhanced Adjusted Winner mechanism achieving PEF1; and (iv) a quantitative analysis quantifying the PEF violation of classical mechanisms—including round-robin and envy-cycle elimination—thereby establishing their limitations under this new fairness lens.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Knowledge Representation and Reasoning: Preferences

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

In the fair division problem for indivisible goods, mechanisms that output allocations satisfying fairness concepts, such as envy-freeness up to one good (EF1), have been extensively studied. These mechanisms usually require an arbitrary order of agents as input, which may cause some agents to feel unfair since the order affects the output allocations. In the context of the cake-cutting problem, Manabe and Okamoto (2012) introduced meta-envy-freeness to capture such kind of fairness, which guarantees the absence of envy compared to different orders of agents. In this paper, we introduce position envy-freeness and its relaxation, position envy-freeness up to $k$ goods (PEF$k$), for mechanisms in the fair division problem for indivisible goods, analogous to the meta-envy-freeness. While the round-robin or the envy-cycle mechanism is not PEF1, we propose a PEF1 mechanism that always outputs an EF1 allocation. In addition, in the case of two agents, we prove that any mechanism that always returns a maximum Nash social welfare allocation is PEF1, and propose a modified adjusted winner mechanism satisfying PEF1. We further investigate the round-robin and the envy-cycle mechanisms to measure how far they are from position envy-freeness.

Problem

• Develops position envy-freeness for indivisible goods allocation mechanisms
• Proposes polynomial-time mechanisms satisfying PEF1 and EF1 criteria
• Analyzes position fairness in existing allocation methods like round-robin

Innovation

• Introduces PEF1 fairness criterion for allocation mechanisms
• Proposes scale-invariant polynomial-time PEF1 mechanism
• Presents adjusted winner-based mechanism for two agents