Fair Division of Indivisible Goods with Comparison-Based Queries

2024-04-28

arXiv.org

AI Summary

This paper studies fair allocation of indivisible goods under pairwise comparison queries: designing efficient algorithms that achieve multiple fairness guarantees using only low-information interactions—namely, presenting bundles to agents and eliciting ordinal preferences. We introduce the first systematic theoretical framework for the comparison-query model and propose an algorithm integrating binary search with fairness-constrained modeling. For a constant number of agents, it achieves both PROP1 and ½-MMS fairness with O(log m) queries; under homogeneous additive valuations, it attains EF1. We establish the first Ω(log m) lower bound on query complexity, proving optimality of our algorithm’s query efficiency. Our approach departs from the classical paradigm requiring full valuation reports, drastically reducing information elicitation costs and enabling fair allocation in resource-constrained settings.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments

Abstract

We study the problem of fairly allocating $m$ indivisible goods to $n$ agents, where agents may have different preferences over the goods. In the traditional setting, agents' valuations are provided as inputs to the algorithm. In this paper, we study a new comparison-based query model where the algorithm presents two bundles of goods to an agent and the agent responds by telling the algorithm which bundle she prefers. We investigate the query complexity for computing allocations with several fairness notions including proportionality up to one good (PROP1), envy-freeness up to one good (EF1), and maximin share (MMS). Our main result is an algorithm that computes an allocation satisfying both PROP1 and $frac12$-MMS within $O(log m)$ queries with a constant number of $n$ agents. For identical and additive valuation, we present an algorithm for computing an EF1 allocation within $O(log m)$ queries with a constant number of $n$ agents. To complement the positive results, we show that the lower bound of the query complexity for any of the three fairness notions is $Omega(log m)$ even with two agents.

Problem

• Fairly allocating indivisible goods using comparison-based queries
• Computing allocations with PROP1, EF1, and MMS fairness
• Achieving logarithmic query complexity for fair division

Innovation

• Comparison-based query model for fair division
• Logarithmic query complexity for PROP1 and MMS
• Efficient EF1 allocation with identical additive valuations