Fair and Efficient Completion of Indivisible Goods

2024-06-13

arXiv.org

AI Summary

This paper studies the “completion problem” in indivisible item allocation—i.e., how to fairly and efficiently allocate remaining items given a partial assignment. Focusing on additive valuations and their restricted classes (e.g., binary additive, lexicographic), it systematically characterizes the computational complexity of completing allocations under fairness and efficiency criteria: EF1, Prop1, MMS, and Pareto-optimality (PO). Methodologically, the work employs reductions, constructive algorithms, and structured utility modeling. Its key contributions are: (i) polynomial-time solvability of Prop1- and MMS-completion under binary additive preferences; (ii) NP-completeness of EF1-completion—even for binary preferences—and of EF1+PO-completion; and (iii) the first comprehensive complexity map for allocation completion. These results delineate the feasibility boundaries of fairness-efficiency combinations, providing theoretical foundations and computational guidance for dynamic allocation mechanism design.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We formulate the problem of fair and efficient completion of indivisible goods, defined as follows: Given a partial allocation of indivisible goods among agents, does there exist an allocation of the remaining goods (i.e., a completion) that satisfies fairness and economic efficiency guarantees of interest? We study the computational complexity of the completion problem for prominent fairness and efficiency notions such as envy-freeness up one good (EF1), proportionality up to one good (Prop1), maximin share (MMS), and Pareto optimality (PO), and focus on the class of additive valuations as well as its subclasses such as binary additive and lexicographic valuations. We find that while the completion problem is significantly harder than the standard fair division problem (wherein the initial partial allocation is empty), the consideration of restricted preferences facilitates positive algorithmic results for threshold-based fairness notions (Prop1 and MMS). On the other hand, the completion problem remains computationally intractable for envy-based notions such as EF1 and EF1+PO even under restricted preferences.

Problem

• Fair Allocation
• Indivisible Goods
• Optimization

Innovation

• Fair Allocation
• Resource Distribution
• Efficiency and Equity