Fair and Efficient Completion of Indivisible Goods

TL;DR

This paper investigates the computational complexity of completing partial allocations of indivisible goods fairly and efficiently, revealing that some fairness notions are tractable under certain preferences while others remain hard.

Contribution

It provides a comprehensive complexity analysis of the fair division completion problem for various fairness and efficiency notions, highlighting differences under restricted preferences.

Findings

Completion problem is harder than standard fair division.

Threshold-based fairness notions are tractable with restricted preferences.

Envy-based notions remain computationally intractable even with restrictions.

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.