Fair Division with Strictly Increasing Valuations: A Tight Threshold for Two-Agent EF1 and PO

2026-07-25

AI Summary

This study investigates the compatibility between envy-freeness up to one item (EF1) and Pareto optimality (PO) in the allocation of indivisible goods among two or more agents with strictly increasing valuations. For the two-agent case, the authors precisely characterize the threshold: an EF1 and PO allocation always exists when there are at most seven items, yet a counterexample with eight items demonstrates that EF1 and PO may be incompatible. Moreover, even under restrictions on the placement of zero marginal values, deciding the existence of an EF1 and PO allocation remains NP-hard for three agents. The analysis combines combinatorial constructions, complexity reductions, and normalized submodular valuation functions to delineate the structural boundaries governing the trade-off between fairness and efficiency.

Abstract

We study whether strictly positive marginal values restore the compatibility of envy-freeness up to one good (EF1) and Pareto optimality (PO) for indivisible goods. For two agents, we identify the exact threshold in the number of goods. Every instance with at most seven goods and strictly increasing valuations admits an allocation that is both EF1 and PO, without any submodularity assumption. In contrast, we construct an eight-good instance with normalized, integer-valued, strictly increasing, submodular valuations in which every EF1 allocation is strictly Pareto dominated. Thus, eight goods are necessary and sufficient for a two-agent counterexample. Finally, we strengthen the three-agent NP-hardness result of Chandramouleeswaran and Nimbhorkar (2026): deciding whether an EF1 and PO allocation exists remains NP-hard for normalized, integer-valued, monotone submodular valuations even when zero marginals are confined to eight fixed agent-good pairs, all involving a single agent.

Problem

• fair division
• envy-freeness up to one good
• Pareto optimality
• indivisible goods
• strictly increasing valuations

Innovation

• EF1
• Pareto optimality
• strictly increasing valuations
• indivisible goods
• computational hardness