Nonexistence of Simultaneously EF1 and Pareto Optimal Allocations for Submodular Valuations

2026-07-20

AI Summary

This study investigates whether allocations of indivisible goods can simultaneously satisfy envy-freeness up to one good (EF1) and Pareto optimality (PO) under monotone submodular valuations. The authors present the first counterexample demonstrating that EF1 and PO are incompatible even for two agents with unweighted coverage valuations—a strict subclass of submodular functions—thereby extending prior non-existence results from subadditive to coverage valuations. Furthermore, through a combinatorial construction and a complexity-theoretic reduction, they prove that deciding the existence of such allocations is NP-hard. This work delineates the infeasibility boundary of EF1+PO allocations in broader submodular settings and provides theoretical foundations relevant to practical applications such as chore division.

Abstract

The existence of allocations of indivisible goods that are simultaneously fair (envy-free up to one item (EF1)) and efficient (Pareto optimal (PO)) when agents have monotone submodular valuations has been a longstanding open problem. We settle this question negatively by giving an example with two agents where no allocation is simultaneously EF1 and PO. We also show that determining the existence of such allocations is NP-hard for monotone submodular valuations. Our example uses (unweighted) coverage valuations, which is a strict subclass of monotone submodular valuations. Since EF1+PO allocations are known to always exist for additive valuations via the maximization of Nash Social Welfare (Caragiannis et al. (ACM TEAC 2019)), and for matroid-rank valuations (Benabbou et al. (ACM TEAC 2021)), nonexistence was known only for monotone subadditive valuations (Caragiannis et al. (ACM TEAC 2019)). Our work moves the nonexistence frontier to unweighted coverage valuations. We also show that the example we designed for goods also proves nonexistence of EF1+PO in general, for chores with unweighted coverage costs, by interpreting the valuations as disutilities.

Problem

• EF1
• Pareto optimality
• submodular valuations
• indivisible goods
• fair division

Innovation

• EF1
• Pareto optimality
• submodular valuations
• coverage valuations
• NP-hardness