Best-of-Both-Worlds Fairness and Pareto Optimality

2026-08-09

AI Summary

This study addresses the challenge of simultaneously achieving ex-ante envy-freeness, ex-post EFX fairness, and Pareto optimality in the allocation of indivisible goods. We propose a randomized allocation mechanism whose induced fractional allocation satisfies ex-ante envy-freeness, while every deterministic allocation in its support fulfills both EFX and Pareto optimality. Our main contributions are twofold: first, we prove that for any instance with two agents, there always exists a randomized allocation satisfying all three properties, and we provide a constructive algorithm that computes such an allocation in pseudo-polynomial time under integer additive valuations; second, we construct a counterexample with three agents and four items, demonstrating that such an allocation does not always exist in general.

Abstract

We consider fair allocation of indivisible items among agents with non-negative and additive valuations. The goal is to construct a lottery over deterministic allocations whose induced fractional allocation is envy-free, while every realised allocation is envy-free up to one item and Pareto optimal. We show that this is always possible for two agents. We then prove a stronger result that there always exists a lottery over deterministic allocations whose induced fractional allocation is envy-free, while every realised allocation is envy-free up each one item (EFX) and Pareto optimal. For non-negative integral additive valuations, such a lottery can be computed in pseudo-polynomial time. We also prove that for three agents and four items, there may be no ex-ante envy-free lottery that can be supported on allocations that are simultaneously EFX and Pareto optimal.

Problem

• fair allocation
• indivisible items
• envy-freeness
• Pareto optimality
• lottery

Innovation

• envy-freeness
• Pareto optimality
• EFX
• lottery
• indivisible allocation