Near-Optimal Best-of-Both-Worlds Fairness for Few Agents

2026-02-16

AI Summary

This work addresses the design of algorithms for fair allocation of indivisible goods among three agents that simultaneously achieve ex-ante and ex-post fairness. We propose the first Best-of-Both-Worlds (BoBW) mechanism for three agents: its randomized outcome satisfies ex-ante proportionality, every deterministic allocation in its support satisfies ex-post Epistemic EFX (EEFX), and each agent receives at least 9/10 of their maximin share (MMS). Key technical contributions include an FPTAS for approximating MMS partitions, a construction of EEFX allocations, and an extension from a two-agent subroutine—combining EFX and FPTAS—to the three-agent setting. For two agents, we further achieve ex-ante envy-freeness, ex-post EFX, and (1−ε)-MMS. The algorithm runs in polynomial time, establishing the first near-optimal BoBW fairness guarantee for three agents.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Fair Division
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We consider the problem of fair allocation of indivisible goods among agents with additive valuations, aiming for Best-of-Both-Worlds (BoBW) fairness: a distribution over allocations that is ex-ante fair, and additionally, it is supported only on deterministic allocations that are ex-post fair. We focus on BoBW for few agents, and our main result is the design of the first BoBW algorithms achieving near-optimal fairness for three agents. For three agents, we prove the existence of an ex-ante proportional distribution whose every allocation is Epistemic EFX (EEFX) and guarantees each agent at least $\tfrac{9}{10}$ of her MMS. As MMS allocations do not exist for three additive agents, in every allocation at least one agent might not be getting her MMS. To compensate such an agent, we also guarantee that if an agent is not getting her MMS then she is EFX-satisfied - giving her the strongest achievable envy-based guarantee. Additionally, using an FPTAS for near-MMS partitions, we present an FPTAS to compute a BoBW distribution preserving all envy-based guarantees, and also preserving all value-based guarantees up to $(1-\varepsilon)$. We further show that exact ex-ante proportionality can be restored when dropping EEFX. To do so, we first design, for two agents and any $\varepsilon>0$, a Fully Polynomial-Time Approximation Scheme (FPTAS) that outputs a distribution which is ex-ante envy-free (and thus proportional) and ex-post envy-free up to any good (EFX), while guaranteeing each agent at least a $(1-\varepsilon)$-fraction of her maximin share (MMS). We then leverage this two-agent FPTAS algorithm as a subroutine to obtain, for three agents, the FPTAS guaranteeing exact ex-ante proportionality. We note that our result for two agents essentially matches the strongest fairness and efficiency guarantees achievable in polynomial time, and thus might be of independent interest.

Problem

• fair allocation
• indivisible goods
• Best-of-Both-Worlds
• EFX
• MMS

Innovation

• Best-of-Both-Worlds
• EFX
• MMS
• FPTAS
• fair division