[Paper Review] Near-Optimal Best-of-Both-Worlds Fairness for Few Agents

Moshe Babaioff, Gefen Frosh|arXiv (Cornell University)|Feb 16, 2026

Game Theory and Voting Systems0 citations

TL;DR

The paper designs near-optimal Best-of-Both-Worlds (BoBW) fairness algorithms for few agents, proving ex-ante proportionality with every allocation in the support being EEFX and at least 9/10 of MMS for three agents, and providing optimal poly-time BoBW results for two agents.

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 $ frac{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.

Motivation & Objective

• Motivate fair division of indivisible goods under additive valuations and the BoBW framework.
• Achieve ex-ante proportionality while ensuring strong ex-post fairness across all allocations in the support.
• Provide near-optimal MMS guarantees for three agents within an IMMX (MMS-satisfied or EFX-satisfied) framework.
• Deliver polynomial-time approximations (FPTAS) that preserve fairness guarantees where possible.
• Extend BoBW results to two agents with optimal polynomial-time guarantees.

Proposed method

• Construct a distribution over at most six deterministic allocations that is ex-ante proportional and whose allocations are IMMX (each agent is MMS-satisfied or EFX-satisfied).
• For three agents, ensure every allocation in the support is EEFX and guarantees at least 9/10 of MMS for each agent.
• Define IMMX as a regime where each agent either receives MMS or is EFX-satisfied, providing a robust trade-off between share-based and envy-based guarantees.
• Develop an FPTAS to compute near-MMS partitions and preserve envy-based guarantees and most value-based guarantees.
• Leverage a two-agent optimal BoBW result as a subroutine to obtain exact ex-ante proportionality in the three-agent setting via an FPTAS.
• Provide an optimal poly-time BoBW algorithm for two agents that guarantees ex-ante envy-freeness (hence proportional) and ex-post EFX with at least (1-ε) MMS.

Experimental results

Research questions

• RQ1 Can a BoBW distribution be found for three additive agents that is ex-ante proportional and whose every allocation is EEFX while guaranteeing at least 9/10 of MMS for each agent?
• RQ2 Is there a polynomial-time algorithm that achieves BoBW guarantees for two agents that are ex-ante envy-free and ex-post EFX with at least (1-ε) MMS?
• RQ3 How can MMS partitions be approximated without destroying ex-ante proportionality and ex-post fairness properties?
• RQ4 Can IMMX be achieved in two- and three-agent settings, and what are the trade-offs between MMS and EFX in these settings?
• RQ5 What is the impact of replacing exact MMS with (1-ε)-MMS partitions on the BoBW guarantees?

Key findings

• Existence of a BoBW distribution for three additive agents that is ex-ante proportional and whose every allocation is EEFX and guarantees each agent at least 9/10 of MMS.
• In every deterministic allocation in the three-agent support, one agent is EFX-satisfied and receives proportional share, one is EFX-satisfied and receives at least 9/10 MMS, and one is EEFX-satisfied and receives at least MMS.
• Introduction of IMMX: every agent is either MMS-satisfied or EFX-satisfied within the BoBW allocations.
• An FPTAS can compute a BoBW distribution preserving EEFX and all envy- and near-MMS guarantees, with exact ex-ante proportionality achievable when dropping EEFX.
• A polynomial-time BoBW algorithm for two agents achieves ex-ante envy-freeness, ex-post EFX, and at least (1-ε) of MMS for each agent.
• Result for two agents essentially matches the strongest polynomial-time fairness guarantees in this setting.