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.

翻译：我们考虑在具有可加性估值（additive valuations）的代理之间公平分配不可分割物品的问题，目标是实现双世界（Best-of-Both-Worlds, BoBW）公平性：即一个在事前（ex-ante）公平的分配分布，并且该分布仅由在事后（ex-post）公平的确定性分配所支撑。我们专注于少数代理的BoBW问题，主要成果是设计了首个针对三个代理实现近似最优公平性的BoBW算法。对于三个代理，我们证明存在一个事前比例公平（ex-ante proportional）的分布，其中每个分配都是认知EFX（Epistemic EFX, EEFX）的，并且保证每个代理至少获得其最大最小份额（MMS）的 $\tfrac{9}{10}$。由于对于三个具有可加性估值的代理不存在MMS分配，因此在每个分配中至少有一个代理可能无法获得其MMS。为了补偿这样的代理，我们还保证：如果一个代理未获得其MMS，则她是EFX满足的——这给予了她基于嫉妒的最强可实现保证。此外，利用针对近似MMS划分的完全多项式时间近似方案（FPTAS），我们提出了一个FPTAS来计算一个BoBW分布，该分布保留所有基于嫉妒的保证，并且将基于价值的保证也保留至 $(1-\varepsilon)$。我们进一步证明，当放弃EEFX时，可以恢复精确的事前比例公平性。为此，我们首先为两个代理和任意 $\varepsilon > 0$ 设计了一个完全多项式时间近似方案（FPTAS），该方案输出一个分布，该分布在事前是无嫉妒的（因而是比例公平的），在事后是至多一件物品的嫉妒（EFX）的，同时保证每个代理至少获得其最大最小份额（MMS）的 $(1-\varepsilon)$ 比例。然后，我们利用这个双代理FPTAS算法作为子程序，为三个代理获得保证精确事前比例公平性的FPTAS。我们注意到，我们针对两个代理的结果本质上匹配了在多项式时间内可实现的最强公平性和效率保证，因此可能具有独立的研究价值。