We consider the problem of fairly allocating a set of indivisible goods among agents with additive valuations. Ex-ante fairness (proportionality) can trivially be obtained by giving all goods to a random agent. Yet, such an allocation is very unfair ex-post. This has motivated the Best-of-Both-Worlds (BoBW) approach, seeking a randomized allocation that is ex-ante proportional and is supported only on ex-post fair allocations (e.g., on allocations that are envy-free-up-to-one-good (EF1), or give some constant fraction of the maximin share (MMS)). It is commonly pointed out that the distribution that allocates all goods to one agent at random fails to be ex-post fair as it ignores the variances of the values of the agents. We examine the approach of trying to mitigate this problem by minimizing the sum-of-variances of the values of the agents, subject to ex-ante proportionality. We study the ex-post fairness properties of the resulting distributions. In support of this approach, observe that such an optimization will indeed deterministically output a proportional allocation if such exists. We show that when valuations are identical, this approach indeed guarantees fairness ex-post: all allocations in the support are envy-free-up-to-any-good (EFX), and thus guarantee every agent at least 4/7 of her maximin share (but not her full MMS). On the negative side, we show that this approach completely fails when valuations are not identical: even in the simplest setting of only two agents and two goods, when the additive valuations are not identical, there is positive probability of allocating both goods to the same agent. Thus, the supporting ex-post allocation might not even be EF1, and might not give an agent any constant fraction of her MMS. Finally, we present similar negative results for other natural minimization objectives that are based on variances.

翻译：我们考虑在具有可加性估值的主体之间公平分配一组不可分割物品的问题。事前公平性（比例性）可以通过将所有物品随机分配给一个主体而轻易实现。然而，这种分配在事后极不公平。这催生了"两全其美"（BoBW）方法，旨在寻求一种随机分配方案，既满足事前比例性，又仅由事后公平的分配方案（例如，满足"除一件物品外无嫉妒"（EF1）条件，或提供最大最小份额（MMS）的某个恒定比例）所支撑。常被指出的是，将所有物品随机分配给单一主体的分布无法实现事后公平，因为它忽略了主体估值的方差。我们研究了通过最小化主体估值的方差和来缓解此问题的方法，并受限于事前比例性约束。我们分析了所得分布的事后公平性特性。支持此方法的观察是：若存在比例性分配，该优化问题确实会确定性地输出这样的分配。我们证明，当估值相同时，该方法确实能保证事后公平性：支撑集中的所有分配均满足"除任意物品外无嫉妒"（EFX）条件，从而保证每个主体至少获得其最大最小份额的4/7（但非全额MMS）。负面结果是，当估值不同时，该方法完全失效：即使在仅有两个主体和两件物品的最简场景中，当可加性估值不同时，存在将两件物品分配给同一主体的正概率。因此，支撑性的事后分配可能甚至不满足EF1条件，且可能无法给予主体其MMS的任何恒定比例。最后，我们对其他基于方差的自然最小化目标给出了类似的负面结果。