[论文解读] On Best-of-Both-Worlds Fairness via Sum-of-Variances Minimization

Moshe Babaioff, Yuval Grofman|arXiv (Cornell University)|Jan 23, 2026

Game Theory and Voting Systems被引用 0

一句话总结

论文分析在事前成比例性下最小化代理人价值方差和以实现BoBW公平性，显示在相同估值（两个代理人）时的优点强，但在非相同估值以及n≥3个相同估值时存在显著负面影响。

ABSTRACT

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.

研究动机与目标

• 在事前成比例性下动机化并形式化SoV目标作为BoBW方法。
• 在估值相同的情况下，刻画最小化SoV的分配的事后公平性性质。
• 研究非相同估值以及超过两个代理人时SoV最小化的表现。
• 展示SoV在不同变体与目标下的计算困难性与基本局限性。

提出的方法

• 定义事前成比例分布和分配之间的SoV目标。
• 将SoV最小化分布的支集表征为到事前成比例份额向量的距离最小化。
• 证明在估值相同且n=2时，SoV最小化分布在事后是MMS-公平的（因此事后EFX成立），并指出计算此类分布的NP-hard性。
• 扩展到n≥3个相同估值，显示某些情况下事后MMS公平性失效但事后EFX成立，意味着常数MMS近似。
• 为非相同估值给出负面结果，显示EF1/EFX在支集内可能失败，即使在2代理人、2商品的情形下也如此，并分析其他基于方差的目标。

实验结果

研究问题

• RQ1 在事前成比例性下最小化方差和是否能保证相同估值下的事后公平性（EF1/EFX或MMS）？
• RQ2 当估值相同时，且代理人多于两个时，SoV最小化分布在事后体现出何种公平性特征？
• RQ3 非相同估值是否削弱SoV最小化分布的事后公平性，即使在最简单的两代理、两商品情形？
• RQ4 在事前成比例性下寻找SoV最小分布是否存在计算或复杂性障碍？
• RQ5 其他基于方差的目标是否同样无法保证事后公平性？

主要发现

• 对于具有相同估值的两个代理人，每个SoV最小化分布在事后都是MMS-公平的（因此事后EFX成立）。
• 对于n≥3个相同估值，SoV最小化分布在事后是EFX成立的，在n=3时达到2/3的MMS近似，在n≥4时为4/7的MMS近似；3代理人的MMS近似上限不超过275/304 ≈ 90.4%。
• 在两个相同代理人下、在事前成比例性约束下计算SoV最小分布是NP困难的。
• 在估值非相同的情形下存在一个两代理人、两商品的实例，其中每个SoV最小化分布都把两件商品中的一个分配给单一代理人并因此破坏事后EF1（从而EFX），且不保证常数MMS。
• 对于其他自然的方差基目标（最大方差、最大标准差、标准差之和、方差的方差等）也存在类似的负面结果。
• 当估值不同的时候，SoV方法并不保证事后公平性，即使在事前成比例性下亦然。