We introduce and formalize the notion of resource augmentation for maximin share (MMS) fairness for the allocation of indivisible goods. Given an instance with $n$ agents and $m$ goods, we ask how many copies of the goods should be added in order to guarantee that each agent receives at least their original MMS value, or a meaningful approximation thereof. For general monotone valuations, we establish a tight bound: an exact MMS allocation can be guaranteed using at most $Θ(m/e)$ total copies, and this bound is tight even for XOS valuations. We further show that it is unavoidable to duplicate some goods $Ω(\ln m / \ln \ln m)$ times, and provide matching upper bounds. For additive valuations, we show that at most $\min\{n-2,\lfloor\frac{m}{3}\rfloor(1+o(1))\}$ distinct copies suffice. This separates additive valuations from submodular valuations, for which we show that $n-1$ copies may be necessary. We also study approximate MMS guarantees for additive valuations and establish new tradeoffs between the number of copies needed and the approximation guaratee. In particular, we prove that $\lfloor{n/2}\rfloor$ copies suffice to guarantee a $6/7$-approximation to the original MMS, and $\lfloor{n/3}\rfloor$ copies suffice for a $4/5$-approximation. Both results improve upon the best-known approximation guarantees for additive valuations in the absence of copies. Finally, we relate MMS with copies to the relaxed notion of 1-out-of-$d$ MMS, showing that improvements in either framework translate directly to the other. In particular, we establish the first impossibility results for 1-out-of-$d$ MMS. Our results highlight the power and limits of resource augmentation for achieving MMS fairness.

翻译：我们针对不可分割物品分配中的最大最小份额（MMS）公平性，提出并形式化了资源增广的概念。给定包含 $n$ 个智能体和 $m$ 件物品的实例，我们探究需要增加多少件物品的副本，才能保证每个智能体至少获得其原始 MMS 值或其有意义的近似值。对于一般单调估值函数，我们建立了紧确界：使用至多 $Θ(m/e)$ 个总副本即可保证精确 MMS 分配，且该界即使对于 XOS 估值函数也是紧的。我们进一步证明某些物品不可避免地需要复制 $Ω(\ln m / \ln \ln m)$ 次，并给出了匹配的上界。对于可加估值函数，我们证明至多 $\min\{n-2,\lfloor\frac{m}{3}\rfloor(1+o(1))\}$ 个不同副本即可满足需求。这区分了可加估值与子模估值——对于后者我们证明可能需要 $n-1$ 个副本。我们还研究了可加估值下的近似 MMS 保证，并建立了所需副本数量与近似保证之间的新权衡关系。特别地，我们证明 $\lfloor{n/2}\rfloor$ 个副本足以保证对原始 MMS 的 $6/7$ 近似，而 $\lfloor{n/3}\rfloor$ 个副本足以实现 $4/5$ 近似。这两个结果都改进了无副本情况下可加估值的最佳已知近似保证。最后，我们将带副本的 MMS 与松弛的 1-out-of-$d$ MMS 概念联系起来，证明任一框架的改进都能直接转化到另一框架。特别地，我们首次建立了 1-out-of-$d$ MMS 的不可能性结果。我们的研究结果凸显了资源增广在实现 MMS 公平性方面的能力与局限。