Fair Division via Resource Augmentation

TL;DR

It is proved 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, which improves upon the best-known approximation guarantees for additive valuations in the absence of copies.

Abstract

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.

Figures (2)

• Figure 1: Illustration of the augmenting path argument in the proof of \ref{['lem:important']}. The element $g \in G$ (whose corresponding node is $\mathcal{Y}_g$, $g=y_1$) is replaced with $g'$ (whose corresponding node is $\mathcal{X}_{g'}=\mathcal{X}_4$, $g'=x_4$). We write $\mathcal{X}_i$ / $\mathcal{Y}_i$ rather than the cumbersome $\mathcal{X}_{x_i}$/$\mathcal{Y}_{y_i}$. It is purely an example that the path nodes are associated with a contiguous prefix of the goods, e.g., $\mathcal{X}_1$ could point to $\mathcal{Y}_5$.
• Figure 2: Running example for $\mathsf{BagFill\hbox{-}with\hbox{-}Copies}$. There is an MMS allocation without copies for this instance. All MMS values are one. For agent 1 we have $v_1(\{g_1, g_2, g_3\})=1.1$, $v_1(\{g_4,g_5,g_6\})=1.1$, $v_1(\{g_7, g_8, g_9\})=1$ and $v_1(\{g_{10}, g_{11}, g_{12} \})=1.1$. For agent 2 we have $v_2(\{g_1, g_2, g_3\})=1.1$, $v_2(\{g_4,g_5\})=1$, $v_2(\{g_6.g_7, g_8, g_9\})=1$ and $v_2(\{g_{10}, g_{11}, g_{12} \})=1.1$. By construction the MMS partitions consist of contiguous runs of $g_1,\ldots,g_{12}$.

Theorems & Definitions (86)

• Proposition : Lower bound for submodular valuations, \ref{['prop:submod-lb']}
• Theorem 2.1
• proof
• Proposition 2.3
• proof
• Theorem 2.4
• Lemma 2.4
• proof : Proof sketch
• proof : Proof of \ref{['thm:alg_for_monotone']}
• Theorem 2.5