Improving Approximation Guarantees for Maximin Share

TL;DR

The paper advances MMS-based fair division by (i) proving a constructive existence of a $1$-out-of-$4\lceil n/3\rceil$ MMS allocation via a bag-filling approach on ordered, normalized instances, (ii) introducing a unified $T$-MMS framework that generalizes both ordinal and multiplicative MMS relaxations, and (iii) analyzing an agent-priority ranking algorithm (RBF) to obtain ex-post and ex-ante fairness guarantees, including best-of-both-worlds results and tight upper bounds on achievable thresholds. It also establishes a near-tight algorithmic landscape through a tight example and obliviousness-based limitations, clarifying the trade-offs between ex-post guarantees and ex-ante expectations. The work provides practical constructive methods and theoretical limits for fair division with indivisible goods, with potential impact on auction design, resource allocation, and multi-agent systems requiring robust MMS approximations. Overall, it delivers meaningful improvements in ordinal MMS, a versatile general framework, and precise performance bounds for deterministic, priority-based allocation schemes.

Abstract

We consider fair division of a set of indivisible goods among $n$ agents with additive valuations using the fairness notion of maximin share (MMS). MMS is the most popular share-based notion, in which an agent finds an allocation fair to her if she receives goods worth at least her ($1$-out-of-$n$) MMS value. An allocation is called MMS if all agents receive their MMS values. However, since MMS allocations do not always exist, the focus shifted to investigating its ordinal and multiplicative approximations. In the ordinal approximation, the goal is to show the existence of $1$-out-of-$d$ MMS allocations (for the smallest possible $d>n$). A series of works led to the state-of-the-art factor of $d=\lfloor3n/2\rfloor$ [Hosseini et al.'21]. We show that $1$-out-of-$4\lceil n/3\rceil$ MMS allocations always exist, thereby improving the state-of-the-art of ordinal approximation. In the multiplicative approximation, the goal is to show the existence of $α$-MMS allocations (for the largest possible $α< 1$), which guarantees each agent at least $α$ times her MMS value. We introduce a general framework of "approximate MMS with agent priority ranking". An allocation is said to be $T$-MMS, for a non-increasing sequence $T = (τ_1, \ldots, τ_n)$ of numbers, if the agent at rank $i$ in the order gets a bundle of value at least $τ_i$ times her MMS value. This framework captures both ordinal approximation and multiplicative approximation as special cases. We show the existence of $T$-MMS allocations where $τ_i \ge \max(\frac{3}{4} + \frac{1}{12n}, \frac{2n}{2n+i-1})$ for all $i$. Furthermore, we can get allocations that are $(\frac{3}{4} + \frac{1}{12n})$-MMS ex-post and $(0.8253 + \frac{1}{36n})$-MMS ex-ante. We also prove that our algorithm does not give better than $(0.8631 + \frac{1}{2n})$-MMS ex-ante.

Figures (8)

• Figure 1: For all $k \in [d]$, we define $C_k := \{k, 2d-k+1\}$.
• Figure 2: Bag initialization
• Figure 3: An illustration of which group each bag belongs to.
• Figure 4: The first $2\ell + t + 2\ell^* -2n/3$ goods are marked with red, and the goods considered in Lemma \ref{['expowerful']} are marked with blue.
• Figure 5: The goods considered in in Lemma \ref{['k-t']} are marked with red and the goods in Lemma \ref{['k+l']} are marked with blue.

Theorems & Definitions (105)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Definition 1
• Definition 2
• Lemma 2
• Proposition 1
• Proposition 2
• proof