[Paper Review] Improving Approximation Guarantees for Maximin Share

Hannaneh Akrami, Jugal Garg+2 more|arXiv (Cornell University)|Jul 24, 2023

Parallel Computing and Optimization Techniques4 citations

TL;DR

This paper introduces a novel $(\alpha,\beta,\gamma)$-MMS framework that generalizes both ordinal and multiplicative approximations in fair division of indivisible goods. It proves the existence of $1$-out-of-$\lceil 4n/3\rceil$ MMS allocations, improving the prior state-of-the-art bound of $\lfloor 3n/2\rfloor$, and establishes a randomized allocation with ex-ante approximation factor exceeding 0.785, surpassing the previous best $\alpha = \frac{3}{4} + \frac{3}{3836}$. The approach leverages a refined analysis of bundle partitions and value redistribution to achieve stronger guarantees under additive valuations.

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 $d=\lfloor3n/2 floor$ [Hosseini et al.'21]. We show that $1$-out-of-$4\lceil n/3 ceil$ 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.

Motivation & Objective

• To address the limitation that maximin share (MMS) allocations do not always exist for $n \geq 3$ agents with additive valuations.
• To improve the state-of-the-art ordinal approximation factor $d$ in $1$-out-of-$d$ MMS allocations, which guarantees each agent at least the value of their $d$-MMS partition.
• To strengthen the multiplicative approximation factor $\alpha$ in $\alpha$-MMS allocations, where each agent receives at least $\alpha$ times their MMS value.
• To unify and generalize both approximation paradigms via a new framework: $(\alpha,\beta,\gamma)$-MMS, which allows different agents to receive different fractions of their MMS values.
• To design a randomized allocation algorithm that achieves a high ex-ante approximation factor while guaranteeing at least $3/4$ of the MMS value ex-post for all agents.

Proposed method

• Introduce a generalized fairness framework: $(\alpha,\beta,\gamma)$-MMS, where $\alpha$ fraction of agents receive at least $\beta$ times their MMS value, and the remaining $1-\alpha$ receive at least $\gamma$ times their MMS value.
• Prove that $(2(1-\beta)/\beta, \beta, 3/4)$-MMS allocations always exist for any $\beta \in (0,1]$, by constructing a balanced allocation via iterative bundle reassignment and value redistribution.
• Use a $d$-normalized and ordered instance transformation to reduce the problem to a canonical form where valuations are scaled and sorted, enabling robust analysis under small perturbations.
• Apply a case analysis on bundle removals and partition reassignments to show that MMS values of agents do not decrease after removing a good set $S$, under different structural assumptions on the MMS partition.
• Leverage the flexibility of choosing which $2(1-\beta)/\beta$ fraction of agents receive $\beta$-MMS value to design a randomized allocation with improved expected performance.
• Establish a connection between $1$-out-of-$d$ MMS and $\alpha$-MMS via a sequence of normalization and ordering transformations, proving that any $1$-out-of-$d$ MMS allocation in the transformed instance yields a valid allocation in the original.

Experimental results

Research questions

• RQ1 Can the ordinal approximation factor $d$ in $1$-out-of-$d$ MMS allocations be improved beyond the prior best bound of $\lfloor 3n/2\rfloor$?
• RQ2 Can a unified framework be developed that captures both ordinal and multiplicative approximations in MMS fairness?
• RQ3 What is the best possible ex-ante approximation factor achievable via a randomized allocation that guarantees at least $3/4$ of the MMS value ex-post?
• RQ4 Does the existence of $1$-out-of-$\lceil 4n/3\rceil$ MMS allocations imply a stronger multiplicative approximation guarantee than previously known?
• RQ5 Can the analysis of MMS value stability under small perturbations be leveraged to design robust approximation algorithms?

Key findings

• The paper establishes that $1$-out-of-$\lceil 4n/3\rceil$ MMS allocations always exist, improving the prior state-of-the-art bound of $\lfloor 3n/2\rfloor$.
• It proves the existence of $(2(1-\beta)/\beta, \beta, 3/4)$-MMS allocations for any $\beta \in (0,1]$, generalizing both ordinal and multiplicative approximations.
• By choosing $\beta = \sqrt{3}/2$, the framework yields a randomized allocation that guarantees each agent at least $3/4$ of their MMS value ex-post.
• The same randomized allocation achieves an ex-ante approximation factor of $\frac{17\sqrt{3} - 24}{4\sqrt{3}} > 0.785$, surpassing the previous best $\alpha = \frac{3}{4} + \frac{3}{3836}$.
• The analysis shows that MMS values of agents are stable under small perturbations when using the $1$-out-of-$d$ MMS framework, unlike the $\alpha$-MMS notion which is sensitive to such changes.
• The paper demonstrates that the $1$-out-of-$d$ MMS framework is robust and can be transformed into a valid allocation in the original instance via normalization and ordering techniques.