Concentration and maximin fair allocations for subadditive valuations

2025-02-19

AI Summary

This work studies fair allocation of indivisible goods under subadditive valuations, aiming to guarantee each agent a higher fraction of their maximin share (MMS). Prior algorithms achieve only an $O(1/(log n log log n))$ MMS approximation ratio. To overcome this limitation, we first establish a tight concentration bound for random sampling from subadditive valuation functions: the median value of a uniformly random subset is at least $frac{2}{3}$ of the expectation minus $frac{11}{12}$ times the maximum value of any single item. Leveraging this characterization, we design an improved greedy allocation algorithm that achieves an MMS approximation ratio of $1/(14 log n)$—the best-known theoretical guarantee for subadditive (and submodular) valuations. This result significantly advances the state-of-the-art lower bound for fair allocation in subadditive settings.

Technology Category

• Game Theory and Economic Paradigms: Fair Division
• Multiagent Systems: Mechanism Design
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Fairness, privacy, and diversity in economic environments
• Web Mining and Content Analysis: Web data quality in the era of algorithmically-generated content

Abstract

We consider fair allocation of $m$ indivisible items to $n$ agents of equal entitlements, with submodular valuation functions. Previously, Seddighin and Seddighin [{em Artificial Intelligence} 2024] proved the existence of allocations that offer each agent at least a $frac{1}{c log n loglog n}$ fraction of her maximin share (MMS), where $c$ is some large constant (over 1000, in their work). We modify their algorithm and improve its analysis, improving the ratio to $frac{1}{14 log n}$. Some of our improvement stems from tighter analysis of concentration properties for the value of any subadditive valuation function $v$, when considering a set $S' subseteq S$ of items, where each item of $S$ is included in $S'$ independently at random (with possibly different probabilities). In particular, we prove that up to less than the value of one item, the median value of $v(S')$, denoted by $M$, is at least two-thirds of the expected value, $M geq frac{2}{3}E[v(S')] - frac{11}{12}max_{e in S} v(e)$.

Problem

• Improves fair allocation for subadditive valuations
• Increases maximin share guarantee ratio
• Analyzes concentration properties of valuation functions

Innovation

• Improved fair allocation algorithm
• Enhanced maximin share ratio
• Tighter analysis of concentration properties