Multilevel Fair Allocation with Matroid-Rank Preferences

TL;DR

This paper introduces two algorithms for multilevel fair resource allocation in hierarchical agent structures with matroid-rank utilities, balancing fairness and efficiency.

Contribution

It proposes the first polynomial-time top-down algorithm with theoretical guarantees and extends Yankee Swap to multilevel settings, enhancing fairness and efficiency.

Findings

The top-down algorithm guarantees fairness and efficiency.

The extended Yankee Swap preserves fairness in practice.

Algorithms perform well in hierarchical resource allocation scenarios.

Abstract

We introduce the concept of multilevel fair allocation of resources with tree-structured hierarchical relations among agents. While at each level it is possible to consider the problem locally as an allocation of an agent to its children, the multilevel allocation can be seen as a trace capturing the fact that the process is iterated until the leaves of the tree. In principle, each intermediary node may have its own local allocation mechanism. The main challenge is then to design algorithms which can retain good fairness and efficiency properties. In this paper we propose two original algorithms under the assumption that leaves of the tree have matroid-rank utility functions and the utility of any internal node is the sum of the utilities of its children. The first one is a generic polynomial-time sequential algorithm that comes with theoretical guarantees in terms of efficiency and fairness. It operates in a top-down fashion -- as commonly observed in real-world applications -- and is compatible with various local algorithms. The second one extends the recently proposed General Yankee Swap to the multilevel setting. This extension comes with efficiency guarantees only, but we show that it preserves excellent fairness properties in practice.