Multilevel Fair Allocation
Maxime Lucet, Nawal Benabbou, Aurélie Beynier, Nicolas Maudet
TL;DR
This work addresses multilevel fair allocation in tree-structured organizations by modeling leaves with matroid-rank utilities and internal nodes with additive welfare. It develops two algorithms: SMA, a polynomial-time top-down method with formal guarantees of multilevel utilitarian optimality and multilevel $\Psi$-maximization (via a bridge between $v$ and an estimated $\hat{v}$), and MGYS, a fast multilevel extension of Yankee Swap that guarantees multilevel utilitarian optimality but only heuristically preserves multilevel fairness. Theoretical results establish the properties and complexity of both approaches, while numerical experiments show MGYS substantially outperforms SMA in speed and scales to large hierarchies with strong empirical fairness. The methods enable principled, scalable, and fair resource allocation across hierarchical organizations such as departments, laboratories, and research groups.
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.