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.

翻译：我们引入了具有树状层次结构主体间关系的多级资源公平分配概念。虽然在每一级可以局部地将问题视为代理向其子节点进行分配，但多级分配可理解为一种追踪过程，反映了该迭代操作持续进行直至树中叶子节点的特性。原则上，每个中间节点都可能拥有独立的局部分配机制，主要挑战在于设计能保持良好公平性与效率属性的算法。本文在假设叶子节点具有拟阵秩效用函数且任意内部节点效用等于其子节点效用之和的条件下，提出了两种原创算法。第一种是通用的多项式时间序贯算法，在效率和公平性方面具有理论保障。该算法采用自上而下的运作方式（这在现实应用中普遍存在），且兼容多种局部算法。第二种算法将近期提出的通用洋基交换扩展至多级场景。该扩展仅提供效率保障，但我们在实践中证明其仍能保持优秀的公平性。