We study the fair allocation of indivisible goods and chores under ordinal valuations for agents with unequal entitlements. We show the existence and polynomial time computation of weighted necessarily proportional up to one item (WSD-PROP1) allocations for both goods and chores, by reducing it to a problem of finding perfect matchings in a bipartite graph. We give a complete characterization of these allocations as corner points of a perfect matching polytope. Using this polytope, we can optimize over all allocations to find a min-cost WSD-PROP1 allocation of goods or most efficient WSD-PROP1 allocation of chores. Additionally, we show the existence and computation of sequencible (SEQ) WSD-PROP1 allocations by using rank-maximal perfect matching algorithms and show incompatibility of Pareto optimality under all valuations and WSD-PROP1. We also consider the Best-of-Both-Worlds (BoBW) fairness notion. By using our characterization, we show the existence and polynomial time computation of Ex-ante envy free (WSD-EF) and Ex-post WSD-PROP1 allocations under ordinal valuations for both chores and goods.

翻译：我们研究在序数估值下，针对权利不平等的代理人，如何公平分配不可分割物品与杂务。通过将问题简化为二分图中的完美匹配问题，我们证明了对于物品与杂务均存在加权必然比例直至一项（WSD-PROP1）的分配，并给出了多项式时间计算方法。我们完整刻画了这类分配，将其作为完美匹配多面体的角点。利用该多面体，我们可优化所有分配，以找到物品的最小成本WSD-PROP1分配或杂务的最有效WSD-PROP1分配。此外，通过使用秩最大完美匹配算法，我们展示了可排序（SEQ）WSD-PROP1分配的存在性与计算，并证明了在所有估值下的帕累托最优性与WSD-PROP1不相容。我们还考虑了“两全其美”（BoBW）公平概念。基于我们的刻画，我们证明了在序数估值下，对于杂务与物品均存在事前无嫉妒（WSD-EF）与事后WSD-PROP1分配，并给出了多项式时间计算方法。