AI 中文总结
研究在主体间公平分配不可分割物品，物品兼具商品与杂务性质。核心方法是引入概率性霍尔型矩阵分解，结合连续极小极大对偶性与有偏流网络。主要贡献是能同时实现事前无嫉妒和事后至多一件物品无嫉妒。
AI 中文摘要
我们研究了在具有可加效用的主体间公平分配不可分割物品的基本问题。在我们的模型中，一个物品可以是对某些主体产生非负效用的商品，同时是对其他主体产生负效用的杂务。我们采用两全其美的视角，目标是构建一种随机分配，事前完全公平，同时事后也基于近似公平的分配。本文研究的公平概念是无嫉妒（EF）及其著名的放宽形式——至多一件物品的无嫉妒（EF1）。我们的主要结果是可以同时实现事前EF和事后EF1。为此，我们引入了一种新颖的概率性霍尔型矩阵分解，将商品和杂务的分数分配复杂地关联起来。我们通过结合连续极小极大对偶性——经由西昂极小极大定理——与精心设计的有偏流网络来解决这个分解问题。
英文摘要
We study the fundamental problem of fairly dividing indivisible items among agents with additive utilities. In our model, an item can be a good yielding non-negative utilities to some agents and simultaneously a chore yielding negative utilities to others. We take the best-of-both-worlds perspective and our goal is to construct a randomized allocation that is exactly fair ex ante while also being supported on ex post approximately fair allocations. The fairness notions examined in this paper are envy-freeness (EF) and its well-known relaxation envy-freeness up to one item (EF1). Our main result is that ex-ante EF and ex-post EF1 can be achieved simultaneously. To achieve this, we introduce a novel probabilistic Hall-type matrix decomposition that intricately correlates the fractional assignments of goods and chores. We resolve this decomposition problem by combining continuous minimax duality -- via Sion's minimax theorem -- with carefully designed biased flow networks.
