The stable and fair division of profits/costs is a central concern in economics. The core, which ensures stability, has long been the gold standard for profit/cost sharing in cooperative games. Shapley and Shubik([SS71])'s classic work on the assignment game revealed that core imputations can be disproportionately favoring certain agents. Recent work ([Vaz24]) gave leximin and leximax core imputations for this game, achieving better fairness properties. We explore these fairness notions for the cores of three cooperative games: the max-flow game, the minimum spanning tree (MST) game, and the bipartite $b$-matching game. For all three games we give examples to show that an arbitrary core imputation can be excessively unfair to certain agents. Leximin and leximax core imputations are natural extensions of the widely used max-min and min-max fairness notions. We show that finding such imputations in the core is NP-hard for the max-flow and MST games, and likely so for $b$-matching as well. To address this, we introduce the concept of Dual-Consistent Core (DCC) imputations, which are characterized by solutions to the dual linear programs. We give polynomial time algorithms for computing leximin and leximax DCC imputations for all three games. These games have numerous applications and these imputations will provide a more fair way of distributing profit among agents for them.

翻译：利润/成本的稳定公平分配是经济学中的核心议题。确保稳定性的核心（core）长期以来一直是合作博弈中利润/成本分摊的黄金标准。Shapley和Shubik（[SS71]）关于指派博弈的经典研究揭示了核心分配可能不成比例地偏袒某些参与者。近期工作（[Vaz24]）为该博弈提出了词典序最小（leximin）与词典序最大（leximax）核心分配，实现了更优的公平性。本文探讨了三种合作博弈的核心的上述公平性概念：最大流博弈、最小生成树（MST）博弈以及二分图$b$-匹配博弈。针对所有三种博弈，我们给出实例表明任意核心分配可能对某些参与者极度不公。词典序最小与词典序最大核心分配是广泛使用的最大-最小与最小-最大公平概念的自然扩展。我们证明，对于最大流和MST博弈，在核心中寻找此类分配是NP难的，对于$b$-匹配博弈很可能也是如此。为解决此问题，我们引入了对偶一致核心（Dual-Consistent Core, DCC）分配的概念，其特征由对偶线性规划的解刻画。我们给出了为所有三种博弈计算词典序最小与词典序最大DCC分配的多项式时间算法。这些博弈具有广泛的应用，而此类分配将为这些应用中的参与者提供更公平的利润分配方式。