Leximin and Leximax Fair Core Imputations for the Max-Flow, MST, and Bipartite $b$-Matching Games

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.