Equitable Core Imputations for Max-Flow, MST and $b$-Matching Games

Abstract

We study fair allocation of profit (or cost) for three central problems from combinatorial optimization: Max-Flow, MST and $b$-matching. The essentially unequivocal choice of solution concept for this purpose would be the core, because of its highly desirable properties. However, recent work [Vaz24] observed that for the assignment game, an arbitrary core imputation makes no fairness guarantee at the level of individual agents. To rectify this deficiency, special core imputations, called equitable core imputations, were defined - there are two such imputations, leximin and leximax - and efficient algorithms were given for finding them. For all three games, we start by giving examples to show that an arbitrary core imputation can be excessively unfair to certain agents. This led us to seek equitable core imputations for our three games as well. However, the ubiquitous tractable vs intractable schism separates the assignment game from our three games, making our task different from that of [Vaz24]. As is usual in the presence of intractability, we resorted to defining the Owen set for each game and algorithmically relating it to the set of optimal dual solutions of the underlying combinatorial problem. We then give polynomial time algorithms for finding equitable imputations in the Owen set. The motivation for this work is two-fold: the emergence of automated decision-making, with a special emphasis on fairness, and the plethora of industrial applications of our three games.