LP-duality theory has played a central role in the study of the core, right from its early days to the present time. The 1971 paper of Shapley and Shubik, which gave a characterization of the core of the assignment game, has been a paradigm-setting work in this regard. However, despite extensive follow-up work, basic gaps still remain. We address these gaps using the following building blocks from LP-duality theory: 1). Total unimodularity (TUM). 2). Complementary slackness conditions and strict complementarity. TUM plays a vital role in the Shapley-Shubik theorem. We define several generalizations of the assignment game whose LP-formulations admit TUM; using the latter, we characterize their cores. The Hoffman-Kruskal game is the most general of these. Its applications include matching students to schools and medical residents to hospitals, and its core imputations provide a way of enforcing constraints arising naturally in these applications: encouraging diversity and discouraging over-representation. Complementarity enables us to prove new properties of core imputations of the assignment game and its generalizations.

翻译：LP-对偶理论从早期至今一直在博弈核心的研究中发挥着核心作用。Shapley和Shubik于1971年发表的论文对指派博弈的核心给出了刻画，在该领域具有范式性的影响。然而，尽管有大量后续研究，基本空白依然存在。我们利用LP-对偶理论的以下模块填补这些空白：1）全幺模性（TUM）。2）互补松弛条件与严格互补性。TUM在Shapley-Shubik定理中扮演着关键角色。我们定义了若干指派博弈的推广形式，其线性规划表述具有TUM性质；利用这一性质，我们刻画了它们的核心。Hoffman-Kruskal博弈是其中最具一般性的形式。其应用包括学生与学校匹配、医学住院医师与医院匹配，其核心分配提供了一种方法，能够施加这些应用中自然出现的约束：鼓励多样性并防止过度代表。互补性使我们能够证明指派博弈及其推广形式的核心分配的新性质。