  LP-duality theory has played a central role in the study of the core, right
from its early days to the present time. However, despite the extensive nature
of this 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. Our
exploration of TUM leads to defining new games, characterizing their cores and
giving novel ways of using core imputations to enforce constraints that arise
naturally in applications of these games. The latter include: 1. Efficient
algorithms for finding min-max fair, max-min fair and equitable core
imputations. 2. Encouraging diversity and avoiding over-representation in a
generalization of the assignment game. Complementarity enables us to prove new
properties of core imputations of the assignment game and its generalizations.
