LP-Duality Theory and the Cores of Games

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: \begin{enumerate} \item Total unimodularity (TUM). \item Complementary slackness conditions and strict complementarity. \end{enumerate} 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: \begin{enumerate} \item Efficient algorithms for finding {\em min-max fair and max-min fair core imputations}. \item {\em Encouraging diversity and avoiding over-representation} in a generalization of the assignment game. \end{enumerate} Complementarity enables us to prove new properties of core imputations of the assignment game and its generalizations.