LP-Duality Theory and the Cores of Games

TL;DR

This paper explores LP-duality theory to address gaps in the understanding of the core in cooperative games, introducing new classes of games, properties, and algorithms for core imputations.

Contribution

It introduces new game classes based on total unimodularity, characterizes their cores, and develops algorithms for fair and diverse core imputations.

Findings

Defined new games using total unimodularity.

Developed algorithms for min-max fair and equitable core imputations.

Proved new properties of core imputations in assignment games.

Abstract

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.