Cores of Games via Total Dual Integrality, with Applications to Perfect Graphs and Polymatroids

TL;DR

This paper extends LP duality theory to analyze the cores of certain combinatorial games using total dual integrality, providing new insights into perfect graphs and polymatroids, and proposing a top-down allocation method.

Contribution

It introduces the use of total dual integrality to characterize core solutions in games related to perfect graphs and polymatroids, and proposes a novel top-down allocation approach.

Findings

Core solutions correspond to optimal dual LP solutions.

Established total dual integrality for stable set, clique, and matroid games.

Proposed a top-down allocation method for sub-coalitions.

Abstract

LP-duality theory has played a central role in the study of cores of games, right from the early days of this notion to the present time. The classic paper of Shapley and Shubik \cite{Shapley1971assignment} introduced the "right" way of exploiting the power of this theory, namely picking problems whose LP-relaxations support polyhedra having integral vertices. So far, the latter fact was established by showing that the constraint matrix of the underlying linear system is {\em totally unimodular}. We attempt to take this methodology to its logical next step -- {\em using total dual integrality} -- thereby addressing new classes of games which have their origins in two major theories within combinatorial optimization, namely perfect graphs and polymatroids. In the former, we address the stable set and clique games and in the latter, we address the matroid independent set game. For each of these games, we prove that the set of core imputations is precisely the set of optimal solutions to the dual LPs. Another novelty is the way the worth of the game is allocated among sub-coalitions. Previous works follow the {\em bottom-up process} of allocating to individual agents; the allocation to a sub-coalition is simply the sum of the allocations to its agents. The {\em natural process for our games is top-down}. The optimal dual allocates to "objects" in the grand coalition; a sub-coalition inherits the allocation of each object with which it has non-empty intersection.