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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 paperof Shapley and Shubik \cite{Shapley1971assignment} introduced the "right" wayof exploiting the power of this theory, namely picking problems whoseLP-relaxations support polyhedra having integral vertices. So far, the latterfact was established by showing that the constraint matrix of the underlyinglinear system is {\em totally unimodular}.  We attempt to take this methodology to its logical next step -- {\em usingtotal dual integrality} -- thereby addressing new classes of games which havetheir origins in two major theories within combinatorial optimization, namelyperfect graphs and polymatroids. In the former, we address the stable set andclique 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 preciselythe set of optimal solutions to the dual LPs.  Another novelty is the way the worth of the game is allocated amongsub-coalitions. Previous works follow the {\em bottom-up process} of allocatingto individual agents; the allocation to a sub-coalition is simply the sum ofthe allocations to its agents. The {\em natural process for our games istop-down}. The optimal dual allocates to "objects" in the grand coalition; asub-coalition inherits the allocation of each object with which it hasnon-empty intersection.

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