  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.

Computer Science and Game Theory;Data Structures and Algorithms;Theoretical Economics