  We present a family of submodular valuation classes that generalizes gross
substitute. We show that Walrasian equilibrium always exist for one class in
this family, and there is a natural ascending auction which finds it. We prove
some new structural properties on gross-substitute auctions which, in turn,
show that the known ascending auctions for this class (Gul-Stacchetti and
Ausbel) are, in fact, identical. We generalize these two auctions, and provide
a simple proof that they terminate in a Walrasian equilibrium.
