  We propose monotone comparative statics results for maximizers of submodular
functions, as opposed to maximizers of supermodular functions as in the
classical theory put forth by Veinott, Topkis, Milgrom, and Shannon among
others. We introduce matrons, a natural structure that is dual to sublattices
that generalizes existing structures such as matroids and polymatroids in
combinatorial optimization and M-sets in discrete convex analysis. Our monotone
comparative statics result is based on a natural order on matrons, which is
dual in some sense to Veinott's strong set order on sublattices. As an
application, we propose a deferred acceptance algorithm that operates in the
case of divisible goods, and we study its convergence properties.
