Product-Mix Auctions and Tropical Geometry
For economists and mathematicians, it connects tropical geometry to auction theory, but the result is primarily theoretical and incremental.
The paper provides a new proof of the Unimodularity Theorem for product-mix auctions using integer programming, and formulates a new sufficient condition for competitive equilibrium with two product types, linking generalizations to the Oda conjecture.
In a recent and ongoing work, Baldwin and Klemperer explored a connection between tropical geometry and economics. They gave a sufficient condition for the existence of competitive equilibrium in product-mix auctions of indivisible goods. This result, which we call the Unimodularity Theorem, can also be traced back to the work of Danilov, Koshevoy, and Murota in discrete convex analysis. We give a new proof of the Unimodularity Theorem via the classical unimodularity theorem in integer programming. We give a unified treatment of these results via tropical geometry and formulate a new sufficient condition for competitive equilibrium when there are only two types of product. Generalizations of our theorem in higher dimensions are equivalent to various forms of the Oda conjecture in algebraic geometry.