Tropical Geometry and Mechanism Design

We develop a novel framework to construct and analyze finite valued, multidimensional mechanisms using tropical convex geometry. We geometrically characterize incentive compatibility using cells in the tropical convex hull of the type set. These cells are the sets of incentive compatible payments and form tropical simplices, spanned by generating payments whose number equals the dimension of the simplex. The analysis of the collection of incentive compatible mechanisms via tropical simplices and their generating payments facilitates the use of geometric techniques. We use this view to derive a new geometric characterization of revenue equivalence but also show how to handle multidimensional mechanisms in the absence of revenue equivalence.