Application of the discrete separation theorem to auctions

The separation theorem in discrete convex analysis states that two disjoint discrete convex sets can be separated by a hyperplane with a 0-1 normal vector. We apply this theorem to an auction model and provide a unified approach to existing results. When p is not an equilibrium price vector, i.e., aggregate demand and aggregate supply are disjoint, the separation theorem indicates the existence of excess demand/supply. This observation yields a refined analysis of a characterization of competitive price vectors by Gul and Stacchetti (2000). Adjusting the prices of items in excess demand/supply corresponds to Ausubel's (2006) auction.