The Polyhedral Geometry of Truthful Auctions

TL;DR

This paper characterizes the geometric structure of truthful multi-unit auction mechanisms, revealing they correspond to regular subdivisions of the unit cube, and uses this to design robust mechanisms with stable allocations.

Contribution

It provides a complete geometric characterization of difference sets in truthful auctions and introduces mechanisms with stability properties based on this geometry.

Findings

Difference sets correspond to regular subdivisions of the unit cube.

Mechanisms can be constructed to be robust to small type changes.

The geometric approach enables new insights into auction design.

Abstract

The difference set of an outcome in an auction is the set of types that the auction mechanism maps to the outcome. We give a complete characterization of the geometry of the difference sets that can appear for a dominant strategy incentive compatible multi-unit auction showing that they correspond to regular subdivisions of the unit cube. This observation is then used to construct mechanisms that are robust in the sense that the set of items allocated to a player does change only slightly when the player's reported type is changed slightly.