Characterization of incentive compatible single-parameter mechanisms revisited

TL;DR

This paper revisits the foundational characterization of incentive compatible single-parameter mechanisms, providing a simpler proof of uniqueness and extending the results to multi-dimensional cases for auctions.

Contribution

It offers an elementary proof of the uniqueness of Myerson's Lemma and generalizes the characterization to multi-dimensional auction settings.

Findings

Elementary proof of Myerson's Lemma

Unified presentation for two classes of allocation functions

Extended characterization to multi-dimensional auctions

Abstract

We reexamine the characterization of incentive compatible single-parameter mechanisms introduced in Archer & Tardos(2001). We argue that the claimed uniqueness result, called `Myerson's Lemma' was not well established. We provide an elementary proof of uniqueness that unifies the presentation for two classes of allocation functions used in the literature and show that the general case is a consequence of a little known result from the theory of real functions. We also clarify that our proof of uniqueness is more elementary than the previous one. Finally, by generalizing our characterization result to more dimensions, we provide alternative proofs of revenue equivalence results for multiunit auctions and combinatorial auctions.